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Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite

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Abstract

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ① (this symbol is called grossone). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts (Gutman et al. in Found Sci 22(3):539–555, 2017; Gutman and Kutateladze in Sib Math J 49(5):835–841, 2008; Kutateladze in J Appl Ind Math 5(1):73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics.

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Notes

  1. Khwarismi International Award, assigned by The Ministry of Science and Technology of Iran, 2017; Honorary Fellowship, the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Degree of Honorary Doctor from Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, 2013; Pythagoras International Prize in Mathematics, Italy, assigned by the city of Crotone (where Pythagoras lived and founded his famous scientific school) and the Calabria Region under the high patronage of the President of the Italian Republic, Ministry of Cultural Assets and Activities and the Ministry of Education, University and Research, 2010; Lagrange Lecture, Turin University, Italy, 2010; etc.

  2. In section 8, the authors of Gutman et al. (2017) inform the reader that before appearing in Foundations of Science their paper has been 5 times rejected by The Mathematical Intelligencer.

  3. A numeral is a symbol (or a group of symbols) that represents a number that is a concept. The same number can be represented by different numerals. For example, symbols ‘4’, ‘four’, ‘IIII’, and ‘IV’ are different numerals, but they all represent the same number.

  4. Recall that numerical computations are performed with floating-point numbers that can be stored in a computer memory. Since the memory is limited, mantissa and exponent of these numbers can assume only certain values and, therefore, the quantity and the form of numerals that can be used to express floating-point numbers are fixed. Due to this fact, approximations are required during computations with them because an arithmetic operation with two floating-point numbers usually produces a result that is not a floating-point number and, as a consequence, this result should be approximated by a floating-point number. In their turn, symbolic computations are the exact algebraic manipulations with mathematical expressions containing variables that have not any given value. These manipulations are more computationally expensive than numerical computations and only relatively simple codes can be elaborated in this way.

  5. A logical fallacy (see https://www.yourlogicalfallacyis.com) is a flaw in reasoning. Logical fallacies are like tricks or illusions of thought, and they are often very sneakily used by politicians and the media to manipulate people. This and the following footnotes explaining the meaning of fallacies were taken from https://www.yourlogicalfallacyis.com.

  6. You could say that this is the mother of all biases, as it affects so much of our thinking through motivated reasoning.

  7. Complex subjects require some amount of understanding before one is able to make an informed judgement about the subject at hand; this fallacy is usually used in place of that understanding.

  8. It is entirely possible to make a claim that is false yet argue with logical coherency for that claim, just as it is possible to make a claim that is true and justify it with various fallacies and poor arguments.

  9. Many ‘natural’ things are also considered good, and this can bias our thinking; but naturalness itself does not make something good or bad.

  10. By exaggerating, misrepresenting, or just completely fabricating someone’s argument, it is much easier to present your own position as being reasonable, but this kind of deceitfulness serves to undermine rational debate.

  11. Notice that the finiteness of the number of symbols in the numeral is necessary for executing practical computations since we should be able to write down and store values we execute operations with.

  12. It should be noticed that the astonishing numeral system of Pirahã is not an isolated example of this way of counting. In fact, the same counting system, one, two, many, is used by the Warlpiri people, aborigines living in the Northern Territory of Australia (see Butterworth et al. 2008).

  13. For instance, ①-based numerals can be used for working with functions and their derivatives that can assume different infinite, finite, and infinitesimal values and can be defined over infinite and infinitesimal domains. The notions of continuity and derivability can be introduced not only for functions assuming finite values but for functions assuming infinite and infinitesimal values, as well. Limits \(\lim _{x \rightarrow a} f(x)\) are substituted by expressions and f(x) can be evaluated at concrete infinite or infinitesimal x in the same way as it is done with finite x. Series are substituted by sums having a concrete infinite number of addends and for different number of addends results (that can assume different infinite, finite or infinitesimal values) are different as it happens for sums with a finite number of summands. There are no divergent integrals, limits of integration can be concrete different infinite, finite or infinitesimal numbers and results can assume different infinite, finite or infinitesimal values. A number of set theoretical paradoxes can be avoided, etc.

  14. Notice that the paper (Sergeyev 2015a) does not say that the rank (9) is ‘the best one’. It is just one more way to rank countries that can be useful in certain situations.

  15. Notice that this point of view implies that the first competition is more important than the second one, etc. violating so the principle of equality of all sportive disciplines.

References

  • Adamatzky, A. (2006). Book review: Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, 2003. International Journal of Unconventional Computing, 2(2), 193–194.

    Google Scholar 

  • Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M. S., & Sergeyev, Y. D. (2017). A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic. Mathematics and Computers in Simulation, 141, 24–39.

    Article  Google Scholar 

  • Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences of the United States of America, 105(35), 13179–13184.

    Article  Google Scholar 

  • Caldarola, F. (2018). The Sierpinski curve viewed by numerical computations with infinities and infinitesimals. Applied Mathematics and Computation, 318, 321–328.

    Article  Google Scholar 

  • Cococcioni, M., Pappalardo, M., & Sergeyev, Y. D. (2018). Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm. Applied Mathematics and Computation, 318, 298–311.

    Article  Google Scholar 

  • Crowell, B., & Khafateh, M. http://www.lightandmatter.com/calc/inf/.

  • D’Alotto, L. (2012). Cellular automata using infinite computations. Applied Mathematics and Computation, 218(16), 8077–8082.

    Article  Google Scholar 

  • D’Alotto, L. (2013). A classification of two-dimensional cellular automata using infinite computations. Indian Journal of Mathematics, 55, 143–158.

    Google Scholar 

  • Davis, M. (2005). Applied nonstandard analysis. New York: Dover Publications.

    Google Scholar 

  • De Cosmis, S., & De Leone, R. (2012). The use of grossone in mathematical programming and operations research. Applied Mathematics and Computation, 218(16), 8029–8038.

    Article  Google Scholar 

  • De Leone, R. (2018). Nonlinear programming and grossone: Quadratic programming and the role of constraint qualifications. Applied Mathematics and Computation, 318, 290–297.

    Article  Google Scholar 

  • Gaudioso, M., Giallombardo, G., & Mukhametzhanov, M. S. (2018). Numerical infinitesimals in a variable metric method for convex nonsmooth optimization. Applied Mathematics and Computation, 318, 312–320.

    Article  Google Scholar 

  • Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.

    Article  Google Scholar 

  • Gutman, A. E., Katz, M. G., Kudryk, T. S., & Kutateladze, S. S. (2017). The mathematical intelligencer flunks the olympics. Foundations of Science, 22(3), 539–555.

    Article  Google Scholar 

  • Gutman, A. E., & Kutateladze, S. S. (2008). On the theory of grossone. Siberian Mathematical Journal, 49(5), 835–841.

    Article  Google Scholar 

  • Iudin, D. I., Sergeyev, Y. D., & Hayakawa, M. (2015). Infinity computations in cellular automaton forest-fire model. Communications in Nonlinear Science and Numerical Simulation, 20(3), 861–870.

    Article  Google Scholar 

  • Kutateladze, S. S. (2011). A pennorth of grossone. Journal of Applied and Industrial Mathematics, 5(1), 73–75.

    Article  Google Scholar 

  • Linnebo, Ø. (2017). Philosophy of mathematics (Princeton Foundations of Contemporary Philosophy). Princeton: Princeton University Press.

    Google Scholar 

  • Lolli, G. (2002). Filosofia della matematica. L’eredità del Novecento. Bologna: Il Mulino.

    Google Scholar 

  • Lolli, G. (2012). Infinitesimals and infinites in the history of mathematics: A brief survey. Applied Mathematics and Computation, 218(16), 7979–7988.

    Article  Google Scholar 

  • Lolli, G. (2015). Metamathematical investigations on the theory of grossone. Applied Mathematics and Computation, 255, 3–14.

    Article  Google Scholar 

  • Margenstern, M. (2011). Using grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers, Ultrametric Analysis and Applications, 3(3), 196–204.

    Article  Google Scholar 

  • Margenstern, M. (2015). Fibonacci words, hyperbolic tilings and grossone. Communications in Nonlinear Science and Numerical Simulation, 21(1–3), 3–11.

    Article  Google Scholar 

  • Montagna, F., Simi, G., & Sorbi, A. (2015). Taking the Pirah a seriously. Communications in Nonlinear Science and Numerical Simulation, 21(1–3), 52–69.

    Article  Google Scholar 

  • Pardalos, P. M. (2006). Book review: Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, 2003. Journal of Global Optimization, 34, 157–158.

    Article  Google Scholar 

  • Prokopyev, O. (2006). Book review: Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, 2003. Optimization Methods and Software, 21(6), 995–996.

    Article  Google Scholar 

  • Rizza, D. (2017). A study of mathematical determination through Bertrands Paradox. Philosophia Mathematica. https://doi.org/10.1093/philmat/nkx035.

  • Rizza, D. (2016). Supertasks and numeral systems. In Proceedings of the 2nd international conference on “numerical computations: Theory and algorithms”, Vol. 1776. AIP Publishing, New York.

  • Robinson, A. (1996). Non-standard analysis. Princeton: Princeton Univ. Press.

    Book  Google Scholar 

  • Sergeyev, Y. D. (2013). Arithmetic of infinity, 2nd ed. Edizioni Orizzonti Meridionali, CS, 2003.

  • Sergeyev, Y. D. (2007). Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers. Chaos, Solitons & Fractals, 33(1), 50–75.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2008). A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica, 19(4), 567–596.

    Google Scholar 

  • Sergeyev, Y. D. (2009a). Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge. Chaos, Solitons & Fractals, 42(5), 3042–3046.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2009b). Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Analysis Series A: Theory, Methods & Applications, 71(12), e1688–e1707.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2010a). Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. USA patent 7,860,914.

  • Sergeyev, Y. D. (2010b). Counting systems and the First Hilbert problem. Nonlinear Analysis Series A: Theory, Methods & Applications, 72(3–4), 1701–1708.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2010c). Lagrange lecture: Methodology of numerical computations with infinities and infinitesimals. Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino, 68(2), 95–113.

    Google Scholar 

  • Sergeyev, Y. D. (2011a). Higher order numerical differentiation on the Infinity Computer. Optimization Letters, 5(4), 575–585.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2011b). On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function. p-Adic Numbers, Ultrametric Analysis and Applications, 3(2), 129–148.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2011c). Using blinking fractals for mathematical modelling of processes of growth in biological systems. Informatica, 22(4), 559–576.

    Google Scholar 

  • Sergeyev, Y. D. (2013). Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer. Applied Mathematics and Computation, 219(22), 10668–10681.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2015a). The Olympic medals ranks, lexicographic ordering, and numerical infinities. The Mathematical Intelligencer, 37(2), 4–8.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2015b). Un semplice modo per trattare le grandezze infinite ed infinitesime. Matematica nella Società e nella Cultura: Rivista della Unione Matematica Italiana, 8(1), 111–147.

    Google Scholar 

  • Sergeyev, Y. D. (2016). The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area. Communications in Nonlinear Science and Numerical Simulation, 31(1–3), 21–29.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2017). Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems. EMS Surveys in Mathematical Sciences, 4(2), 219–320.

    Article  Google Scholar 

  • Sergeyev, Y. D. (2018). Numerical infinities applied for studying Riemann series theorem and Ramanujan summation. AIP Conference Proceedings, 1978, 020004. https://doi.org/10.1063/1.5043649.

    Article  Google Scholar 

  • Sergeyev, Y. D., & Garro, A. (2010). Observability of Turing machines: A refinement of the theory of computation. Informatica, 21(3), 425–454.

    Google Scholar 

  • Sergeyev, Y. D., & Garro, A. (2013). Single-tape and multi-tape Turing machines through the lens of the Grossone methodology. Journal of Supercomputing, 65(2), 645–663.

    Article  Google Scholar 

  • Sergeyev, Y. D., Kvasov, D. E., & Mukhametzhanov, M. S. (2018). On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Communications in Nonlinear Science and Numerical Simulation, 59, 319–330.

    Article  Google Scholar 

  • Sergeyev, Y. D., Mukhametzhanov, M. S., Mazzia, F., Iavernaro, F., & Amodio, P. (2016). Numerical methods for solving initial value problems on the Infinity Computer. International Journal of Unconventional Computing, 12(1), 3–23.

    Google Scholar 

  • Shapiro, S. (2001). Thinking about mathematics: The philosophy of mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Shylo, V. P. (2006). Book review: Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, 2003. Cybernetics and Systems Analysis, 42(6), 183.

    Google Scholar 

  • Trigiante, D. (2007). Book review: Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, 2003. Computational Management Science, 4(1), 85–86.

    Article  Google Scholar 

  • Vita, M. C., De Bartolo, S., Fallico, C., & Veltri, M. (2012). Usage of infinitesimals in the Menger’s Sponge model of porosity. Applied Mathematics and Computation, 218(16), 8187–8196.

    Article  Google Scholar 

  • Zhigljavsky, A. (2012). Computing sums of conditionally convergent and divergent series using the concept of grossone. Applied Mathematics and Computation, 218(16), 8064–8076.

    Article  Google Scholar 

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Acknowledgement

The author thanks four unknown reviewers for their valuable comments. The author thanks Prof. Daniel Moskovich, Ben-Gurion University of the Negev, Beer-Sheva, Israel for providing a preliminary list of logical fallacies present in Gutman et al. (2017).

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    Yaroslav D. Sergeyev

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Appendix

Appendix

The ①-based methodology is one of the possible views on infinite and infinitesimal quantities and Mathematics, in general. It is added to other existing philosophies of Mathematics such as logicism, formalism, intuitionism, structuralism, etc. (their comprehensive analysis can be found, e.g., in Linnebo 2017; Lolli 2002; Shapiro 2001). Three postulates and an axiom that is added to axioms for real numbers form the methodological platform of the proposal. They are given below to make this paper self-contained. A special attention in the ①-based methodology is paid to the fact than numeral systems that are among our tools used to observe mathematical objects limit our capabilities of the observation. A detailed discussion on this methodological platform can be found in Sergeyev (2017).

Methodological Postulate 1

We postulate existence of infinite and infinitesimal objects but accept that human beings and machines are able to execute only a finite number of operations.

Methodological Postulate 2

We shall not tell what are the mathematical objects we deal with; we just shall construct more powerful tools that will allow us to improve our capacities to observe and to describe properties of mathematical objects.

Methodological Postulate 3

We adopt the principle ‘The part is less than the whole’ to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).

The Infinite Unit Axiom The infinite unit of measure is introduced as the number of elements of the set, \({\mathbb {N}}\), of natural numbers. It is expressed by the numeral ① called grossone and has the following properties:

Infinity Any finite natural number n is less than grossone, i.e., \(n <~\textcircled {1}\).

Identity The following relations link ① to identity elements 0 and 1

$$\begin{aligned} 0 \cdot \textcircled {1} = \textcircled {1} \cdot 0 = 0, \quad \textcircled {1}-\textcircled {1}= 0,\quad \frac{\textcircled {1}}{\textcircled {1}}=1, \quad \textcircled {1}^0=1, \quad 1^{\textcircled {1}}=1, \quad 0^{\textcircled {1}}=0. \end{aligned}$$
(12)

Divisibility For any finite natural number n sets \({\mathbb {N}}_{k,n}, 1 \le k \le n,\) being the \(n\hbox {th}\) parts of the set, \({\mathbb {N}}\), of natural numbers have the same number of elements indicated by the numeral \(\frac{\textcircled {1}}{n}\) where

$$\begin{aligned} {\mathbb {N}}_{k,n} = \{k, k+n, k+2n, k+3n, \ldots \}, \quad 1 \le k \le n, \quad \bigcup _{k=1}^{n}{\mathbb {N}}_{k,n}={\mathbb {N}}. \end{aligned}$$
(13)

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Sergeyev, Y.D. Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite. Found Sci 24, 153–170 (2019). https://doi.org/10.1007/s10699-018-9566-y

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