Selected stories from mathematics and physics

1: SELECTED ‘STORIES’ FROM MATHEMATICS AND PHYSICS This book describes some instants of the conceptual history of mathematics and physics, yet the philosophical arguments of the ideas involved in some of them, for example the roots of geometry ,of analysis , of Galois theory, of vectors, space and physical approach of the continuum.These are presented in a technical theory but with a philosophical colour. The following ‘stories from mathematics and physics’ are some of the most important moments in the history of mathematic and physics. They are presented in a technical language but with a philosophical colour, trying so to make 2: their meaning easier for students and for those who are ‘interested’ in the meaning of al these things. http://www.mpantes.gr TABLE OF CONTAINS PREFACE PART I: STORIES FROM MATHEMATICS……5 THE ORIGIN OF METRIC GEOMETRY»…………5 THE ISOMORPHISM OF INVERSION AND THE RELATIVITY GEOMETRY……………………………………………………...8 THE CUBIC EQUATION…………………………….20 THE LIBERATION OF ALGEBRA, PEACOCK……….27 THE NEW ALGEBRA OF HAMILTON, THE QUATERNIONS …………….37 OF 3: THE ABSTRACT ALGEBRA AND THE UNSOLVED PROBLEMS OF ANTIQUITY ………………………………………………………………………………………………………..47 THE MATHEMATICAL AND PHILOSOPHICAL CONCEPT OF VECTOR………………53 THE LOGICAL FOUNDATION OF ANALYTIC GEOMETRY………63 THE INFINITE. THE SERIES (TAYLOR), THE ANALYTICAL FUNCTIONS 70 THE CONSTRUCTIVE STRAGEDY OF MATHEMATICS……..80 THE ACTUAL INFINITY IN CANTOR’S SET THEORY………85 THE DETERMINISM AS AN OPERATIONAL CONCEPT IN PHYSICS …………………………96 THE MATHEMATICAL FORMS OF PHYSICS, THE TENSORS……………………………..96 PART II : STORIES OF PHYSICS……………96 TIME IN PHYSICS……..122 SPACE IN PHYSICS ………..125 THE NATURAL MOTION IN PHYSICS…..136 THE MATHEMATICAL AND PHYSICAL CONCEPT OF 4-VECTOR IN SPACETIME……. MAXWELL’S FIELD AS AN OPERATIONAL CONCEPT …………………….146 AN INTRODUCTION IN SPECIAL RELATIVITY……………..163 THE PHYSICS OF THERMAL RADIATION…………….175 4: My story : THE QUANTIZATION OF SPACE AND TIME…….187 END PREFACE Mathematics and physics The Greeks were highly proficient in geometry, furnishing mathematical geniuses of the first rank, ntably Euclid, Appolonius and Archimides. The outstanding mathematician of the scientific Renaissance in the seventh century was Newton In a broad way we may classify mathematical advances under two headings. First new methods are discoveredmethods which renderpossiblethe solving of problems formely insilutable and open up new avenues of research. Secondly,considerable advances may ensue from the application of known methods. Analytical geometry, the calculus, the thery of groups, vector analysis and tensor analysis illustrste new methods. Newton deduction of the law of the inverse sqare from Kepler’s laws is an example of the successful application of the known method to the particular problem. When the subject matter pertains to the physical world the maehematics are said to be of the applied variety;and 5: when it is selected without regard for the world of the physical experience, the mathematics are called pure. On the basis,theoretical physics and mechanics must be regarded as pranches of applied mathematics, whreas the theory of functions and the theory of groups are reoresentative of pure mathematics…. So it is evident tha the progress of physical theories proceed by progressive mathematical approximations. The mathematical equations from the scaffolding of thephysical theory. Thus the general theory of relativity could never have arisen hadit not been for the prior discoveries of Gauss and Riemann in not Euclidean geometry, and almost all theories draw upon the calcumus. As meaning we say ‘the substantive content that gives reasonable coherence to a concept or action’ The more of the ‘stories’ have been published in Academia edu in recent years. PART ONE: STORIES FROM MATHEMATICS 1.1 THE ORIGIN OF METRIC GEOMETRY From my book: the relativity of geometry and the space ( in Greek) 6: The word “geometry” comes from the Greek geometrein (geo-“earth” and metre in to “measure”; geometry was originally the science of measuring land. Geometers classify geometric properties into two categories, the metric properties, in which the measure of distances and of angles intervenes, and the descriptive properties in which such measure is unessential.(H.Eves) The Pythagorean theorem is a metric property. As an example of a descriptive property we might mention the remarkable “mystic hexagram” theorem of Blaise Pascal (1623-1662): if a hexagon be inscribed in a conic then the points of intersection of the three pairs of opposite sides are collinear, and conversely, if the points of intersection of the three pairs of opposite sides of a hexagon are collinear, then the hexagon inscribed in a conic. The origins There is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognized as rigid) with respect to one another and to parts of the earth. In fact the very word geometry points this point conclusively. D'Abro imagines (the evolution of scientific thought) that for the first geometers who built the pyramids and performed measurements on the earth's surface, two were the concepts of fundamental importance without which the metric geometry would be impossible : A.that B.the of rigid definition bodies, of equality as we guaranteed call by these today rigid and bodies. 7: Today as rigid objects we mean those which maintain the same size and shape wherever displaced in space (regardless of position) and it was natural to consider them as standard for spatial measurements. This discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement probably never have arisen and metrical geometry would have been impossible. But with the discovery of objects which were recognized as rigid it was only natural to appeal to them as standards of spatial measurements. Measurements conducted in this way would soon have proved that between any two points a certain species of line called the straight line would yield the shortest distance; and this in turn would have suggested the use of the straight measuring rods. Henceforth two straight rods would be considered as equal or congruent if, when brought together, their extremities coincided. As for physical definition of straightness, it could have been arrived in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. For instance, two rods would be recognized as straight if after coinciding when placed lengthwise, they continued to coincide when one rod was turned over on itself. Finally parallelograms would be constructed by forming a quadrilateral with four equal rods, and parallelism would thus have been defined. This rigid and straight measuring rod was the tool to generate the metric geometry. Today we call it as Euclidean rod. Equipped in this way, the first geometricians (Egyptians, Babylonians) were able to execute measurements on the earth’s surface and later to study the geometry of solids, or space geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Now, an empirical science is necessarily approximate and geometry as we know it today is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180o and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Geek mathematicians Thales, Democritus, Pythagoras and finally by Euclid. 8: And then suddenly appeared a perfectly reasonable plan to the geometrical behavior of material bodies! The Euclidean geometry: this beginning point of human science! The logic embraced empirical intuition and together formed the duo of scientific development. Today we say that the Euclidean rod is a model of Euclidean “straight line” This is the miracle described by Aristotle in the phrase "it seems that the logic confirms the phenomena, and phenomena logic." But this is a matter of another story. George Mpantes http://www.mpantes.gr 1.2 THE ISOMORPHISM OF INVERSION AND THE RELATIVITY OF GEOMETRY . Abstract The one –to-one correspondence of the geometrical inversion, inverts the Euclidean plane to Ideal plane which contains all the points of the first except one (the center of inversion) plus one point at infinity. The inverse of Euclidean straight lines in Ideal plane, is a system of circles passing from the center of inversion, which complete the axioms of Euclid. So these circles are the Ideal straight lines of the Ideal plane. Now the inversion becomes an isomorphism and the geometry of the two spaces are identical except for superficial differences in terminology and notation. Contents 1. the isomorphism 2. Inversion : the transformation of the plane to itself 9: 3.Definitions of the Ideal geometry 4. The axioms of Euclid’s straight line 5. The ideal geometry and the axioms of Euclid 6. “Ideal geometry” is imaginary but mathematically consistent 7. Comment the isomorphism Anecdote: a mathematician was asked if he believes in God. Answer: Yes, via an isomorphism Removing our interest in the area of geometry, then all the above are translated in what is known from the theory of surfaces: two surfaces E1 and E2 are called isomorphic if it is possible to define a one-to-one correspondence of all the points of E1, on the points of E2 so that each "straight line" of E1 corresponds in a "straight line " on E2. Then the geometries of the two surfaces are identical: each proposition in one (geometry of E1) applies to another (the geometry of E2). In this result, there are the bases of the Euclidean models of the non-Euclidean geometries, so the Ideal geometry of this article, is the first trial. In the sequel, we shall set up a one-to-one correspondence between the points of the Euclidean plane into itself, proving that this correspondence is an isomorphism. Inversion : the transformation of the plane to itself . 10: Let C a fixed circle of center O and radius r, and let A be any point in the plane of C. Then the point A’ on the ray OA such that OA.OA’=r2 is called the inverse of A with respect to circle C. the construction is evident in fig. 1. We add to the plane a single ideal point at infinity. If P≡O, then P’ is taken as this Ideal point. Circle C is called the circle of inversion , point O the centre of inversion , and r2 the power of inversion. There is set up a one-to-one correspondence between the points of the plane of C; to every point there is a corresponding point , the points of the curve c will invert into the points of a curve c΄ , called the inverse of c. According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. We can prove the following theorems concerning this transformation of inversion Th.1 Th. 2 if P’ is the inverse of P , then P is the inverse of P’ A point inside the circle of inversion inverts into a point outside the circle of inversion; a point outside the circle of inversion inverts into a point inside of the circle of inversion; a point on the circle of inversion inverts into itself. If A coincides with O then as A΄we consider a point in infinity. Th.3 that the necessary and sufficient condition that two shapes are inverse, is any two pairs of corresponding points not collinear, are concyclic. Consider the points A, B and the inverse A΄ , B’ for inversion (O, r2)1. (Fig. 2) is then OA.OA = OB.OB = r2 so the four points as long as they are not collinear, are con-cyclic. Conversely, it is easily demonstrated that two shapes, between which there is a correspondence such that any two pairs of corresponding points to be con-cyclic, then the shapes are homologous to an inversion. 1 The circle (O,r) is not in the figure. 11: Th.4. a straight line through the center of inversion inverts into itself Th.5 A straight line that does not pass through the center of inversion, inverts into a circle that does not pass through the center of inversion Th.6 a circle orthogonal to the circle of inversion inverts into itself Th. 7 Two intersecting circles C΄ and C΄΄ orthogonal to the same circle C, are intersecting at points P and P’ which are inverse relative to the circle C. (Fig. 3) . Th. 8. the inverse of a circle that does not pass through the center of inversion, is a circle that does not pass through the circle of inversion, and homothetic to this. Th.9. any circle through a pair of inverse points P and P’ cuts the circle of inversion orthogonally. 12: Th.10. The inverse of a circle that pass from the center of inversion O, is a straight line parallel to the tangent of the circle at O. (fig.5) Th.11. a given circle may be inverted into itself by the use of any given exterior point as center of inversion. Th.12. in an inversion, the angle between two intersecting curves is equal to the corresponding angle between the two inverse curves. A transformation that preserves angles between curves is called conformal transformation. So, inversion is a conformal transformation. definitions of the Ideal geometry Definition of the genus of science (objects of the system) We shall examine now the representation of ordinary plane geometry by the geometry of a system of circles through a fixed point O, with the results of the above transformation of the plane on itself (the inversion with center O and radius 1). It is convenient to speak for the plane of the straight lines and the plane of circles , as two separate planes (the second as Ideal plane). We have seen that to every straight line in the plane of the straight lines , there corresponds a circle in the plane of circles. We shall call these circles Ideal straight lines. The Ideal points will be the same as ordinary points , except that the point O will be excluded from the domain of the Ideal points, plus a point at infinity. As angle of Ideal lines, we define the angle of the archetype straight lines through the inversion, as inversion preserves angles between two inverse curves. Definition of the length of an ideal segment. Without harming the generality, we consider the radius of inversion r = 1 viz.OA.OA΄=1 In Fig. 5 13: we have OA.OA΄ = OΒ.OΒ΄ = 1. The triangles OAB, OA΄B΄ are similar. therefore A΄B΄ / AB = OA΄ / OB Α΄Β΄=ΑΒ.ΟΑ΄/ΟΒ=ΑΒ.ΟΑ΄ΟΑ/ΟΒ.ΟΑ=ΑΒ/ΟΑ.ΟΒ finally finaly Α΄Β΄= ideal length of ideal ΑΒ= ΑΒ/ΟΑ.ΟΒ If points A, B, A’, B’ are collinear the same formula applies to the distance of A’, B’ The axioms of Euclid’s straight line The 1. A axioms straight line can of be drawn from . Euclid any point are: to any point 2 A finite staight line can be produced continously in a straight line. For any two points A, B there is always another C to B is "between" A and C. The meaning of " betweenness" , is basic for Euclidean geometry. 3. A circle may be described with any center and distance. It ensures that the "distance" in the plane (space), will be unchanged for a segment that is moved from one place to another. 4. All right angles are equal to one another 5. The most famous axiom in the history of science: for every like l, and for every point P that does not lie on l there exists a unique line m through P that is parallel to l. 14: The ideal geometry and the axioms of Euclid If we prove that the correspondence of inversion is an isomorphism of the Euclidean plane to the Ideal plane, then the geometries of the two planes will be identical. The properties of the set of circles could be established from the knowledge of the geometry of the straight lines, and every proposition concerning points and straight lines in the one geometry could at once be interpreted as a proposition concerning points and circles in the other. The first axiom Any two different Ideal points A,B determine the Ideal line A, B (fig.7), just as in Euclidean geometry, as three points (O,A,B) define a unique circle. So the first axiom of Euclid is valid in Ideal plane. The 2o and the 4o axioms of Euclid in Ideal plane . In figure 8 the infinite extension of AB creates an infinite extension of the ideal line from O, as are needed infinite such segments to arrive from P1 to B. This phenomenon makes hold the second axiom of Euclid for the ideal line: thereis not a sast point on this, it is a opened line. The securing of betweeness and the axioms of order, result from the exemption of O from the ideal points of the ideal plane. 15: For the fourth axiom we say: as in inversion the angles are preserved (conformal), the axiom for the right angles in Euclidean plane will also hold for the “right angles” in the Ideal plane” . For the 5o axiom. Ideal parallel lines If we have an Ideal line BΓ and an Ideal point A not on the line, we define the “parallel Ideal line” to ΒΓ, from A, the circle which touches at O the circle coinciding with the given line, and also passes through the given point A. So the two Ideal lines touch each other at O , which is not an Ideal point, will be Ideal parallel lines and the second (the circle is unique) will passes through A. So the fifth axiom of Euclid holds on the Ideal straight lines. For the third axiom: the ideal displacements. The length of a segment must be unaltered by displacement. This leads us to consider the definition of Ideal displacement. Any displacement may be produced by repeated applications of reflection; that is by making the image of the figure in a line (or in a plane in the case of solid geometry). So what is the Ideal reflection in an Ideal line? Definition of Ideal reflection : 16: The inversion about any circle of the system is equivalent to reflection of the Ideal points and lines, in the Ideal line which coincides with the circle of inversion.( Bonola p .247) theorem 13: the Ideal length of an Ideal segment is unaltered by inversion of the segment to any circle of the system. Every such inversion inverts a circle of the system in a circle of the system as the inverse of O is the same O. Let γ3 be any circle of the system (the circle of inversion) and let Γ its centre. Then inversion changes an Ideal line into an Ideal line. Let the Ideal segment AB of γ1 inverts into the Ideal segment A’B’ of γ2 . then 𝑖𝑑𝑒𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ𝐴𝛥 𝛢𝛥 𝛢΄𝛥 𝛢𝛥. 𝛰𝛢΄ = / = 𝑖𝑑𝑒𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝛢΄𝛥 𝛰𝛢. 𝛥 𝛰𝛢΄. 𝛰𝛥 𝛢΄𝛥. 𝛰𝛢 From similarity of the triangles ΓΑΔ , ΓΑ΄Δ και ΟΑΓ ,ΟΑ΄Γ we have ΑΔ/Α΄Δ=ΓΑ/ΓΔ=ΓΑ/ΓΟ=ΟΑ/ΟΑ΄ So Ideal length ΑΔ =Ideal length Α΄Δ, similarly holds: Ideal length ΒΔ= Ideal length ΒΔ΄ so subtracting we have Ideal length ΑΒ =Ideal length Α΄Β΄. (this proof is in the memorandum of Elementer -Mathematics). Wellstein, in the Enkyklopadie The Euclidean image is in fig.6a der 17: So finally we have proved that any Ideal displacement of an Ideal line segment, does not alter it’s Ideal length, so in Ideal geometry holds the third axiom of Euclid. More details and additional properties for the inversion we can find in the geometry of the Jesuits. Ideal geometry is imaginary but mathematically consistent . Finally we set up a one-to-one correspondence between the points of the Euclidean plane into itself, the inversion, having a new space of two dimensions, the Ideal plane. There the images of the Euclidean straight lines were a system of circles but they completed the axioms of Euclid. Then the two spaces are isomorphic. So it is possible to “translate” every proposition in the ordinary plane geometry into a corresponding proposition in this Ideal geometry. We have only to use the words Ideal points, Ideal lines, Ideal parallels etc. in the ordinary points, lines, parallels. But we can invent relations which were unknown in the systems of circles, as we see through the figure 10. The system of translation of the concepts will be like this (fig.10) Εuclidean The sum of the System of circles in «triangle» ΑΒΓ the sum of “angles” 18: angles of a triangle is π In orthogonal triangle ΑΒΓ hold the Pythagorean theorem Every segment has a unique middle. In every orthogonal triangle the median from the right angle is the half of is π. ςχ.10 In figure 10 we have (ΑΒ2/ΟΑ2ΟΒ2)+(ΑΓ2/ΟΑ2ΟΓ2) =ΒΓ2/ΟΒ2ΟΓ2. In every segment ΒΓ fig. 10 there is a unique Μ such as ΒΜ/ΟΒ.ΟΜ=ΓΜ/ΟΓ.ΟΜ. In every “orthogonal triangle” ΑΒΓ there is a Μ on ΒΓ as ΒΜ/ΟΒ.ΟΜ=ΜΓ/ΟΜ.ΟΓ and ΑΜ/ΟΑ.ΟΜ== 1/2ΒΓ/ΟΒ.ΟΓ hypotenuse Thale’s theorem Θεώρημα του Θαλή. In figure 11 holds ΑΒ.ΟΓ/ΟΑ.ΒΓ= Α΄Β΄.ΟΓ΄/ΟΑ΄Β΄Γ΄ Comment The interpretation of the Ideal plane is not an interpretation of the world round us, as it is rejected by the observations, till now. The same happened with the Riemannian geometry of the curved space, whose curvature we never “saw”, but we had to accept it as a reality if we want to interpret the reality of the conclusions of general relativity. 19: After the isomorphism through the inversion, the Euclidean straight line and geometry, loses it’s absolute character supported from experience, and we have the beginning of the great revolution in mathematics, that of non-Euclidean geometries. The truth of the Ideal plane is an hypothetical truth, an Aristotelian form, in the realm of potential reality. The two isomorphic spaces changed the perspective of mathematics, separated them from the accepted set of initial statements (material axiomatics) which were linked with the intuition, and led to a deeper study and refinement of the axiomatic procedure (formal axiomatics). How could the Ideal geometry be real? the straight line for us should be every circle in the figure 13. We must imagine we are tiny and every segment ds, looks as straight line. Also the rays of light, trace out the circles of the system, with the point O being a black hole. If we lived in such physical conditions our geometry should be Euclidean. SOURCES 1. Ευκλείδειοσ Γεωμετρία ΟΕ΢Β 1975 ………………ΑΠ. ΚΑΝΕΛΛΟΤ. 2. Θεωρητική γεωμετρία………………………………….Π. ΣΟΓΚΑ 3. Τριγωνομετρία…………………………………………… Γ. ΖΟΤΡΝΑ 4. Γεωμετρία Ιηςουιτών ………………Υ.G.M εκδόςεισ Καραβία 1952 5. Γεωμετρία Λομπατςέφςκυ…Αθήνα 1973…………΢, ΠΑΠΑΥΛΩΡΑΣΟΤ 6. Τα θεμέλια τησ Γεωμετρίασ: Μετάφραςη από την έβδομη Γερμανική Έκδοςη(Leipzig 1930) ελληνική έκδοςη Σροχαλία. ΢ΣΡΑΣΗ΢ ΠΑΠΑΔΌΠΟΤΛΟ΢ 7. Non Euclidean Geometry : Dover PUB. ……..ROBERTO BONOLA 8. Euclidean and non Euclidean Geometries W. H. Freeman and Company N.Y by MARVIN JAY GREENBERG 9. Foundations and fundamental concepts of mathematics H.Eves Dover GEORGE MPANTES Serres Greece www. mpantes. gr 20: . the cubic equation Historicalnote The historical roots of the study of polynomials up to the second degree are in ancient Babylon, where the time of the hanging gardens, the Babylonians were able to solve quadratic equations (x2+ bx + c = 0). Quadratic equations were treated systematically in the works of the Baghdad scholar al-Khwarizmi, -our Known formula     2  4 a - which were later translated into Latin, inspiring mathematical 2a progress in Europe for centuries. His name is the origin of the world algorithm. Moreover the word algebra is derived from the title of one of his works, al-Jabr. The purpose of mathematics was to find similar formulas for third and higher degree equations. After seven centuries of fruitless efforts, the Scipio del Ferro and Niccolo 21: Fontana, working independently, found a formula for solving the cubic equation (pressed) x3 + px + q = 0. ……..(1) This formula was known after Cardano published it in his book "Great Art» (Ars Magna) in 1545, and attaches it to Fontana. In the same book there is a reduction method of solving a fourth degree equation in solving third degree equation. It has been historical the controversy of Cardano, del Ferro and Ludovico Ferrari, for the authorship of these formulas. These techniques that were used for the cubic and quartic equations were systematized, already in Cardano’s time, so that they could be applied to equations of the fifth degree. But after three hundred years of failure , mathematicians began to suspect that perhaps there were no such formulas after all. He question was resolved in 1826 by Niels Henrik Abel (1802-1829) who showed that there cannot exist general solution formulas for equations of the fifth and higher degree, that involve only the usual arithmetic operations and extraction of roots. One say that such equations cannot be solved in radicals. A generalization of Abel’s approach, which was applicable to all polynomial equations was found a few years later by the twenty -year-old Evariste Galois (18111832) who invented the criteria that allow one to investigate any particular equation and determine whether it can be solved in radicals. A feature of the history of solving polynomial equations, is the dramatic difficulties of life of its protagonists (Stein). Cardano’s son was executed for murder while his other son was imprisoned for criminal activities. Cardano himself was imprisoned for heresy, the Ferrari was poisoned, the historians say by his sister .- When o Fontana was a child, a soldier tore the face with his sword thing that affected his speech so I was given the nickname Tartalia (stutterer). Abel died at the age of 27 years from tuberculosis and Galois was killed in a duel 21 years old. 22: Σhe cover page of Cardano’s book , “Ars magna” Cardano’s method Having the identity (α+β)3=α3+β3+3αβ(α+β) we put in (1) χ= α+β so χ3+pχ+q=0 → (α+β)3+p(α+β)+q=0 …….(2) but (α+β)3-3αβ(α+β)-(α3+β3)=0 ….(3) and in order to satisfy also the (2) we must have α3β3=- p3 27 and -q=α3+β3 so α3 and β3 have to be solutions of the quadratic p 3 κ2+ qκ - ( ) 3  0 ………..( 4) which is called resolvent of the initial (1). 23: In this point, Cardano who ignored the complex numbers, supposed that it’s solutions were real, so q p   ( ) 2  ( ) 3 >0. Then (4) has the solutions 2 3 a3   q q p  ( ) 2  ( ) 3 .......... ..(5) 2 2 3 q 2 q 2 p 3  3    ( ) 2  ( ) 3 .......... ..( 6) And we have the cubic formula of Cardano the real solution of the initial (1), χ =α+β viz x 3  q q p  ( )2  ( )3  2 2 3 - 3 q q p  ( ) 2  ( ) 3 .......... (7) 2 2 3 Today with the knowledge of algebra of complex numbers the equations 5 and 6 have more solutions also the initial (1). So α3, β3 give (ω is a complex cubic root of 1) a1  3  with corresponding 1  2  3   3  3   2 p p  3a1 33  2  with  p p   1  1 2 3a 2 3a1  3  with p  1 3a3 so all the solutions of (1) are x1  3  q q p  ( )2  ( )3  2 2 3 3 - q q p  ( ) 2  ( ) 3 .......... (7) formula Cardano 2 2 3 24: x2   3  q q p  ( ) 2  ( )3   2 2 2 3 x3   2 3  3 - q q p  ( ) 2  ( ) 3 .......... (8) 2 2 3 q q p q p q  ( ) 2  ( ) 3   3 -  ( ) 2  ( ) 3 .......... (9) 2 2 3 2 2 3 The solution (8) and (9) are two centuries later of Cardano’s time. Investigating of cubic equation The discriminant D of a polynomial is a number, calculated from the coefficients of the polynomial, and which is zero if, and only if, the polynomial has one (or more) double root. This is possible because D can be expressed in terms of the polynomial roots. For example for example in ax2 + bx + c the known discriminant D = b2-4ac, written and D=α2(ρ1-ρ2)2. Using discriminant we can draw conclusions about the nature of the polynomial roots without resolve. In cubic polynomial the discriminant term is used with some ambiguity. While the actual discriminant is D=(χ1-χ2)2(χ2-χ3)2(χ3-χ1)2, q 2 p 3 we have used in solving Cardano the cubic discriminant   ( ) 2  ( ) 3 There is an expression of √Δ in terms of the three solutions χ1, χ2, χ3 , in Cardano’s method, as √Δ=1/2(α3-β3)= 1 ( x1   2 x 2  x3 ) 3  1 / 54( x1  x 2   2 x3 ) 3  54 ( 1 / 18( 2   )( x12 x 2  x1 x 22  x 22 x3  x 2 x32  x1 x32  x12 x3 )  1 i 3 ( x1  x 2 )( x 2  x3 )( x1  x3 ) 18 25: 18 Viz. 3 i q2 p3   ( x1  x 2 )( x 2  x3 )( x1  x3 )  ..... 4 27  4 p 3  27q 2  ( x1  x 2 )( x 2  x3 )( x1  x3 ) with D= -4p3-27q2 =(χ1-χ2)2(χ2-χ3)2(χ3-χ1)2 the discriminant of the initial cubic (1), x3+px+q=0 . evidently is D=-108 Δ and the investigation for the roots is reduced in the investigation of D as follows A. if D=0 the initial cubic (pressed) has either three equal roots , or one double, all of them real. Β. If the quantity α3=  q q p  ( ) 2  ( ) 3 is a real number so if Δ>0 D<0, some 2 2 3 of the terms of (χ1-χ2)2(χ2-χ3)2(χ3-χ1)2 is negative so two roots are complex conjugate (so their difference is imaginary). So we have one real root the ρ. Γ. If the quantity α3=  q q p  ( ) 2  ( ) 3 is a complex number Δ<02, D>0 2 2 3 we have three discrete roots. Now the cubic type needs the cube roots of a complex number. I say go with a quadratic with complex-valued roots and take the cube roots of these complex numbers. The Cardano noticed this, called the polynomial «irreducible» (casus irreducibilis), but did not go to the complex numbers. Why all the roots are real? Because if we had complexvalued roots, then the D <0. All the roots are real, but w require complex numbers to be expressed in radicals! 2 The paradoxes of the cubic formula with the square roots of negative numbers, was the phenomenon that focused mathematicians in the study of complex numbers. 26: Finally the solutions of the pressed cubic equation is the combination of the cubic roots of the resolvent. If D=0 a double root or all them equal. If D<0 ,one real root and two complex If D>0 three discrete roots. Example 1.1 χ3-3χ+2=0 . Solution: A root the 1 so, (χ-1)(χ2+χ-2) ρίζεσ 1,1,-2 From the investigation of Cardano is p=-3 q=2 D=0 so real roots, two equal Example 1.2 χ3 -6χ+ 4=0 . Solution: a root the number 2 and becomes (χ-2)(χ2+2χ-2) with solutions 2, 1+√3, -1-√3. From the investigation of Cardano: είναι D= >0 and α3= -2+2i = √8(ςυν3π/4 +i ημ 3π/4). So the cubic roots of -2+ 2i α1= 6 8 (ςυνπ/4+iημπ/4)=1+i με β1=2/α1=2/(1+i)=1-i α2 = (1+i) j α3=(1+i) j2 με β2=2/(1+i) j=(1-i) j2 με β3=2/(1+i) j2 so χ1=α1+β1=(1+i)+(1-i)=2 χ2 = α2+β2=…= -1-√3 χ3=α3+β3= -1+√3 27: χ3+3χ+2=0. example 1.3 Solution: it has not a rational solution, D=-216<0 so we have one real and two complex solutions we have α3=-1+√2 ….. x1  3  1  2  3 1  2 (Cardano) x2   3 - 1  2   2 3 1  2 x3   2 3 Example 1.4 solutions 3 -1±3i. 1 2   3 1 2 χ3+6χ-20=0 has a solution the number 2 and the complex from Cardano’s formula we 10  108  3 10  108 . So this must be equal to 2 so 3 have the real solution. 10  108  3 10  108 =2 (it is verified with a pocket calculator). Also from χ3+2χ-3=0 we have x3 3 3 5 11 3 3 5 11    1 2 6 3 2 6 3 Also the χ3-7χ+6 =0 has solutions 1,2 και -3 and Cardano’s formula gives 1  400 3 1  400 (6  ) (6  ) where the radicals of the negative numbers 2 27 2 27 give some of the numbers 1,2,-3). These were the paradoxes of the cubic formula for the mathematicians in Cardano’s era. 4. THE LIBERATION OF ALGEBRA FROM ARITHMETIC, PEACOCK Abstract . 28: Peacock described the artificial numbers of his time in logical symbols instead of their intuitive interpretation with pictures of the real world. This was done through the axiomatization of the operations of arithmetic , where the symbols of the operations have no other meaning than the giving from the axioms i.e addition means some axioms. This new algebra is the “symbolical algebra” the first step to “abstract algebra”. His fundamental logical principle was the “principle of permanence of equivalent forms”. introduction Algebra considered with reference to it’s principles has received very little attention, and consequently very little improvement , during the last century. I regard it’s completeness as an independent science ….Peacock In the early nineteenth century, algebra was considered simply symbolized arithmetic. In other words instead of working with specific numbers , as in arithmetic , in algebra letters were used that represent these numbers. The positive integer and the four artificial forms of numbers constitute the “number system” of algebra viz the negative, the fraction, the irrational and the imaginary. In the mid 19th century, almost simultaneously with the liberation of geometry from the fifth axiom, starts the liberation of algebra from its tie to arithmetic, becoming a purely formal hypothetico-deductive study, as geometry. The discovery of non-Euclidean geometry , greatly influenced the development of axiomatic method in algebra, “the movement of axiomatization in mathematics”, but another factor was the recognition , first by the British mathematicians, (Analytical society of Cambridge) of the existence of structure in algebra. They conceived that the axiomatic 29: foundation of algebra means axiomatic foundation of it’s number system. The numbers in algebra were as the straight lines in geometry. This liberation brought in mathematics, new algebras beyond the arithmetical algebra of positive integers, and eventually new mathematical systems i.e sets equipped with various structures which had no classical analogues. We had the algebras of Lie, of Jordan, of Hamilton, of Grassman, the quaternions, the hypercomplex numbers, as we had the geometries of Lobatchewsky, Riemann etc. We even had the algebraic structures of groups, fieds, rings etc. So in 1870 appeared a new area of mathematics, called until the mid-twentieth century 'area of modern mathematics. " The most renowned representatives of “Analytical community of Cambridge” were Herschre, Babbage, Morgan and Peacock, who focused to the foundations of algebra. In his book "A Treatise on Algebra 1840" promoted the idea that the algebra if addressed properly, it is a productive science as geometry, and attempted to place algebra on a rigorous footing, basing the axiomatic thinking in algebra (Euclid of algebra). It is what he calls symbolic algebra. The number system before Peacock . Not one of the artificial numbers was accepted until its correspondence to some actually existing thing had been shown, the fraction and irrational , which originated in relations among actually existing things, naturally making good their position earlier than the negative and imaginary , which grew immediately out of the equations , and for which a “real” interpretation had to be sought… But the necessity remained of justifying this acceptance by purely algebraic considerations, this was first accomplished , though incompletely by G.Peacock. Henry B.Fine For example the irrational numbers could be thought of as points on a line, and as to their utility there could be no question. Hence ,though there was no logical basis for irrational numbers they were accepted in the number system. 30: The troublesome and intuitively unacceptable elements were the negative numbers and the imagine numbers. They were attacked and rejected in the 19th century with the same virulence as in previous centuries. Morris Klein The opposition to the idea of negative numbers was founded on viewing arithmetic as based on the concept of quantities that can be observed. It reminds us the intuition of geometrical ideas of Euclid. In 1758 the British mathematician Francis Maseres was claiming that negative numbers "... darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple" . Maseres and his contemporary, William Friend took the view that negative numbers did not exist. However, other mathematicians around the same time had decided that negative numbers could be used as long as they had been eliminated during the calculations where they appeared Negative numbers were accepted that stated amounts as debt, a year before, opposite direction etc, but as Klein reports they were discharged as roots of an equation, from the greatest mathematicians. They were false results, (Carnot) fictitious (Cardan), false (Descartes), inconsistent or vague solutions (Morgan), in the sense that they represent numbers less than nothing, so meaningless ... .Newton ". By the beginning of the 19th century Caspar Wessel (1745 - 1818) and Jean Argand (1768 - 1822) had produced different mathematical representations of 'imaginary' numbers, and around the same time Augustus De Morgan (1806 - 1871), George Peacock (1791 - 1858) William Hamilton (1805 - 1865) and others began to work on the 'logic'of arithmetic and algebra and a clearer definition of negative numbers, imaginary quantities, and the nature of the operations on them began to emerge. A modern view of algebra. The algebra of positive numbers . 31: Algebra is two things: a set of elements and the operations on this set. of numbers. That means the basic laws of the operations and their consequences; example: Let us take as number system the positive integers, and the operations the usual addition and multiplication symbolized with + and x . These two binary operations performed on the set of positive integers posses certain basic laws. We have Law Ι. α+β=β+α commutative law of addition Law ΙΙ. α+(β+γ)=α+β+γ associative law of addition Law ΙΙΙ. αxβ=βxα commutative law of multiplication Law ΙV. αx(βxγ)=αxβxγ associative law of multiplication Law V. αx(β+γ)=αxβ+αxγ the (left) distributive law for multiplication over addition. The five basic laws of the operations and their consequences constitute an algebra applicable to the set of positive integers. These statements are symbolic , and it is conceivable that they might be applicable to some other set of elements for example in even positive integers , in rationals in real numbers with the usual addition and multiplication, all the polynomials with positive integers coefficients, the set of pairs of positive integers (a , b ) with operations as ( a ,b ) + ( c , d ) = (a + c , b + d ) and (a , b ) x ( c , d ) = (ac , bd) , etc it is easily proved that the five properties above, apply to all of the above sets, and in many others. That is to say that , there is a common algebraic structure (the five basic properties and their consequences) attached to many different systems. The five basic properties may be regarded as postulates for a particular type of algebraic structure and any theorem formally implied by these postulates would be applicable to each of the examples given above, or to any other interpretation satisfying the five basic properties.. These are as Euclid’s axioms for the straight line , and algebra of positive integers seems like a deductive system. So algebra severed from arithmetic as geometry from the Euclidean straight line. 32: This is the modern view of algebra (the axiomatization of algebra) and the earliest glimmerings of the above modern view of algebra appeared about 1830 in England with the work of George Peacock .(H.Eves) The symbolical algebra, the permanence of forms The symbolists in England with G. Peacock made the distinction between what he called “arithmetical algebra” and “symbolical algebra”. The former is regarded by Peacock as the study that results from the use of symbols to denote ordinary positive integer numbers, (so it was on firm ground, Morris Kline) together with signs for the operations , like addition and subtraction, to which these numbers may be subjected. Now, in “arithmetical algebra” certain operations are limited in their applicability. For example , in subtraction ,a-b, we must have a greater than b. The symbolic algebra was the first attempt of axiomatic foundation of algebra in the first half of the 19th century, modeled on the Euclidean geometry . In this algebra the symbols of the operations have no other meaning from that given by the laws. 33: Peacock’s “symbolical algebra” adopts the operations of “arithmetic algebra” but ignores their restrictions . Thus subtraction in “symbolical algebra” differs from the same operation in “arithmetical algebra” in that it is to be regarded as always applicable. The justification of this extension of the rules of “arithmetical algebra” to “symbolical algebra” was called, by Peacock, the principle of the permanence of equivalent forms (H.Eves) Peacock’s symbolical algebra is a universal “arithmetical algebra” whose operations are determined by those of “arithmetical algebra” so far as the two algebras proceed in common , an by the principle of permanence of equivalent forms in all other cases. The principle of equivalence is a logical principle and played a historical role in development of algebra of negatives and complex numbers “All the results deduced in arithmetical algebra, whose expressions are general in form but particular in value, are correct results likewise in symbolic algebra where they are general in value as well in form….”Peacock” We understand this difficult principle in the example of the symbolical definition of subtraction in the new algebra. Arithmetical subtraction . To subtract b from a is to find a number to which if b be added , the sum will be a. the result is written a-b; by definition , it identically satisfies the equation Law VI. (a-b) +b= a Theorem VII: if a+c=b+c → a=b Obviously subtraction is always possible when b is less than a, but then only. Unlike addition , in each application of this operation regard must be had to the relative size of the two numbers concerned. From the laws Ι-VI we can prove all the equations properties which follow: 1.α-(β+γ)=α-β-γ=α-γ-β 2.α-(β-γ)=α-β+γ 3.α+β-β=α 34: 4.α+(β-γ)=α+β-γ=α-γ+β 5.αx(β-γ)=αxβ-αxγ For example 5. is proved αxβ-αxγ=αx(β-γ+γ)-αxγ = αx(β-γ) +αxγ-αxγ =αx(β-γ) law VI law V property 3. The symbolical subtraction, . If a<b what does the relation VI mean? Then a-b is not a number, but a symbol of algebra. The meaning of the equation of arithmetical algebra changes. Now equation is any declaration of the equivalence of definite combinations of symbols, equivalence in the sense that one may be substituted by the other and now (a-b)+b=a may be an equation whatever the values of a,b but now is a definition of this symbol. (symbolical definition). The numerical definition is subordinate to the symbolical definition , being an interpretation of which it admits when b is less than b. But from the standpoint of the symbolic definition , interpretability –the question whether a-b is a number or not – is irrelevant; only such properties may be attached to a-b, by itself considered, as flow from the generalized equation (a-b)+b=a (Henry Fine). Negative number is a symbol that admits of definition by a single equation of a very simple form viz.,( a-b)+b=a, with the laws I-VI and other theorems as 1-5 governing this symbol. Subtraction means no more than any process which obeys the laws I-VII. And the principle of permanence of form justifies logically the view that the processes in algebra have to be based o a complete statement of the body of laws or axioms which dictate the operations used in the processes. Indeed the above properties 1-5 are the properties of the symbol a-b as for the number a-b, and the result a-b , as defined for all values of a,b , led to definitions of the two symbols 0 ,-d , zero and the negative. 35: For example Let us see the introduction in algebra of a new symbol , zero, which contribute greatly to the simplicity and power of it’s operations.(Henry Fine) When b is set equal to a , in the general equation (a-b)+b=a It takes one of the forms (a-a)+a=a (b-b)+b=b It may be proved that a-a=b-b for (a-a)+(a+b)=(a-a)+a+b law II since a-a)+a=a and (b-b) +(a+b)=(b-b)+b+a=b+a Laws I, II since (b-b)+b=b therefore a-a=b-b theorem VII a-a is therefore altogether independent of a and may properly be represented by a symbol unrelated to a. The symbol which has been chosen for it is 0, called zero . the critique . But the deepest meaning of the symbolical algebra and the principle of permanence, is the symbolic form of the equations. The negative is a symbol for the result of an operation which cannot be effected with actually existing sets of things, which is therefore, purely symbolic. “The equation, the fundamental judgment in all mathematical reasoning , becomes a mere declaration regarding two combinations of symbols , that in any reckoning one may be substituted for the other (Henry Hine)” But the weakness of Peacock’s work is the logical principle of permanence of equivalent forms. This principle does not answer why the various types of numbers have the same properties with positive integers. This is a hypothesis ad hoc, to validate what was empirically correct but not logically based . The results of these calculations 36: actually were right when each number (negative , real or complex ) was replacing the letters . But these numbers were not actually understood nor their properties were reasonably disclosed .If Peacock knew quaternions and Hamilton’s algebra (no commutative property in multiplication) will not have established the principle of permanence, since the letters representing the quaternions have not all the properties of real and complex numbers. The quaternions invalidate the principle of permanence! What soon became evident after the introduction of quaternions , is that there is not one algebra but many. This principle has not predicted the new algebras, but symbolic algebra paved the way for the algebraic research in general. But Peacok was a pioneer as Leibnitz . His idea was a revolutionary idea of correspondence axiomatic bases in number systems. .Peacock’s Symbolic algebra was the beginnings of 'abstract algebra' which was a movement of algebra as generalized arithmetic in a purely formal (formal) algebra. The symbolic algebra underlined the importance of the structure over the meaning and acknowledged what has been formulated as “principle of mathematical freedom”. This principle implies that the algebra deals with arbitrary symbols, meaningless, mathematicians construct their operating rules and interpretation, follows rather than precedes the algebraic manipulations. " (Patricia R Allaire, Robert E. Bradley 'Symbolical Algebra as a foundation of calculus, internet) For recapitulation of Peacock’s work Dubbey summarises as the main thesis of Peacock's ideas in the book (Dubbey, Babbage, Peacock, and modern Algebra, Historia Math.(3) 1977 (internet) 1. Algebra had previously been considered only as a modification of arithmetic 2. Algebra consists of a manipulation of symbols in a way independent of any particular interpetation 37: 3. arithmetic is only a special case of algebra –a “science of suggestion” as Peacock put it 4. the sign = is to be taken as meaning “is algebraically equivalent to” 5. the principle of permanence of equivalent forms. But the really big idea in Peacock’s work is the decoupling of numbers with reality, as before him the numbers were not accepted until their correspondence to some actually existing thing had been shown. But numbers are symbols of mind, exist only in our mind, and as such can describe things of our imagination far from any intuition, as the non-Euclidean straight lines. Mathematics are not experiments of physics, mathematics live in mathematical fancy.. Dedekind said that numbers are a free creation of human mind and Peacock was the first who tried to express numbers as such creations. All the axiomatic movement of modern mathematics aimed at limiting of intuition in the logical reasoning, and Peacock essentially moved in this direction. On liberating numbers form reality he liberated algebra from arithmetic. These symbols (the numbers) are based on a complete statement of the body of laws and axioms which dictate the operations used in the processes. The operations define the numbers and not the numbers the operations. 5. the new algebra of Hamilton, the quaternions introduction 38: In the twentieth century, Hamilton’s quaternions were gradually rejected by most mathematicians, when the tools of vector analysis and matrix algebra were sufficiently developed and propagated. Only a small minority of researchers continued to see their value. Ironically, however, the basic concepts of vector analysis were generated by Hamilton's quaternions. Numbers are the cognitive tools of measurement and as tools, they are configurated according to the measured magnitudes. Other numbers measure indivisible sizes (sheep), others that are divided into equal parts, others sizes that are only approached, others the infinite (aleph ). For Hamilton this is exactly what happens, for him the symbols of algebra had to stand for something “real” not necessary material object but at least mental constructs; it is necessary, Hamilton claimed, to look “beyond the signs to the things signified” (Kleiner). So Hamilton constructed numbers-the quaternions-to count ... the rotations in space! . but their properties differed from those of real and complex numbers. This marked the genesis of a non-commutative algebra. This abstraction begins with the symbolic algebra of Peacok- as we saw in a previous article- which was objected by Hamilton. He characterized this as a philological algebra and later wrote for Peacock: “the author designed to reduce algebra to a mere system of symbols and nothing more. So I refused in my own mind, to give the high name of science to the results of such a system”. The next steps of mathematical generalization through abstraction were: (A) Hamilton's algebra where the form of the numbers and axioms themselves changed, and then B) the abstract algebra where the numerical sets became abstract structures, the algebra embedded the formalism of Hilbert, where we did not know exactly what we are talking about. The structure of the group could depict numbers, transpositions, other numbers, transformations, etc. It was a new type algebra, the abstract algebra. In this algebra Hamilton's algebra became a non-commutative multiplicative ring. 39: Complex numbers as geometrical operators. Hamilton was fascinated by the role of C in the two dimensional geometry, where we can use complex arithmetic to do a geometric operation. The best way to approach the creation of Hamilton is it’s elegant negotiation of complex numbers as pairs of real numbers. For mathematicians of his time considered the complex numbers as numbers of the form a + bi with a, b real and i2 = -1. The addition and multiplication of complex numbers held ( as is known ) manipulating a + bi as a linear polynomial in i, and substituting i2 where appeared to -1. Thus after a+bi is well defined by two real a, b Hamilton represents the complex number with real ordered pair (a,b). He stated that two such real pairs ( a,b ),( c,d ) are equal if and only if a=c and b=d Addition and multiplication of two such numerical pairs will be (a,b)+(c,d)=(a+c,b+d),and (a,b ).( c,d ) = (ac - bd ,ad + bc ) as (a+bi)(c+di)=(ac-bd)+i(ad+bc) With these definitions we can show that the addition and multiplication of ordered pairs of real numbers is 1. commutative, 2. associative and 3. multiplication is distributive over addition. It should be noted that the system of real numbers is embedded within the system of complex numbers. This meant that if any real r identified by numeral pair (r, 0) then the correlation is maintained by the operations of addition and multiplication of complex numbers because (a ,0 ) + ( b,0 ) = (a + b,0) ( a, 0) ( b,0) = ( ab, 0) To achieve the old form of a complex number from the form of Hamilton, we observe that each complex (a , b ) can be written 40: (a,b) = (a,0) + (0,b) = (a,0) + ( b,0 ).(0,1) = a + bi , where (0,1) denoted by the symbol i and (a ,0), (b,0) the real a and b. Eventually we see that i2 = (0,1) (0,1) = (-1,0) = -1 But the system of the complex numbers is a very appropriate numerical system to study vectors and spins in the plane. The relationship of complex numbers with the geometry are known from high school. The complex numbers are two-dimensional. While the real correspond in points on a straight line, the complex correspond to points in the (complex) plane. The complex x+yi corresponds to the point with coordinates ( x,y). But what is the geometric role of the mysterious i, which is known to us as imaginary number? It's role is related to the rotation in the plane. Hamilton knew that rotations in R2 could be computed efficiently by thinking R2 as the complex numbers, in which case rotation counterlochwise through θ was simply multiplication by the comple number eiθ. That is: The rotation in the plane, as shown in the figure, it is the movement of a figure, eventually of a point which is defined by a pair of coordinates, round a fixed point of the plane. Rotations are related to a reference system, because ultimately the calculation of the rotation is to calculate the new coordinates of the points, after the rotation. To calculate the rotation in two dimensions, there are two methods that of matrices and that of complex numbers (we are considering) . Let's consider a point on the plane , e.g. (1,1) and we are looking for the geometric position of its rotation in 1800: It is (-1,-1), and this is equivalent to multiply the number (1+1i ) by -1 . What happens if we rotate the image of (1,1) at 90o ? (always in the positive direction ) Then is equivalent to multiplying by i say 41: i.(1+1i) = -1+I, viz i.(a + bi) =- b + ai If we perform a second rotation by 90o then have i.i. (1 + i) = -1 (1 + i)= -1-i which is the result of rotation in 180o as we saw in the beginning. The number i, beyond that allows all polynomial equations to have a solution, it also gives a great tool for the rotations in the plane. Multiplying it by the coordinates of a point that has been written in complex form, we have the coordinates after rotation by 90o that is a purely geometric effect. In the case of a rotation by an arbitrary angle θ, we have the Euler’s formula 𝑒 𝑖𝜃 = 𝜍𝜐𝜈𝜃 + 𝑖𝜂𝜇𝜃 . So the point Α(x,y) → z=x+yi and the image of z’ =eiθ.z is the image of Α after rotation in an angle θ. We finally conclude that the rotations in the plane depend on a parameter, the rotation angle θ, and as the complex numbers operators of rotation satisfy the commutative law, the same would occur with the rotations in the plane. The order of two successive rotations in the plane does not change the result of the final rotation. The complex numbers are so two-dimensional numbers that count the vectors and their rotations in the plane. The new algebra of quaternions . Having defined complex numbers as vector in the plane, it was natural to Hamilton to inquire whether an algebra of triplets (as he called them) would be possible vectors in three space. Since the complex numbers were fundamental in many branches of mathematics and their applications, he considered the task of finding a similar algebra for triplets to be of vital importance. Hamilton’s idea was to define a multiplication on R3 that would work the same magic for rotations in 3-space, but he couldn’t find one. He worked fifteen years trying to find a multiplication for triplets, that fulfils the properties 1,2,3 for the complex numbers, and finally he had an inspiration. He tried looking in R4 instead, still he could not find a “true” multiplication, but he discovered that if he was willing to give up the commutative law for 42: multiplication, then everything would work out well. These properties which differed from those of real and complex numbers, marked the genesis of a non-commutative algebra. In these investigations he was led to a set of ordered quadruples (instead of ordered pairs) of real numbers (a,b,c,d) having both real and complex numbers embedded within them. Calling these quadruples quaternions he defines the equality (α,β,γ,δ)=(e,f,g,h) if and only if α=e,β=f,γ=g, δ=h. addition (α,β,γ,δ)+(e,f,g,h )=(α+e, β+f, γ+g, δ+h) Multiplication (α,β,γ,δ).( e,f,g,h)=( αe-βf-γg-δh, αf+βe+γh-δg, αg+γe+δf-βh αh+βg+δe-γ f) Now the reals and complex numbers are embedded in quaternions as we can identify the the real number μ with quaternion ( μ ,0,0,0 ) and the complex (a,b) with (a,b,0,0). Then we have (α,0,0,0)+(β,0,0,0)=(α+β,0,0,0) (addition of reals) (α,0,0,0).(β,0,0,0)=(αβ,0,0,0) (multiplication of reals (α,β,0,0)+(γ,δ,0,0)=(α+γ,β+δ,0,0) (addition of complex) (α,β,0,0).(γ,δ,0,0)=(αγ-βδ,αδ+βγ,0,0) (multiplication of complex) It can be shown that the addition of quaternions is commutative and associative and the multiplication is associative and distributive over the addition. But the commutative law for multiplication ceases to be valid. For example, for quaternions (0,1,0,0 ) and ( 0,0,1,0 ) We have (0,1,0,0).(0,0,1,0)=(0,0,0,1) while (0,0,1,0) (0,1,0,0) = (0,0,0 , -1) = - ( 0,0,0,1 ) the 43: If we symbolize the quaternion units ( 1,0,0,0 ) , (0,1,0,0) (0,0,1,0) (0,0,0,1) with 1 i, j, k, (as 1 in natural numbers) we can verify the previously mentioned acts, i2 = j2 = k2 = -1, ij = -ji = k, jk =-kj =i, ki =-ik = j with the table in the figure. Well, we have new numbers and a new algebra! It is said that the idea of deleting the commutative law of multiplication, came to Hamilton in a flash after fifteen years of fruitless meditation and thinking, (this time was necessary to challenge an axiom, the Euclidean twenty centuries) on a bridge in Dublin. He was so shocked by the unexpected idea, that he noted the main points of the above acts on a stone of the bridge (figure) . We can write the quaternion (a , b , c , d ) in the form a + bi + gj + dk as the complex in a+bi. Then two quaternions written in this form can be added and multiplied as polynomials of j,j,k and the result to write in the same form by the above multiplication table . Hamilton proposed a geometric interpretation of the fantastic triad bi + gj + dk , by considering the coefficients b,c,d as the rectangular coordinates of a point in space. The oriented line from the top of the system at the point (b,c,d) was named by Hamilton, vector (pure quaternion). So quaternions are mutant sizes, may be numbers or vectors or number-vectors! Hamilton strongly believed that quaternions were hiding the secret for a full description of the laws of universe. Indeed Maxwell formulated the laws of electromagnetism in quaternions form. We still can define the conjugate quaternion q*, the length and the unit quaternion as: q=w+xi+yj+zk. q*=w−xi−yj−zk. ∥q∥2=w2+x2 +y2 +z2 =qq*=q*q 44: The quaternion q is a unit quaternion when w2+x2 +y2 +z2 =1 The geometry of quaternions How can a quaternion, which lives in R4, operate on a vector which lives in R3? First we note that a vector 𝑢 is a pure quaternion whose real part is zero ie.q=(0, 𝑢) Because the vector part of a quaternion is a vector in R3, the geometry of R3 is reflected in the algebraic structure of the quaternions. It turns out that for a given axis and given angle of rotation we can easily make a quaternion and vice versa a quaternion can readily be read as an axis and rotation angle. These are much more difficult to compute with matrices or with Euler's corners. Theorem3: for any vector v of R3, and for any unit quaternion q=ςυνθ/2+𝑢 ημθ/2 the action of the operator Lq(v)=qvq* οn v, is equivalent to a rotation of the vector v through an angle θ about 𝑢 as the axis of rotation. It is similar to the known plane action of rotation z '= eiθ z. For example, the point (1,0,0) is transferred with a rotation about the y-axis by 900 at (0,0, -1). This is the geometry of the quaternions which makes them numbers with geometric action. Incorporating real, complex and vectors in space, the quaternion is a mathematical hybrid in form (quadruples) and meaning (rotation in space). 3 Quaternion and rotation(OM S 477/577)Notes Yan-Bin-Jia .Sep. 8 2016 internet 45: epilogue the Grassmann algebra where instead of considering just ordered sets of quadruples of real numbers, Grassmann considered ordered sets of n real numbers. It is correct to say that mathematicians have studied well over 200 such algebraic structures. As written by Americans algebraists Garett Birkhoff and Saunder MacLane “modern algebra has exposed for the first tim the full variety and richness of possible mathematical systems." (Howard Eves). Something similar happened in the elliptical and hyperbolic geometries. Comment. The numbers do not determine the binary operations but the operations define the numbers, as in geometry the straight line determine the space! Lines and numbers are symbols of mind, addressed to it, and will be defined in mind, viz. by axioms. As Lobatchewski freed the geometry from the Euclidean fifth axiom, Hamilton and Grassmann freed the algebra from arithmetic, with the only tool, the logical consequence. Hamilton's quaternion or Grassman's algebra and various other publications define the dawn of a new era. If they are destined to remain mere monuments of gentleness and sharpness of their writers or are meant to become powerful tools in discovering new truths, it is perhaps impossible to predict. 46: References . Number system of Algebra (Henry B. Fine, internet) Αrithmetical and Symbolical Algebra’ (Peacock, internet) Abstract Algebra : P.H.Nidditch Foundation and fundamental concepts of mathematics, Howard Eves The loss of certainty (Morris Klein) Πωσ τα μαθηματικά εξηγούν τον κόςμο (Σζέιμσ ΢τάιν, Αυγό) A short account of the history of mathematics (Rousse Ball, Dover) Η ιςτορία των μαθηματικών (Richard Mankiewics, εκδόςεισ Αλεξάνδρεια) Mathematics the loss of certainty (Morris Klein Dover) Israel Kleiner (2007): A history of abstract algebra, Birkhause Boston Ronald Solomon(2003): Abstract algebra, American Mathematics society. Needham Tristan,(1998): Visual Complex Analysis Clareton press Oxford 47: 6. THE ABSTRACT ALGEBRA AND THE UNSOLVED PROBLEMS OF ANTIQUITY Key words algebraic structure of constructible numbers with ruler and compass, field, field extension, the degree of field extension, the algebraic number, Geometric image construction, Trisection of angle , Doubling cube, Squaring the circle introduction 48: What is characteristic of pure mathematics is its irrelevance to immediate or potential application. Some pure mathematicians argue that there is a potential usefulness in any mathematical development and no one can foresee its actual future application. We shall see an example. With the elementary theory of fields of abstract algebra, we can understand why the Greeks were unable to solve the famous three problems, that of the trisection of the angle, doubling the cube and squaring the circle, using only the compass and ruler. The conceptual and chronological distance between the geometry of Euclid and abstract algebra is huge, how they are connected? With the common algebraic structure of constructible numbers with ruler and compass, and that of the field. For example: what do they have in common, the set of polynomials with real coefficients, all of the free vectors, all matrices mxn m, n Є N, the set of real functions with a common domain, the set of complex numbers, etc. It's all vector spaces over the field of real numbers, so have a common axiomatic basis. All theorems we prove in spaces, apply to all of the above sets, if we define appropriately the operations. The geometry of the Greeks standardized by Euclid , was originally based on the concepts of point , straight line and circle, and for this reason, tools for the study of geometry was the ruler for the construction of lines and compass to construct circles . The ruler was not calibrated because that did not account for distances but scratched lines between two points. Later began studying shapes that are not constructed with only ruler and compass , but the three unsolved problems mentioned above, raised in Euclidean ' genus ' of ' Elements'4. Why does in this genus we could bisect an angle with ruler and compass , but no trisect it? ? Or why we could construct a segment equivalent to α√ 2 ( α given ) and not to 4 ? Its construction difficult or impossible? The solution of Menechmos on the problem of doubling the cube using parabola and hyperbola, could not be drawn with a ruler and compass 49: The answer will be given through the theory of the field extension. In abstract algebra, we need some concepts for the three problems : the concept of field, of field extension , of the degree of field extension , and of the algebraic number. The set of rationals Q is a field ( satisfies some axioms , has a structure), if we add (attach ) the number √ 2 and all numbers with the operations of √ 2 with the rationals, we have the extension of Q, Q (√ 2) containing numbers of the form a + b √ 2 with a, b rationals . The degree of extension is denoted [Q (√ 2): Q] is the degree of the irreducible polynomial over Q, with √ 2 as a root i.e. x2 -2 (degree two, irreducible on Q). Algebraic number, is any number that is the root of a polynomial with coefficients in Q, ie the √ 2 is algebraic because it is the root of x2 -2, o as a root of x3 -2 , etc. Obviously the algebraic numbers are extension of rationals, any rational is algebraic .Every non- algebraic number is called transcendental , ie the number e. The numbers and geometry in Greeks. Greeks arithmetic and geometry were always together because they were considered two different ways of exploring the same number system , so seemed very natural the geometric constructions to perform arithmetic operations. Each number ought have a geometric construction. And as the known numbers were the rationals , the coexistence of arithmetic with the geometry was harmonious. After the shock of the revelation that √ 2 was not rational , and the fact that he had a geometric existence but non arithmetic5, strengthened the belief in the geometric construction of numbers. The numbers were contructible in geometry of rule and compass , and vice versa only numbers that would be presented in the geometrical construction of rule and compass could exist , meaning that we construct geometrically the number a mean that construct the segment length │a│. Within the framework of the three problems , our 5 The √ 2 is the hypotenuse of a right triangle perpendicular sides 1. 50: belief was that the numbers contained in them, clearly exist, so should therefore be a corresponding geometric construction . But all numbers are constructible6 ? Geometric image construction . .The geometry of the rule and compass basic constructions defining constructible points and hence constructible numbers are 1. assuming any two points O ( 0,0) and A ( a, b ) ( with coordinates rational numbers , key points ) 2 . construction of a straight line through the two points or cycle from it’s center and radius 3 . , construction of a point of intersection of two straight lines , 4. construction of the two points of intersection circles and straight or cycles. So produced all constructible points in a problem since the shapes made by rule, and compass is straight lines and circles. Let's take an example of geometrical construction of a number through the above process of constructible Be constructed with ruler and compass points : the square root of any positive integer x. As we will construct a right triangle with hypotenuse (x +1 ) / 2 and vertical side (x -1 ) / 2 Let OA = (x-1) / 2 and OB = (x +1) / 2 M midpoint of OB, then PB = P’ B = x (the PB, PB’ are tangents to the large circle so the OPB rectangle at P etc ...) Here the constructible number x, which corresponds to segment PB defined by a series of constructible points A, B, P, P’ which emerged with the above four possibilities provided by the rule, and compass. Similar constructions with ruler and compass we can remember many of the high school as: If the ends of a line segment are constructible then the midpoint is constructible (join with the ruler the intersections of two circles with centers of the two points and 6 We always mean with the use of the rule and diabetes 51: the radius length equal the length of the section) or If the three peaks parallelogram level are constructible then the fourth is constructible etc. Abstract algebra . Relations 2 and 3 reduce to rational operations. The intersection of a straight line and a cycle or two cycles, are reduced to the solution of quadratic equation , ie the square root extraction . So considering that the integers are easily constructible , it is known from A. high Each school rational constructions is to show that constructible B. if a > 0 is constructible then and √ a similarly is constructible . ( design radius of a circle (a +1 ) / 2 and the center ( (a +1 ) / 2,0 ) and from the point A (1.0 ) draw perpendicular to the x-axis that meets the circle at B. The segment AB is √ a). C. if a, b is constructible similarly are and a ± b , ab , a / b ( b ≠ 0) , ie the constructible numbers with compass and ruler are a field ( these are the conditions) , which contains the rationals , is an extension of rational since it contains the √ a. An example of constructible number is Theorem 1, if k is a constructible real number then k is the algebraic on Q , and the degree of extension Q [( n ) / Q] is a power of 2 . This theorem is a necessary condition for the existence of a constructible number k and applying it, we can show that there is no geometrical construction for the three problems of antiquity using only compass and ruler. . Trisection of angle . The first solution was Hippias by using the squared curve. We show that there is no always geometric construction for trisection angle θ using the rule and diabetes. Knowing an angle is equivalent to know the cosine of the angle . So on to trisect the angle 3θ have to construct the solution of the equation cos3θ = 4cos3θ – 3cosθ . ( 1 ) If the angle θ=200 so cos3θ = 1/ 2, the equation (1 ) becomes 8x3 - 6x -1 = 0 and the polynomials is irreducible with a real root a = cos20 0 and degree 52: of extension and [Q ( cos200 ): Q] = 3 therefore cos200 is not constructible , i.e. the angle 200 is not constructible . ( Theorem 1 ) . If the angle 3θ = 900 ( 1 ) becomes 4x3 - 3x = 0 which is irreducible , and also know that the 30 ° angle is reducible . Doubling cube . Solutions given by Hippocrates , Archytas Menechmou etc. If x is the edge of cube with twice the volume of the cube with edge 1 then x3 = 2.13 ie x= . To construct, the point ( appears impossible since , 0 ) must be constructible . But from the theorem 1 is the root of the irreducible polynomial x3 -2 and thus [Q ( ): Q] = 3. (theor. 1). Squaring the circle The problem is the construction of a square with an area equal to the area of a circle 1. We investigate if the number √ π is constructible . But from a classical result that was shown by the F.Lindemann in 1882, we know that the number p (pi) is transcendental over Q, ie it doesn’t satisfy some polynomial equation with rational coefficients ( not algebraic on Q, Theorem 1 ) . Hence the √ π is transcendental too, so it is not constructible with ruler and compass . Therefore, we can not square the circle using only rule and compass. Books : “Introduction to Galois theory”:Adrew Baker (University of Glaskow) “Algebraic extensions of fields” : Paul J. McCarthy Dover books “Field and Galois theory” :J.S.Milne (inernet) “Web sites” :Wikipedia, Wolfran Mathdord, Planet Mathword, Proofwiki 53: 7.The mathematical and philosophical concept of vector Historical summary The vectors The transformation equations of coordinates the transformation equations of vectors Euclidean Geometry and Newtonian physics . Philosophical comments, Aristotle Historical summary “There is un unspoken hypothesis which underlies all the physical theories so far created, namely that behind physical phenomena lies a unique mathematical structure which is the purpose of theory to reveal. According to this hypothesis , the mathematical formulae of physics are discovered not invented, the Lorentz transformation , for example ,being as much a part of physical reality as a table or a chair”. ( RELATIVITY: THE SPECIAL THEORY J.L.Synge p.163) Indeed in our example the physical phenomenon is the force, and the underlying mathematical structure is vector analysis. But looking the historical process, mathematics create their truths independently, discover new entities, and their tendency for generalization goes ahead exceeding the initial physical presuppositions. The new discoveries of the mathematical process return to the physical theory where 54: they create new unifications and generalizations, now for the phenomena of the actual world. Can we trust them? Can mathematics lead the physical theory? The answer seems to be positive, if the measurements agree with the mathematical conjectures (electromagnetic waves!). that is that mathematical structure extends the physical theory. So, for example, with the support of the use of vector methods , we had a development of theoretical physics and by the beginning of the twentieth century, vector analysis had become firmly entrenched as a tool for the development of geometry and theoretical physics. As we look back on the nineteenth century it is apparent that a mathematical theory in terms of which physical laws could be described and their universality checked was needed. Figuratively speaking two men stepped forward in this direction, Hamilton and Grassman. Hamilton was trying to find the appropriate mathematical tools with which he could apply Newtonian mechanics to various aspects of astronomy and physics. Grassman tried to develop an algebraic structure on which geometry of any number of dimension could be based. The quaternions of Hamilton and Grassmann’s calculus of extension proved to be too complicated for quick mastery and easy application , but from them emerged the much more easily learned and more easily applied subject of vector analysis. This work was due principally to the American physicist John Willard Gibbs (1839-1903) and is encountered by every student of elementary physics. The vectors . What is behind the physical phenomenon of the velocity; of the force; there is the mathematical concept of the vector. This is a new concept, since force has direction, sense, and magnitude, and we accept the physical principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces . These are the basic physical indications for the mathematical treatment, for “vector7 geometry”, where 7 The term vector was introduced from Hamilton 55: the term vector denotes a translation or a displacement a in the space.8 The statement that the displacement a transfers the point P to the point Q (“transforms” P into Q ) may also be expressed by saying that Q is the end-point of the vector a whose starting point is at P. if P and Q are ant two points then there is one and only one displacement a which transforms P to Q. We  shall cal it the vector defined by P and Q and indicate it by PQ . There are two fundamental operations, which are subject to a system of laws, viz. addition of two vectors (the translation which arises through two successive translations (law of parallelogram), and multiplication of a vector by a number (is defined through the addition). These laws are A.Addition: a+b=c with the properties a+b=b+a (a+b)+c=a+(b+c) If a and c are any two vectors , then there is one and only one value of x for which the equation a+x=c holds B. Multiplication b=λ.a with the properties (λ+μ)a=(λa)+(μa) λ(μa)=(λμ)a 1.a=a λ(a+b)=(λa)+(λb) In elementary physics , a vector is graphically regarded as a directed line segment , or arrow. This is the translation or the displacement described by Weyl. So in elementary physics , vector was something apparent, something concrete and intuitively simple. It was geometrical. In theoretical physics it became an idea, something cerebral, connected with algebra. The first was a sketch of the second. This is the course of mathematics. The formula for algebraic vector was the old bold Cartesian binding of 8 This definition is from Weyl, (Space, time, matter) 56: geometry with algebra viz this of a picture with the abstract and compact truth of numbers, a good combination between intuition and rigor, through concepts, in the center of which was the well-known coordinate system, one of the more significant generalization of mathematics. By means of a coordinate system, a set of ordered triples of real numbers can be put into one-to-one correspondence with the points of a three dimensional Euclidean space. However many aspects of modern-day science cannot be adequately described in terms of a three-dimensional Euclidean model. The ideas of vector analysis when expressed in a notational fashion are immediately extendable to n-dimensional space and their physical usage is amply demonstrated in the development of special and general relativity theory. With the change of the figurativeness of the points, change also the description of the vector. The set {A1, A2, A3} of all triples (A1, A2, A3), (A1΄, A2΄, A3΄) etc., determined by orthogonal projections of a common arrow representation on the axes of the associated rectangular Cartesian coordinate system is said to be a Cartesian vector. Many triples means many systems, but all these represent the same Cartesian vector, which has a family of arrows as its geometrical representative. The binding of orthogonal projections with the law of parallelogram is the base of all the formalism of vector analysis.9 A Cartesian vector (A1, A2, A3), (3-tuple), can be represented graphically by an arrow, with it’s initial point at the origin and it’s terminal point at the position with coordinates (A1, A2, A3), but it is not the only possible arrow representation. An arrow with initial and terminal points (a,b,c) and (A,B,C) such that A1=A-a, A2=B-b, A3=C-c can be considered a representative of a 3-tuple. A Cartesian vector with respect to a coordinate system, is characterized by a magnitude , a direction and a sense , and its components in any coordinate system satisfy the algebraic laws of the triples, viz the laws 1 and 2 for the vectors, expressed algebraically , if we define a=(a1,a2,…..an) b=(b1,b2,….bn) i.e (a1,a2,…..an)+(b1,b2,….bn)=(a1+b1, a2+b2+……an,bn). 9 See my article “the mathematical forms of nature, the tensors” 57: λ.a=λ(a1,a2,…..an)= (λa1,λa2,…..λan) . Now an analytical treatment of vector geometry is possible, in which every vector is represented by it’s components and every point by its coordinates. How all these triples, (A1, A2, A3), (A1΄, A2΄, A3΄) etc., are related? The transformation equations of coordinates A fundamental problem of theoretical physics is that formulating universally valid laws relating natural phenomena. Because the transformation idea is of such importance, the development of vector geometry and later of vector analysis is build around this. A rectangular Cartesian coordinate system10 imposes a one-to-one correspondence between the points of Euclidean three-space and the set of all ordered triples of real numbers. A second rectangular Cartesian system brings about another correspondence of the same point . What is the nature of those transformations that relate such coordinate representations of the three-space? The specific transformations of coordinates for our example in the development of vector analysis, are called translations and rotations. They are linear transformations and they connect orthogonal Cartesian systems. All linear transformations have the characteristic that the fundamental relations (A) and (B) are not disturbed by the transformation viz they hold for the transformed points and vectors : α΄+b΄=c΄ b΄=λ .a΄……… DEFINITION 1. The transformation equations that relate the coordinates ( x1 , x 2 , x 3 ) and (x 1 , x 2 , x 3 ) in rectangular coordinate systems , the axes of which are parallels are x j  x j  x0j .......... .......... ....(1) 10 We examine this particular case in our example. 58: where ( x 01 , x 02 , x 03 ) represent the unbarred coordinates of the origin of the barred system O’. These are called equations of translation. The Cartesian vector concept is employed in obtaining them. DEFINITION 2. The transformation equations that relate the coordinates ( x1 , x 2 , x 3 ) and (x 1 , x 2 , x 3 ) in rectangular coordinate systems, having a common origin and such that there is no change of unit distance along coordinate axes, are related by the transformation equations x j  ckj x k .......... ..(2) where transformations the coefficients a kj are direction cosines satisfying the conditions  3 j 1 c kj c pj   k p of 59: These are called equations of rotation. The transformations of the coordinates (2) are a subset of the linear or affine transformations, with the general form  x1   c11  2  2  x    c1  x3   c3    1 c12 c 22 c 23 c31  x 1    c32  x 2 .........( 3) c33  x 3  where apply the conditions of orthogonality, they are the orthogonal transformations that connect orthogonal Cartesian systems with common origin and are produced from the vectorial behavior of the vectorial units (bases) in the axes of the two systems . Physically they describe, as we have mention, the rotation of an orthogonal Cartesian system. The orthogonal transformations fulfill the first unification of geometry (the Euclidean metrical geometry in every orthogonal system) and as the geometry is a fundamental branch of physics, this unification will be the model of the unification of physical laws in all the systems.(universality) But what about the vectors? What is their deepest behavior in the scene of coordinate systems? the transformation equations of vectors We have seen that a Cartesian vector (A1, A2, A3) can be represented graphically by an arrow, but The components of this arrow, transform under rotation, as the coordinates . Proof: If the transformation (2) is applied to the coordinates of P0 and P1 , the coordinate differences { x1j  x 0j } satisfy x1j  x0j  ckj x1k  ckj x0k  ckj ( x1k  x0k )......... .......... ..(4) that is the transformation (2). 60: A corresponding verification of the statement holds for translations, where the vector components remain unaltered. So we have the definition of the Cartesian vector under the light of both transformations: A Cartesian vector (A1, A2, A3), is a collection of ordered triples , each associated with a rectangular Cartesian coordinate system and such that any two satisfy the transformation law Aj  x j k A .......... .......... .......... ......( 5) x k where the partial derivatives are the coefficients c i j of the linear transformation (3), of coordinates. We must notice that every component of the vector in the new system is a linear combination of the components in the initial variables. So if all components of the vector are zero in the initial system they will be also zero in the new variables. This is the more important property of vectors: a vectorial equation holds in every rectangular Cartesian system (for our paradigm), if it holds in one! This is the root of the universality of the physical or geometrical laws, as we see in the end of the article. Newton’s law is universal because it is written in vectorial form. It’s invariance in translation is the mathematical acceptance of the Newtonian principle of relativity. The scalars . A second concept which has evolved in the development of vector analysis is that of the scalar. The definition of scalar states that it is a quantity possessing magnitude but no direction. Such entities as mass, time, density and temperature are given as examples. But for mathematics, the prize example is the real number, as it does not have to be associated with magnitude. From a historical point of view scalar is a quantity invariant under all transformations of coordinates (Felix Klein). Whether a given algebraic form is invariant depends on the group of transformations under consideration. Again the scalars , as vectors, are associated with coordinate systems and transformations. 61: Euclidean Geometry and Newtonian physics . The mathematical investigation showed that our Known geometrical vector (arrow) has hidden qualities which are raised by their correlation with coordinate systems: The laws of it’s transformation. The vector concept received much of its impetus from this fact, so it plays a fundamental role in many aspects of geometry and physics. This mathematical result underlies the principles of relativity of Newton and Einstein, that would be ungrounded without the mathematical discovery of the transformation theory of the vectors and (later) of tensors. Magnitude and angle are fundamental to the metric structure of Euclidean space. They are scalar invariants under the transformations of the orthogonal Cartesian set. The inner product transforms 3 3 3 3 j 1 j 1 j 1 k 1  P j Q j   (crj P r )(csj Q s )   crj csj P r Q s   r s P r Q s   P k Q k and the distance of the points X 1j , X 0j (through 4) ( x11  x01 ) 2  ( x12  x02 )  ( x13  x03 ) 2  ( x11  x01 ) 2  ( x12  x02 ) 2  ( x13  x03 ) 2 These formulas carry out the first unification of metrical Euclidean geometry. An observer who measures a distance and an angle in a orthogonal Cartesian system uses the same formulas and finds the same results with somebody else who measures the same magnitudes in another orthogonal Cartesian system, which subsists a translation or a rotation of the first. The question of finding those entities (as distance and inner product) that have an absolute meaning transcending the coordinate system, is of prime significance. This gives us a direction as to which of the concepts considered in the framework of rectangular Cartesian systems should be generalized as well as how to bring about the generalizations. This is the criminal point of the universality of the physical laws. Moreover, the availability of the Cartesian systems of reference will be valuable when considering the special theory of relativity, in a later article. In vector formalism, we will now show the covariance of Newton's law in linear systems with given origin ( rotation) . 62: In the system K we have Fk  d (m k ) dt x k Multiply by and summing with respect to k (from 5) we have x r x d x k d Fk  (m k  k )  Fr  (m r ) dt xr xr dt So the form of the equation remains the same in the new system ( covariant ) , and the mathematical formalism demonstrates that the laws of Newton have a universal application in Euclidean space, where we can adjust orthogonal Cartesian systems. The physical laws are invariant in form in all the orthogonal Cartesians systems (my article “covariance and invariance in physics”) Philosophical comments, Aristotle . The quotation of Synge about the discovery of mathematical structure of vector analysis , where the vector is as much a part of physical reality as a table, is very poetic, having construct a reality of visible and invisible objects, as the vector and the table. Vector exists as the table but it is invisible! Aristotle in his ideas of the theory of knowledge (Analytica posterioria) says that “the knowledge of a fact differs from the knowledge of a reasoned fact” . The theoretical foundations of the systems of this deductive reasoning, account of first principles where are the bases of every science. …. The scientific knowledge through reasoning is impossible if we do not know the first principles. ….it is clear that in science of nature as elsewhere we should try first to determine questions about the first principles …..αληθείσ και πρωταρχικέσ και άμεςεσ και πρωθύςτερεσ και αιτίεσ του ςυμπεράςματοσ,the first basis from which a thing is known……as regards their existence must be assumed for the principle ,(what a straight line is , what a triangle is ..)but proved for the rest of the system, by logical reasoning. How are these first principles to be established? ….they are arrived by the repeated visual sensations, which leave their marks in the memory. Then we reflect on these 63: memories and arrive by a process of intuition (νουσ) at the first principles ….if there in not something intelligible behind the phenomena, there is not science for anything, science is not created from senses…. So vector is an intelligible creation, a first principle, that is a conviction, a support of the deductive reasoning. This reasoning constructs a mental logical reality in our brains, which is the human’s way of comprehension. Mathematics are neither discoveries nor inventions. Mathematics are creations , as poems, but logical creations, based on the rules of deductive reasoning. But in fact, their first principles are founded in nature. Sources Herman Weyl (space,time, matter,Dover) H.Eves (foundations and fundamental concepts of mathematics,Dover) J.L Synge (Relativity: the special theory, Noth Holland publising Company Amsterdam New York Oxford) Robert C.Wrede (introduction to vector and tensor analysis, Dover) Aristotle (Analytica posterioria, internet 8. THE LOGICAL FOUNDATIONS OF ANALYTICS GEOMETRY 64: (The royal road in geometry11) Apollonius The first four books of the Conics survive in the original Greek, the next three only from a 9thcentury Arabic translation, and an eighth book is now lost. Books I–IV contain a systematic account of the essential principles of conics and introduce the terms ellipse, parabola, and hyperbola, by which they became known. Apollonius’ conics is almost as famous as the Elements of Euclid. Kepler was able to establish his three laws of planetary motion only because Apollonius had supplied so much information about the conic The idea The story of the connection of arithmetic with geometry is dew to the great deal of progress had been made in algebra during the later half of the sixteen century and the early part of the seventeenth. Cardan, Tartalia, Viète, and Descartes and Fermat themselves had extended the theory of the solution of equations, had introduced symbolism, and had established a number of algebraic theorems and methods. And both Descartes and Fermat, working independently of each other, saw clearly the potentialities in algebra for the representation and study of curves. Their basic thought was that algebra should be used to characterize any curve and as the means of deducing facts about the curve. Hence in some way, numbers had to be brought into the picture. That is the idea of analytic geometry. 11 When asked by king Ptolemy for a short but to geometric knowledge, Euclid is said to to have replied , “there is no royal road ih geometry. 65: The idea of locating points with numbers was old. For example Ηipparchus had introduced latitude and longitude to locate points on the surface of the earth.. How this idea was born again? Fermat and Descartes decide that mathematics needed new methods of working with curves. What is relevant here with Descard, is his general concept of method and his success in introduction method in geometry by means of algebra. As an appendix to his Discourse on the method he published his geometry. Descartes complained that the geometry of the Greeks was so much tied to figures “that it can exercise the understanding only on condition of greatly fatiguing the imagination. He also apprediated that the methods of Euclidean geometry –speaking as the king Ptolemy- are exceedingly varied and specialized, particularly in the study of the conic sections and the few other curves explored in Greek geometry. Also since Fermat had also participated in the advancement of algebra, he, too, became aware of the potentialities in algebra for the investigation of geometry. Moreover the practical applications (projectiles) and such informations is provided by algebra. Descartes and Fermat , who were interested in curves lying in a plane, introduced two perpendicular lines or axies and agreed to represent any point in the plane by it’s distances from the two axes. To know a curve means mathematically to know some property that characterizes all the points of the curve. The circle can be defined as the set of al points that at the same distance from the center. According to the Pythagorean theorem of Euclidean geometry we the equation x2+y2=r2 an equation, an algebraic statement that holds for each point on the circle and for no other points. 66: The parabola as was defined by Apollonius can be defined as the set of all points that are equidistant from a given point (focus) and a given line(directrix) so in the figure here we have PS=PM so (𝑥 − 𝑎)2 + 𝑦 2 = 𝑥 + 𝑎 𝑎𝑛𝑑 Finally y2=4ax In such a process we could ask ourselves whether the analytical geometry method is sufficient to study and prove every problem and theorem that arises from the development of the Euclid's plane basis. This question is a difficult problem, and it is transferred to Hilbert's modern formal axiomatic and metamathematics, which is not mentioned in the analytical geometry textbooks. But there is a logical gap here-how to predict that all the theorems of geometry have an algebraic solution - can we look at them one by one? - it reminds us of the series in the era before Taylor, where the convergence of each series was neglected and its study was downgraded. In my book "The Relativity of Geometry and Space" (Janus) www.mpantes.gr, we have seen the concepts of interpretation of an axial basis - interpreting the original terms of the system in some way transforming them in terms of some understanding the concept of its model - that is, an interpretation of the system for which the axioms of the system are valid,- and the concept of the isomorphism of the axiomatic baseswhen the axioms of one are logical consequences (theorems) of the axioms of the other. Now we will look at the interpretation and a model of Euclidean axioms in algebra and the isomorphism that binds the two models, geometric and algebraic. The algebraic interpretation of the primitive Euclidean terms 67: Our task is to assign algebraic meanings to the primitive terms point, line on, and congruent (as applied to segments and to angles) that will convert each of Euclids postulates into a theorem of algebra. By a point we mean any ordered pair of real numbers.(the coordinates of the point) By a line we mean any equation in the two variables x and y of the form ax+by+c=0 a,b,c real numbers and a,b not both 0. A point is on the line if the coordinates of the point satisfy the equation of the line. We say a point is on a line if and only if the coordinates of the point satisfy an equation of the line. We say the segment denoted by (x1 ,y1 ),(x2 ,y2) is congruent to the segment denoted by (x3,y3) (x4,y4) if and only if (x2-x1)2+(y2-y1)2=(x4-x3)2+(y4-y3)2 We say finally that the measure of an angle A denoted by (x2 ,y2 ),(x1 ,y1), (x3,y3) is given by cosA= 𝑥 2 −x 1 x 3− x 1)+( y 2 −y 1 (y 3 −y 1 ) (𝑥 2 −𝑥 1 )2 +(𝑦 2 −𝑦 1 )2 (𝑥 3 −𝑥 1 )2 +(𝑦 3 −𝑦 1 ) 2 So two angles A and A΄ are congruent if and only if cosA=cosA΄ when Α΄ is the angle denoted by the point (x΄2 ,y΄2 ),(x΄1 ,y΄1), (x΄3,y΄3) The algebraic model of the Euclidean axioms With the acceptance of these algebraic interpretations of the primitive terms of Euclidean postulate set we may now convert each postulate of the set into an algebraic statement . this can be shown , by the methods of algebra alone, so each postulate becomes a theorem of algebra. Axiom 1 there is one and only one line passing any two given distinct points 68: To verify it in our interpretation we must show A. A. that there is an equation in the form ax+by +c=0 (a,b not both zero, all reals) which is satisfied by two distinct points x1 ,y1 ),(x2 ,y2) of the variables x,y. We consider the equation (y2-y1)x-(x2-x1)y+(x2y1-x1y2)=0 ….. (1) Substitution of x1 ,y1 ),(x2 ,y2) in the equation (1) shows that this equation is satisfied by the pairs (x1 ,y1 ) and (x2 ,y2).but this equation is of the desired for. B. But we must show that, to within a constant nonzero constant factor , this is the oly equation of the desired form satisfied by the distinct pairs of values (x1 ,y1 ) and (x2 ,y2). To this end, suppose (x1 ,y1 ) and (x2 ,y2) satisfy the equation ax+by+c=0 …..(2) with a,b,c reals and a,b, are not both 0. Then we have ax1+by1+c=0 Or by subtraction και ax2+by2+c=0 (3) a(x2-x1)+b(y2-y1)=0 (4) Now suppose a≠0 then the equation (3) become x1+By1+C=0 x2+By2+C=0 (5) where B=b/a C=c/a since a≠0 we cannot have y2-y1=0 for otherwise (4) would reduce to a(x2-x1) or (x2-x1)=0, a situation which is impossible Solving equations (5) simultaneously for B and C we have 𝑥 −𝑥 1 𝐵 = − 𝑦2 2− 𝑦 1 C= 𝑥 2 𝑦 1 −𝑥 1𝑦 2 𝑦 2−𝑦 1 which result in analogies a:b:c =(y2-y1) :- ( x:2-x1) :( x2y1-x1y2) 69: and, except for a possible constant nonzero factor equation (2) becomes our equation (1). A similar argument can be carried out if instead of supposing a≠0, we suppose b≠0. Thus postulate 1 becomes in our interpretation, a theorem of algebra. Axioms 2,3,4 5 are long and are in Eves page 96. For example in parallel axiom we can prove that if A(x1 ,y1 ) is a point and m is the line ax+by+c=0, then the unique line through A that does not intersect m is given by ax+by-(ax1+by1)=0 The isomorphism-analytic method of geometry. Thus, finally, the algebraic model of the Euclidean base is valid, it is another model of the Euclidean base. But in my same book we saw: the categorical postulate set P: it is caterorical if every two models of the set are isomophic. Yet the Euclidean axiom system is categorical. Therefore, the geometric (our known geometry) and the algebraic model of the Euclidean system are isomorphic, i.e they are identical, we can replace one geometry with the other by replacing the original terms and the relations from the archetype to the images, one becomes the translation of the other. This enables us to translate every geometric theorem into a corresponding algebraic statement and step by step to translate the geometric proof of the geometric theorem into an algebraic proof of the corresponding algebraic statement. Here we have to make clear that this isomorphism is the meaning of analytic geometry and not the algebraic model of the geometry we constructed. It is the translation that the isomorphism installs from geometry to algebra and from algebra to geometry that produces the final geometric result. That is why we say that analytical geometry is a method rather than a branch of mathematics. The student who is more proficient in formalistic algebraic process and thoughts, than in geometry meditation, prefers analytical geometry, is for him 70: the royal road to the geometry that Euclid thought did not exist, and which today, after the discovery of computers, became a royal avenue for geometry. Relative books “Foundations and Fundamental concepts of mathematics” Howard Eves Dover “Mathematics and the Physical world” Morris Kline dover «Συνοπτική ιςτορία των μαθηματικών» Dirk J.Struik Δαίδαλοσ «η ςχετικότητα τησ γεωμετρίασ και ο χώροσ» Γιώργοσ Μπαντέσ www.mpantes.gr George Mpantes Serres Greece 17/5/17 THE INFINITE, THE SERIES (TAYLOR) AND THE ANALYTICS FUNCTIONS The infinite, the series (Taylor) and the analytic functions introduction numerical series The sum and the limit The history of series . 71: Series of functions Juethadera, Taylor series, analytic functions Some results Comment Introduction . Infinity is omnipresent in mathematics agenda, hidden or apparent, in all eras. We know how much attention and perspicacity Greeks handled the concept of irrational number, associated with the idea of infinity. But even in the simplest numbers, the natural, the infinity is behind their unlimited succession , in rational numbers behind their decimal form. The first contact of mathematical practice with infinity was in the infinite sums of numbers - now called series- which occurred through direct questions about the continuity of concepts, such as space, or time , i.e if a distance can be divided in two, four, eight .....ad infinitum parts, where it is easy to put aside reality and renegotiate the question in terms of abstract objects which are infinitely divisible: the numbers (the thinkable ghost behind phenomena, Aristotle) . Numerical series . Suppose we take the algebraic sum of one million finite numbers. The sum is clearly finite . But if we assume that we consider an infinite succession of numbers, we have an infinite series . Now we can not speak of the sum of the terms since infinity is never exhausted. We can not speakof the sum of an infinite series , since infinity is never axhausted . But we may consider magnitude which plays the same role with respect to an infinite series as the sum does to a finite 72: succession of terms. This magnitude is the one towards which the sum of the first n terms of the infinite series tends when n is made to increase indefinitely, It is a non-algebraic concept, called the limit of the infinite series. Example 1 . We ½ ΢ +1 shall is / find 4 the +1 the limit / 8 sum ΢ = 1/2 +1 / 4 +1 / 8 + ... 1/2n ΢ of (1) of +1 the / first the series 16 +,,,,,,,,,,, n terms which will calculate in algebra. = 1-1/2n…….(2) Assuming now that n increases infinitely, the fraction tends to zero and have the limit of ΢, the limit S = 1, the series converges, the limit number is finite. The sum and the limit . Important differences distinguish the sum of a finite number of numbers from the limit of an infinite number of numbers. Let us mention a few. We Know that the sum of a billion numbers, some of whish are positive and others negative has exactly the same value whether we effect the summation in the order given or in any other order. But as was proved by Dirichlet in the nineteenth century , this conclusion is not necessary correct when we are dealing with infinite series: the limit may have one value or another according to the way in which the terms are ordered. But there are and other distinctions between the sum and the limit. Thus the sum of a finite number of numbers is always finite and , as just explained, is perfectly well determined. On the other hand, the limit of an infinite series may be finite or infinite, , and sometimes the series has no limit. In the two latter cases the series is said to diverge . For example, if we have the unending series 1 +1 +1 +1 + ........ the limit is infinite , the series diverges . Similarly , if we have a finite sequence of terms such as 1-1 +1-1 +1 or 1-1+1-1+1-1 the sum will have the value 0 or 1 according to whether the number of terms is odd or even. But if we are dealing with an infinite series of this type , there is no definite limit , for we have a kind of oscillation between 0 and 1 – the series diverges. 73: As an illustration of a series the limit of which is well defined and finite, we may mention the series 1+1/2+1/4+1/8+….. its limit is 2. The series 1-1/2+1/3-1/4+1/5-….. also converges, but here we must be careful not to change the order in which the terms are written. If we change the order of the terms ,we may find that the series diverges. Today with the notation we mean the series and with the notation we mean the limit of series. The History of series . The first series appeared is that of Zeno (example 1) ½ +1 / 4 +1 / 8 +1 / 16 +…… which caused the historical paradox of dichotomy. What was the source of the paradox? Zeno saw the limit as a sum within the Pythagorean culture of distinctness, he did not conceive the meaning of limit, (here it was 1, but for Zeno it was infinite as we don’t stop adding terms) he did not understand infinity through mathematics. A cavalier treatment of infinite can lead to absurd results! The first who historically calculated a limit of a series was Archimedes. He proved that the area of a parabolic segment ABCDE is 4/3 the triangle ACE having the same base and vertex. For this purpose, he constructed continuously an 74: unlimited sequence of inscribed triangles, starting from the initial ACE of area E, adding triangles between the existing. He conceived that the area of the parabolic segment will be covered after infinite such triangles, and by geometrical tricks, (δια των γεωμετρούμενων), he did not the concept of limit, he found that E(1+ 1/4+ 1/16+ 1/64+……….)=4/3E or 1+ 1/4+ 1/16+ 1/64+……….)=4/312 This process was a conceivable experiment in the field of geometric intuition. The experiment does not end but we watch it’s course as we want, even we imagine this. Hence we perceive the result of the experimen.. Later, this geometrical process was replaced by numbers, and perfected when we completed our knowledge about the real numbers Until the 18th century the procedures of infinity were treated without rigor and precision , perhaps as reported by Struik, because of uncontrolled enthusiasm ( history of mathematics Dirk Struik ) . Known limits of series in the period before Cauchy were (Leibniz) and (Euler) So maddening infinity initially disappears thanks to our own ingenuity , and the results were strange . …As Newton , Leibniz, the several Bernoullis, Euler , d’ Alembert, Lagrange, and other 18th-century men struggled with the strange problem of infinite series and employed them in analysis, they perpetrated all sorts of blunders, made false proofs, and drew incorrect conclusions; they even gave arguments that now with hindsight we are obliged to call ludicrous….. (Morris Klein ). " ..... But questions linger centuries answered , preparing us for a long and deep mystery ." Cauchy, ( ultimately in 19th century ) observed that between the infinite and the infinitesimal there is an interdependence . If we look at the sequence 1,1 / 2, 1/3, ¼, 1/5, ....... will …. These equations are no less certain than the others…albeit we Mortals whose 12 reasoning powers are confined within narrow limits , can neither express nor so conceive all the terms of these equations , as to Know exactly from these the quantities we want…..Newton 75: observe that with increasing the order of terms, they are closer to zero . Approaching zero does not end ( infinitesimal ) as does not end the succession , (infinite ) " ... when the successive values given in a variable, approach indefinitely a fixed value differing from it as one wishes, the latter is called limit of all others .... " noted Cauchy and translated this definition in a rigid formulation : Definition 1. say that the number λ is the limit of the sequence xn if for every e > 0 there is n0 ЄN such as for n > n0 we have | xn - λ | < e. The sequence converges to λ and we now have a strict convergence criterion for all sequences. Based on this definition, a calculus of limits developed for series, (root test of Cauchy, ratio test of D’Alembert, comparison test ..). These tests stopped the tricks for each series separately, we can manipulate satisfactorily many problems in the convergence of series, that up Euler was the weak point of mathematics. Series of functions . Series of functions differ from numerical series in that the successive terms are functions of a magnitude x whose precise numerical value is not specified. Thus the infinite series f1(x)+f2(x)+f3(x) +……….. where f1(x), f2(x), f3(x),…..are known functions of x, represents a series of functions. The functions now define numbers, and our series of functions becomes a numerical series. By changing the value ascribed to x , we define one numerical series after another. For example a series of functions is x-x3/3+x5/5-x7/7+…….. for x=1 we have the numerical series 1-1/3+1/5-1/7+……. For x=2 we have 2-8/3+32/5-128/7+…….. In most cases the numerical series converges if certain values of x are substituted, whereas it diverges (i.e. has no limit or becomes infinite) when other values of x are taken. Suppose for argument’s sake that the series converges for all values of x of the interval (a,b), and diverges for all other value of x. The limit of the series then has a welll-determined value only 76: when x is situated between a and b, and usually this value will depend on the value assigned to x in the interval. The limit of the series thus defines a function of x. But note that it is only when x is situated between a and b (where the series converges) that the series defines a function; elsewhere the series is meaningless and defines nothing. 13 Jyesthadera . A first occurrence of the phenomenon of series of functions comes from India, one hundred years before the results of the Europeans. The Indian Jyesthadera (1500 AD ) expressed the number π as a numerical series. Previously demonstrated algebraically the expansion: The trick is algebraic and piecemeal, we know nothing deeper. He replaces the parenthesis from the identity: Jyesthadeva thereby turned a simple fraction into an infinite series. But he did not suspect that the effects of formula 2 apply for certain values of x. If we set x = 1 in both sides we have that ½ = 1-1 +1-1 +1-1 ........ which is not true as stated . For x = -1 have the result 1/0 = ∞ and x = 1/2 we have 2/3 = 1-1/2 +1/4-1 / 8 +1 / 16 ....... which may be true or not. I.e. For different values of x the result of Jyesthaderan leads to "truth or mystery or nonsense." (David Perkins) Today we know that the formula ( 2 ) is valid for -1 < x <1, he could not know this, but in his proof, the geometric entanglement of the formula in a figure, ensured incidentally that π is in the above range. So the formula was correct for x=π although Jyesthadera did not note it. 13 In 1828 Abel wrote: divergent series are the invention of the devil and is unorthodox to rely on them to prove anything. 77: With formula (2 ) and geometry Jyesthadera proved the surprising result in the numerical series : (3) “….Who would suspect this link between the constant π and a series involving the odd numbers?..... David Perkins”. Taylor series , analytical functions . This process of Jyesthaderan was the task of life of Brook Taylor ( www.mpantes.gr, οι μεταμορφώςεισ τησ ςυνάρτηςησ), who calculated the necessary and sufficient condition that a function be developed in power series, ie a polynomial with infinite terms (long polynomial), that is a technical and difficult subject , the known as Taylor series. If f has infinitely many derivatives at the point where x=a, we conclude with Taylor that the power series generated by f at x=a is f ( x )  k  0  f ( k ) (a) ( x  a) k .......... .......( 4) k! But the second half of the transformation was to study the convergence of the resulting series . This was ignored by Taylor. The complete study of power series was by the founders of the complex analysis (Cauchy, Riemann, Abel, Weierstrass) a century after their construction. The series (4)always converges for the value x=a since it reduces to the first term. But the power series may also converge for some other value of x. Mathematical analysis shows that in this case the series will necessary converge for a continuous range of values of x and that the point x=a occupy the centre of the range. Within this range of convergence, the power series defines a function of x which can be shown to be a continuous function, all of whose derivatives are also continuous. A power series of this last kind (which converge in an interval) constitutes a Taylor series so that a continuous function defined by a power series is what we have called an analytic function of x. Example: the function y=sinx ……(5) 78: Substituting the formula (4) for the coefficients in Taylor expansion we have y  x x3 x5 x7    .......... .....( 6) 3! 5! 7! Yet we can’t equate the right sides of (5) and (6) before we study the convergence of the series. For some x could the two members give different results! Today we conclude (ratio test) that the Taylor series in (6) converges for all x ЄR, that is the domain of the function. Now the two expressions (5) and (6) are equivalent and the function y=sinx is called the analytic function. Taylor considered it for granted for all the functions. But all functions known to the mathematicians of the 18th century happened to be analytic. Thanks to this coincidence, (analytical ) function became a mathematical tool capable highlights the corollaries of mechanical motions for point masses and rigid bodies and solve the differential equations describing the laws of these phenomena . So the successful resolution of many problems in physics and celestial mechanics, which would not happen without the Taylor series, due to the fact that the functions involved in these areas were analytic . The c o r r e c t r e s u l t s w e r e i n c i d e n t a l because the convergence conditions were applied to the problem, unnoticed by the scholars . It is really a fact that the progress of physical theories is dependent of the mathematical development of the day. But while the rigor is essential for mathematics is often preferable not to require in early stages. An early insistence on rigor could strangle discovery. History shows that the details can wait ... .. "what serves our admiration for the work of the builder if we can not appreciate the design of the architect? Poincare » Some results The analytical function described relations (there are many such in the world of engineering) which are not directly expressed with elementary expressions but with successive approximations, with both practical results (however accurate ) getting some terms of the series, also theoretical results by resorting to limit. In this sense the work of Brook Taylor is one of the most important works of Mathematics. The extension of the capabilities of the function after it’s development in Taylor series (analytical function) There was no way to express the initial of y= sinx/x or is y  ex 2 obvious. 79: There were computational problems for example the value of cos1 0 As cosx=1-x2/2+x4/4-x6/6+…. and for x=π/180 from the two first terms we have cos10=1-0,00015=0,99985. We can find approximately the solution of the equation cosx-2x2 with the previous series or the indefinite integrals of sinx/x , or e-x2, as and differentiation of functions, because technically it is easier to work with the simpler terms of the series. But the more substantial contribution of analytic functions was to solve differential equations with great practical and theoretical importance. The equations of Airy, Legendre, Hermitte, Bessel, are equations that their solutions can not be written in terms of known functions such as polynomials, exponential or trigonometric, but only be expressed in a power series. For example, the differential equation of the pendulum could not be resolved, the physical description were at an impasse . Comment The graph of an analytic curve called analytical curve. This has an almost metaphysical status. The mathematical analysis shows that if we know exactly a part of it, no matter how small, we can design the whole, we know the whole route! If two analytical curves coincide at a small fraction of them arbitrarily then necessarily coincide in their entire length. Here we can diagnose the doctrine of causality in physical systems. If we know the initial conditions we can describe their evolution. That is described here is the uncanny ability of mathematics to describe nature, analytical curves involved in the doctrine of causality of natural systems. Sources . Οι μεταμορφώςεισ τησ ςυνάρτηςησ Γιώργοσ Μπαντέσ (www.mpantes.gr) Modern mathematical analysis Murrey Protter (Addison-Wesley publishing Company) Μαθηματικά Β Λυκείου (ΟΕΔΒ) (από τα καλύτερα βιβλία ςτο Λύκειο) A short account of the history of Mathematics: W.W. Rouse Ball Calculus and its origins , David Perkins (Mathematical association of America) Γενικά Μαθηματικά (Σόμοι Α, Β) Ν. Αναςταςιάδησ, Θες/νίκη Θεωρία Μιγαδικών ΢υναρτήςεων Ν. Οικονομίδησ, Θες/νίκη ΢ΣΟΙΦΕΙΑ ΑΝΩΣΕΡΩΝ ΜΑΘΗΜΑΣΙΚΩΝ ΕΥΗΡΜΟ΢ΜΕΝΩΝ, ΢Σ. ΒΤΖΗ 1966 The Taylor Series, by P. Dienes, Dover Publications 80: Infinite sequences and series by, Konrad Knopp, Dover Publications A short account of the history of Mathematics by, W.W.Rouse Ball , Dover Publications Mathematics and the physical world, by, Morris Kline, Dover Publications Mathematics the loss of certainty by Morris Kline, Dover ΢υνοπτική Ιςτορία των Μαθηματικών Dirk Struik Δαίδαλοσ, Ζαχαρόπουλοσ The rise of the new physics , by A.D’ Abro , Dover 9. THE CONSTRUCTIVE STRAGEDY OF MATHEMATICS …..it is true that whoever mathematician is not a kind of poet will never be a perfect mathematician......Weierstrass We read in text books: the logical gap in Mathematics initiated by the paradoxes of Zeno , finished with the construction of the real numbers by Dedekind, on 19o century. Yet we read in the mathematical literature: : .... O Dedekind defined the system of real numbers as a collection of all “cuts” of rational numbers (Leo Corry 1996), ...... and each such cut that doen’t correspond to rational number , defines an irrational number (Landau 1917), ......... if along with Dedekind we conceive a real number as a .. (Weyl 1919) ....... we identify the real numbers with certain sets , called Dedekind’s cuts ( Maddy 1992) ..... in any given section represents a certain rational or irrational number (Dedekind 1872), construction of real numbers with the Dedekind’s cuts (Wikipedia), …specific creation of new irrational numbers .... (Dedekind, Continuity and rational numbers) and we have many words , definition , design, representation , identification , construction , to describe the logical extension ( Dedekind’s cuts) of number system of rationals , on another ( reals ) . From all these words, seems at first that the new numbers is a human act , a free creation of the mind , as says Dedekind. What exactly is meant? Mathematical contruction . 81: We have a clean criterion for when we may say something “exists” for mathematics in Euclid’s Elements , where a figure , such as a circle or a square, will exist, only when we have drawn it. We have the idea of a square, and according to that idea, we frame a definition. But a definition does not assert that what we have difined, exists. (see the commentary on the definitions on Euclid’s Elements. That corresponds to the operationalistic school, in Physics (P.Bridgman). It is only when we present the logical steps to draw (construct) a square, that shows it is more than just an idea. The square that exists for mathematics is the square we have actually produced. As with everything in life that begins as an idea, we must bring it into reality. This reality for the square is the reality of the ruler and the compass. This example shows the difference between the existential approach to the constructive approach of mathematics. In school mathematics, a very small part is in the first category, no one cares if the sum of two numbers exists, just goes and finds their sum. The constructive approach differs from existential that is not satisfied to work with ideas whose justification arises from an existential proof, but insists that the objects corresponding to these ideas are constructible in some mathematical reality. The reality of the geometry is attributed better to the word construction , the rule and compass are realities. We construct the square, the equilateral triangle, and all that we remember from the geometry of the school. The equilateral triangle is not constructed in elliptic geometry , so it doesn’t ‘exist’ there , the hyperbolic line ‘exists’ as the geodesics of a pseudosphere, etc. The same structures of concepts were preserved in the great revolution of algebra in abstract algebra which was a movement of algebra as generalized arithmetic in a purely formal (formal) axiomatic algebra. Again the requiring mental manipulations that transform the meaning of the definition in operational concept by Bridgman ( treatise Bridgman, « The logic of modern physics » ), in a concept of mathematical reality. Such manipulations are exactly the construction of real numbers of Dedekind, as in the construction of the Euclid square. The proper word for attributing meaning to the expansion of a number system that represents the concept of this mathematical construction is the same as the geometry , the construction of the new number system , but now change the meaning of reality. 82: The construction of new numbers. As the reality of the square is located in the level, straight and circle ( the manipulations made with ruler and compass ) , now the reality of the numbers is 1. the positive integers and 2. their axiomatic basis, ie the laws governing the addition and multiplication of positive integers. ( basic operations of positive integers , we met in nature, from which all began ) . ….Successive extensions of the numerical scheme is justified on the one hand by our desire to extend our operations applicable to a system in reversal operations , or to more general , and on the other from the goal to have a numerical system that is "closed" for these operations .. Dedekind. These laws define the operations and then the new operations define the various artificial numbers, as typical results. Each new number system produced by the former, finally by positive integers with a interference –extension of properties of the old operations, the new axiomatic basis , of the new set of numbers.. These operations make up the numbers , not conversely!. Well the actualization of the complete ordered field in Dedekind, will start from the reality of rationals , with an extension of their operations ( as we extended substraction in positive integers when a < b ), to fulfill the axiom of continuity. …just as the negative and rational numbers resulted by a new creation and the rules of the operations of these numbers must and can be traced back to the rules of acts of positive integers , so we should try to fully define the irrational numbers only through the rationals, in my work ….i show that in the field of rational numbers we can identify a phenomenon, the cut, which can be used to fill this field with a particular creation of new irrational numbers , and I prove that the resulting field of all real numbers, holds the property which I assume as the foundation of continuity. I also show that the addition and all operations of real numbers are defined with all the rigor and that on this basis, the proposals form the backbone of arithmetic can be proven with all the rigor .... (Dedekind, Continuity and rational numbers internet) 83: That is the logical steps of the construction of Dedekind, (this creation- construction) were the "cuts" with the whole equipment, the Cantor’s (another specific creation) were the sequences of Kauchy similarly etc. The best known constructions are (they exist in all books of modern analysis or set theory) a. construction of Weierstass, who never published it but presented in his lectures and posted by students with b . The construction of Dedekind, with the different variations. cuts of rational numbers, and c. The construction of the Cantor with rational sequences of Cauchy (proof of Cantor, http://www.math.nus.edu.sg/ ~ urops / Projects / RealNumbers.pdf). The last two were published almost simultaneously in 1872. But there are too many other mathematical structures that have been proposed as new structural interpretations of real numbers. Each generation is reviewing real numbers, in light of its own math standards. Philosophical reference: This article was associated with the philosophical deal for the existence of mathematical objects. What is the relationship of mathematics with reality? Where are the mathematical objects? Mathematicians create as poets or discover as Columbus? In Greeks, we have the two eternal rivalries: the Platonic and Aristotelian forms. Plato argued that the objects of mathematics are in a eternal and unchanging realm of ideas, so we're not looking any relation to reality we know. The mathematical objects such as numbers and geometric objects are not created, not destroyed and can not be changed. It is the knowledge of eternal being and not those once vanishes and once born ........ (Πολιτεία Plato). We can immediately observe that the eternal world of Plato is located in our imagination, since only there time is not flowing and the changes stop, which exempts it from the physical concepts, since it is not defined operationally . 84: The other answer was given by Aristotle, and we basically witnessed in the history of real numbers . As we saw the irrationals exist in an abstract chain of deductive reasoning , but starting from sizes that can not be measured with the rationals (diagonal of a square with perpendicular sides equal to one ) . Born in our minds , the way we think , because there is the Pythagorean theorem , but start from things that touch the senses. It's what we call Aristotelian forms, not Platonic forms, if we had not met the diagonal of the square , we shouldn’t create the irrational numbers . .... Aristotle postulated some mental capacity of abstraction with which (mathematics ) objects created or otherwise produced or captured by consideration of physical objects ... objects that are obtained through abstraction , there are no pre - or independently from - the objects of which have been obtained. Note that arithmetic and geometry are verified literally with such an interpretation , in which the missing link is an explanation of abstraction (Stewart Shapiro: Thoughts on mathematics University of Patras ) We observe Shapiro's words over the words of mathematicians at the beginning of the article. This classifies Aristotle on top of constructivism. The reality of numbers goes back in the sheeps we were (once upon a time) mapping with the fingers of our hands. The others created, produced or captured. This view of Aristotle is the beginning of the theory of the constructivism , ie the step-by-step construction of the mind, of mathematical objects. For fans of this school the Platonic view is ontology of metaphysical nature. "If existing has not the meaning of constructing then necessarily it has metaphysical significance (Υίλων Βαςιλείου)). This view was investigated by α whole school called intuitive school , led by Dutch LEJBrower. Bibliography Calculus and its origins : David Perkins Dedekind:, Continuity and rational numbers Essays on the theory of numbers, (διαδίκτυο, Morris Klein: Mathematics and the physical world Henry B Fine: the number system of Algebra (internet) Eric Schehter : What are the “real numbers” really? (internet) Thoughts on mathematics Steward Shapiro George Mpantes mathematician, mpantes on scribd . 85: THE ACTUAL INFINITY IN CANTORS SET THEORY 1.The actual infinity Aristotle-Cantor . 2.The origins of Cantor’s infinity 3. The natural infinity 4.The mathematical infinity 5. A first classification of sets 6. Three notable examples of countable sets 7. The 1-1 correspondence, equivalent sets, cardinality . 8.The theory of transfinite numbers 9. Existence and construction The theory of Cantor, introduces us to the mathematical study of infinity . The definition of the Definition Cantor set is : 1 . A set is a gathering together into a whole of definite distinct objects of our perception or of our thought which are called elements of the set…G Cantor 1. The actual infinity, Aristotle-Cantor . The above definition in the case of infinite sets , establishes the concept of active or actual infinity, namely the existence of an infinite set as a mathematical object (in whole), on par with the numbers and finite sets . 86: The concept of actual infinite reaches from Greek antiquity and was rejected by Aristotle . " ..... There are no infinite objects , wrote Aristotle , and the infinite totalities of objects do not form objects of study . Infinite totalities of objects can be studied only internally ... ( potential infinity) there is no actual infinity, viz infinity as a whole , whose parts would exist simultaneously , otherwise we would not comprehend nor the continuity of time (the time would have start and end) , nor divisibility of sizes ", which means that for Aristotle the only meaningful study is the study of potential infinity ( Anapolitanos 2005 ) He wrote that: …infinity does not exist in the form of an infinite solid or an infinite magnitude perceived from senses ... infinity exists only potentially, while actual infinity as a subject , is the result of a mental leap , which is a process not allowed…. Gauss later (1831) said that : in mathematics infinite magnitude may never be used as something final:infinity is only a façon de parler, meaning a limit to which certain ratios may approach….. Let's look at the mathematical version of the above : We consider the natural numbers ( 0,1,2,3 , .... n, n +1, ..... ) . Potential infinite in this case means that one can continue to write terms infinitely without never reaching to an end of the succession, the infinity of numbers is tested internally. So for Aristotle, the collection of natural numbers is potential infinity as there is not a greater last natural , but not an actual infinity as it does not exist as an integrated entity " ( Aristotle) But in mathematics of Cantor, set is a certain object whose clarity arises from the fact that it has exactly the elements that indicates, and if the natural numbers are a set N , then the above procedure is considered finished , and N is actual infinity the same for rational numbers , so for real numbers . Here we say : the natural numbers is potential infinity because there is no greater number , but the Cantor’s set of natural numbers is actual infinity because it exists as an integrated entity . Thus an expression “actual infinite set " is redundant , saying only infinite set . The concept of Cantor’s set is a development exceeded the physiology of the human brain , which in the route of it’s growth has learned thinking linearly (especially 87: regarding time ) , the only way to think infinity is a line that never ends , or a thing that exists forever , how to conceive a line without end but finished ! We forget intuition , we did and other times , let us remember the space-time, the continuum , etc. Descardes said that infinity is recognizable but not comprehensible. In Aristotle's question "where the human being can sustain the controversial conceptual leap of objectification the infinity ? "The answer is : in the concept of Cantor’s set. In the sequence 1,2,3,4,5…we can write a billion, a trillion,…..terms but all of them are aleph (null) terms. 2.The origins of Cantor’s infinity . The origin of infinity has to do with it’s trading. Cantor makes a tripartite distinction of infinity in three contexts : "….first when it is realized in its most complete form, in a fully independent supernal being , in Deo, whereat I call Absolute Infinity , (that means God ,) the highest perfection of God is the ability to create an infinite set , and His boundless magnanimity prompted him (Cantor) to create it ." We are in the heart of metaphysics ! The infinite set created by God , and him (Cantor) discovered it ! secondly when displayed in the eventual world of creation ( ie the natural infinity) , third when the mind conceives it in abstracto, as a mathematical magnitude , cardinal or ordinal number. ( the mathematical infinity ). I want strongly to dissociate the Absolute Infinite from what I call transfinite ( will see the meaning of the term) , ie integrated infinite of the second and third kind, which are clearly delimited , can grow more, and therefore are associated with the finite ….Cantor ) . 3. The natural infinity. Is there the infinite "out there" in nature ? Here things are clearer to the reader since neither the individuality of matter , nor later of electricity and energy advocate with the idea of infinite divisibility as it is contradicted by the experience of physics and chemistry ( Hilbert , for infinity ). 88: However, there are three areas in which our world appears to be unblocked and therefore infinite . It seems that the time is not possible to have an end , also the space , and yet it seems that every spatial or time interval can be divided indefinitely . But, recent opinions in physics verify the views of the operational definitions of physical concepts , ( The Logic of Modern Physics, Bridgman) , without exception , even in mental (i.e mathematical concepts, as the mathematical continuity), so the assertion that time is infinite is a mere statement, such as the statement of classical mechanics, that time is global… the concepts of science are defined by sets of operations , and not by arbitrary definitions of philosophical type , they are in a Platonic world only in our minds , Bridgman is the modern Aristotle for the new realities of physics (relativity). What experiment has proved that time is infinitely divisible? It is readily apparent that experiments and infinity are incompatible since the life of the experimenter is finite . In all theories of physics the problem of infinity is responsible for creating impasses . Even Zeno 's paradoxes arise from trying to connect infinity by the motion of natural bodies . These ( theories ) start to accept infinity initially and then slowly pooping out this concept to be able to produce equations to correspond to reality. The vision of the Dirac was just that: to throw out the infinities of the equations studied ( ' unification of forces” Abdus Salam). That seems as the second infinity of Cantor , in the eventual world of creation, the natural infinity, doesn’t exist just as Aristotle considered .. 4.The mathematical infinity : Until the nineteenth century mathematicians systematically abstained from the concept of actual infinity . So when Cantor introduced actually infinite sets he had to advance his creations against conceptions held by the greatest mathematicians of the past. The infinitesimals which founded calculus is an example of the contradiction between the two concepts: they were treated as both potential and actual infinity. Potential because they are constantly decreasing quantities without end . Actual because they take part in operations and behave as zero. The unending process of operation ( reduction ) that produces them, is considered finished . Restoring logical consequence in Calculus is done using only the potential infinity . This is basically the meaning of expulsion of infinitesimals from the scope of Analysis : is dismissed the actual infinity. Indeed the 89: great of analysis , as Cauchy, Weierstrass and others talked about the infinite small and the infinitely large just internally , through permissive acts of limits, ie as something that is born , formed , for an infinite process without ever achieved identification . They traded the potential infinity ( eliminate the infinitely large and infinitely small and reduced the proposals referred to them in relations between finite sizes , Hilbert ) , ie the rigorous foundation of calculus is on mathematics of potential infinity . Only in the work of Dedekind the infinite sets become self-consistent logical entities , ie the mathematical study of actual infinity has begun. This was completed by Cantor . 5. A first classification of sets . Two sets are called equal A = B if they contain the same elements. Sets are divided into finite and infinite, An infinite set called countable set if we can construct an 1-1 correspondence between the elements of the set and the natural numbers. 6. Three notable examples of countable sets . theorem 1 : the set of all rational is countable. The principle behind the above proof is called first method of diagonals (Kamke). Consider first the positive rational numbers . We imagine lines as follows : in first line all integers in order of magnitude , ie all rational with denominator 1, second line the numbers of the first line divided by two, third likewise divided by 3 etc . Thus we have the sequence of numbers in the order that the line meets omitting the numbers we get, is clear that in the sequence will appear every positive rational only once 1, 2, 1 / 2, 1 / 3, 3 , 4 ,3 / 2, 2 / 3, 1 / 4 , .... And if we denote this sequence by { a1 , a2 , ...... } is obvious that { 0, - a1 , a1 , - a2 , a2 , ....... } are all the rational numbers and the set theorem 2 : is countable . 90: the set of all algebraic numbers is countable . It is possible , however, that all infinite sets are countable . Separating sets in countable and non- countable acquires significance only after proving the existence of non countable sets . Their existence follows from the following theorem of Cantor theorem 3 : the set of real numbers in the interval [0,1 ] is not countable . ( uncountable ). The proof is based on a method the second method of diagonals of Cantor difficult and imaginative. So the infinite sets are subdivided into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets). So he proved the surprising result that the set of whole numbers is equivalent to the set of rational numbers but less than the set of all real numbers. We can’t enumetate the real numbers. Corollary : the set of transcendental numbers is not countable . (We presuppose the existence of transcendental numbers). 7. The 1-1 correspondence, equivalent sets, cardinality . Now the next question is : is it possible a further division of the class of noncountable sets ? Cantor raised this question in the following manner truly intelligent (Kamke): « Is it possible to generalize the concept of number , so that in each set corresponds one of the generalized these “numbers” as somehow typical of the " number of elements "? If this could happen then it will automatically result a classification of infinite sets , depending on their " number of elements" . Cantor , needed new numbers to measure the infinity. The 1-1 corresponcence . Cantor’s basic idea to distinguish the infinite sets was the one-to-one correspondence . We can set up the following one-to-one correspondence between the whole numbers and the even numbers 1 2 3 4 5 …… 2 4 6 8 10….. 91: Each whole number corresponds to precisely one even number its double and each even number corresponds to precisely one whole number its half. Hence Cantor concluded that the two sets contain the same number of objects. Now an infinite set can be put into one-to-one correspondence with a proper subset of itself. In this correspondence, Cantor saw that infinite sets could obey laws that did not apply to finite collections , as quaternions could obey new laws that did not hold for real numbers. This correspondence is a thought experiment , a process we saw several times in science (space –time, continuum). We get conclusions applicable equally to finite or infinite sets. Definition 2 two sets A and B have the same power or are equivalent if and only if they can have one to one correspondence with each other eg A = {1,2,3 ....... 24} B = {letters of the alphabet} . The concepts: equivalence and correspondence are the drivers of Cantor in his research. We mention a few examples of equivalent sets. 1.For points any two finite intervals T1 and T2 always happens T1 ~ T2 2.A semi-line 3.The sets and of an points entire [0,1], line are (0,1], equivalent [0,1), with (0,1) an are interval. equivalent 4.The interval [0,1] is designated as the continuum. The intervals, the semi-lines and the lines are equivalent to each other, especially with the continuum.(fig1,2) 5. there are infinite sets that are not equivalent to each other, for example, the continuum and a countable set. But this does not preclude that all uncountable sets are equivalent to each other. Therefore, the following proposal acquires fundamental importance theorem 4 : there are infinite sets that are neither countable nor equivalent to the 14continuum. (Second method the diagonal) The cardinality . 92: Cantor gave the concept of cardinality ( cardinal number ) with a very sophisticated way ( Katerina Gikas N ) , in which if we take a random collection of discrete objects and remove all the physical properties of each object , then remains "something" called cardinality or cardinal number of the objects in the collection. A trio of trees and a trio of apples have a common property which we call three. The cardinality so appear as a characteristic of sets . All the sets that have the same cardinal number with { a, b } are said to have cardinal number two , all sets with the same cardinal number as the set { a, b , c } are said to have three cardinal number , etc. and we denote cardinal numbers one, two, three , ... with 1,2,3 .... The cardinality of the set, no matter how abstract seems this definition, agrees to the case of finite sets with the number expressing the number of their elements. Even we will get the sense of the crowd, when move from finite to infinite sets ( Transfinite numbers , an extension of R). .... and the mathematicians who incorporate various branches of mathematics within set theory are justified to identify the finite cardinality with natural numbers for the purpose of this activity ... (Moshe Machover). 8.The theory of transfinite numbers . Just as it is convenient to have the number symbols 2,4,8,etc to denote the number of the elements of finite sets , so Cantor used symbols to denote the number of elements in infinite sets (their cardinals). These symbols are known as transfinite numbers. The set of the whole number and sets that can be put into1-1 correspondence with it, have the same number of objects and this number is denoted by ‫אּ‬0 (aleph null, aleph the first letter of the Hebrew alphabet ). Since the set of all real numbers proved to be larger than the set of whole numbers , he denoted this set by a new number c, the cardinal number of continuum15. 15 Before Cantor mathematicians accept a single infinite , denoted by the symbol ∞, and implied the 'number ' of elements of sets such as the natural numbers or the real numbers 93: To compare cardinalities, we introduce the relationship “is smaller than” by the definition: cardA <cardB iff A is equivalent to a subset of B and B is not equivalent to any subset of A e.g. cardQ <cardR. Cantor discovered and proved operations of an arithmetic with transfinite numbers, If n these is a finite cardinal are: ( natural number ) : Note that subtraction and division are not defined in this arithmetic . There are not zero , unit , inverse and negative . The well-known theorem of Cantor tells us that the power set (the set of subsets ) of a set is greater than the whole. If the set has n elements of the power set has 2n Cantor by considering all the possible subsets of the set of integers, was able to prove that 2‫= אּ‬c. Also he proved . Cantor's theory provides an infinite sequence of transfinite numbers and there is evidence that an unlimited number of cardinals larger than c, there is in reality. 1 = The 2 = The size size of the power set of a set with size aleph zero of the power set of a set with a size aleph one, 3 = The size of the power set of a set with a size aleph two ..... We take this way the sequence of the first infinite transfinite numbers 94: But are there transfinite numbers between ‫אּ‬0 and c; the name ( aleph 1 ) was given for such a cardinal . The belief was that there was not such a number, ie there is no set with cardinality between the two . This belief is known as the " continuum hypothesis ", and was not possible to be proved . Finally it has been shown that is not supported by the usual axioms of set theory , but is usually taken as an additional axiom (Paul Cohen). At this point the theory bifurcates, here applies the continuum hypothesis there is not true ! The situation is analogous to the parallel postulate of Euclidean geometry . In addition to the transfinite numbers already described which are called transfinite cardinal numbers , Cantor introduced transfinite ordinal numbers symbolized by ω. The distinction is rather delicate, it is another infinity different than ‫אּ‬, is subject in different operations and laws i.e ‫אּ‬+1=1+‫אּ‬ ω+1≠1+ω, but it is beyond this description. Set Theory, as developed by G. Cantor, is often termed naive as it was based on the intuitive notion of sets and their properties. But this was the third crisis in the foundations of mathematics, after the irrational numbers and infinitesimals. Began to appear reasonably paradoxes and contradictions in the Cantor building, and to avoid such inconsistencies and keep Set Theory contradiction-free, mathematicians came up with several axiomatic systems, of which the one known as Zermelo-Fraenkel (ZF) became the most popular. This story, however, is beyond the limits of this article . 9. Existence and construction . The two schools of mathematical philosophy , constructivism and realism , carry in centuries the differences between Aristotle and Plato . In the case of actual infinity Aristotle does not admit infinite sets as a totality, or a seven –sided regular polygon. The Platonists (Cantor was one) on the other hand believe that this exists in some objective word independent of man, who discovers them. ….After so many centuries may be gathered that the root of the disagreement lies beyond logic and mathematics, in obscure psychological differences pertaining to the various kinds of minds. There is not hope of the one ever convincing the other, so that nothing is to be gained by perpetuating endless controversies…D’ Abro.. 95: But the problem of existing question is transferred as problem in the value of existing proofs. Cantor proved that there were more real numbers than algebraic numbers. Hence there must exist transcendental irrational numbers. But this existence proof did not enable one to name, much less calculate, even one transcendental number. Also Gauss , proved that every nth degree polynomial equation with real or complex coefficients had at least one root. But the proof did not show how to calculate this root. Many mathematicians regarded the mere proof of existence as worthless. They wanted a proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object, say that the proof of existence should enable mathematicians to calculate the existing quantity. Such proofs they call constructive proof. But if we do believe in mathematical reasoning we need not to distinguish between the existence and construction proofs. Existence or constructive proofs are finally mathematical proofs, and mathematical reality is the mathematical proof. Indeed the first predisposes to a strict construction, (after we have secured the existence). But in actual infinity can not occur an constructive proof, we can not construct some aleph number. Actual infinity divides mathematics in two parts keeping for itself just the existential part of math, we can’t palpate actual infinity but only to photograph it. It is a mathematical reality just for Platonists, without a connection with the real world round us. Sources . Amalia Christina N. Babili (διπλωματική εργαςία): Σο μαθηματικό άπειρο , τα παράδοξα και ο νουσ Γενικά Anastasiadis: μαθηματικά Howard Eves: foundations and fundamental concepts of mathematics, Dover . Moshe James Stein Machover: How set math theory explains and their the world limitations. Avgo Katerina Gikas N: Θεμελίωςη του ςώματοσ των πραγματικών αριθμών , ιςχύσ και διάταξη αυτού, διπλωματική εργαςία), 96: Robert L.Vaught E. Kamke: :set theory theory: (versions an introduction) Karavias 1962) Patric Suppes: Axiomatic set theory (Dover) Hilbert D’Abro: the rise of new physics Dover Morris Klein:mathematics the loss of certainty Dover THE MATHEMATICAL FORMS OF NATURE THE TENSORS The first mathematical forms of nature that man discovered were the numbers. The strong premature impression of the concept of form of beings, that later proclaimed by Aristotle, appears in the doctrine of the Pythagoreans "Everything is number ." Now, the simplest mathematical form of nature, towards the homogeneous and isotropic space is, as we see, the vector . Later this form proceeded deeper, in spaces not homogenous and not isotropic, in surfaces, in curved space and in other media, extending the concept of vector in this of tensor . For example, the description of the pressure and tension at each point of an elastic medium which has undergone deformation is given by a second order tensor 16. ( vectors are first class tensors ). Willard Gibbs linked tension with tensor since it was impossible to express the situation in a threedimensional space of an elastic medium with only three components. Now the stress tensor is the mathematical form of the deformation of the elastic medium. What happens with the “homogeneous and isotropic” space ? We see in the example that the coordinates of the force is transformed in the same manner as the coordinates of the position. The change of the coordinates of a specific change of the reference system is not arbitrary but obey 16 “Concepts and methods of theoretical physics” by Robert Bruce Lindsay Dover p. 298 97: certain mathematical rules17 . This conclusion is based on the physical assumptions of Newtonian mechanics to the nature of the force , which is the result of experiments and observations. So the form of the vector was found in nature and was developed in mind by mathematics ( the form can not be separated from things, despite except with intellectual energy ... Aristotle Physics193b 5) This particular behavior of components , enables us to discover a criterion of objectivity. The components themselves do not have absolute existence, do not transcend the reference system. They represent but partial aspects of vectors mere modes or shadows varying as they do, with our system of reference. But in the vector can be assigned an objective existence which exceeds the reference system , since the components in another system obey strict mathematical rules. If they were not linked together by the stringent mathematical rules we have referred to , we should to recognise that we could not contemplating the same vector when we passed from the first coordinate system to the second. Thus the set of numbers (1.2 , -3 ) by itself is not any form . To obtain mathematical and physical meaning therefore must not only determine whether these numbers are the components relative to a base vectors ( reference system ) , but also which of the infinite number of bases was chosen. The tendency of mathematics for generalization through abstraction is always ahead of us . The vectors were considered as tensors of first order (relative to the reference system ) while the scalar sizes ( as classical mass) zeroorder tensors. And tensors ? (Always relative to a reference system ) is mathematically sizes dependent of components, which as in vectors but more complex, have specific rules of change in a change of the reference system. It is one of the "higher genera " of Aristotelian forms . Let now pass to one of the most important characterisics of vectors (and tensors). The fact that their components are submitted to the same rules of change when we pass from one coordinate system to another, proves that if two 17 The rules of the orthogonal linear transformations of Cartesian system. 98: vectors (tensors) situated at the same point of space are equal that is if the components of the one are equal to the components of the other in our coordinate system, this equality will inevitably endure in any other coordinate system. In other words, the equality of two vectors (tensors) in a point of space constitutes an equality global, which a change in our coordinate system can never destroy. So the equations between vectors (tensors), often called vector equations and tensor equation exhibit the remarkable property of remaining unaffected by a change of coordinate system (covariance). More generally, the tensor calculus is the study of geometric entities and algebraic forms that do not depend on the reference system. There are many philosophical views on the remarkable correspondence between mathematical reasoning and behavior of the physical world, but none of them is final . But certainly we need to understand that a new mathematical language , more comprehensive and complex , is not something without substance, although the same conclusions can be drawn without using new symbols ( Maxwell wrote his equations without the use of vectors ) . But the development of language contributes to the development of thinking, the more comprehensive the language becomes the more deep becomes our penetration in natural world , mathematical symbols , - language- is not a typical theme . The electromagnetic four - tensor of special relativity with respect to orthogonal transformations of space-time , (which was implanted in electromagnetism, as vectors in classical mechanics, after too many attempts) , reveals the close relationship between the electric and magnetic aspects of the electromagnetic field, since they participate as components of ( tensor ) which gives this union an objectivity. It is the mathematical form of electromagnetic reality. The mathematics generalize, the deeper understanding of the nature! Special relativity with the tensor writing is a special case of general relativity . Indeed today one of the constant pursuit of geometry and physics is to find the widest possible sets of transformations through which physical geometric laws remain unchanged, ( same format ) and then is proving that they are reduced to more and more general unifying principles . 99: TENSORS AND GEOMETRY OF RECTILINEAR SYSTEMS . To understand all the foregoing, it will give the easiest example of tensors, these of linear reference systems, as well as and the geometric reality installing to such systems, changing the formulas of rectangular Cartesian systems we know from school. THE GROUP OF LINEAR TRANSORMATIONS . Even the three variables χ1,χ2, χ3 transforming in a new set, x 1 , x 2 , x 3 under the following lineal transformation x 1  C11 x1  C 21 x 2  C31 x 3 x 2  C12 x1  C 22 x 2  C32 x 3 x 3  C13 x1  C 23 x 2  C 33 x 3 where C ij constants, in matrix form we have  x 1   C11  2  2  x    C1  x3  C3    1 C 21 C 22 C 23 C 31  x1    C 32  x 2  C 33  x 3  With the summation convention we have x i  C ij x j .......... .......... .......... .....(1.1.1) The matrix ( C ij ) is called matrix of transformation18 and it’s determinant Cij ,determinant of transformation. It is C ij  x i .......... .......... .......... .......... (1.1.2) x j If we suppose C ij ≠0 then matrix of (1.1) is reversed so we have 18 The transformations (1.1) form a group. 100:  x1   c11  2  2  x    c1  x3   c3    1 c12 c22 c23 c31  x 1    c32  x 2 .......... .......... ...(1.1.3) c33  x 3  where the matrix (c ij ) is the ( C ij ). So applies Cmr csm   sr .......... .......... .......(1.1.4) And (1.3) is written xi  cij x j .......... .......... .........(1.1.5) With cij  xi .......... ........(1.1.6) x j and Ci j cij  1.......... (1.1.7) DEFINITION OF VECTOR19 AND TENSOR . We have the functions α1,α2, α3 (components) of variavles χ1, χ2, χ3. The triplet (α1,α2, α3 ) or αr with r=1,2,3 is called contravariant vector or contravariant tensor of order one for the transformations (1.1.1) when for a linear transformation (1.1) of variables, the αr are transformed as the variables xr viz ar  x r k a  Ckr a k .......... .......... .........(1.2.1) x k Or in matrix form  a 1   C11  2  2  a    C1  a 3   C3    1 C21 C22 C23 C31  a1    C32  a 2 .......... ......(1.2.2) C33  a 3  Similarly the triplet of functions (α1, α2, α3) or αr is called covariant vector20 or covariant tensor of first order, for 19 The vector should not for the time to connect with the known vector geometry, it will be in the next chapter. Now I mean it as an ordered triplet of numbers transformed with the particular mode 101: the transformation (1.1.1) if in a transformation (1.1.1) χr the αr tranform ar  x k ak  crk ak .......... .......... ........(1.2.3) x r Or in matrix form 1  a1   c1    1  a2    c2  a   c1  3  3 c12 c22 c32 c13  a1    c23  a2 .......... .......... ...(1.2.4) c33  a3  the matrix (c ij ) being the inverse of ( C ij ). c i j So we have a2  c12 a1  c22 a2  c23 a3 or a 2  C12 a1  C22 a 2  C32 a 3 ΣENSORS OF HIGHER ORDER. Having χr and λr and multiplying every component of the first with every component of the second we have a system a mn  x m n with nine components a11 , a12 , a13 ..... In the new components (1.1.1) we have a rs  (Cmr xm ).(Cnsn )  Cmr Cns amn .......... ..(1.3.1) The system a mn is called contravariant tensor of second order as the transformations (1.1.1) or of type  2   if in a 0 linear change of variables it is transformed as in (1.3.1) Similarly, we call a covariant tensor of second order or of type 0   , a similar system-product of two covariant vectors that is  2 tranformed 20 The terms covariant and contravariant were introduced by James Sylvester. In Cartesian orthogonal systems the two concepts coincide. 102: ars  crmcsnamn .......... .......... .......(1.3.2) Having defined the covariant and contravariant tensors relative to linear transformations, the mixed tensor (as in linear transformations) is defined as a set Tstr of quantities that (in a linear transmormation of variables) they transform as Tstr  Tnpm 1 x r x n x p  Cmr csnctpTnpm .......... .(1.3.3) of type   m s t x x x  2  2 Or Tmnrs  Ckr Cls cmp cnjTpjkl .......... ....(1.3.4) of type    2 GEOMETRY AND TENSORS – introduction To understand the concept of tensor, we will follow their applications in the extension of geometry in rectlinear systems, which is the main mission. How that will formulate the laws of Euclidean geometry (which we learned in school and probably we did not notice that referred to rectangular Cartesian systems) on any rectlineal systems. Now tensors and vectors will get geometric significance. The transition from the theory of tensor to geometry of linear systems will be based on the following assignments. . 1. The variables xi of the previous chapter, will now be considered as coordinates of a point in three-dimensional space in linear reference systems . 2.. I know that the coordinates of two rectlineal reference systems with common origin are associated with linear transformations. Vectors in oblique (not orthogonal) systems will be covariant or contravariant . They are defined by the laws of transformation of the components of the previous chapter , that now acquire geometric significance. 103: THE DISTANCE Consider two points x r , x r 1 2 orthogonal system and BETWEEN TWO POINTS . y1r , y 2r P1, P2 with coordinates respect to Cartesian the rectilinear coordinates to another oblique. It is known that d 2  (12 ) 2  ( y11  y 12 ) 2  ( y12  y 22 ) 2  ( y13  y 23 ) 2 If y1r  cmr x1m , και y2r  cmr x2m then d 2  [c1mc1n  cm2 cn2  cm3 cn3 ]( x1m  x2m )(x1n  x2n ) 3 If g mn   c mr c nr g nm .......... .......... .....( 2.3.1) r 1 then d 2  gmn ( x1m  x2m )(x1n  x2n )......... ....( 2.3.2) Especially the distance of P (xr) from the origin is d 2  gmn xm xn .......... .......... .......... .......( 2.3.3) The ( 2.3.3 ) is the Pythagorean theorem to any linear reference system if and only if gmn is a tensor! Because then the distance d remains unchanged x i x j x m i x n j x  g ij x i x j  d 2 x g mn x x  g ij m j n i x x x x m n The tensor gmn is the mathematical form of the relationship between two reference systems , which is the link and ensure the flow of the laws of geometry from system to system , although it is difficult to imagine it as the natural state because of its many components. Is the cosines of the angles formed by the new axes with old ones, that is a real physical situation , so a "superior genus " ofAristotelian form . THE INNER PRODUCT OF VECTORS . Consider two vectors Ar , Br with magnitudes A and B. If θ the angle of their directions then we can prove ( from the relationship AB2 = OA2 + OB2 2OA.OB.coshθ) the relation 104: gmn Am Bn  AB .......... .......... ....( 2.6.1) where gmn is the former tensor that defines the distance d2 in the oblique system . This invariant is the inner product of vectors Ar , Br , at oblique system , the extension g mn Am B n of the known size of orthogonal systems . (Second member ) The invariance of the inner product is obtained directly from the x i x j x m i x n j V  g ij G iV j relations G g mnG V  g ij m j n i x x x x m n The production of relations that carry the metric geometry of orthogonal systems known from school , in oblique systems, continues with starring always the metric tensor . Later in curved spaces ( initially in surfaces ) tensors are expressed relative to the curvilinear coordinates , the meaning is the same but the math is more PART II: STORIES FROM PHYSICS 1. II 1. THE DETERMINISM IN PHYSICS (mechanics, quantum mechanics and chaos) complex . 105: Abstract . In this article, the characterization of a theory as determinist, is defined by the predictability that produces in the results of theory. We have distinguished corresponding three areas of measurements in physics with theories of deterministic status (determinism, uncertainty, randomness) a. the measurements in the macrocosm which reveal the causal (deterministic) class of Newton's cause and result, even believed that governs the macroscopic world, b. the measurements in the microcosm where is emerging the class of "not accurate" , statistical and indeterminate. c. the iterated measurements on non-linear systems, and chaotic non- deterministic class, viz randomness. Predictability is the degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. Contents the concept of causality, philosophical and scientific the differential causality of Newton the mental mechanism of differential causality The determinism of quantum mechanics The course of determinism , the chaos The non-linear systems A model of chaotic behavior, the repeating functions Determinism and chaos . 106: The philosophical and scientific concept of causality . The basic idea of the concept of causality emerged in our minds from the directly observed regularities of nature. The sun rises every morning and sets every night, the succession of the seasons, the procession of stars in the night sky, even the more complex motions of the moon and planets, was the tinder for deeper observations, that created deeper concepts. In the 6th century in Ionia developed just this new concept, according to which the universe is comprehensible, because it has an internal order, because in nature there are regularities that allow the disclosure of it’s secrets and of its function. Nature is not entirely unpredictable, because there are rules (natural laws) that must obey. In this order and admirable character of the universe, the ancients gave the name World, viz beauty (ςτολίδι). Nικόλαοσ ΢πύρου professor of the Physics Department of the Aristotle University of Thessaloniki This concept of natural laws of regularities, is inherent to the philosophical doctrine of determinism: The determinism is the philosophical idea which particularly affected scientific thought from ancient times to today. It accepts the existence of causality, the universal and causal relevance of all phenomena, a general predictability of the universe.(Theodoridis) ”Nothing occurs at random, but everything for a reason and by necessity." Democritus Tied in this sense of determinism is the concept of cause: 107: ... a concept somewhat vague and ambiguous. The most common use means something which produces a phenomenon or a change (old anthropomorphic view) or a phenomenon firmly inseparably bound to another, so when this occurs, (the cause) follows the other (result, newest positive and scientific view) ..Theodoridis I will not analyze this metaphysical concept of causality which is unscientific and does not result from the methodology of science. But philosophical fashions change at least twice a century. (article the methodology of science vs. metaphysics) https://www.scribd.com/doc/260025743 The concept of cause “itself” is of the same status with the concept of matter of Aristotle; looking for the cause of the cause …we shall end up nowhere, as we know from the methodology of science that the substance of the world escapes us and it’s structure is our own construction. David Hume challenged this metaphysical doctrine of causality: ….According to Hume, there is no contradiction if one claims that for a cause does not imply the result that is attributed. All events seem entirely loose and separate. One event follows another; but we never can observe any tie between them. They seem conjoined but never connected. And as we can have no idea of anything, which never appears to our outward sense or inward sentiment, the necessary conclusion seems to be, that we have no idea of connexion or power at all, and that these words are absolutely without any meaning, when employed either in philosophical reasoning, or in private life. Neither the relation of cause and effect nor the idea of necessary connection is given in our sensory perceptions; both, in an important sense, are contributed by our 108: mind. The causal relationship and dependence is therefore a subjective creature…(Hume on the perception of causality David R.Shranks) In this article, our idea is to investigate causality in physics, not in nature. In nature we see regularities and the method to study these regularities, is cognitive. Determinism thus is attached to the nature by our axiomatic system, which we install on things, doing science. Causality is man-made. The scientific causality In this article we define scientific causality as the predictability of a physical theory, translated according to the scientific theories of mechanics, quantum mechanics and chaos theory. Now it becomes an operational concept which refers on the experimental confirmation of the mathematical apparatus of the relevant scientific theory. Confirmation is to verify the predictions with experimental measurements. The core thus of the scientific causality is mental (our mind produces determinism), but causality emerges from the experiment. The mental mechanism of causality is the axiomatic basis and deductive reasoning of Aristotle, through the shape Premises,→ Aristotelian logic→ conlusion Article: The axiomatic method: Euclid, Hilbert https://www.scribd.com/doc/161365902 and one saying that the world is deterministic imply that is predictable at least in principle. So, It seems that scientific causality exists for human beings, because there is the logical causality, 21 21 Many philosophers have recently inclined toward what a recent collection of papers calls “causal republicanism” the view that “although the notion of causality is useful, perhaps indispensable , in our dealings with the world, it is a category provided neither by God nor by physics , but rather constructed by us”…(A.Eagle, pragmatic causation) 109: The evolution of living beings would not happen if they could not watch the causality in nature. In lower organisms this may be simple photochemical reactions, but in human evolution the secret is hidden in the "Analytica posterioria" of Aristotle. For Aristotle, logic is the instrument (the "organon") by means of which we come to know anything. He proposed as formal rules for correct reasoning the basic principles of the categorical logic that was universally accepted by Western philosophers until the nineteenth century. Aristotle further supposed that this logical scheme accurately represents the true nature of reality. “…. It seems that logic, confirms phenomena and phenomena logic ... on Sky 270v4”. Thought and reality are isomorphic, so the natural causality is reflected on logic which can help us to understand the way things really are. This phrase expresses one of the underlying regularities in nature (the hidden regularities are superior than the obvious, Heraclitus) and this discussion has never end. Newton’s determinism (differential determinism) . The Newtonian laws of classical mechanics have traditionally been regarded, and theoretically are , infinitely accurate in their predictions, ie approach the infinite number of decimal digits (in practice they are as accurate as the experimental verification), so as to allow a check of a strict causality in this system: the same causes corresponding same results. Generalizing from mechanics to all physical systems , we may formulate this doctrine of causality as follows: ….The evolution of every physical system is controlled by rigorous laws. These taken in conjunction with the initial state of the system (assumed to be isolated)) determine without ambiguity all future states and also al past ones. .So the entire history of the system throughout time, is thus determined by the laws and by the initial states …..(A. D 'Abro) (these are the premises and the axioms of Aristotle’s scheme of science) 110: In Newtonian system the cause of the mechanical universe is the force, and the result is the change of motion. This scheme does not include the inertial motion, which is without cause. The strict laws are the laws of mechanics and of gravitation. One of the greatest Newton’s achievements is the discovery of the method of representation of a natural law, and thus of a causal chain, through mathematics. We talk about instantaneous description of the phenomenon of motion, how cause and effect operate at an instant , in an infinitesimal interval (in calculus), for instantaneous velocity, for instantaneous acceleration, and this is because in nature everything is changing22, stability is an exception, the velocity of a falling body to earth is variable, the speed of the planets around the sun is so variable, however, in a infinitesimal time or spatial interval all changes "freeze" the curve is straight, the forces are stable, motion is inertial, etc. and we have the simple effect of the composition of stable forces with the rule of the parallelogram and composition of motion of Galileo. So the motion of every moment is the cause of the motion of the next, since the phenomenon now is repeated with new initial conditions, but with the same law. The moon because of inertia, would move in tangential trajectory towards infinity. The gravity pulls it to the earth, but inertia does not leave it to fall on earth. The composition of the two forces each time. gives the part of the track in next moment, an endless tiny zig-zag. A more precise presentation would require us that the relation which constitutes the law is arrived at by a limiting process. If we set x and υ the measure of position and velocity at any instant t the consecutive states of a mechanical system are defined by (χ0,υ0) ,(χ1, υ1)…….(χn, υn)….. The permanent relation to which we have referred in the text is of the form 22 Later we will understand that everything in nature is nonlinear (chaos theory), the Newtonian linearity is the exception. 111: d 2x our known m 2  F (t ) dt where F is a function which defines the force, is the same for all values of n , and hence is the same at all instants of time. in the beginning of every time interval , so the phenomenon will be repeated. The relation (1) I a differential equation. If we wish to obtain a knowledge of the states that this system will assume after some finite interval of time, we shall have to repeat the foregoing procedure an infinite number of times in succession. The difficulty is overcome by Newton’s discovery of the method of integration. Thanks to this powerful method it s always possible, at theory at least, to follow the causal chain over finite intervals. …In order to give his system mathematical form , Newton had frst to discover the concept of the differential coefficient, and to enunciate the laws of motion I the form of differential equations – perhaps the greatest intellectual stride that it has ever granted to any man to make….Einstein When the law governing the evolution of the system is known and its mathematical transcription, the corresponding differential equation is obtained we are in position to to derive a knowledge of the evolution of a system from any given initial state. Thus if A represents the initial state, the differential equation , by expressing the relationship between A and the next state B , enables us to obtain the Knowledge of B. Since we know the state B , a second implication of the different equation yields the state C and so on. We must remember however that , that these states occur at instants of time that are separated only by infinitesimal intervals. All these are contained in the functioning of the differential equation, which is the “organon” of rigorous causality The 'mechanism' of differential causality. To understand this process as a mechanism, we would yield some pages of Newton's Principia, which describes the differential causality and the use of his laws, to 112: derive an actual effect. It is the equivalent of the method of exhaustion in geometry, a thought experiment in motion23. The proposal demonstrated is the known law of the areas in the central motion. Now the applied force can be the force of gravity, and the motion, the motion of a planet around the sun: The areas swept by the line joining the mobile with the center of force, lie in the same plane and are commensurate with the times of removal. This is a law of Kepler, who (Kepler) observed macroscopically from Earth. Newton, however, interpreted it. What is this interpretation? It is that he imagined rational mental processes in motion, the infinitesimally intervals of time, which determines the macroscopic phenomenon, an invisible mechanism of causality. The physical causality, causality is reduced to the logic of deductive reasoning underlying mathematics. The roots then of this differential causality is the mathematics of calculus, which we analyzed in the article "the tiny quantities in mathematics, Leibniz's infinitesimals, the limits of Cauchy." The basis of reasoning (calculus) is to consider motion in infinitesimally intervals which intersect the orbit infinitesimally segments in which we study the area to be swept, regardless of the others! Now we will see this mechanism of differential causality. (Figure 1 Principia) Suppose the first infinitesimal period of time the body removes the segment AB. Whatever kind of motion it performs as a whole, however, in this infinitesimal (very small) time, the motion can be seen as inertial by the Newtonian sense. The smaller dt (hence the piece of track), the more justified this identification. This is reminiscent of the fact in the calculus that the infinitesimal parts of a curve are linear and the exerted force will appear in the end of this dt. The inertial motion of infinitesimal portion is the cell of arbitrary motion in direction and measure. This motion is "indelibly entered" in the body, which can perform two motions (Galileo) and hence is entered in the differential equation of motion (as we saw before), in the form of the initial conditions. Position A and the inertial motion at A (speed) are the initial conditions of the system for AB. 23 The thought experiments were introduced by Galileo, abandoned by science in the 18th and 19th century, and reverted with Einstein. 113: In the second infinitesimal portion would occur the same, but WITH NEW INITIAL CONDITIONS . Because as body reaches B we assume that a centripetal force is exerted on it, that causes it to continue to move the portion BC, instead of Bc. Incise the cC parallel to BS which meets the BC in C. At the end of the second infinitesimal portion of time, the body will be in C, in the same plane with the ASB. (Euclidean stereometry) The same will happen to every infinitesimal portion of the track resulting mobile keeps track line ABCDEF. Readily apparent that the geometric area SAB = SBC = SCD = SDE = SEF and adding e.g the SADS and SAFS have one to another, the same relation are as times elapsing. Now, if the number of these triangles increase, with dt and hence the AB, BC, etc. tend to zero (the word tend to have special meaning in the differential calculus) the final perimeter ABCDEF will be a curve and also the centripetal force with which the body is drawn sequentially from the tangent of this curve, will act continuously. And any volatile surfaces are coplanar and analogous of the time elapsed. This entire process is shortened by the differential formalism (differential equations) which produces the result: The angular momentum of the mobile in the central motion is constant. The uncertainty of quantum mechanics. Today , less than three centuries after Newton made his momentous discovery , doupts are being cast on the validity of the rigorous causal connections of classical science. The attack is due not to the impossibility of testing the doctrine in practice, but to totally new discoveries in the subatomic world , where the mysterious quantum phenomena become noticeable. The novelty resulting from the discoveries of the quantum theory is that we now have reasons to suspect a definite theoretical impossibility, which would render illusory 114: any attempt to test rigorous causal connections . The quantum theorists under the lead of Born, Heisenberg , Bohr and Dirac , agree with classical scientists in recognizing that the practical difficulties of testing causal connections may be disregarded. Indeed it is very difficult on operating on perfectly isolated systems ; first of all , because no such systems exist; and secondly , if we grant the existence of such systems , we cannot observe their internal workings with out disturbing them and thereby destroying their isolation. In addition , human measurements are necessarily imprecise. These difficulties were not regarded as fatal to the doctrine , because approximately isolated systems could be found , and our measurements could be so refined that they would not perceptibly disturb the magnitudes to be measured (D’Abro, the rise of the new physics). But the quantum theorists are adamant in their claim that the recently discovered theoretical impossibility , cannot be dismissed so lightly. The developments of the quantum theory indicate that the “uncertainty relations” discovered by Heisenberg , prohibit us in principle from effecting simultaneous accurate measurements of so-called conjugate magnitudes. This very general rule applies in particular to position and momentum, or velocity, to time and energy, to the electric and the magnetic intensities at the same point in a electromagnetic field. Thus if the position of the centre of a particle is measured with accuracy , the unpredictable disturbance , entailed by the measurement itself, causes the particle’s momentum to be vague. The classical contention that by, exercising sufficient care , we may reduce the disturbance indefinitely is here no longer valid, for the essence of the uncertainty relations is that the limit we might hope to attain is not vanishing but is finite. Obviously , if this principle is accepted , the state of a mechanical system, involving as it does a simultaneous knowledge of the positions and momenta of the various masses, cannot be known with accuracy. Consequently , a test of rigorous causality is impossible in quantum mechanics the same conclusion may be extended to all departments of physics. The source of this theoretical impossibility unsuspected by classical science , must be sought in the very nature of things. It is intimately connected with Planck’s constant h. the theoretical impossibility would vanish if this constant , the value of which is finite, were to be infinitesimal , as classical science had implicitly assumed. So the "hard" deterministic model is not applicable but only if we consider the probabilistic uncertainty a part of the deterministic image of the Universe. Moreover , probabilities are a priori, produced by theories. In this case, however, the idea that the 115: universe acts "as would act anyway" is not applicable. In the hypothetical case that the history - or part of history - was repeated, quantum uncertainty would create a different version of the story. It is important to understand that the limitations imposed by the uncertainty relations are theoretical , and not practical. Thus the uncertainty relations do not interfere with our measuring , as accurately as we choose, the position alone or the momentum alone of a particle. They only prevent us from executing simultaneous measurements with accuracy. According to the quantum theorists , however, practical difficulties in measurement may be waived aside exactly as they were in the classical critique of the causal doctrine. We are called to decide whether a concept which cannot be tested or a magnitude which cannot be measured , in principle (in contra-distinction to in practice) should forthwith be classed as meaningless and cease to play any part in a theoretical discussion. The dilemma is not entirely new , for it arose in the theory of relativity. There , for reasons which have nothing in common with practical experimental difficulties , a velocity through the stagnant ether cannot be detected. Absolute velocity is thereby dismissed because it cannot be observed in principle. In view of Einstein’s attitude towards absolute velocity , we might expect him to adopt a similar one with respect to vigorous causality. But he does not do. Einstein and Planck both retain a belief in rigorous causality , extending it even to living matter. But those who have contributed most to the development of quantum mechanics resist Einstein’s views and insist that rigorous causality is a myth. We have now an entirely new reality in nature around us. It is a word far removed from our every day experience. And what about logical causality? It still exists and instead the differential formalism gave us the mathematical treatment of the possibilities, to manipulate the new reality. We don’t forget that the mathematical truth is an hypothetical truth, we can create mathematical models for every situation in nature. We had created mathematical models for the ether! The differential calculus was the model of classical causality. «……With indeterminacy corrupting the experiment and dissolving causality , all seems lost. We must wonder how there can be a rational science. We must wonder how there can be anything at all but chaos. The detailed determinacy claimed by classical science, is replaced by the exactitude of probabilities, even though it seems paradox. But quantum mechanics has discovered precise and 116: wonderful laws governing the probabilities, so science overcomes its handicap of basic indeterminacy….» Banesh Hoffmann A compromise between causality and uncertainty, attempts David Ruelee in his book “Chance and chaos” in free translation …If you want to say that quantum mechanics is deterministic, so it is: the Schrödinger equation clearly provides the time evolution of the amplitudes of probabilities. If you want to say that quantum mechanics is probabilistic, you can say: the only provisions that gives, concern possibilities… The chaos theory . Determinism quite recently, has undergone the biggest challenge. There are phenomena that are unpredictable, viz their evolution does not show any regularity. These phenomena are called chaotic. Such phenomena are fluid flows, the motion of billiard balls or the evolution of the weather, even our solar system. Chaos theory, is regarded as a new leap in science, made prohibitive the prediction of a phenomenon for a long time. In particular, in the solar system we can not predict what will be the position of the planets after one hundred million years. (Hadjidimitriou) Yet all these phenomena are governed by the same deterministic laws of Newton. So why the motion is not, in this case, regular and therefore predictable? The story of chaos is a mathematical story and is a property of some solutions of a system of non-linear equations. It has the origins on the three-body problem, studied by Poincaré, who revealed chaos in the Solar system. He understood that very small effects can be magnified through iteration. Postulated "A minimal cause that escapes attention can cause a significant effect". Who can doubt that the motion of the pair of Earth-Moon is not affected by the pull of the Sun or Jupiter etc.? The problem is (and it just raised by Poincaré) that making the simple step from two to three bodies (for example by trying to include the effects of the Sun on the Earth-Moon system) equations of Newton, because of formal mathematical reasons, can not be solved: a set of approximations is needed to "get closer" an answer (the series describing the orbits of three interacting celestial bodies, not only converge in some preset solutions, but instead diverge!). . Each approach is less 117: than the previous and Poincaré hoped that after a potentially infinite number of such corrections to get the correct answer. But the result was shocking! There were a few orbits, for which a small gravitational pull from a third body, after iterations, could be disorientated the motion of the planet, even to expel it, out of the planetary system. .. Poincare disclosed that chaos , or the possibility of generating chaos is the substance of a non-linear system, and that even a fully determinate system, as rotating planets, could have unspecified evolution. In one aspect he understood how micro effects could be magnified through iteration. He distinguished that a simple system, can end in an uncontrolled and remarkably complex behavior ...... .Turbulent mirror The non-linear systems . A linear process is one in which, if a change in any variable at some initial time produces a change in the same or some other variable , twice as large a change at the same initial time will produce twice as large a change at the same latter time. You can substitute “half” or “five times” or a “hundred times” for “twice” and the description remains valid. Edward Lorenz in the “Essence of chaos” A linear system is a dynamical system whose evolution is a linear process. All systems that are not li near are called non-linear systems. In these systems , the change in a variable at a initial time can lead to a change in the same or a different variable at a later time that is no proportional to the change at the initial time. Nature is intrinsically non-linear and nonlinearity is rather the rule than the exception. ….“it does not say in the Bible that all laws of nature are expressible linearly ….Enrico Fermi ….Using a term like non-linear science is like referring to the bulk of zoology as the study of non-elephants animals ….Stanislaw Ulam 118: For a linear system , we can combine two solutions , and the result is a solution for the system. Here is based the reducibility of classical science (the perception that the world is an aggregation of parties). The above property is called linearity and it makes the linear systems mathematically tractable. We can break up a linear problem into little pieces , solve each piece separately and put them back together to make the complete solution. Nonlinear systems in the other hand cannot be broken up into little pieces and solved separately. They have to be dealt with in their fully complexity. They are ubiquitous and their behavior differ qualitatively from the behavior of linear systems, they can display a variety of behaviors including chaos. The linear systems were studied for a long time although they are an exception, because all linear problems are solvable and nonlinear problems are seldom exactly solvable. Before the advent of computers , almost nothing could be said about the behavior of nonlinear systems. Also nonlinear systems of interest, are approximately linear for small perturbations about a point of equilibrium. The nonlinear equations concern specifically discontinuous phenomena such as explosions, sudden breaks In materials, or tornados. Although they share some universal characteristics, nonlinear solutions tend to be individual and peculiar. In contrast to regular curves from linear equations, the graphic representation of nonlinear equations shows breaks, loops, recursions all kinds of turbulences. Using nonlinear models, on can identify critical points in the system at which a minute modification can have a disproportionate effect (a sufficient condition for chaos). A model of chaotic behavior , iterative processes . ….Chaos is not related necessarily with complex systems and abstract concepts. We can find chaotic behavior in simple systems and thereby study the indeterminate, in its most basic form. Even in the most simple quadratic equations we have chaos, if simply we handle them as iterative functions, ie. if we have every so a feedback of it’s value from the previous result (continuous synthesis of function with itself). (Anastasia Karakosta) The nonlinear quadratic y = rx (1-x) has rich chaotic behaviour. 119: This will give us all the mathematical properties of chaos, when translated into a demographic model published by biologist Robert May, on the standards of the work of Velhulst24. It is called logistic map, and in the reproducibility of manipulation takes the form χν+1 =p χν (1-χν) …………(1) It seems to be generally acknowledged now that this logistic function is the basis of modern chaos theory , although Verhulst himself had absolutely no idea that something like that lay hidden in his formula. This is a prototype of a nonlinear iterative process where we calculate the evolution of a population , by starting with some initial population x0 (between 0 and 1) and applying the formula again and again thus obtaining the sequence of the values of the population, x1 ,x2 , x3 ,and so on. where xν is the number of the population between v and v + 1 period (year, decade, etc.). In the formula , p is the growth coefficient, i.e this period the number of births is p times bigger of the previous period and x0 the initial population. These are the initial conditions, and as we shall see, tiny difference in initial parameters will result in a completely different behavior of a complex system. Even the smallest deviation –say in the hundredth or thousandth decimals- from the initial value of x0, may have a significant effect in the end result , in a totally different evolution. It is important to notice that also our computers which work with a fixed number of decimals , are subject to this type of unpredictability , however powerfull they may be…. (Marcel Ausloos, Michel Dirickx)) At the moment when the system becomes chaotic the size of the population at each step in the iteration will be different from it’s value at any of the previous steps. There is no stability or regularity any more. Let us follow this evolution of a system. When carrying out this iteration scheme, one finds that the resulting evolution. When p<1 the population sequence tends to 0 independently of the exact value of p (the births are less than deaths). For 1<p<3 the population sequence tends to the limit value 1, for p<2 this happens in a monotone way, but for 2<p<3 in an oscillatory way. 24 Verhulst’ principle was even applied to economics and sociology. The work of Verhulst received so much attention all of a sudden :it’s application in chaos theory. 120: For 3<r<3,5699 the sequence is periodic25 and it’s period depends on p. first we have an oscillation with period 2 (maximum and minimum) , then an oscillation between 4 different local extremes (period 4) and subsequently with period 8, perio16 and so on. Now the system can go from orderly to chaotic behavior. For most values or 3,5699<p<4 the sequence shows no regularity any more , it is chaotic. We see that the main characteristic of a chaotic system is this extreme sensibility to a change of the initial conditions. Two sequences with almost identical values for x0 , will at first behave in a virtually identical manner , but they suddenly diverge so that from then on there is no correlation between the two oscillations. A similar sensitivity is also observed with respect to a change in the growth parameter p. Order and chaos coexist in nonlinear systems but chaos occurs when the system reaches in a critical value . This is the route to chaos. Determinism and chaos . As we have unpredictable linked determinism with predictability, chaotic behavior is and therefore non-deterministic. ... in fact, motion is always deterministic, i.e the same starting situation always corresponds to the same final state. If we were able to just know the initial state with immense precision, and further if we were able to perform numerical calculations with absolute accuracy (for example in the 20th decimal place, without approximation), then we would not have problem in the predictability of a phenomenon, and so between regular and chaotic motions. But it never is possible to do measurements with absolute precision, e.g the position and velocity of a body with absolute precision. ... .. (Hatzidimitriou) This view of professor Hatzidimitriou refers in causality itself, in causality as an axiom, where the principle of causality is non refutable, i.e, not confirmed by experience, since it is an axiom that precedes experiences. For example, this principle is accepted a priori in physics in a simple form, it reads: “Every effect has a cause.” That is that chaotic systems are deterministic but not predictable 25 Periodic sequence with period p is when we have an+p=an for all n 121: But this is metaphysics, as there is nothing accepted a priori in methodology of physics. In practice, we can never specify (or know) the initial conditions exactly (Heinenberg). So there will always be some uncertainty in the initial conditions in dynamical systems , as two kettles of soup heat under the same conditions, they will behave differently. But it makes sense to characterize the behavior of a system in terms of its response to this uncertainty. Basically, a chaotic system is one in which any uncertainty in the state at time t=0 leads to exponentially larger uncertainties in the state as time goes on, and a non-chaotic system is one in which any initial uncertainty in the state decays away or at least stays steady with time. In the former (chaotic) case, given that we can't know the initial conditions to infinite precision, there will always be some time, after which, the predictions of the behavior of the system become essentially meaningless - the uncertainty becomes so large that it fills up most of the state space. This is effectively similar to the behavior of a truly non-deterministic (e.g. quantum) system, in that our ability to make predictions about it, is limited, chaotic systems are non-deterministic, but random . In this connection the scientific determinism is illustrated in the diagram: Newtonian Newtonian differential determinism + causality uncertainty → Nonlinear systems + iteration → randomness rigorous principles determinism → uncertainty Sources . Η Αριςτοτελική κίνηςη ςτη ςύγχρονη φυςική www mpantes. gr Mathematics and the physical world, Morris Klein, Dover Φώροσ και ςχετικότητα, Υρανςουάζ Μπαλιμπάρ, Εςτία H ελαφρότητα του είναι, Frank Wilczek (Κάτοπτρο) Σhe rise of the new physics A.D’ Abro (Dover) Ειςαγωγή εισ την κβαντομηχανικήν Γ.Ι. Ανδριτςόπουλοσ Η Κβάντωςη του χώρου και του χρόνου (άρθρο, mpantes on scribd) Ο ταραγμένοσ καθρέφτησ Gohn Briggs , F. David Peat KATOΠΣΡΟ ΦΑΟ΢ ΢ΣΟ ΗΛΙΑΚΟ ΢Τ΢ΣΗΜΑ Ιωάννησ Δ. Φατζηδημητρίου (διαδίκτυο) Logistic map and the route to chaos: Marcel Ausloos, Michel Dirickx Α primer on Determinism Door John Earman (klwver the language of science) Great Physicists from Galileo to Einstein George Gamow (Dover) Σύχη και χάοσ David Ruelle (Σραυλόσ) Γνωριμία με τα φράκταλσ Διπλωματική εργαςία Αναςταςία Καρακώςτα 122: QED R.Feynman Tροχαλία Non linear systems for beginners, Lui Lam TIME IN PHYSICS(Aristotle,Newton,Einstein Aristotle laid the foundations of the diachronic study of the concept of time, connecting it with the movement. Time said, is connected with movement , but time is not movement. On the other hand it is obvious that without movement and change there is to time…. Time is real as movement is real too, time is interwoven with the affairs of the physical world. So to understand the nature of time, we should understand that there is not time without moving (Υυςικά Δ11) Time is not moving but without moving there is not time. The exact relation of moving and time is that time describes a number which refers to the movement , itshelf is not movement. . “because this is time, the numbering of movement according to before and after”. But the movement too is a measure of time, movement is a measure of time and time is a measure of movement. What is the most appropriate movement for the measurement of time in nature? This is the smooth circular motion of the sky, which he has proved in Υυςικά, in his geocentric system, is eternal at constant speed, and therefore can become the world clock, because “it’s number is the best known”. The uniform circular motion, in the highest degree can be used as a measure of the number , “because it is the most easily accessible in knowledge, “time seems to be one cycle because of the extend of such an orbit, and such an orbit serves as a displacement measure that counts”. Aristotle then counts time with the movement of the sky but did not identify with it, “because if there were many of heavens, the movement of any of them , without distinction, 123: would constitute the time so there would exist several times” which is impossible, because time is the same in all places in every instant, time is universal. Newton . This perception heralds the absolute time of Newton which flows uniformly and it is universal, but independently of the motion of bodies, as he was unable to notice a movement uniform and eternal for the counting. The movement of the stars was not uniform and ideal, that Newton knew well (the tides were causing delays in the rotation of the earth, the geocentric system was rejected) so it could not be a measure of time. Now the motion associated with the time measurement, is ideal and theoretical (mathematical time), is the Newton’s first law, which ensures the characteristics of Newtonian time: “time flows uniformly, everywhere and eternally, because of the principle of inertia!” Absolutely free bodies would move with absolutely constant speed along a straight line i.e running equal distances in equal times. The flow of Newtonian time can therefore be recognized by reading the subdivisions of the rule from which this body passes. But this movement is not seen anywhere, is fantastic, so the time is disengaged from real movement, is associated with a fantastic move and becomes metaphysical, is not defined operationally. So the time of Newton flaws evenly and eternally ,even if there is no movement in nature, becomes absolute. “the absolute time in astronomy differs from the relative, because natural days are actually unequal, is possible to be no uniform motion that can accurately measure time…Newton comment ” Here he recognizes the lack of observation of inertial motion. “…The absolute true and mathematical time, from it’s own nature, flows uniformly without regards to external things, and is called duration. The relative obvious and common time is a perceptible and external measure of duration through the movement, and is often used istead real time , such as one day , one hour, one year…comment in Principia” 124: The time we measure is not the genuine. Whether the movement is rapid or missing completely, time will continue to run. Time exists and centuries will roll, even if it stops any movement in the world. This is not the time of Aristotle. But while the uniform flow of time in Aristotle and Newton acquires physical meaning of the corresponding movements, another assumption of the picture for the time, remains completely hypothetical: that of the universality. Why two clocks at different points in space and different kinetics show the same time? It automatically follows from a principle of mechanics? What ensures the synchronization of clocks in the Newtonian edifice? It is the exact analogous of the Euclidean straight: why does there only a parallel to it? There is no answer in the Euclidean system. Synchronization of clocks is a prerequisite for Newtonian construction. It is an ambiguity which no one seemed to dispute before Einstein. Einstein With Einstein we are going back to the operationally appointment of time. Einstein was looking for another movement that can measure time. It is the strange moving of light: “…but it is useful for the theory to choose the procedure that we know something for sure. This applies to the propagation of light in vacuum, to a greater extent than any other that we thought, thank to discoveries of Maxwell… If we consider a light beam passing through an inertial system, it’s speed in this system is the same independently of the relative motion of the source and the system and independently of the direction of the beam..” This movement will measure the time of Einstein. Time measuring, using like clockwork this movement of the light beam, negated it’s universality. Two events simultaneous for one observer, were not simultaneous for another (relativity of simultaneity). This entangled time with space in one reality (space- time) and abolished the concept of absolute space (the space is the same for all) since two observers in relative motion defined as simultaneously different group events, that different spaces. When later (general relativity) proved that the movement of light (orbit) was affected from a gravitation field, then the time and hence the space were connected, via 125: the movement of the light, more closely to the material bodies, since they were the source of the field. Larger masses mean greater time dilation, always due to the light movement. Epilogue The concept of time in physics is connected with the natural movement which is the measure, as conceived by Aristotle. For Aristotle is the movement of the sky: time flows evenly everywhere, is universal, but is time relative to the motion of bodies, operational and it will not exist if there is no movement in world. In Newton is the imaginary inertial movement: time flows uniformly and universally, but it is absolute, it doesn’t depend on the bodies and flows in an empty space. It is a metaphysical concept. In Einstein is the movement of light, which is strange and specific, so time doesn’t flow universally and uniformly, it is local, contracts and dilates, engages with space, is the overturn to the traditional concepts of space and time. SPACE IN PHYSICS(Aristotle ,Newton,Einstein Motion and space . What does the sentence “a body moves” mean? We say that is changing position relative to some other bodies. But this does not answer the question . Body moves or move the others ? Makes sense such a question ? Is the earth that rotates on its axis from west to east or the stars 126: rotate from east to west ? If the earth rotates , then how can measure the motion of other bodies ? But , if earth offers us a frame of reference for motion, then we can ask whether another system is available for the reference of motion too . Is there in space a really stationary system or a really stationary material body, as to which other bodies “really” move ? (Absolute motion) . If there is not something like this, then all motions should refer to bodies (relative motion) but a really stationary body does not exist as there is not an immobile system of reference. All motions are apparent (as related to other bodies). If so , the question of whether the earth or the stars revolve , is meaningless , merely different but equivalent expressions of the phenomenon . (relative motion) So in every physical theory for motion, the fundamental problem is this global reference of motion somewhere in space, of a particular aspect namely whether it is absolute or relative. Such questions confronted all those were studying the phenomenon of motion , Aristotle, Descartes , Newton , Leibniz , Mach , Einstein . The response of each researcher in this question produces his views on space, since the motion is described in spatio-temporal terms. Absolute motion means absolute space, relative motion means relative space. Aristotle (the relative motion) . The natural motions for Aristotle are , from, toward and around a point O, which coincides with the center of the earth. So, does Aristotle use the space with absolute or relative meaning? As this point O, coincides with the center of the earth , then theory does not use absolute quantities , the natural motion of the material bodies depends on their relative position to another body , the earth . But Aristotle writes that 127: ….If someone transposes the earth where there is now the moon, then each part to the earth would not be transported , but will remain in the place now stands. On the Heavens 310 A 21 That is the point O is not the center of the earth , but earth just removed toward this, because earth is heavy, and they coincided . That means, point O is not identified with a body. But back to On heavens, ( II , 130), he indicates that the center O is defined by the outer sphere of the universe, the ethereal sphere of distant stars, being it’s center . So physical laws describe the motion relative to other bodies , in particular to the mathematical center of distant stars, that is the earth. The Aristotelian motion is relative and refers to the system of distant stars . “He has found and defined the universal and ultimate frame of reference…..Julian Barbour” (The Discovery of Dynamics) The concept of Aristotle’s space . The existence of place is held to be obvious from the fact of mutual replacement. when the water has gone out as from a vessel, air is present; and at another time another body occupies this same place.” (208Physics IV, 1) For the relationship space- motion, he observed that both require reciprocal one another , because the motion is possible only in relation to something. " The space would not be investigated if there was not a local displacement ." As Aristotle’s motion was related with bodies (relative) , so does the space itself, ie the space exists because there are bodies to which relates every motion, there is no motion without space or space without motion . In this way , he first pointed out the relativity of space. The space , where the motion is performed , is the "place " of 128: Aristotle’s (τόποσ) and as motion occurs relative to the stars, so space should be understood as to the stars . The space is the immediate motionless limit of bodies. ….place is not part of the thing , is the innermost motionless boundary of the surroundings ( Φυςικά Δ4 ).26 Thus space is associated with bodies , it’s characteristics are produced by the reference of motion of bodies found therein. These positions of Aristotle is the idea that drives the concept of space in the entire path of physics. Reference of motion gives us the concept of space. His view of the space is completed by the ingenious conception of the denial of a physical existence of the empty space or void27 . “Again, just as every body is in its place, so, too, every place has a body in”. Aristotle denied that there could be literally empty space. In physical terms , therefore he considered the world as a holistic single entity , " then nothing can be converted into nothingness” . The absence of the ability of observation of the void in the physical world, leads him safely to the denial of the existence of a void, and that motion presupposes a friction p between different substances . ( ……supporters of void - writes Aristotle- say that void is υ= B / p) “devoid of body ." Why think that the place is something independent of the bodies ( the place despite the bodies ) and that every sensible body is located in a place that exists independently of the body . But so consider the space a kind of vessel which is complete when it contains matter ( " δοκεί δε πλήρεσ είναι , όταν έχει τον όγκον ου δεκτικόσ εςτίν") and empty otherwise. And even argue that void is necessary , because otherwise there would be no local motion ( Metaphysics 214 b 12-216 a 26): 26 Descardes criticized Aristotle’s definition because there are often no such “innermost motionless boundary” and yet things are not said to lack a place. Nevertheless he apparently abides by the same ordinary conception, for he characterizes motion or change of place” in the true sense” as “the transport of a part of matter from the neighbourhood of those that touch it immediately, and that we regard as being at rest , to the neighbourhood of some others…Roberto Torreti The void –that is not void- is described in modern physics in two forms, the space-time and the quantum void (phenomenon Casimir). 27 129: Here is the big objection of Aristotle: if the void exists there will be no motion , because there would not be a direction to which the body would in preference move, the void as such implies no difference . "Why stop the body here and not there ? " If void exists, continues Aristotle , would force something strange , the subsequent principle of inertia ( which he rejects introducing the “first mover” ) , which is impossible , since it has proved that is impossible the motion in an infinite straight line, is impossible to travel the infinity . Even the speed of a body in a void, would be infinite because of the zero resistance of the void in motion. (from the equation υ = B / p, p=0). Therefore, if there was the void, there would not be natural motion (we can not conceive infinite speed). What exists in space beyond the outer sphere? The tension for the atomists is: the void. If it’s nothing but void out there, then how is anything located? Aristotle said: … so outside the universe (where there is no motion) there is no place, outside of everything there is nothing, heaven does not exist inside another thing. There is not reality outside the sky , because the void is non- being “…. These two concepts, represented respectively by the finite Aristotelian cosmos and infinite Euclidean space, were both developed with great precision and detail long before the end of Greek antiquity and lived on cheek by jowl until almost the middle of the seventeenth century…” Newton . The void , however , and not a body , ie the space without bodies, was the entity to which Newton related the "real " motion . Having not been able to identify a body really motionless, he ... imagined it ! space In the " Comment " which is an introduction to the “Principia” states his views on which I will mention briefly . 130: …….Space is something separate from the bodies and exists independently of the existence of bodies. This space will exist there even if all bodies disappear from the universe . There was even there before the bodies . It is the conceivable container of the material universe . Absolute space is by nature without any relation to anything external , and remains always similar to himself and motionless ....... while relative space is a movable part or the absolute dimension we sensed from his relationship with the bodies and that layman is confused with the motionless space.. But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define [definimus] all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred. . Scholium Newton '. That is, we measure motions in relative space but the real existing space is the absolute one. This fantastic construction , eternal , immovable , unchanging e.t.c which regulates the motion in the physical world , seemed to many that was protecting the concept of God as the divine substance that reveals it’s infinity on the double infinity of absolute space and time ( ubiquitous present and eternal ) . The interpretations of Newtonian theory in this direction were many , as in Aristotle. So the system was endowed further with a social efficiency ( religious), just as in Aristotle. “As for the absolute void , where they would like to arise a being with a divine fiant, its origins are purely human J. Milhau” Einstein. Einstein returned to Aristotle's views on space after 2300 years, considering the space as something that identifies with the Aristotelian place (τόπο) . Now , however, everything changes . The reference of motion is not made in respect 131: of some body or bodies of the universe , but as to any material body . This relativity of motion is the only reality for reporting motion , since neither Aristotelian center of the world exists , nor even the absolute Newtonian space . The laws of motion should remain unchanged, regardless of the reference system. In particular because of it’s radical engagement with time ( relativity of simultaneous) , space is different for different observers ( length contraction ) , since each observer sees as simultaneous different groups of events . They agree only about what events there are, not about where or when these events take place. Each observer has his own space and several spaces are associated with the Lorentz’s transformations. It is the complete overthrow of the classical concepts ! As has been shown in special relativity is not possible to conceive of absolute space and absolute distances , but only if the simultaneous is absolute. Even, the relationship of space and material bodies is the central idea of general relativity and was verified by observation. There bodies define the geometry of space, and the space is leading bodies in their orbits . Einstein again raises the Aristotelian question of whether can the word place has an independent meaning of what we call material object ....If ( says) give a negative answer to question (so does himself) we conlude that the space ( or place ) is actually not but only a kind of ordering of material bodies .... we can not conceive any place without a material body ... since we accept such a concept for the space , it becomes impossible to talk about void space , no longer has meaning .... ….The revolution we carry out, uses no absolute space .... Einstein The "revolution of Einstein appears as a “dethronement” of the perceptions of Newton. For this fact Einstein seems to apologizes from Newton. (Newton, verzeih 'mir). Epilogue . Aristotle’s space is finite, and takes it’s meaning from the material bodies of the world and their motions. These motions are referred in the sphere of the stars , that is surrounding the world. The inner of this conceived sphere is the space. 132: Newton’s space is extended in infinity as Euclid’s straight line, and is a fantastic motionless container where are located and moving –if existing- the material bodies. Measuring their motions we use a local system, but this is the space for the affairs of measuring, the “real space” is an entity, actually exists, is the space to which the bodies are really at rest or moving. Einstein’s space takes it’s meaning of the material bodies as in Aristotle, it doesn’t exist without bodies , but it is not unique for all observers. It is engaged with time in the absolute space-time but two observers in relative motion have different measurements of space. It is the complete overthrow of Aristotle and Newton. Source : Aristotelian “motion” in modern Physics (www.mpantes.gr, N.code:5632/29-1-08) THE MATHEMATICS 28 IN PHYSICS . Introduction , homogeneous and isotropic space, , the second law of Newton, the mathematical forms of nature, the formalism of tensors in geometry and physics. 28 Here the forms have the Aristotelian meaning. 133: INTRODUCTION . One of the central tenets of physics and geometry is that their laws are applicable in every region of the Euclidean space. In other words, if someone is running an experiment here (measurement ) and has a certain effect, then someone else performing the same experiment elsewhere , should retrieve the same results. The two experimenters by running our hypothetical experiment , it is reasonable to use a coordinate system . Usually the two systems are not identical , since the execution of the experiment is in another place and with different orientation (relative to the stars ) of axes . But we expect the " same results" in both experiments ! What exactly is meant by this proposal ? Not necessarily mean that the two experimenters will find the same numbers in all measurements, for example the numbers that identify the position of a point (the coordinates) are different in different coordinate systems . What we really expect is that the physical or geometrical laws ( equations ) that appear to apply to a system will apply to another. We say that in fact the natural and geometric laws do not depend on the choice of coordinate system. That is, if a law of nature ( Newton's second law , Pythagorean theorem ) is verified in a coordinate system then automatically will be verified in any other system . Where results, however this possibility of formulating laws ? It is described in the cosmological doctrine ( for Popper) for the homogeneous and isotropic space. ……Homogeneous why testimonies of observations in different parts of the site does not change , and isotropic because the same happens for observers investigating in any direction in space .... Wikipedia The bases of this doctrine are revealed in what we call “mathematical forms of nature” a phenomenon that we shall detect to the following example. 134: THE SECOND LAW OF NEWTON . Let us take a Cartesian system ( orthogonal ) axes. The coordinates of a particle is given then by the triad (x,y,z) and if the projections of the force ( think a pair of charged particles , one fixed to the origin O and the other at the point P , in which the force is applied ) to the axes of a Cartesian system are (Fx,Fy,Fz) then the motion of the particle satisfies according to Newton's second law the three equations: ….(1) Consider a new coordinate system by rotating the original system around an axis at a particular angle . Suppose that the shift is on the axis z and therefore the coordinates in the two systems are linked through the ( linear ) transformation shift : …(2) 135: . Let us calculate the projections of force in the new coordinate system . This requires the determination of the projections of the forces to new axes that can only be achieved if we accept the principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces . We will then have that Fz=Fz΄ as the z-axis is stable. The force on the axis x΄, the Fx΄ will be given by the projection on this axis of the force FX exerted along x axis, and the projection of the force Fy exerted along y axis . Similarly we calculate the Fy΄ . Therefore projections of power are transformed as follows : . ………..(3) What form has Newton's law in the new system? Differentiating the ( 2 ) and making use of ( 3 ) we have Therefore the form of the equation remains unchanged, Newton's law applies to the new coordinate system . This is called covariance of Newton's law as to the rotation of the rectangular reference system . If now we denote the three components of force by the symbol 136: and the acceleration with Newton's second law can be written as ( vector form) and now applies to all Cartesian systems with the same origin 29 . But because in a ' transfer ' of the system , the coordinates of the vectors do not change , the vector equation still applies. The location and direction then of the system leaves the vector equations unchanged in form. THE NATURAL MOTION IN PHYSICS (Aristotle,Newton,Einstein) Aristotle: heaven, first mover Newton: absolute space, vis insita Einstein: space time, metrical field The natural motions. Throughout the history of physics, the source of natural motion was an active intangible, a metaphysical assumption in the scene of motions. What is a natural motion? It is an unforced motion. In physics it was described by the laws of motion, as referred to tracks of unconstrained bodies. 29 The appellation “invariance” would tend to make us believe that neither side had varied at all, which would , of course, be incorrect. So it is customary to speak of the covariance rather than of the invariance of vector and tensor equations. It expresses the fact that both sides have varied in exactly the same way. 137: In Aristotle every physical body has a natural place to which it naturally moves and in which it naturally rests. Natural motion is an inherent element of the body, it’s form. A straight motion from the center of the Universe was ascribed to the elements air and fire. A straight motion towards the center was given to the elements earth and water, and a circular motion around the same center was attributed to ethereal heaven. It was uniform, circular and eternal. In Newtonian mechanics and special relativity the natural motion was a uniform rectilinear motion, following it’s first law of motion (inertia, no force acting on it). In general relativity a natural motion may be linear or curved depending on which frame we take our description. There the law of motion is that every free particle moves as to take the extremal interval - length between two events30. In relativity we can only get greatest intervals by having least distances. So the “straight line” aspect of inertial motion is automatically included when we describe it in maximum interval terms. Aristotle’ s heaven . …in one way we call heaven the essence of the ultimate circulation of the universe….to another way we call heaven the body which is in continuity with the ultimate circulation of the universe to which the sun and moon and some of the stars are contained. ….Again in another way we call heaven the body which is comprehended by the last circulation….(On the heaven chapter ΙV) . Among the natural motions , Aristotle noticed one that was different from the others. It was the motion of the celestial spheres, ie the stars. So the idea that the motion of material bodies could be described as a motion of bodies as seeking their natural place (Aristotle, Physics ), could not be applied to the stars , whose movement is clearly cyclical and eternal . 30 It is the Wheeler’s principle of extremal aging but in different words. 138: But different natural motions means different bodies and thus Aristotle stated that as their motions are ruled by principles other than those of bodies in the sublunary sphere, so the heavenly bodies and the heaven, on which the stars are considered fixed , are not composed of the four elements, earth , water, air, fire, but a fifth element the ether, ( from the verb aitho meaning burn ) whose natural motion was circular and eternal, with all the qualities that he had discovered in the circular motion. These properties will determine the heavens ontologically. ( On the heaven 269 b 1 ). According to Aristotle, this essence (pemptousia, the ether, the fifth (pempto) essence), is the material that constitutes the transparent concentric celestial spheres on which are located the stars, and exhibits many properties: it is un-generated, un-aging, incorruptible, eternal, constant and unchanging. All these are properties of absolute space, and space-time, the later heavens. So heavens and everything in it, is immaterial forms. ( On heavens 278 b 12). Matter for Aristotle is the possibility of organization, the potentiality, and form, it’s actuality, these are the basic concepts of Aristotle’s ontology. . But what is a form without matter? It is like weight without body, wave amplitude without wave , is an immaterial existence of properties without possibilities of change, a new entity. Today we should say: it is mathematics! It seems that Aristotle did regard them as living beings with a rational soul as their form. Something like a mathematical space that can be Euclidean, elliptic, having dimensions and properties but without bodies . But unlike the mathematical space, the heaven should be active as moving the stars , it is a pure Aristotelian form, one end of nature the other is pure (prime ) matter, must be an active intangible, operates without mechanisms, without the factor of physical necessity that characterizes every other change . The concept of active intangible returns in the physical space-time of general relativity that we will analyze below, as in absolute Aristotle’s first mover . space of Newton. 139: What is the Aristotelian unmoved mover ? From his considerations of the nature of motion (in physics book 8 ) Aristotle concludes that there must be a logically first unmoved mover in order to explain all other motion. He argues that the motion is eternal, motion is not imparted from nothing, but from some part of the whole that is already in motion. In such cases , the motion of the part that moves the other parts of a thing requires a mover. He concluded (Physics 8.5) that must there be something that imparts motion without itself being moved. If there were no unmoved mover, there could be no motion , because a moved mover requires a cause of it’s own motion and no infinite regress is possible. Aristotle determines that there is only one unmoved mover (Physics 258b 21). This is the first mover . What are "natural" qualities of this first mover? I give the conclusions , because the discussion is very long in book 8 in Physics. 1. There may be not something that moves itself . Because this would need a cause of it’s motion . Going ever backwards from effect to cause , the process would continue ad infinitum. So the first mover is unmoved ( b Physics 258 5) 2. . A first mover that moves everything without moving itself should be rather attractive despite repellent . 3. un unmoved mover causing eternal motion must likewise be eternal 4. it is also without magnitude, since infinite force cannot reside in a finite magnitude (and can be no infinite magnitudes) 5. Having no magnitude means that the first mover is indivisible, having no parts. 140: The unmoved movers are finally, themselves, immaterial substances, (separate and individual beings), having neither parts nor magnitude. Where is located the first mover? In the first heaven, the outmost sphere of fixed stars, It is clear then that there is neither place, nor void, nor time, outside the heaven. Hence whatever is there, is of such a nature as not to occupy any place, nor does time age it; nor is there any change in any of the things which lie beyond the outermost motion; they continue through their entire duration unalterable and unmodified. The first mover on heaven is a concept of the same epistemological status with the absolute space of Newton. It is the immaterial source of natural motion in the world . . This immaterial form of activity must be intellectual in nature and it cannot be contingent upon sensory perception if it is to remain uniform. What does this mean? The hitting of the ball with the racket is the efficient cause of motion of the ball . But the milk in a dish causes the movement of the cat in another way . The cat moves aiming milk, which acts as a final cause rather than efficient . Milk acts on the “intelligence” of the cat, and cat responds to this action. All these seem mythology in modern physics . It is very " natural" the snap-action of the gravitational force, without any intermediate, while the "intelligence" in motion seems unthinkable . So the first mover is something that we define, something inconceivable, as if it was existing in nature, otherwise we can not get to the source of natural motion. It is the metaphysical assumption in the interpretation of natural motion motion. 141: But such a mythology developed in general relativity in which space-time ( the heaven ) determines the motion of bodies through the intangible curvatures of the metric field, that faithfully correspond to the first mover . Is there a mechanism here? The space-time is filled with " grooves " into which bodies move. The grooves are not materials like furrows in the earth's soil. These are mental, grooved on geometric texture of space-time made by the masses who are nearby , but not mechanically, is a brain structure so the motion is like a mentally planned movement. " Space and time are not conditions - conditions in which we liv , but ways in which we think of …” Einstein Newton’s heaven and first mover . Newton’ heaven was his absolute space! It is the metaphysical assumption in Newton’s system. We analyzed this concept in article “the space in physics”. The natural motion in Newton’s system arises from his First law: When viewed in an inertial reference frame, an object either is at rest or moves at a constant velocity, unless acted upon by an external force. This is the inertial motion, the natural motion for Newton. Into the absolute space of Newton's, inertial motion corresponds to the absence of applied force, that is the real force due to interaction with other material bodies . But then what it’s cause of ? What causes it ? there is one possible answer : the absolute space .. since this is the only " object" that is present in the material body (Francois Balibar) It is confirmed here that the absolute space must be considered (Einstein) as a physical entity, it exerts forces , in contrast with the space of geometry. So we can assume that the effect of absolute space stems from a "force " of a different nature from the applied forces involved in the formula F=m.a This is why Newton calls it vis insita (often translated as inertial force , or as fantastic or hypothetical force). Vis insita is the first mover. 142: The vis insita of the absolute space of Newton generates and sustains the inertial motion, the only 'natural' motion of Newton’s world , as the first mover of Aristotle’s heaven generates and sustains the natural movement of the stars . The absolute space exerts a ‘pseudo-force” on bodies to make spurt the phenomenon of motion . This force, vis insita , is a force that doesn’t not cause acceleration! as the first mover that acts as a final cause on the “intelligences” of bodies. The two descriptions are equivalent for the deep consciousness of man, for the “mind's eye”, both have emerged logically, changing only the Kuhnian paradigm. Finally, even though his physics (Newton’s) captured much of what we experience physically, the reality it describes for heaven and first mover turned out not to be the reality of our world. Especially when we tried to measure the relative velocity of the earth as to heaven, Newton’s heaven collapsed and was rejected by physics. Now his metaphysical assumptions must be replaced by other metaphysical assumptions, this is the history of physics, (article Physics, metaphysics, Aristotle, Duhem) But, “There is always a logical and conceivable principle of motion which is itself independent of the motion”. Later, Einstein’s space-time with it’s metric field, was another paradigm of the two entities. Einstein’s heaven: the space-time . This new reality was that space and time, as physical constructs, have to be combined into a new mathematical/physical entity called 'space-time', because the equations of relativity show that both the space and time coordinates of any event must get mixed together by the mathematics, in order to accurately describe what we see. Because space consists of 3 dimensions, and time is 1-dimensional, space-time must, therefore, be a 4-dimensional object. It is believed to be a 'continuum' because so far as we know, there are no missing points in space or instants in time, and both can be subdivided without any apparent limit in size or duration. So, physicists now routinely consider our world to be embedded in this 4- 143: dimensional space-time continuum, and all events, places, moments in history, actions and so on are described in terms of their location in space-time. In space-time the world becomes a four-dimensional geometry. So we solve geometric problems and then translate the solutions to spatial and temporal effects. When we examine a particular object from the stand point of its space-time representation, every particle is located along its world-line. This is a line that stretches from the past to the future showing the spatial location of the particle at every instant in time. This world-line exists as a complete object which may be sliced here and there so that you can see where the particle is located in space at a particular instant. Once you determine the complete world line of a particle from the forces acting upon it, you have 'solved' for its complete history. This world-line does not change with time, but simply exists as a timeless object. Similarly, in general relativity, when you solve equations for the shape of space-time (it’s metric field), this shape does not change in time, but exists as a complete timeless object. You can slice it here and there to examine what the geometry of space looks like at a particular instant. Examining consecutive slices in time will let you see whether, for example, the universe is expanding or not. General relativity . General relativity is the revolutionary assumption of Einstein that gravity is a consequence of the fact that space-time is not flat, as had been believed, but curved . It is the geometric version of gravity. General relativity puts two questions : 1. How the curvature of spacetime determines the natural motion of matter ? 2. How the presence of matter affects the curvature of space-time ? The first answer is the geodesic equation, ie the description of the shortest interval of space-time. It is the curve joining those two events which has the maximum possible length in time — for a timelike curve — or the minimum possible length in space — for a spacelike curve. The geodesic is the " straight " line in curved spaces. It’s 144: form ( the equation ) is dependent on the curvature of the space and is given in differential geometry. The second is Einstein's field equations . These are ten equations in which one member has the characteristics of matter and the other the curvature, in the form of the famous metric tensor gmn that determines the geometry of space-time, it’s metric and hence the geodesic. The geodesic equation and the field equations are associated with our wellknown Principle of Least Action. So the space –time is causally efficacious in the sense that space-time causes the distribution of matter and energy in the universe which in turn affects the curvature of space-time. The metric field “makes” the space for bodies to be able to move , so “acts geometrically” on bodies, viz mentally, as the milk in the intelligence of cat. This is a new active intangible. The metaphysical assumption for the source of natural motion. The new law of motion is that all free bodies follow geodesics of space-time , regardless whether located in areas close to the masses or in remote locations, even in inter-stellar space. This is the generalized principle of inertia . The motion close to the sun or in inter-stellar space is the same " free". The gravitational phenomena that seemed to determine the motion in heaven, are geometric phenomena. Here are not existing gravitational forces. So what is a natural motion? the unforced motion which takes place on the track of a geodesic. But again is an inherent element of the body (Aristotle), as this same element installed it’s geodesics. Here the idea of natural motion is closer to Aristotle than Newton! The metric field that corresponds to the first mover is “located” in space- time, as the first mover on heaven. 145: The ontology of space-time . The ontological status of space –time is the same as Aristotle’s heaven and Newton’s absolute space: it is a metaphysical assumption, an abstract entity that help us to grasp the reality. “…. If there is not something conceivable behind phenomena there is no science for nothing …Aristotle Physics” The heaven and first mover, the absolute space and vis insita , the space-time and it’s metric field, these are the sources of natural motion. That is always, there is a logical and conceivable principle of motion which is itself independent of the motion. So we see that Aristotle’s features permeate the whole story of the motion . All of them are an element of immateriality that human mind introduces in a material world. “… What moves the bodies lies within them, does not come from something external , and in the same time, do not move the bodies themselves , but the nature, …. Physics 255 A 6 “ In general relativity this Aristotelian picture of the motion is described in mathematical terms , where while the motion of bodies is seemingly shelf-motion in fact is a shelf-motion moving outside and where the chain " mover - moving " implies a " stationary mover " What is the metric field of space-time ? Aristotle would say it is the first mover but with mathematical form . The curved space-time is not a body , it is Aristotelian form , and something unusual happens, as in heavens. Usually we attach forms in the bodies of our experience , now we rendered form where there was no body , it is something recognizable only by the human mind which physics has never included in it’s description . it is a pure form, pure mathematics for today. The same happened with Newton’s absolute space The curvature of the surface of curtain is a form attached to the curtain so it is recognizable by the mind, but in space-time curvature which is the curtain ? The characteristics of the first mover of Aristotle in what body referred ? There is no body , nor there , nor here, only properties which are recognized there by the logic and here by mathematics. Everything in space time as in heaven are immaterial forms. 146: The effort of physics to attribute a physical existence to space-time, always loyal to external material reality, has not produced anything yet. (Aristotle would say that the form is not material ). It should then the bodies be regarded as packages of distorted space-time, but gravitons, hypothetical particles that interpret the gravitational interaction are massless, ... www. Physics 4 Source: My book: Η Αριςτοτελική «κίνηςη» και η ςύγχρονη φυςική www mpantes.gr MAXWELL’S FIELD AS AN OPERATIONAL CONCEPT operationalism, mathematical field, Faraday’s field, Maxwell’s field Introduction Electricity is a very deep phenomenon and natural philosophy is evenly divided in dealing with it. Electric theory described a number of electric forces from the Coulomb’s force 147: to the forces of Ampere, Grassman, Biot-Savart, Laplace, Weber, Gauss, Clausius , Lorentz having two paths for the study of electricity, which appear to represent deeper differences that reach to the roots of human ability for description and understanding of nature. In one it connects electricity with bodies, and studies mutual interactions of charged bodies from distance without intermediate, and the other the electric interaction is by contact through a medium. The contact should restore the ether, after replaced by the "field". We call them particle and field electricity respectively. Along the way, the view of action by contact, prevailed, but the medium of the contact was abolished, ( the ether) , we talked about waves without medium of propagation, it led to the notion of immaterial field unrelated to bodies, a super reality that lies behind everything, Aristotelian and conceivable without any operationalist verification. Physics endeavored much with the atomism on electricity, an undeniable reality, because the electric field theory that had dominated , had refused it from the beginning! The electric charge was finally accepted as a deformation of the ether, the deformation occurs on a particular material particle, and later when the ether proved nonexistent, electron was a condensation of the ' field ' that was immaterial . Wavered physics between mathematics symbols and physical entities , in propagating potentials and propagation velocity of electric action , devised equations whose mathematical treatment created endless debates about the nature of electricity. Eventually the medium theories were verified by means of electromagnetic waves of Hertz, but after several years we learned that the waves were not as water waves in the sea, they were as material waves accompanying the electron, they accompanied the photon, they were probability waves and the light was a swarm of particles, the photons. But I think that in effort to understand this edge area of nature, the area of electricity and light, we put to a test the entire philosophical background of scientific inquiry: The main dipole that emerges in the study of the electric theory is the dipole , concept – reality, and ONE (philosophical) relation of them is operationalism. 148: Operationalism . Operationalism is the view that all theoretical terms in science must be defined only by their procedures or operations. Operationalism is based on the intuition that we do not know the meaning of a concept unless we have a method of measurement for it. It is commonly considered a theory of meaning which states that “we mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations” (Bridgman 1927,). Bridgman’s operational analysis explicitly acknowledged that concepts were inevitably linked to human experience and that they were equivalent to the actions involved in the formulation and use of the corresponding terms. “ The logic of modern physics”31 . “….Hitherto many of the concepts of physics have been defined in terms of their properties. An excellent example is afforded by Newton's concept of absolute time. The following quotation from the Scholium in Book I of the Principia is illuminating: …I do not define Time, Space, Place or Motion, as being well known to all. Only I must observe that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into Absolute and Relative, True and Apparent, Mathematical and Common. (1) Absolute, True, and Mathematical Time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called Duration” . Now there is no assurance whatever that there exists in nature anything with properties like those assumed in the definition, and physics, when reduced to concepts of this character, becomes as purely an abstract science and as far removed from reality as the abstract geometry of the mathematicians, built on postulates. It is a task for experiment to discover whether concepts so defined correspond to anything in nature, and 31 Percy Bridgman publ. MacMillan (New York) Edition, 1927. 149: we must always be prepared to find that the concepts correspond to nothing or only partially correspond. In particular, if we examine the definition of absolute time in the light of experiment, we find nothing in nature with such properties. The new attitude toward a concept is entirely different. We may illustrate by considering the concept of length: what do we mean by the length of an object? We evidently know what we mean by length if we can tell what the length of any and every object is, and for the physicist nothing more is required. To find the length of an object, we have to perform certain physical operations. The concept of length is therefore fixed when the operations by which length is measured are fixed: that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined. In general, we mean by any concept nothing more than a set of operations; the concept is synonymous with a corresponding set of operations. If the concept is physical, as of length, the operations are actual physical operations, namely, those by which length is measured; or if the concept is mental, as of mathematical continuity, the operations are mental operations, namely those by which we determine whether a given aggregate of magnitudes is continuous. It is not intended to imply that there is a hard and fast division between physical and mental concepts, or that one kind of concept does not always contain an element of the other; this classification of concept is not important for our future considerations. We must demand that the set of operations equivalent to any concept be a unique set, for otherwise there are possibilities of ambiguity in practical applications which we cannot admit. It is evident that if we adopt this point of view toward concepts, namely that the proper definition of a concept is not in terms of its properties but in terms of actual operations, we need run no danger of having to revise our attitude toward nature. For if experience is always described in terms of experience, there must always be correspondence between experience and our description of it, and we need never be embarrassed, as we were in attempting to find in nature the prototype of Newton's absolute time. Furthermore, if we remember that the operations to which a physical concept are equivalent are actual physical operations, the concepts can be defined only in the range of actual experiment, and are undefined and meaningless in regions as yet untouched by experiment. It follows that strictly speaking we cannot make statements at all about regions as yet untouched, and that when we do make such statements, as we 150: inevitably shall, we are making a conventionalised extrapolation, of the looseness of which we must be fully conscious, and the justification of which is in the experiment of the future…..….. Also the term "true" or "false" can be attributed to a sentence only "operationally" ie only with the description of measurements or operations needed to verify whether or not the proposal is true ...” 32 But what are operations? To take the simplest example, the operation of counting is a mental operation, but it is an integral part of many “physical” procedures. He called such crucial non-physical operations “paper-and-pencil” operations. Bridgman lamented that it was the “most wide spread misconception with regard to the operational technique” to think that it demanded that all concepts in physics must find their meaning only in terms of physical operations in the laboratory. Later he gave a rough classification of operations into the instrumental, mental/verbal, and paper-andpencil varieties.. This view is tested exactly in the concept of field. Bridgman himself denies it’s physical existence and states that we have not even found, manipulations linked to the objective substantiation of the field, which to convey it in the reality of physics. The reality of the physicist is an operationalist reality, other realities are not in physics research. They are studied in the class not in the lab, the class leads us straight to the Platonic forms. Conclusion . It took a long time for the philosophers of science to accept that any theoretical concept used in a physical theory was not required to have its counterpart in our experience (logical positivism). But we say that if we reject operationalism from natural philosophy, then the guide of the understanding of the world are mathematics, we have not anything else, and this was the case of Maxwell’s field. If the experimental results agree with the math results, then the concepts …. follow. We create concepts ad hoc nor understood nor 32 Bridgman, Percy Williams. 1927. The Logic of Modern Physics. New York: Macmillan. 151: measured, basically we dress up maths with physical concepts, talking about logical positivism, a new intuition etc. This happens throughout the course of the electric field in electricity. The course of the mathematization of the world has started from there, even though Bridgman rejects this development: ….we are convinced that purely mathematical reasoning never can yield physical results, that if anything physical comes out of mathematics it must have been put in in another form. Our problem is to find out where the physics got into the general theory 33Bridgman The concept of the field before Maxwell . Electrostatic starts from the electric force of Coulomb, who in 1785 , with the help of the torsion scales which devised himself, verified Priesley 's law that: “The repulsive force between two small balls charged with the same kind of electricity, is inversely proportional to the square of the distance of their centers”. .... Whatever the cause of electricity, we can explain all phenomena on the assumption that there are two electric fluids, portions of the same fluid is repelled and attracted with the portions of other fluid ........ Coulomb The Coulomb force then mentioned among the portions of the fluid , ie between molecules of the electric fluid , called charges and is known from high school: The type of this force that resembled the type of Newton's gravity, gave in electricity an old mathematical background to solve practical problems . The two central mathematical concepts of electrostatics is the field and the potential. These help us to solve practical problems of calculation of electric power in the bodies , and ... there it stops. In each point of a charge, we define a vector E ( force exerted at each point of the fluid round the charges ) .Thus the force exerted between the loads q1 33 The Logic of modern physics p169 152: and q2 is F = Eq2 where E is the field of q1 and the force F, Coulomb’s force. But beware , we measure the force and we define the field. Next , the mathematical result of the electrostatic field, is a function of the potential U associated with the intensity from the time of Lagrange by of calculating the electrostatic field with much simpler steps from the calculation of E. This magnitude is numeric and it’s physical ' interpretation ' is that any electric or other preservative system, is considered that it is accompanied by an imaginary energy storage, the potential , from which it is pumped the energy that moves charges through ( 2 ) . When the system generates work, it finds it on the storage of potential so the potential energy decreases in the amount of energy produced. The opposite happens when we give the system energy outside. The field then before Maxwell is a simple mathematical representation, just as was the Newtonian gravitational field. Has no status independent of its source, is not material and does not connects evolutionary the events in space and time. This mathematical approach of the field will not remain the same today as the field has become the reality of the natural world. In field electromagnetism theory constructed by the Faraday and Maxwell, fields are basic physical entities and may be examined without reference to material bodies . Today mathematical expressions Poisson and Gauss ( potential and field ) became realities without changing anything in the formulas. The pulsed electric and magnetic fields can be moved in space in the form of radio waves and other types of electromagnetic radiation. The pre-history of Maxwell’s electric field . Michel Faraday revolutionized physics in 1830 with its lines of force and the field as a physical entity . He managed to support the overthrow of this on a careful experimental basis. Between 1864 and 1873 James Clerk Maxwell achieved a similar breakthrough only with clear thinking34. The starting point of his theory, were the 34 I believe that with Maxwell’s work starts this "the now widely noticeable arrogance of mathematical physicists to give priority in formalism against experimental events”…George Galeczki 153: experimental results of Coulomb, Ampere, and Faraday that until 1864 expressed the laws of electromagnetism in integral form, according to the mathematics of their time and the ending point were the electromagnetic waves. With the formulation of Coulomb's law, which resembled Newton's gravitational force that led to the mechanical triumphs from earthly to heavenly things , the ' electricians ' set themselves the target to build the electric theory on the same philosophy of action at a distance, with no assumption about intermediate medium, believing in similar developments in electricity These researchers described all phenomena , grounding a complete electrical theory, considering the electricity consists of charged particles , rendered magnetism an electrical phenomenon exactly as we now interpret it, interpreted the induction with their electrons , measured the forces between currents, between moving electrons, between accelerated electrons, ... and then were forgotten. Eventually the British school of physics dominated, giving a strong and coordinated ideological struggle. It’s central philosophy was the Maxwell field which were implementing the program of electric philosophy of Faraday, on the idea of action by contact through a medium. But the reader should understand that the discoveries of the first half of the 19th century, that began in France and spread to Germany, centered Gauss and his students (Weber, Riemann..), was not a secondary stream in physics. It remained for almost the entire 19th century, the central line of thinking. We will begin recounting our story with the philosophy of the great British experimenter Michael Faraday. There are the lines of gravitation force , those of electrostatic induction, those of magnetic action I don’t perceive in any part of space whether (to use the common phrase) vacant of filled with matter in which they are exerted…Faraday35 The man who processed and refined with mathematical modeling of the views of Faraday, Maxwell wrote: 35 “Experimental resresearches in electicity” 154: …We are unable to conceive of propagation in time, except either as the flight of a material substance through space or as the propagation of a condition of motion or stress in a medium already existing in space …Maxwell36 So there is a reality in the area round of charges and currents in the work of Maxwell and Faraday ..in theories of action by contact, the intensity of the field is a reality that exists even though the reactive charges are removed ...... P. Hertz even .... The theory of Maxwell continues to yields a shelf existent reality to the vector E independent of the presence of the test (second) charge ...... Abraham-Becker The origin of Maxwell’s field . The origin of Maxwell’s field are the Faraday’s lines of force. These lines are used to explain the action of forces from a distance. Faraday, the great experimenter philosopher, was the first to conceive the concept of the electromagnetic field. He distinguished, a reality of another class from that of matter ( electrotonic state), it was something real that was taking place in electric and magnetic phenomena. It was capable of carrying influences from place to place without being treated like a mathematical structure as the gravitational field of its time. In his opinion, the phenomena of electricity and magnetism should be accessed through the field rather than through the charged bodies and currents . In other words , according to Faraday, when a current flows through a conductor , the most important aspect of this phenomenon was not the electric current , but the fields of electric and magnetic forces distributed to the round from the area of the current. In 1835 for wrote for intermediate particles that their contact transfers electrical activity: .... It seems probable and possible that the magnetic action can be transmitted at a distance , through the action of intermediate particles in a way that similarly the forces of static electricity are transferred in distance. Interfering particles , we assume for the 36 Treatise…p.492 155: moment, that they are in a special situation that I often express ( without much success ) the term electrotonic situation….Faraday This Farady’s view became conviction for many generations of physycists, after the domination of mechanical models in electricity. How is manifested this reality for Faraday? With lines of force which he imagined to fill all space . Physicists were used to illustrate the magnetic forces, scattering iron filings on a piece of paper and observing the lines formed by the action of a magnet underneath the paper .. These lines submitted to Faraday the idea of magnetic lines of force with which , he claimed, we can get informations not only on the direction of the magnetic force but also on its size. The intensity of the field, characterized by the density of field lines per unit area , vertical in it’s direction. ..... I do not perceive in any part of space whether (to use the common phrase) vacant or filled with matter can help myself , anything but forces and the lines in which they are exerted …Faraday37 Later in the study of dielectric media he introduced electric field and electric lines of force, wherein the reduction or the increase of the density of these lines was describing the behavior of dielectric. Faraday’s lines of force were the transport routes of action of the intervening medium in electric and magnetic phenomena. For example how work the lines of force in the law of Coulomb; In Faraday’s view, the medium intervenes between the electrified balls - whatever it is - manifests attractions and tensions which are presented in "squinting " or " contraction " of the lines of force, also in mutual repulsions and lateral pressures ( densification or dilution ) . So when they end up in the other ball , these deformations are transmitted to it and move it . When the field is more powerful the lines of force are more and we have a more intense effect . The lines of force exert mechanical actions! The idea of this spreading action from point to point through a medium with the effect of adjacent and contiguous particles were applicable to every area of physics in Faraday’s theory. 37 Experimental researches in electricity 156: He believed in “possible and probable physical existence of lines of force for gravitation, electrostatics magnetism” He indulged in the speculation that light and radiant heat were tremors of the lines of force: “ a notion which, as far as it is admitted, will dispense with the ether , which in another view is supposed to be the medium in which these vibrations take place” ….Faraday Thus, the lines of force become an independent reality, they are the " nerves " of the medium, which moved charged and magnetized bodies. “…Instead of an inviolable action at a distance between two electrified bodies, the Faraday considered the entire space between the bodies taut and full of mutual driven off loops ..... The concept of dynamic lines are in my opinion one of the biggest services Faradayto science ... J.J.Thomson “..... The view consolidated views of Faraday delivers a real existence in the dynamic lines in the sense that they exist as an independent entity ....” Grimsehl-Tomaschek ....” As a result of the researches of Faraday and Maxwell we regard the properties of charged bodies as due to lines of force which spread out from the bodies into the surrounding medium” ... E.W. Barnes, Scientific theory and religion ) It is better to imagine (to understand Faraday) that these ' energy pathways ' did not appear there simultaneously with the iron filings, but there were like landscape around the magnet, and now drifted filings in their topography . This image, intuitively touch better in our imagination than the idea that around the magnet there is nothing and that all phenomenon is created by placing the iron filings. This is the reason why the medium theories eventually passed through the consciousness of physicists. Instead of an intangible action at a distance between two electrified bodies , Faraday regarded the whole space, between the bodies as full of stretched mutually repelent springs. The conception of lines of force is in my opinion one of the greatest of Faraday’s many great services to Science ….J.J.Thomson 38 38 Today this seems to be becoming rather excessive for celebrating the philosophy of the field rather than the Faraday. The greatest service Faraday's science is vast experimental work in which we met the electrical phenomena, not their interpretations 157: But what are these springs? How they can transfer energy? In Faraday’s field , where mathematics are missing, physical processes become Aristotelian qualitative descriptions : “….According to Ampere the existence of a magnetic field is inseparably connected with the motion f charged bodies. But according to Faraday the magnetic field is associate , not with the motion of charges, but with the motion of the tubes (of lines of force) attached to them;and since the tubes are flexible, there is every reason to suppose that the tubes might move without the charges….N.Campbell, modern electricity theory Bambridge” They are qualitative descriptions as Albert O’Rahily says39: …we obtain the expression of the force -vector at any point. We then draw the tangent-lines of this vector as a useful graph. To strengthen our belief we call them “tubes” . Next we endow them with “flexibility” and declare that they “might move without the charges” from which we started. Then we pick out one prtion of the force on moving charges and call it “magnetic force” and similarly attribute to it flexible independently moving tubes. Finally we declare that this double system of tubes is “the only important theory which has ever been proposed to explain electricity magnetism and light……Alfred O’ Rahily Today we know that the lines of force are not so a service to science, as a contribution to the educational process. The natural assumption which underscores (the ether) has been removed. The situation described in the following passage, brings us back to the operationalism in definition of concepts in physics. …any explanation of this kind which attributes mechanical properties to tubes of force is highly artificial as there is no evidence for their existence. Nowadays physicists are becoming more and more inclined to shun such explanations; so the mechanical explanation of the interaction of electrically charged bodies is rapidly falling into disuse. It still lingers in text-books, however, and it is important to recognise its arbitrariness…Piley 40 39 Electromagnetic theory by Alfred O’ Rahily Dover 40 Electricity Oxford 1933 158: Faraday, being purely experimental with little knowledge of theory, could not perceive an electric field with the purely mathematical way. So he introduced the lines of force and he believed in their material substance. . Maxwell continued and described mathematically exactly the ideas of Faraday's lines of force , the field that intervenes in electrical phenomena . The lines of force became field and were installed as an absolute ( materials in genesis ) reality. …We are unable to conceive of propagation in time, except either as the flight of a material substance through space or as the propagation of a condition of motion or stress in a medium already existing in space …Maxwell41 The school of Cambridge . In prevalence of theories of electric medium which began with the intuitive interpretations of Faraday, the main protagonists apart from him, was the whole "school of Cambridge " ( as Tait , W.Thomson, Heavyside, etc himself Maxwell) which is founded on mathematics of Green, Stokes and W. Thomson, in which (school) dominant belief was that any physical action is based on dynamics. ….I am never satisfied if I can not construct a mechanical model of something. Then only I can understand. And this is why I could not capture the electromagnetic theory ...... I had not a moment of peace or happiness in relation to electromagnetic theory from November 1846 ... W. Thomson Thomson, began to investigate the proportions of electrical phenomena and flexibility. These surveys showed a picture of the spread of electrical and magnetic activity. He made the suggestion that they spread like the spreading the elastic displacement in an elastic solid. He was unable to promote his instructions connecting the ideas of Faraday to the mathematical proportions had invented. So towards the end of his life in 1896 after the failure of login electromagnetism with mechanical models wrote: ….A single word characterizes my efforts on scientific research over the past fifty years. I 41 Treatise…p.492 159: do not know anything more about the electric and magnetic force , the relationship of the ether of electricity and matter than I knew and I taught my students fifty years ago ....W Thomson.. Maxwell was the one who showed the " electrotonic ¨ Faraday's situation can be represented by mathematical symbols borrowing ideas from the researches of Thomson . Maxwell received the torch of the " school of Cambridge " whose electrical philosophy presented in December 1864 at the presentation of his book ' A dynamic theory of the electromagnetic field . " He then said: …It happens a tuple of phenomena in electricity and magnetism lead to the same conclusion as that of Optics, namely that there is an ethereal medium that permeates all bodies and is differentiated only by their presence. That the parts of this medium are tucked in motion by electric currents and magnets, that this movement can spread from one part of the medium to another through forces generated from the connections of these parts, that the influence of these forces generated deformation dependent upon the elasticity of these connections , and finally as a consequence of all this, it is possible to show the energy of this medium in two forms , one as kinetic energy of the parts and the other as dynamic energy of its connections as a result of their elasticity …Maxwell So then we see that we are leading to the assumption of a complex structure capable of a multitude of movements , but also having such connections, that the movement of one region depends according to specific relations with the movement of other parts, which moves associated with forces that are born from the movement of the joined elements thanks to their elasticity . Such a mechanism should be the object of the general laws of dynamics. Later Maxwell described the field in a mathematical way, expressing the laws Faraday (induction) Ampere and Gauss, which were expressed in integral form, in a system of partial differential equations, where the theorems of Stokes, Green and divergence were dressed with physical metaphors and changed the world. But still believing that these metaphors were real, as the lines of force of Faraday. Now they constituted the field. So there are two tendencies in Maxwell’s work: the first is the attempt to explain electrical actions by the properties of the hypothetical medium which is their carrier 160: and the second is a purely mathematical description by means of partial differential equations based on the assumption of certain vectors specifying the electric and magnetic state of a body. Electricity accordingly exists in two entirely different forms: the electric substance within the conductor and the electric field in free space. …The electric field is the portion of space in the neighbourhood of electric bodies, considered with reference to electrical phenomena… Maxwell The space all around a magnet pervaded by the magnetic forces is termed the field of that magnet…..Livens The space within which the ether is sensibly disturbed and within which sensible ponderomotive forces are exercised ..is called the electrostatic field….Drude …There is said to be an “electric field” in a region which is traversed by lines of force…Bragg ..In the near action theory the field strength is a reality which exists even when the reacting bodies are removed … Hertz …E is not the actual strength of the electric field at the point where the charge e is situated, but rather the field-strength that would exist at that point if the charge e were not present at all…Planck We must suppose that it E exists at all points about q even when our test charge is not present; but we can prove it’s existence only by bringing the test charge toy Q…White ….... The idea of Faraday's force field -the field- has fundamentally changed our picture of the world ... B. Bavik Maxwell continued and described mathematically exactly the ideas of Faraday’s lines of force, the field that intervenes in electrical phenomena and was installed as an absolute (material in genesis) reality. The essence of the above field approaches in Faraday's perception is the assumption that when there is not second charge the lines of force exist and move towards infinity . But without the second charge they are nonexistent. The flow of Gauss in the Coulomb field is fantastic, it is a mathematical trick to simplify the results. But in Faraday’s scope becomes real, is the material factor that produces the phenomena. How believable are they? 161: …Any statement which is made about the electric field in the neghbourhood of a charged body cannot strictly speaking be taken to mean more than that a second charged body, if placed there , would behave in a particular way…the physical reality of the magnetic pole remains as questionable as that of the electric field ..J. Piley Even more decisively Leech, carries us back to pre- Faraday period. ..... The electric field variables φ and E are not subject to direct observation but their values can be derived from observations in material systems. A clear understanding of this fact could prevent questions about the nature of these variables. Their reality should be attributed as follows: to be considered as mathematical entities whose importance lies in the possibility to use them to describe and predict observable changes in the behavior of exclusively material systems ..... Leech Mathematical entities…but the mathematical entities became basic concepts of real word. ….it follows that the (Maxwell’s) equations form a consistent scheme, independently of the hypothesis from which they have been derived,…independently of any physical interpretations which may be assigned to the various terms of the equations….we may if we please to discard Maxwell’s interpretation. James Jean Here Bridgman says: …there would seem to be no necessary inherent in the requirements of the model itself , that all the mathematical operations should correspond to recognizable processes in the physical system. Nor is there any more any reason why all the symbols appearing in the fundamental mathematical equations should have their physical counterpart, nor why purely auxiliary mathematical quantities should not be invented to facilitate the mathematical manipulations, if that proves possible…Bridgman Well, what is going on? May be the charge produces a change in the state of the surrounding aether, may be the charge extend into the region about it, or is something incapable of description in mechanical terms. But the important point is that if another charge is placed at any point of such space it will be acted on by a force and accelerated. The issue touches on philosophy. In operationalist view this important point is the only point. 162: …I believe that a critical examination will show that the ascription of physical reality to the electric field is entirely without justification. I cannot find a single physical phenomenon or a single physical operation by which evidence of the existence of the field may be obtained independently of the operations which entered into the definition ….i do not believe that the additional implication of physical reality has justified itself by bringing to light a single positive result , or can offer more than the pragmatic plea of having stimulated a large number of experiments , all with persistently negative results…..the electromagnetic field is an invention and is never subject to direct observation. What we observe are material bodies with or without charges-including eventually in this category electrons- their positions, motions and the forces to which they are subject. Bridgman Sources . 1. Electromagnetic Theory : Alfred O’ Rahily. Dover Publications Inc. New York 2. A Treatise on Electricity and magnetism : James Clerk Maxwell, Dover Publications Inc. New York 3. Electromagnetic Waves : F.W.G. White, Wiley & sons Inc. Methuen &Co. LTD. New York –John 4. Lectures on Electromagnetic theory: L. Solymar , Oxford University press 5. Foundations of electromagnetic theory John R. Reitz, Addison Wesley publishing company 6.Aether and Electricity: Sir Edmund Whittaker , Harper Brothers , New York 7. Electrodynamics: Arnold Sommerfield, Academic Press New York and London 8. Electrodynamics and relativity, E.G. Cullwick, Longmans, Green and CO. 9. Electromagnetism: Hohn C. Slater and Nathaniel H. Frank, Dover Publications Inc. New York 10. Σο ηλεκτρομαγνητικό πεδίο ωσ εκτελεςτική έννοια www.mpantes.gr George Mpantes mathematics teacher . 163: AN INTRODUCTION IN SPECIAL RELATIVITY. The hidden harmony is superior to the apparent….Heraclitus Introduction The physical principle The geometrical nature of space-time The transformations of coordinates The geometrical structure of space-time , the four vectors The 4-velocity vector Physical results of 4-velocity introduction 164: This article is a continuation of "the mathematical and philosophical concept of vector” (//www.academia.edu/8188816) which referred to vectors of Euclidean space, we will now consider the vectors in the "world" of Minkowski. We will see that the four-vector in relativity is again a mathematical form of nature-as in classical physics- which resulted from deeper surveys and thinner experiments in areas that were not in our immediate observation, and even caused our intuition. 42 But here these 4-vectors, do not only perform a simple mathematical destination, this of simplicity and economy in writing equations, namely the geometrical unification of immovable reference systems. The 4-vectors unify geometrically, moving reference systems, so discovering hidden natural unifications of magnitudes and principles, which were invisible to the world of 3-vectors. (equivalence of mass and energy)43 The physical principle The physical principle of beginning the investigation of 4-vectors is the second postulate of Einstein, in the foundations of relativity: The velocity of light in empty space is the same in all reference frames and is independent of the motion of the emitting body.44 This axiom, however violates common sense since two observers count everyone at the same speed of a wave, while moving between them. 42 Technically we say that the discovery by Herman Minkowski was that the modified theory of space and time on which Albert Einsein had founded the relativistic electrodynamics of moving bodies, was 4 non other than the theory of invariants of a definite group of linear transformations of R , namely the Lorentz group 43 ) “. ….from now on the ideas of space and time as independent concepts shall disappear and only a union of the two shall be retained as an independent concept” …Minkowski ,Lecture given in Cologne , Sept.21 1908 44 At the time it was made (the axiom) , was a very bold step , for the experimental evidence in favor of this was not then overwhelming , as it is today. Then ,as now , it was contrary to the common sense, this axiom accepted the experimental evidence as fact. It left it open the need to explain the paradox of the light spheres…Albert Shadowitz p.9 165: Based on this axiom and the imaginary experiments , Einstein was able to reconcile all the experimental results with the theory,so today is a cornerstone of natural laws. For our theme we will see the consequences of this axiom. Let (Figure 1), two rectangular Cartesian inertial moving parallel in the x-axis with their origins, O and O’ coincide for t = t’ = 0 (standard configuration) and a light emission from a common principle, in the same time . An observer at rest in the S receives a light signal at time t, at a point M (x, y, z) of a sphere of radius ct, while an observer in system S 'receives the signal at point M΄ (x΄, y΄ , z΄) of the corresponding sphere at time t'. Since the speed of light is the same in both systems, the following relations hold x2 + y2 + z2-c2 t2 = 0 and x΄2 + y΄2 + z’2 –c2 t΄2 = 0 or in general, by letting the light emission occur at an arbitrary point, and not in time t=t΄=0 then holds 45: (χ2-χ12)+(y2-y1)2+(z2- z1)2-c2 (t2 –t1 )2 =(χ΄2-χ΄12)+(y΄2-y΄1)2+(z΄2- z΄1)2-c2 (t΄2 –t΄1 )2 , hence coordinate transformations linking the two systems should left unchanged the amount s2 =x2+y2+z2-c2t2 …………(1) .(for dinstances from origin ) or ds2=dx2+dy2+dz2-cdt2 (differential form) The invariance of s2 is the mathematical expression for the constant velocity of light in vacuum for all the observers non accelerated with respect to each other 45 Παπαδημηηράκη Χλίτλια, -Τζοσκαλάς ζελ.289 166: the geometrical nature of space-time Minkowski immediately recognised in the mathematical form of this invariant (1) the expression of a square of a distance in a four dimensional continuum. This distance was termed the Einsteinian interval or simply the interval. The invariance of all such distances implied the absolute character of the metric relations of this four-dimentional continuum, regardless of our motion , and thereby implied the absolute nature of the continuum itself. The continuum was neither space nor time , but it pertained to both, since a distance between two of its points could be split up in space and time distances in various ways , just as a distance in ordinary space can be split up into length ,breath and height , also in various ways. For these reasons it was called space-time. The geometry of this space-time based upon the metric (1) will be called geometry of Minkowski or M-geometry. This metric gives us the "distance" for all observers, of events like one in Athens today and another tomorrow in Paris! This interval was a discovery of great importance to the history of natural philosophy. It was an invariant , representing the square of the spatial distance covered by a body in any Galilean frame , minus c2 times the square of the duration required for this performance (the duration being measured of course, by the standard of time of the same frame). It may be positive, negative or zero (for events on a light ray), and this creates a classificassion in intervals, and later in 4- vectors . This contribution of signs, makes us be careful not to think of the world Minkowski as a straightforward generalization of ordinary Euclidean 3-space to four dimensions, with time as just one more dimension. The space-time has no isotropic properties, in stark contrast with the Euclidean space with it’s positive definite metric. (W.Rindler p.63) As we saw, in interval , the distance of two events is correlated with the distance which light can travel in the time between the two events. So the interval is a operationally defined magnitude , with instrumentation clocks, solid rods and light rays. The metric of spacetime is an operational metric, as the metric performed with solid rods in three-dimensional Euclidean space. ……this does not mean , however that space and time lose their specific individual differences , for, clearly , clocks and measuring rods are quite different types of measuring instruments. This union of space and time , therefore , preserves their specific 167: (Reinhenbach properties…… p.188) What is the role of the light ray? This makes the conceptual union of space and time, an operational union. This union has it’s genesis in the light ray that will link the two events, since the motion of the light created the interval(the second postulate of relativity). The light ray creates the union, joining the two measurements of the rod and the clock, making the light protagonist of the universal metric. Hence the presence of space-time continuum began to be distinguished, as physics proceeded to study the world of high speeds. The next problem is to determine the geometry of this mysterious continuum. In the first place , it may appear strange that measurements with clocks can be coordinated with measurements with rods. This difficulty , need not arrest us ; for although dt is a time which can be measured with a cock, yet cdt , being the product of a velocity by a time , is a spatial length since it represents the spatial distance covered by light in the time dt. For this reason we may consider our four-dimensional continuum to possess the qualifications of an extensional space. In tensor calculus we learn that knowing the metric tensor of a reference system, which is characteristic for each system, we can extract all the geometric features of the space and to establish the analytic geometry of the system, the metric tensor is a function that shows us how to calculate the distance between two points in a given space. In flat spaces with rectilinear coordinate systems the metric tensor is independent of the position of points. In curved spaces (curvilinear coordinates) a metric tensor is a function of position, it determines the metric in an infinitesimal region around it, the geometry becomes differential . In our reference system, it is easy to see (from the metric element 1) that the metric tensor is g mn 1  0  0  0  0  1 0 0 .......... ...( 2) 0 1 0  0 0  1 0 0 168: So we have a “flat space" and rectilinear coordinates. The only flat continuum which is now known, is the Euclidean space, but space-time although it is flat, it is not strictly Euclidean: the coefficient of c2t2 is -1 instead of 1 that we would have in a four-dimensional Euclidean space. Thus, space-time was characterized (by the terminology of Hilbert) as a pseudo-Euclidean and measured by means of pseudo-Euclidean metric, which is the geometry of the flat space-time, the M(inkowskian) -geometry. The transformations of coordinates . The invariance of the interval in all frames of reference means that the transformations of coordinate of systems, should guarantee this behavior. Fortunately, there is a unique transformation that satisfies the above condition while maintaining the linarity of the relations between the coordinates (space homogeneous and isotropic, homogeneous time), called "Lorentz transformation" x   ( x' t ' ), y  y' , z  z' , t   (t'  The Lorentz transformations (are the  x' c2 ), , , , , , , , , , , , , , , , , (3) corresponding of orthogonal linear transformations of Euclidean geometry) are essential in the theory of relativity, which could be called "theory of Lorentz transformations» (Synge). The first service is that substituting in (1) the coordinates of (3) we have that x 2  y 2  z 2  c 2 t 2  x ' 2  y ' 2  z ' 2 c 2 t ' 2  s 2 Yet the appearance of the space-coordinate x , in the transformation of the time is the mathematical expression of the relativity of simultaneity, etc. But a key geometrical feature is that they connect space-time systems with a common origin. It means that in M –geometry, the Lorentz transformation describe a "rotation" of the space-time system, unlike Galilean transformations describing a translation of systems in the Euclidean space. This difference in the form of transformations, converts 169: the invariance of Newton's laws in covariance of relativistic laws. The geometry is the deeper reality. The geometrical structure of space-time, the 4- vectors ………If there is not something inconceivable, beyond phenomena (καθέκαςτα) ), but all were sensible , we would not have science for any thing, except only if one says that the sense is science. (Aristotle, Metaphysics 999 b 1) This phenomenon of the space-time linear element, Minkowski has incorporated into an elegant calculus of a new geometry. The need of transition to a new geometry for the physical description comes from the first axiom of relativity: all inertial frames are equivalent for the formulation of all physical laws. Hence all physical laws should be written in vector language but now the "space" of phenomena was space-time. That is the new geometry would be four-dimensional . This is the geometry of Minkowski. But 4- vectors have a deeper geometrical characteristic with physical meaning: via these, the natural laws that are independent of motion are produced, thus we can recognize by their form alone whether a given or proposed law is Lorentz invariant without having to apply a transformation. This has great heuristic value. Moreover , by automatically combining such entities as space and time, momentum and energy, electric and magnetic field, etc. the formalism illuminates some profound physical interconnections. So the production of 4-vectors precedes the equation of the law. The production of 4-vectors in Minkowski space is the entire course of the theory of relativity. How can we explain this fact? The geometry of space time produce new physical phenomena! But we have seen this situation in geometry of 3-space. As in evolution of geometry of ordinary space, we had new truths for this space, now we have new theorems in geometry of space-time. For example in Euclidean geometry we have the theorem of Gauss that “there is not an upper limit in the area of a triangle” . It is a profound geometrical interconnection in space’s structure, a new truth. In the evolution 170: of geometry of space-time, we have new 4-geometrical discoveries, but now they are physical events. Eventually relativity became a geometric theory, "the geometry of flat space-time." The formalism of tensor calculus in geometry remains, although intuition is lost in fourdimensional world. The Lorentzian 4-vectors are the basic mathematical objects of this geometry. So far we have met the metric tensor gmn, the relative position vector xr, and the displacement vector dxr .Their essence as the known 3-vectors, is that they can be defined independently from any system (now) of spatio-temporal coordinates. For space-time, as for the static Euclidean space, the vector is an oriented line segment (arrow) . There is the initial and terminal point, it’s magnitude, sense , direction, and all known concepts from the vector geometry. Still, in connection with the linear spatiotemporal reference systems, we should define four numbers, each gives the length of the projection of the arrow in the respective axis of the system. These four numbers form the 4-vector. The three of them refer to spatial directions and the fourth in the "direction" of time, all referred in this reference system. If the magnitude of this 4vector is invariant with respect to Lorentz transformations, then it is an original geometric object in Minkowski spacetime, as a 3-vector in Euclidean 3-space. In order to understand the new vectorial reality in M- space we will proceed as follows: In connection with an event O as an origin of the coordinate system, the world of facts (x, y, z, t), can be divided in a Lorentzian invariant way into two parts , which are characterized by s2 <0 s2> 0 These x2 (past y2 future) (intermediate regions + and are + z2-c2 region) separated t2 (A) = by 0 (light (B) the cone cone) (C) on which the world lines of the light rays lie.. If one lets the startind point of a vector coincide with the origin of the coordinate system , the vector is called space –like if its end point lies in world region (B) , time- like if it is in (A). It is called a null vector (vector of magnitude zero) if it lies on the cone (C). 171: Space-like events can not influence the observer directly, as never happens x> ct, these events do not happen in the reality of our world (are located elsewhere, ). The region occupied by events of type (B) do not stand in an absolute temporal relationship with O, so these events can not have a causal connection with the origin. Still, if we consider that the speed of light is maximum in nature, then the path of body with mass, must be time-like, that is the cosmic line be located entirely inside the light cone. Only world points in the region (A) can have causal connections with the origin. All these are shown in figure of the cone of light in the system (x O, ct), who is analyzed in every book of relativity46. We will simply note that the diagram of the cone is the Euclidean model of the pseudo-Euclidean metric. We illustrate the space-time geometry (M-geometry) with Euclidean shapes (E-geometry). Their relationship the vector geometry points us an in R3 was an invariant analogy as, theory of orthogonal transformations, now the M -geometry is an invariant theory of Lorentz transformations, The events in space-time are points in diagram, the cosmic body lines are straight lines, the locus of points with constant M- distance from the origin is E-hyberbolas, the Lorentz transformations (in the figure) can not be looked upon as a rotation of the coordinate system , but as a transformation of one system of conjugate diameters of the hyperboloid x2 + y2 + z2 - (ct) 2 = constant, into another47, the M-rotation, 46 For easy visualizations of four dimensions, two space coordinates are often suppressed 47 This interpretation of the Lorentz transformation ,and also the terminology employed here , occur first in Mikowski’s work (Pauli) 172: the M-orthogonal directions (OA,OB in figure) form E-equal angles with the null lines, etc. The 4-velocity vector. As an example of composition of a four vector referring to the world of motion, we shall define the 4-vector of velocity, the 4-velocity or the cosmic velocity. Such a vector in a change in the Galilean system must to transform as the coordinates (Lorentz) but still be linked (marginally) with known three-vector u, the 3- velocity of the particle. The process of generalizing a known 3-vector by a slight modification ,if necessary, of its three components and the addition of a fourth to form a 4-vector is a most fruitful way of discovering significant 4-vectrors , and through them, the relativistically valid laws of physics.(Rindler) The natural solution is to divide the 4-vector dR = (dr, cdt) with the differential element dτ of proper time, which is an invariant in the Minkowski space-time as the dt in classical spacetime. For this ratio, is easily proven transformations in a change of that satisfies the Lorentz system, in M-space. But from time dilation we know that proper time between successive positions is associated with the coordinate time t dt  d . 1 u2 1 2 c   (u ).d This U of the system with the relationship description gives  dR d    (u ) (r , ct )   (u )(u , c)......( 4) d dt It is apparent from the foregoing description that this is the desired 4-vector speed, transformed as d R in a change of the reference system, and it is time-like vector as its magnitude is –c2. Physical results of 4-velocity When a particle in a system is at rest (u = 0), it’s cosmic speed is (0,0,0, c), that is 173: parallel to the time axis and has a length equal to one unit of time. So even when we sleep, we "run" in the direction of time, with speed c. It's what we say that time never stops. But revelations of the cosmic speed are deeper. With increasing space velocity u of a mobile, increase the temporal dimension of cosmic speed. What does this mean? that the fraction cdt /dτ becomes larger so the ratio dt /dτ increases. This means in turn that the proper time of the mobile becomes increasingly smaller relative to the coordinate time system, i.e. the flow of time becomes increasingly slower for the mobile, relative to the system, thus the mobile is "aging" later . Placing these in our daily experience, we say that the faster we run, the more slowly age with respect to the whole of life on earth. But the findings continue towards a new kinematics: this is a new type of addition of velocities which takes us completely outside of our traditional perceptions. Let a body in inertial system S with velocity u, we find the velocity in the inertial frame S ' in the positive direction of the Ox. We should not confuse γ(u) of the cosmic speed with γ(υ) of the Lorentz transformations, u is the known speed of a mobile in our system. υ is the relative velocity of the two systems, Applying for the 4- velocity Lorentz transformations we have for the first component ux '  ux  .......... .(5) u x 1 2 c (5) is the well-known formula for the addition of velocities in relativity. Observe that if we put c in place of ux we have ux = c ie known proposal for the constancy of the speed of light in all systems (second principle of relativity). The formula (5) differs from the corresponding classical formulae by the presence of the denominators. The classical model can be recover, as might be expected , by letting c → ∞. 174: Even the new formula produce the effect of limiting the speed of light in the universe. Speeds measured in the same system, are added as in classical theory, with the rule of the parallelogram. For the observer of another system, the rule of the parallelogram ceases to be valid. The 4-speed will run its course in relativity, producing new 4-vectors which in turn will change the entire landscape of Newtonian mechanics. books I read 1. Introduction to vector and tensor analysis: Dover B.Y( Robert C. Wrede) 2. Special relativity:Oliver and Boyd Edimburg,( W. Rindler) 3. The meaning of relativity:.Princeton University Press……(Albert Einstein) 4. Relativity and geometry: .Dover N.Y.(Roberto Torreti) 5. Relativity: the special theory: North-Holland publishing Company ...(Synge) 6. ειςαγωγή ςτον τανυςτικό λογιςμό Γ.Μπαντέσ http://www.scribd.com/doc/230022822 7. Η θεωρία τησ ςχετικότητασ ςτο Λύκειο Γ. Μπαντέσ http://www.scribd.com/doc/228965756 8. the philosophy of Space and time Dover, Hans Reichenbach 175: THE CHAPTER OF THERMAL RADIATION …..Light and matter are both single entities and the apparent duality arises in the limitations of our language (Heisenberg) Introduction . Light from hot bodies Classical statistics of radiation Quantum statistics of radiation The calculations with infinity The series and the improper integral The Euler-Maclaurin formula Introduction . For the measuring of thermal radiation, two mathematical models of infinity were proposed, the improper integral and the infinite series. The experimental physics rejected the first and verified the second . The mathematical models are referred in numbers, as mathematics is the science of numbers, where experiments are imaginary, we have the mathematical continuity of numbers and the discontinuity of radiation. In this sense we understand more clearly the Pythagorean dictum: "Everything is number" since the interpretation of nature is investigated through numbers, in radiation the series replaces the integral. 176: All bodies, rocks, snow, oceans ,ourselves, emit radiation. Most of it is invisible to humans, but that does not make them less real. The classical worldview, namely that it consists of matter (particles) and radiation, could be the basis for a description of all natural phenomena. We consider as point-particles (before the discovery of the proton) electrons and light radiation the Maxwell electromagnetic radiation, which is emitted by an accelerated charge or retarded relative to the observer. The energy radiated from a charged particle can be absorbed by another that is located within the electromagnetic field of the first. The mechanism of emission-absorption of radiation by matter, is the mechanism of interaction of radiation with matter, and it’s understanding is important for interpreting the behavior of matter. But classical physics fails to explain adequately this mechanism, and this has motivated the creation of quantum physics. Light from hot bodies. What connects mater and light? It is the heat. It is known that all material bodies become luminous when heated to a sufficiently high temperature. On cosmic scale sun and stars emit light because their surfaces are very hot. We call thermal radiation the electromagnetic radiation emitted by bodies because of their temperature. ( black body radiation physicsgg) 177: The study therefore of the interaction light- matter acquires experimental entity and documentation through the heat, namely the thermodynamic energy exchange between light and matter, therefore is investigated how a body radiates in heat. In this physics of heat and light, experimental spectrum guide is of light, which is emitted by substances either found in high temperatures or subjected to discharges. solids continuous, these of gases discontinuous and The and electrical spectra liquids of are characteristics of the gas. It is a common experience that at comparatively low temperatures , as in the case of room heating units, one gets radiant heat nut no visible light. The surface of a material body that is at a certain temperature T emits electromagnetic radiation, whose spectrum is continuous. The emission is due to thermal excitation of the electrical charges of the substance (heat radiation). How much radiation is emitted (energy is the intense of spectral line) and at what frequency? These are experimentally measured by spectra ,-a glowing electric resistance with temperature above 20 000 emits a bright light which however, looks yellowish as compared with the light of the brilliant electric arc, operating at a temperature between 30000-40000- and described in figure1, (Gamow). We see the observed distribution of intensity u(n) between different wave lengths in the radiation emitted by material bodies at different temperatures. Thus as the temperature goes up, the emitted radiation becomes rapidly more intensive , and reacher in the short wave lengths, tends to zero for very small and large frequencies. umax shows a frequency for which the energy distribution exhibits a maximum, depending on the temperature. By increasing the temperature increases and the frequency that corresponds to the maximum of the distribution. Also it is found experimentally that the location of the peak and the amount of energy transmitted is independent of the surface material. The fact that the curve u (n, T), is independent of the material, in contrast to the linear spectra of gases or the continuous spectra of nonblack bodies, shows that it can be interpreted without any reference to the atomic 178: structure of matter. (Black body)48 This is the phenomenon, and theoretical physics called to investigate, with experimental tool, the spectra. Light emission by hot bodies is subject to two important laws during the second half of the last century: Wien’s law: the wave length corresponding to the maximum intensity in the spectrum is inversely proportional to the (absolute)b temperature of the emitting body Stefan –Boltzmann law : the total of energy emitted by a hot bodyis proportional to the fourth power of it’s (absolute) temperature. However none of the two laws do not solve the basic problem of interpretation of emissivity, the ability born by each ν in the spectrum of a black body at temperature T. For this we need to establish the functional dependence of energy u in terms of ν and T, the u(ν,Σ). The mathematics of thermal radiation Classical statistics of radiation 48 Black body is an ideal body to study the phenomenon in question, which is approximated by the walls of a closed cavity, where we study the radiation emitted, and exiting of a hole small relative to the dimensions of the cavity (Ανδριηζόποσλος)) 179: The classical statistical treatment of the equilibrium of black body radiation in a cavity, considers the radiation from the standpoint of wave motion in a continuous medium . Here the blackbody radiation is modeled as the radiation emitted from oscillating charged particles of the objects surface, which are an homomorphism of standing waves inside the body.. These oscillations are produced by the thermal motions of the charged particles. If we treat each particle as a simple harmonic oscillator , then the energy E of the oscillators can take on any value (classic physics), which is conserved because energy is conserved. Since there are many oscillating charged particles , we need to consider a very large number of identical oscillating systems. This indeed suggests the possibility of accounting for the spectral distribution by some statistical considerations similar to the kinetic theory of gases.(Lindsay) και αποδεικνύεται ότι ο αριθμόσ των ςτάςιμων κυμάτων ανά μονάδα όγκου με ςυχνότητα ν είναι Ν=8πν2/c3 In Planck’s time, it was known that that the probability that a collection of identical systems at the same temperature T but starting from different classical initial conditions would have an energy E, was proportional to the Boltzmann factor e-E/Kt …………(1) where k is the Boltzmann constant. Let’s recall the definition of probability. Generally given N possible events ε1, ε2, ….εN the probability P that an event εn will occur is defined to be P ( n )  number of ways  n can occur total number of ways any event can occur When we concern ourselves in the average behavior of a collection , we are using a method commonly called statistical. Here the particle concept remains as fundamental but we do not describe what each particle of the collection is doing at every instant but describe the average behavior of each. In classical statistical mechanics of continuum distribution, we are forming the average of f(x) for xe[a,b] by multiplying it by the possibility P(x) that f(x) shall lie in the prescribed interval about this value and integrating over all values of the f(x) components viz 180:  f ( x)   f ( x) P( x)dx.......... .......... .(2) This is essentially the definition of a weight average, as each f(x) value carries a weight proportional to the possibility that a system shall have this f(x) value. From (1) the possibility for an oscillator to have energy E is P( E )  e  E / kT   0 e  E / kT .......... .......... .......... ....( 3) Having the results from the improper integrals   0 e  ax dx  1   0 και a   0 xe - x  1 We have for the mean value of the energy of the oscillator   E   E.P( E )  kT .......... .......... .......... .( 4) 0 The energy’s density R of the radiation is R=N . <E> …………………..(5) where N is the number of the oscillating charged particles in unit of volume with frequency ν , and finally from (5) and classical wave theory is R( , t )  8 2 kT .......... .........( 6) c3 It is the Rayleigh-Jeans radiation law and results in an “ultra-violet catastrophy” with infinite total radiated energy since Tν2→∞ as ν→∞, that is all energy should long ago have escaped from matter in a catastrophie burst of ultraviolet radiation. Σhe Ryleigh –Jeans law was derived viewing light as electromagnetic waves governed by Maxwell’s equations , which forced Planck in his “act of despair” to give up the wave model and replace it by the statistics of “ quanta”, viewing light as a stream of particles or photons. 181: Quantum statistics of radiation . Planck’s ingenious idea was to purpose the following radical hypothesis: the energy E could not take on just any value but only certain discrete values called “quanta” of energy. His model was that a lump of matter could be represented by innumerable particles (the oscillators) where all frequencies of oscillation being including. These absorb heat and light energy and give energy off again. The lump of matter absorbs energy by getting warm. An oscillator with Ένασ ταλαντωτήσ with frequency ν can absorb or emit radiation only in quantities Ε=nε, n=0,1,2,…He suggested that higher frequencies motion meant higher energy. Thus he proposed that the energy of quanta be multiplied with the frequency ν, that is En=nhν………………………(7) where the constant of proportionality needed to give energy units he called h , which is now known as Planck’s constant. Let the event εn be a measurement of the energy of the system at temperature T that yields an energy E=nhν. The probability of such an event , according to (1) is P( n )  e  nh / kT  e .......... .......... .......... ....(8)  nh / kT n 0 where n runs from unity to a very large number , which we may take effectively as infinity (Lindsay). The sums then become infinite series and we can write the relation (4) transformed for the discontinuous distribution as    E   P( n ) E n  n 0  nhe  nh / kT n 0  e .......... ...(9)  nh / kT n 0 For e  h / kT  x  1, ,  E  (9) becomes (limits of geometrical power series) h ( x  2 x 2  3x 3  ....) hx(1  x  x 2  ....)' h 1   hx  h / kT .....(10) 2 2 1 x e 1 1  x  x  .......... 1  x  x  ...... 182: This is Planck’s average energy formula from which Planck was able to show that the Planck’s energy density of the emitted radiation from the black-body could be given R( , T )  8h 3 1 .......... .......... ....(11) 3 h / kT c e 1 for hν<<κΣ and the relation ex≈1+x and (11)becomes the Raylegh-Jeans formula, and Planck’s law agrees with experiments in the infinite frequency range. If we integrate (11) we find the total energy density of black body U=At4 (Stefan – Boltzmann law.) The calculations with infinity. The spirit of the work of Planck is based on a mathematical trick, invented by the Greeks, whereby a baffling smoothness ( Banesh Hoffmann) is replaced by a series of minute jerks much more amenable to mathematical treatment. This trick, the foundation of the calculus is a simple one in it’s general aspects. For example if we try to calculate (not to measure) the length of the circumference of a circle we find the smooth circumference offers little mathematical foothold. Thus , we mark the circle into four, eight, 24, 25 , and so on equal parts and join the marks by straight lines as shown. For each of these regular polygons it is possible to calculate the total perimeter, and it is obvious that as we take more and more smaller and smaller sides the total length will come closer and closer to the circumference of the circle. For instance , the total perimeter of the sixteen-sided figure is much closer to the circumference of the circle than the sum of the sides of the square. So the mathematician can calculate the perimeter for a figure of some general number of sides. Then he lets the number of sides in his formula increase without limit, having the 183: perimeter of circle. This was the Greek manipulation of the infinity, in Greek mathematicians there was always a gap between the real (finite) and the ideal (infinite) Aristotle used the infinite as adjective, denying the real (physical) existence of infinity, since this term contains the active infinity which realistic natural philosophy did not accept (Carl Boyer). In commenting the view of mathematicians Aristotle said: ……….In point of fact they do not need the infinite and do not use it. They postulate only that the infinite line may be produced as far as they wish…Hence for the purpose of the proof , it will make no difference to them to have such an infinite instead , while its existence will be in the sphere of real magnitude (Physica iii. 207b) Eudoxus proposed the method of exhaustion in an axiom (Euclid X.1). His work is based at every point on finite , intuitively clear and logically presice considerations . He was a mathematician who was at the same time a scientist with none of the occult or mystic on him. He showed that there is no need to assume the 'existence' of infinitely small amounts, one can reach a size as small wants with continued divisions of a given size. But all Greek mathematicians (including Archimedes) excluded the infinite from their reasoning. The reasons for this ban are obvious: intuition could at the time afford no clear picture of it , and it had as yet no logical basis.(Boyer) The latter difficulty having been removed in the nineteenth century and the former being now considered irrelevent, the concept of infinity has been admitted freelly into mathematics. Indeed mathematics became independent from intuition and reality, only care about the consistency of the systems developed, and the problem is in physicists to find if this system is applied or not, in nature. The leading modern mathematical concepts in the study of thermal radiation is, the infinite series and the generalized integral. Both approaches describe our inability to add together infinitely many things.. But we do this with series and integrals. The series and the integral both represent the limiting behavior of a sum of finitely many terms, i.e. 1. In the case of a series we are interested in what happens to a finite sum as the number of terms increases without bound. 184: 2. In the case of the integral we are interested in what happens to a series of the Riemann sum over [a, b] written as the agent (xi –xi-1) = (b-a) /n of each term decreases tending to zero .In series we have the distinct infinite, the integral range is produced from the continuous infinite (infinite divisibility). We can see now the difference between series and integrals. The integral represents the area under the curve and sums up every value that the curve lies on (not just integer coordinates!). The infinite sum only sums the integer values and is equivalent to an area approximation (a Riemann sum). More clearly, for series we use discrete values summed over integers and for integrals we use values which are in a continuous interval. But all these happen in mathematics, viz in numbers which are the imaginable behind the phenomena. This infinite divisibility we can only imagine as it is completely lost from supervision of the mind in the range of numbers, and the supervision of the senses in geometry. Now in the thermal radiation, it has been excluded from physics. This tending to zero, is an imaginary experiment of the mathematician that leaves the line to divide indefinitely, but follow conceivably the logical consequences of this division, without clearly captures the accuracy, but i m a g I n e s the approach through ever denser divisions since the infinitesimal is not described in a final division. The two concepts these of limit and infinite are tided together, after the foundation of the continuum of real numbers. In the case of radiation, Greek attitude seems more "natural". The radiation is not continuous (like the continuum of real numbers), and it’s distinctness only roughly resembles the infinite series, as we have seen in equation (8) that the number of oscillators is a very large number , which we may take effectively as infinity . What does that mean? The oscillators are of course a large but finite 185: number, but we need to approach a mathematical concept to use it’s model . So we resort to the series, in a really strong physical approach. The numerical The two values concepts are of series connected and with improper the following integrals. sentence: If f (x) is non-negative on [a, ∞) and decreasing with derivatives, (as many times as needed), the two symbols both converge and diverge together, but do not give the same results. Even if we know the numerical result of one, we can not have the result of the other. For example, if we calculate the 1 2   2 6 n 1 n   και   1 1  n(n  1)  1 και  1 dx  1 x2  1 n 1 results dx  ln 2 x(x  1) we do not discern any relationship. The Euler-Maclaurin formula This formula is an important tool in numerical analysis. We provide a strong connection between integrals and sums. It is used to approach integrals by finite sums or to calculate finite sums and infinite series using integrals. This approach is evident if both converge, because the sum is a discrete version of the integral, and have approximately the same shape. The formula of Euler-Maclaurin gives us just how connected the numerical values of the sum and the integral. The first form of formula . For any function f with continuous derivative in [1, n] we have 186: n  f (k )   n 1 k 1 n f ( x)dx   ( x  [ x]) f ' ( x)dx  f (1) 1 where [x] denotes the greatest integer ≤ch (Tom Apostol sel.560) The last two terms represent the error that occurs when the sum of the first member is approached from  n 1 f ( x)dx . The formula is useful because f need not be positive or decreasing. We have variations of the formula when we try to extract more information about the error. For the case of infinite series and generalized integrals, the formula describes the difference between their numerical values, using higher order  e n2 n 0   e  x2 0 dx   1 2 derivatives and Bernoulli’s numbers , but we shall not extend. A result of the formula is (Apostol) Even the process of formula may establish a numerical approximation between the results of classical and quantum laws of radiation, for a suitable function. Conclusion: for the physical processes we do not follow mathematics, but the experiments! Mathematics (the mind) is adapted to experiment, and not vice versa. Sources Διπλωματική εργαςία Αμαλία-Χριςτίνα Μπαμπίλη Η ακτινοβολία του μέλανοσ ςώματοσ (physicsgg) Μαθηματικά και ςτοιχεία ςτατιςτικήσ Γ.Λυκείου (ΟΕΣΒ) Ειςαγωγή Ιωαννίνων) εισ την κβαντομηχανικήν (Γ.Ι.Ανδριτςόπουλοσ Concepts and methods of theoretical physics (Robert Lindsay, Dover) Πανεπιςτήμιο 187: The strange story of quantum (Banesh Hoffmann,Dover)cience and engineering ( Σhe great physicists from Galileo to Einstein (George Gamow,Dover) The history of the Calculus and its conceptual development (Carl B.Boyer, Dover) Statistic physics for Students of Science and engineering (Robert Reed, interνετ, Διαφορικόσ και ολοκληρωτικόσ λογιςμόσ ΙΙ, (Tom Apostol Πεχλιβανίδησ ) THE QUANTIZATION OF SPACE AND TIME George Mpantes Mathematics teacher Abstract: Descriptions of quantum mechanics are conventionally cast in terms of the Copenhagen interpretation. This interpretation was primarily the offspring of Niels Bohr1 and Werner Heisenberg and today an evolved Copenhagen remains the consensus view among most physicists7. The physical existence of space point and temproral instant, viz the discontinuity of space time, gives a new meaning in this Copenhagen’s interpretation of quantum 188: mechanics. The uncertainty and the complementarily principles may be attributed to the space-time’s discontinuity, through the physicisation of the geometrical concept of “point” in space time. K e y w o r d s: The space –time quantum, uncertainty relations, the photon , the matter waves, the complementary principle, the Bohr’s atom Introduc tIon While the methods of quantum mechanics have proven their utility , no consensus exists even to this day on what quantum mechanics really “means”. Some say that the question itself is meaningless, that the mathematics speak for itshelf. Such a meaning has a philosophical background that is based in a philosophical doctrine as old as Aristotle: this of the continuity of space and time. Therefore if we Every orbit of a material particle is composed of a finite number of parts of space lengths and time durations which depend on the particle’s momentum and energy. These are the “point” and the “moment” in space and time respectively for this particle. Introduc tion The two principles that express in qualitative terms the axiomatic basis of quantum mechanics viz the uncertainty principle and the complementarity principle are in some way metaphysics. In spite of the special puzzling features of quantum mechanics, it is possible to visualise to some degree that happens to matter in terms of ordinary space and time. The disappearing of causality in atomic field and the limitation of the classical concept that the behaviour of atomic systems can be described independently of the ways by which they are observed, must be explained in a more physical sense, through the matrix of all the descriptions: the space-time. 189: In this paper we shall attribute a physical sense to the “event” of flat space-time: the point in space and the instant in time. Having linked the geodesic of space-time with the orbit of a free body, it follows that our proposed interpretation of the concepts point in space and instant in time will be related to bodies and their motions. Definitions Thus, if (ε) is the cosmic line of a body between the events A1(x1,t1) and A2(x2,t2), we accept that the distance x2-x1 is equal to a finite number of “points” each of length Sq = h/p, where h is Planck’s constant and p the momentum of the body, namely x2-x1 = n h/p ………………………………………………………………..(1) Furthermore, the difference t2-t1 = n h/E ………………………………(2), where E is the energy of the body/particle and tq = h/E the “instant” in time of the course of the body. The material body cannot trace an orbit of less than sq and the description of its behavior is limited temporally by tq to durations longer than tq. These two quantities are the «point and the instant» of the specific cosmic line. The bodies are moving with jumpings in space and time. They are connected conceptually with the infinitesimals of Leibniz in space and time. If we define dx = sq = h / px and dt = tq, we know that in the interval dx a line’s curvature is not changing and in dt the changes freeze (not the time flow) and thus Calculus captures them as a photo. But for the bodies of our experience, even for the planets and stars, the spatiotemporal quanta are infinitesimals, so disappear from the physical reality, and in order to describe their orbits, we have the space-time continuum that we studied in calculus. Then we have the world of classical mechanics, ie the bodies crossing a continuous space-time. But for very small particles, spatio-temporal quanta grow, become appreciable, and the bodies indicate that the continuum of space-time is discrete in depth, composed of infinitesimal indivisible, and when experimental measurements reach this depth, the discrete is revealed. Here are the origins of the complementary principle. The bodies move in steps, not on wheels. Now we have quantum particles, 190: and we should have another description, not continuous description, even though we still maintained the doctrine of continuity, which produced many philosophical “paradoxes” in the world of the “very small”. From (1) and (2), the quantization of the action follows naturally, since Action = (energy) x (time ), so action = E.ntq = nh nεN …..(3) Uncertain ty RelatIons 6 The physical existence of the “point” of space-time will give a new meaning to the uncertainty principles. It is known that any measurement of a physical magnitude causes an alteration in the state of the system in which the measurement takes place. In particular, the disturbance caused to a microcosmic system is not negligible. This disturbance is caused by the application of a force F on the system, which will act for a spatial interval Δx and time interval Δt. These intervals cannot be as small as we wish: it is in this that the differentiation with classical science is manifested. Δx = nSq and Δt = ntq, however accurate the experimental conditions are. Σherefore the force F, applied for a finite time and space interval will cause a change in the momentum and energy of the body in accordance with the formulas Δp = F Δt ……………………………………………………………….(4) ΔE = FΔx ……………………………………………………………….(5) where the first members are the continuous metric of physics and laboratory and the second, the space and time that are defined by the bodies, as does the curvature in general relativity. Here the changes ΔP and ΔΕ express the gaps of discontinuity in the momentum and energy, which are produced by the discontinuities Δχ and Δt, in the space-time description. But in continuous view of space-time, interpreted them as the experimental measurement uncertainties On eliminating F from (4) and (5) we have 191: Δp.Δx = ΔΕ.Δt, and, since every element of this relation has dimensions of action, we have Δp.Δx = ΔΕ.Δt = nh ……………………………………………….(6) The new meaning of this relation is that since the point in space and the instant in time have finite dimensions, the uncertainties of (6) express the discontinuity which space-time attributes to the history of bodies. These, in the continuous description of space-time, attribute the uncertainty principles of quantum mechanics, which result in the Fourier analysis of the contributions of many waves of the wave packet of de Broglie matter waves, and the verification is done there with ideal experiments. In other words, we have from that indeterminism of quantum mechanics is a fundamental result of granularity or space-time for the bodies moving in. The new interpretations of uncertainty principles now, attach the time and point of the discontinuous changes. ΔΕ.Δth can be interpreted as follows: The alteration of energy by ΔΕ cannot take place instantaneously, but in some finite time n.h/ΔΕ. The least time in which this can occur is Δt=h/ΔΕ. For example, the “most rapid” energy alteration by ΔΕ can be described as follows: at instant t1 we have energy E and at t2 = t1+h/ΔΕ energy Ε+ΔΕ. For the interval t2-t1 the law of conservation of energy is violated, but is restored at instant t2. Similarly the alteration in momentum by Δp requires an interval [in order] to occur. The smallest interval is h/Δp. This is the “point” of alteration of momentum, in the same way as h/ΔΕ is the ‘instant” of alteration of energy. So these alterations are not continuous. The wave theory This new interpretation of the uncertainty principles leads in two significant conclusions in the field of wave theory. The spatio-temporal sizes that define the existence of a wave, which (wave) carries momentum and energy in space and time, so quantizes space-time, is the period T and the wavelength l. The quantization of these spatiotemporal sizes will produce results of the energy E and momentum p of the wave. We can give a new expression for the quantity of energy which is emitted out by the wave over a period and for the amount of momentum in a wavelength.3 192: Let A be a point in the space of propagation of a wave and Eπ be the energy which appears over a period T. Then, from (6) for ΔΕ=Επ and Δt=T we have Eπhv …………………………………………………………………….(7) Namely, the energy of the wave over a period is quantized from hv and thus acoustic phonons are considered to correspond to sound waves. Also from (6) we have p=n h/λ = nhk …………………………………………………………….(8) namely, we correspond momentum with phonons, where k is the wave number of the wave. The relations (7) and (8) are valid for any wave with the elements v, λ. These relations will be of use to us in further calculations. The phot on The existence of the quantum of action was first discerned by Planck via the statistical properties of radiation in 1900, and it was soon put to use by Einstein in 1905 to arrive at the concept of quanta of light. This failure of a classical statistical mechanics when combined with Maxwell theory in the black body radiation has its origins in the quantization of space time which brings the quantization of radiation. The principle of Planck’s principle is not an axiomatic principle. It is produced by quantization of space and time. Planck's law E = n.h.n for the radiation energy, is produced by T = nh / E for the period of radiation, through (7). In what follows we shall interpret the particle nature of light. It will be shown that its particle behavior must be attributed to space-time. The uncertainty relations which follow from the physical definition of the space-time point, will provide a basis for the consideration of the photon. At a point A in the space of propagation of electromagnetic radiation we have the appearance of a given energy Ε=ΔΕ over a period T=Δt. It follows from (6) that E=nh/T=nhv ………………………………………………………………..(9) 193: Definition: we consider that this alteration of electromagnetic energy E over a period is the most rapid that can occur in nature, then it follows from (9) that E=hv, i.e. that the energy per period is hv. Its action is revealed discontinuously, despite the continuous nature of its alteration, since in a lesser time (one period) the principle of conservation of energy is violated: “the instant of alteration is the period”. Thus at the conclusion of each period, the quantity of energy hv, appears to the interactions, giving a basis to the discontinuity of the emission of the beam of light, the particle behavior of light (the photoelectric phenomenon). This image exists at every point of the emission of light. Consequently, the “phenomenon” is propagated at a velocity of c, namely we have the emission of energy hv over a period supplied continually by the propagation of the wave (photon). This image tells us that the photon does not have the same physical basis as the material body which is in motion from point to point. One photon exists in one place and another in another. The photon is an operation of space-time: the continuous variations of the electromagnetic field are converted by space-time into the discontinuous emission of a quantity of energy hv towards the environment. Certainly the propagation of the phenomenon which causes the discontinuous emission of the light beam is equivalent in description to the displacement of the quantity of energy hv, and therefore with the translation of momentum p=h/λ. Now the particle reality is complete. Photon is the microscopic description of Maxwell waves. In modern terms a photon is an elementary excitation of the quantization of the electromagnetic Matter waves, the c o m p l e m e n t a r y p r I n c I p l e An attempt will now be made to describe the motion of a particle in Sq. Sq is not a distance in the orbit of the body, and therefore is connected to the time tq. Consequently its velocity must be c2/u, a fact which destroys the particle image (description). The restoration of the description is performed through (8). 194: The momentum translated by a wave is p=h h/λ and the smallest momentum of the wave is p= h/λ …………………………………………………………….(9) Comparing (9) with (1), it can be seen that the momentum of a body is equal to the momentum of the wave of wavelength λ=Sq. The identity of the descriptions becomes an essential identity, since reality is attributed through the time-space description. As it is impossible to describe motion within Sq with particle characteristics, then the body is described viz becomes a wave which appears for distances compatible with Sq and whose characteristics are λ=h/p, a velocity of c2/u and T=tq. This is Broglie’s material wave, and the quantization of space-time is the root of this phenomenon. From this the wave-packet 2 is constructed, which has a group velocity of u. For example an electron with speed5.106 m/sec travels in space with steps and mass 9,1. 10-31 kg Sq =1,46.10-10 m. this is the quantum of space for this electron. Imagine now the “steps” of a planets. These very very small parts of the space are interpreted in description as the famous continuum of space and time. It is a memory of the magic Units of the Pythagoreans for this separate world, their number is infinite, points are indivisible and their dimensions ...... mystery. Thus with the acceptance that the space-point and the time-instant have dimensions, the wave nature of matter is produced, as the only possibility of description. This mental discontinuity of space time must be considered as real, if we want to be reality the wave-particle description of quantum mechanics. Today we know that the emergence and propagation of forces through these two theories (particles and fields} initially considered as two distinct interpretations. But it turned out to be of no different interpretations, but it is the same theory. When the idea of quanta and quantum theory matured, in which the exchange of particles that are identical to the quanta of transmission energy, not only did put these two theories the one opposite to the other, but the particles and fields into which they act, are treated as interrelated , fields spread with particles and we see that the particles "creates" the space-time description of the field, through the quantization of the data of their spatiotemporal existence. Therefore the fields and particles reconciled! 195: Other results An extension of these ideas lies in the relationship of the motion of bodies, and the mathematical description of this relation. The paradoxes of Zeno due just to the transport of data of motion in numbers -the points of the continuous straight- now not undergo. The path AB of the paradox of dichotomy consists of finite though very large number of mobile’s spatial steps, which are over, and the body reaches B. The mathematical infinity is separated of the physical motion, as was everywhere in every branch of physics. So the mathematical description becomes a map of traffic, with scales etc., But as we know, the maps do not describe the motion, but the path of the motion. In classical mechanics, the differential causality of calculus is an accurate approximation of motion, because for the bodies of our direct experience, the spatiotemporal quanta tend to zero, yielding the concept of infinitesimals of Leibniz. The concept of the mathematical continuum that followed, refers in numbers, which we considered as images of every reality, even of motion. So we remained Pythagoreans for many centuries, believing that "everything is number." The limits of Cauchy, the continuum of Dedekind and Cantor are mathematical discoveries about numbers, so they do not mean anything about the nature, apart from catalytic simulation and approach succeeded. In Bohr’s atom4 The first Bohr’s axiom The belief of Bohr that an electron in an atom moves in certain permissible orbits round the core, stationary stability situations, is interpreted with the new concept of uncertainty principles (6) where the points of the electron path having dimensions, so a closed trajectory specifies an integer number of points. Still any permanent change in the motion of electrons is accompanied by a full transfer from one state to another, which Bohr could not interpret. But a change in the movement means change in spatial quantum hence in the closed track defining a different number of points, that is another path. 196: Even quantization of angular momentum results from the quantization of the track. We have that the periphery consists of integer number of steps that 𝑛 ℎ = 2𝜋𝑟 𝑚𝜐 ά𝜌𝛼 𝑚𝜐𝑟 = 𝑛ℎ 2𝜋 Also the electron jumps "instantaneously" from one track to another, by the addition or loss of energy, disappears and reappears, with infinite speed, another paradox, but the time of the energy change (moment)is too small for experimental verification that gives the impression of instantaneous displaying of the route from one orbit to another (point). Like the infinitesimals of Leibniz’s curve, where the curvature is repealed and study the slope of the infinitesimal. Continuous description in “very small”, produces paradoxes The second Bohr’s axiom . The radiation absorbed or emitted from an individual system with the emission or absorption of radiation frequency n, given by E1-E2 = hn where E1, E2 are the energy values in the two situations, and n is the frequency. Here is the first member of the ΔΕ of (6) and the second the term h / ΔΕ. The new uncertainty principles just describe the second condition of Bohr (the “moment” of change in energy) This resurrection of infinitesimals through the basic considerations of quantization of space-time, is a proposal that reconciles the ancient philosophical controversy of continuous vs distinct, classical and quantum physics, acknowledging the omnipotence of spatio-temporal description, as in general relativity where again a feature of space-time, it’s curvature overturned all our views on the geometry and gravity. References 197: 1. Victor j. Stenger (1995): the unconscious quantum Prometheus Books 2. Leonard J.Schiff (1984): quantum mechanics Mc Graw –Hill international Book Company 3. L.Harris, A.L.Loeb (1963): Introduction to wave mechanics McGraw –Hill book Company 4. Banesh Hoffmann (1959): The strange story of quantum Dover publications 5. Γ.Ι. Ανδριτςόπουλοσ( 1975): Eιςαγωγή εισ την Κβαντομηχανικήν Πανεπιςτήμιο Ιωαννίνων 6. Ramabhadra Vasudevan, K. V. Parthasarthy, R. Ramanathan: Quantum mechanics, a stochastic approach Alpha science international 7. James T.Cushing (1984): Quantum mechanics Historical Contigency and the Copenhagen hegemony ,the University of Chicago press END