Foundations of mathematics

Stéphane Dugowson, quand nous allons de son cours de Catégories pour philosophes à son interview d’Olivia Caramello avec Anatole Khélif, nous fait accomplir un parcours ascendant parmi les termes transcendentaux présélectionnés par Albert Lautman, là où se joue le rapport principal entre la philosophie et les mathématiques.

Mostramos como tanto el teorema de Gödel como las paradojas del mentiroso se deben, no solo a autoalusión, sino al cumplimiento de tres condiciones (denominadas “Islas Gödelianas”). Así mismo señalamos que pueden recibir un tratamiento completo mediante otro valor de verdad “G”=paradójico.

O objetivo deste minicurso é despertar no aluno a inquirição: o que é a Matemática? Sendo que o intuito é fazer o aluno compreender de uma maneira global ao que se refere o termo e os diversos conceitos, assim como apresentar as principais áreas, suas ramificações e aplicações. De uma maneira geral, os alunos num curso de graduação, não só em Matemática, como também em outros ramos da ciência, têm uma série de cursos dispersos, entre os quais não conseguem perceber uma conexão imediata. Com frequência ficam atrelados ao aprendizado de técnicas. Existe ainda no ensino da Matemática uma grande falta de sistematização, conduzindo o aluno a não ter uma visão global e sistemática do assunto. O objetivo deste curso é discutir o conceito envolvido na definição de Matemática, sabendo-se que esta definição não é simples e depende do grau de instrução. Finalizando com a análise da Matemática como: técnica, arte, ciência e filosofia, seguindo a abordagem do filósofo da Matemática L. A. Santaló (1994). Serão abordados os seguintes tópicos principais: a etimologia, história, objetos de estudo, origens, modelos, notação e linguagem da Matemática. Em seguida se discutirá a divisão da Matemática em pura e aplicada, discorrendo sobre as suas definições, objetos de estudo e subáreas. A Matemática pura pode ser subdividida nas seguintes áreas principais: álgebra; geometria; análise e fundamentos. Nas subáreas a abordagem detalhará os diversos nuances da Matemática pura. Na sequência serão discutidas as principais áreas da Matemática aplicada: física Matemática; computação; teoria dos jogos; fractais e teoria do caos; estatística e em outras ciências. Será ressaltada a ampla abrangência da Matemática e sua pertinência em qualquer ramo do conhecimento humano. O fecho será a abordagem da educação Matemática, desde que todas as áreas da Matemática pura e aplicada necessitam de pessoas que possam transmitir os conhecimentos supracitados, exigindo grandes investimentos na formação dos futuros professores tornando está subárea da Matemática um item muito importante.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the challenge to explain the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the challenge to explain their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is unmysterious. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary  implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it.  Unlike Quine’s, this argument begins with the banal observation that there are different candidate interpretations of the predicate ‘could have been the case’.  This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’.  The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ (Clarke-Doane & Hamkins [2017]) claims that the search for the true axioms of set theory is.  We show that the obvious responses to this argument fail.  However, a new response has emerged that purports to prove, from higher order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special.  We distill two lines of reasoning from the literature, and argue that their import depends on premises that a ‘modal pluralist’ should deny.  Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow.  In other words, both presupposes that there is a unique candidate for what higher-order claims could mean.  We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced.  We also sketch the bearing of the resulting ‘Higher Order Pluralism’ on the theory of content.  An upshot is that, if Higher Order Pluralism is true, then there is no stable metatheory from which to characterize higher order reality.

In this dissertation I approach the philosophy of mathematics and its foundations from the point of view of modern cognitive science. The first, introductory part, considers a cognitive foundations program, in general philosophical terms. The second part, which brings in the cognitive-mathematical substance, demonstrates how a cognitive approach can empower the exploration and provide insight on classical issues in the philosophy of mathematics.

В статье рассматривается краткая история развития теории множеств как оснований математики и краткая история становления теории типов. Приводится подробное сравнение теории множеств и теории типов, демонстрируются проблемы теории множества, а также преимущества теории типов как оснований математики. Кроме того, в статье обсуждаются прагматический аспект теории типов как оснований математики и кратко рассматривается современный вариант теории типов, а именно гомотопическая теория типов.

Introduced several new axiomatic systems, that are not less general than group theory, and discovered discontinuous analysis.

In this work I introduce and study in details the concepts of funcoids
which generalize proximity spaces and reloids which generalize uniform
spaces, and generalizations thereof. The concept of funcoid is generalized
concept of proximity, the concept of reloid is cleared from superfluous
details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets). Also funcoids
and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of calculus and discrete
mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids.
Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-dif\-fe\-re\-nti\-able solution of a (partial) differential equation. It's raised the question how do such solutions ``look like'' starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx - \int_a^b f(x)dx = 0$ is defined and true for \emph{every} function.

The concept of continuity is defined by an algebraic formula (instead
of old messy epsilon-delta notation) for arbitrary morphisms (including
funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and
generalization of pointfree funcoids: staroids and multifuncoids.
Also I define several kinds of products of funcoids and other morphisms.

I define \emph{space} as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc.\ all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.)\ in an arbitrary space. Now general topology is an algebraic theory.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.

O que é a Matemática? Esta é a inquirição que pretendemos abordar. A Matemática poderia ser conceituada como técnica, arte; filosofia ou ciência. Complementando com a descrição sucinta das diversas áreas da Matemática pura, da aplicada, e finalizamos com a educação Matemática.

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals 1; @0; !, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems.

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both counts. I will compare Parsons' argument with Brouwer's and defend the latter. I will not argue that Parsons is wrong once his own conception of intuition is granted, as I do not think that that is the case. But I will try to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion that, when developed further, would lead to Brouwer's notion thus analysed, or at least something relevantly similar to it. (This version contains a postscript of May, 2015.)

The present paper is devoted to several major changes in how the foundations of mathematics are understood – changes that have been taking place in the post-Gödelian intellectual landscape. The foundationalist and speculative approaches of the previous historical period first gave way to the “no foundations” approach and, next, to two concurrent trends of “anthropological” and “practical” approaches to the foundational problem. An attempt is made to demonstrate that each of the two recent trends can be interpreted in terms of communication with both human and non-human agents.

Статья посвящена исследованию возможностей применения абдуктивных рассуждений в рамках философии математики для выбора наилучшей теории в области оснований математики. Рассматриваются понятие простоты математического доказательства и связь простоты с убедительностью. В контексте оснований математики проблематизируется критерий простоты оснований, выоявляются сложности его применения, а также намечаются пути дальнейших исследований методологии философии математики.

Рассматривается теоретико-типовое понятие доказательства корректности компьютерных программ. Исследуются особенности этого понятия, а также ряд связанных с ним теоретико-познавательных проблем. Особое внимание уделяется проблеме обозримости и связи априорного и апостериорного. Настоящая статья является первой в серии о понятии теоретико-типового доказательства.

What are the Elements about? What was Euclid trying to accomplish? Why did Euclid use diagrams? Did Euclid produce any constructions at all in the Elements? Does the word axiom or postulate appear anywhere in the Elements? Does the word geometry appear anywhere in the Elements? Did Euclid acknowledge the concept of infinity? Are there any circular definitions? Are there any vague or obscure definitions?

To understand how the operations of arithmetic were transferred from geometry to algebra through the abstract unit, you will need to study the following article:

https://www.academia.edu/69488136/Theory_of_fractions_from_Book_5_of_Elements_for_Dummies

Статья посвящена рассмотрению текущего состояния конструктивистского и структуралистского направлений в философии математики. Обосновывается взаимосвязь указанных направлений в рамках теоретико-типового подхода к основаниям математики через рассмотрение понятий изоморфизма, инварианта и структуры. Особо обсуждаются аксиома унивалентности и современное теоретико-типовое понятие равенства.

Рассматривается понятие математического доказательства в связи с применением компьютеров в математике. Исследуются особенности этого понятия, а также связанных эпистемологических проблем. Особое внимание уделяется проблеме обо- зримости на примере доказательства теоремы о четырех красках. Настоящая ра-бота представляет собой вторую статью в серии о понятии теоретико-типового доказательства.

Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more important than the second one which, in its turn, is incomparably more important than the third one, etc. In this paper, Lexicographic Multi-Objective Linear Programming (LMOLP) problems are considered. To tackle them, traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single-objective function. The latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single-objective one and the search of correct weights can be very time consuming. In this work a new approach for solving LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed. It is shown that a smart application of infinitesimal weights allows one to construct a single-objective problem avoiding the necessity to determine finite weights. The equivalence between the original multi-objective problem and the new single-objective one is proved. A simplex-based algorithm working with finite and infinitesimal numbers is proposed, implemented, and discussed. Results of some numerical experiments are provided.

In her essay Math Matters, Sabine Hossenfelder claims that results in the foundations of mathematics are irrelevant for physics, because physics is science, not mathematics. Her argument is based on the possibility of making a distinction between a scientific and a non-scientific activity or domain. Even though physics and other sciences (like virology or climate science) are relevant in larger non-scientific contexts (like the corona crisis or climate change), the distinction between what is scientific and what is not, remains a case by case matter, not something we can do in principle. The legitimacy of physics to be scientific is entirely apparent for the one who practices it, but is not a particular instance of the more general characteristic ‘scientific’. In the same manner, climate science is entirely apparent for the climate scientist, less so in a larger non-scientific debate. Mathematics is a domain that has established, through a number of definitive foundational theorems (Cantor, Gödel, Turing …) what its own limits and possibilities are. On a general level, no matter in what non-mathematical context it is relevant, mathematics has determined on its own terms, what it can and cannot do. In that sense mathematics – as an example, that a distinction in principle between what is proper to one’s own domain and what is not, can be established – is highly relevant to science, specifically physics.

The paper explores Hermann Weyl's turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl's thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl's continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl's wavering foundational positions. Building on a philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl's motives for considering such a radical shift in the first place. It concludes by showing that Weyl's shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism.

A very simple question is posed: Employing a ball-and-stick modelling system, and given a supply of the component balls and rods, then, treating it as a gedanken experiment, what is the most simple structure that it is possible to construct?

Abstract Frank Thomas Sautter:
In this paper Chateaubriand's approach to solve some problems related to the nature of logic is confronted with the traditional approaches. It is shown that his hierarchy of logical types opens up new possibilities to characterize logical properties and logical truths and that it also sheds some new light on the foundations of mathematics. The order in which a system is exhibited almost always does not correspond to the order in which it has been elaborated. The system proposed by Oswaldo Chateaubriand is no exception (see Chateaubriand 2001). Since it contains many original ideas and original interpretations of traditional ideas which were initially developed to refute the slingshot-argument, especially the variant formulated by Gödel, the first chapter of the book with regard to the order of elaboration is the fourth chapter with regard to the order of exhibition. The diverse parts of the system form an organic whole making it difficult to apprehend them in isolation; this explains also the recurrence of some ideas.

Abstract response:
In §1 I examine the connections between my account of logical properties and Tarski’s account of logical notions. In §2 I briefly present some of my views on modality and the basis for my claim that there are intensional as well as extensional relations between properties. In §3 I compare my views on the nature of logic and of mathematics with Gödel’s views.

В последние десятилетия в математике особую остроту приобрели
эпистемологические проблемы, связанные со слишком большой длиной
доказательства важных математических результатов, а также с большим и
постоянно возрастающим количеством публикаций по математике.
Предполагается, что эти затруднения могут быть разрешены (хотя бы
частично) путем обращения к компьютерным доказательствам. Однако и
компьютерные доказательства оказываются проблематичными с
эпистемологической точки зрения. И относительно доказательств в
обычной (неформальной) математике, и относительно компьютерных
доказательств в равной степени актуальна проблема их обозримости.
Исходя из традиционного понимания доказательства оно обязательно
должно быть обозримым, иначе оно не будет достигать своей основной
цели - формирования убежденности в правильности доказываемого
математического результата. Около 15 лет назад начал развиваться новый
подход к основаниям математики, сочетающий в себе конструктивистские,
структуралистские черты и ряд преимуществ классического подхода к
математике. Этот подход выстраивается на основе гомотопической теории
типов и носит название унивалентных оснований математики. Благодаря
мощному понятию равенства этот подход позволяет значительно сократить
длину формализованных доказательств, что намечает путь к разрешению
возникших эпистемологических затруднений.

[In recent decades, some epistemological issues have become especially
acute in mathematics. These issues are associated with long proofs of
various important mathematical results, as well as with a large and
constantly increasing number of publications in mathematics. It is
assumed that (at least partially) these difficulties can be resolved
by referring to computer proofs. However, computer proofs also turn
out to be problematic from an epistemological point of view. With
regard to both proofs in ordinary (informal) mathematics and computer
proofs, the problem of their surveyability appears to be fundamental.
Based on the traditional concept of proof, it must be surveyable,
otherwise it will not achieve its main goal, the formation of
conviction in the correctness of the mathematical result being proved.
About 15 years ago, a new approach to the foundations of mathematics
began to develop, combining constructivist, structuralist features and
a number of advantages of the classical approach to mathematics. This
approach is built on the basis of homotopy type theory and is called
the univalent foundations of mathematics. Due to its powerful notion
of equality, this approach can significantly reduce the length of
formalized proofs, which outlines a way to resolve the epistemological
difficulties that have arisen.]

It will be shown in this contribution that the Received View on Zeno’s paradoxical arguments is untenable. Upon a close analysis of the Greek sources, it is possible to do justice to Simplicius’s widely neglected testimony, where he states: "In his book, in which many arguments are put forward, he shows in each that a man who says that there is a plurality is stating something contradictory". Thus we will demonstrate that an underlying structure common to both the Paradoxes of Plurality (PP) and the Paradoxes of Motion (PM) shores up all Zeno's arguments. This structure bears on a correct — Zenonian — interpretation of the concept of “division through and through”, by taking the philosophical legacy on the "Being-Now" of his master Parmenides properly into account.

The author offers a solution to the problem of universal characteristic posed by G.W.Leibniz. The triune formula of proportion sets the Rule of physical sense for the creation of any universally valid scientific statements. The universal characteristic shows that the purpose of development of mathematical formalism of physics is the description of biological and social life of the person.

In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge ; Brouwer disagrees on both counts. I will compare…

Mostramos como el Argumento Conjuntista de Cantor, se basa en una concepción epistemológica errónea. Esta concepción que produce paradojas (como las que analizamos en otro artículo del banquero o de la sucesión de polígonos) es refutable de modo similar a como refutamos su argumento diagonal.

The Version of Record of this manuscript has been published and is freely available in Inquiry, https://www.tandfonline.com/doi/abs/10.1080/0020174X.2020.1787222.

‘Open texture’ is the property of concepts or terms that they are not fully defined with regard to unexpected questions. Even though it is a ubiquitous phenomenon, there is no consensus on which concepts are subject to open texture. I present three equivalent definitions of open texture, connecting the concept to the notions of analyticity and algebraicity. I show that the claim that a concept is open-textured is equivalent to the claim that the class of analytic sentences using it can be changed, which in turn is equivalent to the claim that its definition is not algebraic. While this equivalence is interesting on its own, with it I can show the extent of open texture: all concepts that are not algebraically defined mathematical concepts are open textured. Finally, I show that this has unfortunate consequences for attempts to provide a foundation of mathematics.