Mathematical Reasoning

2020 ◽  
Vol 57 (4) ◽  
pp. 74-86
Author(s):  
Vitaly V. Tselishchev ◽  
Keyword(s):  
Cognitive Abilities ◽  
Mathematical Reasoning ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal System ◽  
Logical Structure ◽  
Formal Proof ◽  
Formal Proofs ◽  
Logical Conclusion

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself.

MATHEMATICAL RIGOR AND PROOF

2019 ◽  
pp. 1-41 ◽  
Author(s):  
YACIN HAMAMI
Keyword(s):  
Mathematical Knowledge ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal Proof ◽  
Major Obstacle ◽  
Standard View ◽  
Precise Formulation ◽  
Mathematical Proofs ◽  
Primary Form

Abstract Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

scholarly journals Anti-foundationalist Philosophy of Mathematics and Mathematical Proofs

Studia Humana ◽  
2020 ◽  
Vol 9 (3-4) ◽  
pp. 154-164
Author(s):  
Stanisław Krajewski
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal Proof ◽  
New Approach ◽  
Mathematical Proofs

AbstractThe Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic.

A Priority and Application: Philosophy of Mathematics in the Modern Period

2009 ◽  
Author(s):  
Lisa Shabel
Keyword(s):  
Early Modern ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Natural World ◽  
Pure Mathematics ◽  
Philosophical Account ◽  
Mathematical Ontology ◽  
To Come ◽  
Modern Mathematics

The state of modern mathematical practice called for a modern philosopher of mathematics to answer two interrelated questions. Given that mathematical ontology includes quantifiable empirical objects, how to explain the paradigmatic features of pure mathematical reasoning: universality, certainty, necessity. And, without giving up the special status of pure mathematical reasoning, how to explain the ability of pure mathematics to come into contact with and describe the empirically accessible natural world. The first question comes to a demand for apriority: a viable philosophical account of early modern mathematics must explain the apriority of mathematical reasoning. The second question comes to a demand for applicability: a viable philosophical account of early modern mathematics must explain the applicability of mathematical reasoning. This article begins by providing a brief account of a relevant aspect of early modern mathematical practice, in order to situate philosophers in their historical and mathematical context.

Proof as a Cluster Category

2020 ◽  
Vol 51 (1) ◽  
pp. 50-74 ◽  
Author(s):  
Jennifer A. Czocher ◽  
Keith Weber
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Sufficient Conditions ◽  
Mathematical Practice ◽  
Cluster Category ◽  
Mathematics Educators ◽  
Adequate Definition ◽  
Improve Instruction ◽  
Definition Of ◽  
Necessary And Sufficient

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a collection of necessary and sufficient conditions. We argue that defining the proof category in this way renders the definition incapable of accurately capturing how category membership is determined. We propose an alternative account—proof as a cluster category—and demonstrate its potential for addressing many of the intractable challenges inherent in previous accounts. We will also show that adopting the cluster account has utility for how proof is researched and taught.

Philosophy of Mathematics

Philosophy ◽  
2010 ◽  
Author(s):  
Otávio Bueno
Keyword(s):  
Philosophy Of Science ◽  
Set Theory ◽  
Category Theory ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Mathematical Explanation ◽  
Excellent Work ◽  
Novel Approaches

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

How to believe a machine-checked proof

1998 ◽  
Author(s):  
Robert Pollack
Keyword(s):  
Technical Problem ◽  
Formal System ◽  
Physical World ◽  
Formal Proof ◽  
Fundamental Difference ◽  
Formal Proofs ◽  
Mental Processes ◽  
Current State ◽  
Informal Mathematics ◽  
Abstract Representations

Suppose I say “Here is a machine-checked proof of Fermat's last theorem (FLT)”. How can you use my putative machine-checked proof as evidence for belief in FLT? I start from the position that you must have some personal experience of understanding to attain belief, and to have this experience you must engage your intuition and other mental processes which are impossible to formalise. By machine-checked proof I mean a formal derivation in some given formal system; I am talking about derivability, not about truth. Further, I want to talk about actually believing an actual formal proof, not about formal proofs in principle; to be interesting, any approach to this problem must be feasible. You might try to read my proof, just as you would a proof in a journal; however, with the current state of the art, this proof will surely be too long for you to have confidence that you have understood it. This paper presents a technological approach for reducing the problem of believing a formal proof to the same psychological and philosophical issues as believing a conventional proof in a mathematics journal. The approach is not entirely successful philosophically as there seems to be a fundamental difference between machine-checked mathematics, which depends on empirical knowledge about the physical world, and informal mathematics, which needs no such knowledge (see section 3.2.2). In the rest of this introduction I outline the approach and mention related work. In following sections I discuss what we expect from a proof, add details to the approach, pointing out problems that arise, and concentrate on what I believe is the primary technical problem: expressiveness and feasibility for checking of formal systems and representations of mathematical notions. The problem is how to believe FLT when given only a putative proof formalised in a given logic. Assume it is a logic that you believe is consistent, and appropriate for FLT. The “thing” I give you is some computer files. There may be questions about the physical and abstract representations of the files (how to read them physically and how to parse them as a proof), and correctness of the hardware and software to do these things; ignore them until section 3.1.

Computational Tractability and Conceptual Coherence: Why Do Computer Scientists Believe that P≠NP?

1993 ◽  
Vol 23 (3) ◽  
pp. 349-363 ◽  
Author(s):  
Paul Thagard
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Formal Proof ◽  
Turing Computability ◽  
Mathematical Concepts ◽  
Church’S Thesis ◽  
Conceptual Coherence ◽  
Computer Scientists ◽  
Intuitive Concept ◽  
Church's Thesis

According to Church’s thesis, we can identify the intuitive concept of effective computability with such well-defined mathematical concepts as Turing computability and partial recursiveness. The almost universal acceptance of Church’s thesis among logicians and computer scientists is puzzling from some epistemological perspectives, since no formal proof is possible of a thesis that involves an informal concept such as effectiveness. Elliott Mendelson has recently argued, however, that equivalencies between intuitive notions and precise notions need not always be considered unprovable theses, and that Church’s thesis should be accepted as true.I want to discuss a thesis that is nearly as important in current research in computer science as Church’s thesis. I call the newer thesis the tractability thesis, since it identifies the intuitive class of computationally tractable problems with a precise class of problems whose solutions can be computed in polynomial time. After briefly reviewing the theory of intractability, I compare the grounds for accepting the tractability thesis with the grounds for accepting Church's thesis. Intimately connected with the tractability thesis is the mathematical conjecture, whose meaning I shall shortly explain, that P≠NP. Unlike Church's thesis, this conjecture is precise enough to be capable of mathematical proof, but most computer scientists believe it even though no proof has been found. As we shall see below, understanding the grounds for acceptance of the conjecture that P≠NP has implications for general questions in the philosophy of mathematics and science, especially concerning the epistemological importance of explanatory and conceptual coherence.

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

2018 ◽  
Vol 11 (4) ◽  
pp. 665-704
Author(s):  
YACIN HAMAMI
Keyword(s):  
Mathematical Proof ◽  
Mathematical Practice ◽  
Formal Proof ◽  
Logical Inference ◽  
Descriptive Account ◽  
Study Results ◽  
Philosophical Account ◽  
Mathematical Inference ◽  
Made In ◽  
The Ideal

AbstractThe deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

Proof as a Cluster Category

2020 ◽  
Vol 51 (1) ◽  
pp. 50-74 ◽  
Author(s):  
Jennifer A. Czocher ◽  
Keith Weber
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Sufficient Conditions ◽  
Mathematical Practice ◽  
Cluster Category ◽  
Mathematics Educators ◽  
Adequate Definition ◽  
Improve Instruction ◽  
Definition Of ◽  
Necessary And Sufficient

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a collection of necessary and sufficient conditions. We argue that defining the proof category in this way renders the definition incapable of accurately capturing how category membership is determined. We propose an alternative account—proof as a cluster category—and demonstrate its potential for addressing many of the intractable challenges inherent in previous accounts. We will also show that adopting the cluster account has utility for how proof is researched and taught.

scholarly journals IVAN SLESHYNSKY AS A POPULARIZER OF THE IDEAS OF MATHEMATICAL LOGIC IN UKRAINE

2020 ◽  
pp. 99-107
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Mathematical Reasoning ◽  
Mathematical Proof ◽  
Scientific Work ◽  
The United States ◽  
Late Nineteenth Century ◽  
Common Belief ◽  
Early Twentieth Century ◽  
Mathematical Sciences

The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk (Odesa) University Ivan Vladislavovich Sleshynsky. Sleshynsky (1854 –1931) is a Doctor of Mathematical Sciences (1893), Professor (1898) of Novorossiysk (Odesa) University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin (1881–1882), where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary (1882–1886), Odesa high schools (1882–1892), and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem (1889) proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat (1909). Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof” (1925–1929), the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk (Odesa) University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development.

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