# Mathematical Proof

Edited by Jordan Bohall (University of Illinois, Urbana-Champaign)
 Summary Mathematical proof concerns itself with a demonstration that some theorem, lemma, corollary or claim is true. Proofs rely upon previously proven statements, logical inferences, and a specified syntax, which can usually trace back to underlying axioms and definitions. Many of the issues in this area concern the use of purely formal proof, informal proof, language, empirical methodologies, and everyday practice.
 Key works
 Introductions Horsten 2008
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1. Existentially Closed Structures and Gödel's Second Incompleteness Theorem.Zofia Adamowicz & Teresa Bigorajska - 2001 - Journal of Symbolic Logic 66 (1):349-356.
We prove that any 1-closed (see def 1.1) model of the Π 2 consequences of PA satisfies ¬Cons PA which gives a proof of the second Godel incompleteness theorem without the use of the Godel diagonal lemma. We prove a few other theorems by the same method.
2. Lakatos on proof and on mathematics.J. Agassi - 1981 - Logique Et Analyse 24 (95):437.
3. Goedel Theorem of Incompleteness.I. Aimonetto - 1993 - Filosofia 44 (1):113-136.
4. Are There Viable Connections Between Mathematics, Mathematical Proof and Democracy?D. F. Almeida - 2010 - Philosophy of Mathematics Education Journal 25.
5. Justifying and Proving in the Mathematics Classroom.Dennis Almeida - 1996 - Philosophy of Mathematics Education Journal 9.
6. Aspects Of Proof: Special Issue Of Educational Studies In Mathematics. [REVIEW]Dennis Almeida - 1995 - Philosophy of Mathematics Education Journal 8.
7. Wittgenstein on Mathematical Proof.Alice Ambrose - 1982 - Mind 91 (362):264-272.
8. Almost everyone has an intuitive notion of what a random number is. For example, consider these two series of binary digits: 01010101010101010101 01101100110111100010 The first is obviously constructed according to a simple rule; it consists of the number 01 repeated ten times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be 0 and 1. Inspection of the second series of digits yields no such comprehensive (...)
9. On the Depth of Szemerédi's Theorem.Andrew Arana - 2015 - Philosophia Mathematica 23 (2):163-176.
Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...)
10. Mathematical Proof and Experimental Proof.Sr Arthur H. Copeland - 1966 - Philosophy of Science 33 (4):303-316.
In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...)
11. Abduction and Conjecturing in Mathematics.Ferdinando Arzarello, Valeria Andriano, Federica Olivero & Ornella Robutti - 1998 - Philosophica 61 (1):77-94.
The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in (...)
12. In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classiﬁed under two headings: some (...)
13. Proof Theory. Gödel and the Metamathematical Tradition.Jeremy Avigad - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
14. Understanding, Formal Verification, and the Philosophy of Mathematics.Jeremy Avigad - 2010 - Journal of the Indian Council of Philosophical Research 27:161-197.
The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...)
15. Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. [REVIEW]Jeremy Avigad - 2008 - Philosophia Mathematica 17 (1):95-108.
Published in 1891, Edmund Husserl's first book, Philosophie der Arithmetik, aimed to ‘prepare the scientific foundations for a future construction of that discipline’. His goals should seem reasonable to contemporary philosophers of mathematics: "…through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. 1"But the ensuing strategy for grounding mathematical knowledge (...)
16. Mathematical Method and Proof.Jeremy Avigad - 2006 - Synthese 153 (1):105 - 159.
On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
17. The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.
18. The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof.J. Azzouni - 2013 - Philosophia Mathematica 21 (2):247-254.
The relationship is explored between formal derivations, which occur in artificial languages, and mathematical proof, which occurs in natural languages. The suggestion that ordinary mathematical proofs are abbreviations or sketches of formal derivations is presumed false. The alternative suggestion that the existence of appropriate derivations in formal logical languages is a norm for ordinary rigorous mathematical proof is explored and rejected.
19. That We See That Some Diagrammatic Proofs Are Perfectly Rigorous.J. Azzouni - 2013 - Philosophia Mathematica 21 (3):323-338.
Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of (...)
20. Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented (...)
21. It is conventionally understood that computers play a rather limited role in theoretical mathematics. While computation is indispensable in applied mathematics and the theory of computing and algorithms is rich and thriving, one does not, even today, expect to find computers in theoretical mathematics settings beyond the theory of computing. Where computers are used, by those studying combinatorics , algebra, number theory, or dynamical systems, the computer most often assumes the role of an automated and speedy theoretician, performing manipulations and (...)
22. The Incompleteness Theorems.H. P. Barendregt - 1976 - Rijksuniversiteit Utrecht, Mathematisch Instituut.
23. The Surveyability of Mathematical Proof: A Historical Perspective.O. Bradley Bassler - 2006 - Synthese 148 (1):99-133.
This paper rejoins the debate surrounding Thomas Tymockzko's paper on the surveyability of proof, first published in the "Journal of Philosophy", and makes the claim that by attending to certain broad features of modern conceptions of proof we may understand ways in which the debate surrounding the surveyability of proof has heretofore remained unduly circumscribed. Motivated by these historical reflections. I suggest a distinction between local and global surveyability which I believe has the promise to open up significant new advances (...)
24. First‐Year Secondary School Mathematics Students' Conceptions of Mathematical Proofs and Proving.Savas Basturk - 2010 - Educational Studies 36 (3):283-298.
The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students . The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for (...)
25. Proof in Mathematics ("If", "Then" and "Perhaps"): A Collection of Material Illustrating the Nature and Variety of the Idea of Proof in Mathematics.P. R. Baxandall (ed.) - 1978 - University of Keele, Institute of Education.
26. On the Induction Schema for Decidable Predicates.Lev D. Beklemishev - 2003 - Journal of Symbolic Logic 68 (1):17-34.
We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, \$I\Delta_1\$ . We show that \$I\Delta_1\$ is independent from the set of all true arithmetical \$\Pi_2-sentences\$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of \$\Delta_1-induction\$ . An open problem formulated by J. (...)
27. God, the Devil, and Gödel.Paul Benacerraf - 1967 - The Monist 51 (1):9-32.
28. The Moment of Proof: Mathematical Epiphanies.Donald C. Benson - 1999 - Oxford University Press.
When Archimedes, while bathing, suddenly hit upon the principle of buoyancy, he ran wildly through the streets of Syracuse, stark naked, crying "eureka!" In The Moment of Proof, Donald Benson attempts to convey to general readers the feeling of eureka--the joy of discovery--that mathematicians feel when they first encounter an elegant proof. This is not an introduction to mathematics so much as an introduction to the pleasures of mathematical thinking. And indeed the delights of this book are many and varied. (...)
29. Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical (...)
30. A Note On Interaction And Incompleteness.Damjan Bojadžiev - 2003 - Logic Journal of the IGPL 11 (5):513-523.
The notion of interaction and interaction machines, developed by Peter Wegner, includes the comparison between incompleteness of interaction machines and Gödel incompleteness. However, this comparison is not adequate, because it combines different notions and different sources of incompleteness. In particular, it merges syntactic with two senses of semantic completeness, and results about truth with results about provability and their consequences . The comparison also overlooks structural differences in the way diagonalization produces incompleteness. More generally, the comparison is unlikely because interaction (...)
31. Three Theorems of Godel.Andrew Boucher - manuscript
It might seem that three of Godel’s results - the Completeness and the First and Second Incompleteness Theorems - assume so little that they are reasonably indisputable. A version of the Completeness Theorem, for instance, can be proven in RCA0, which is the weakest system studied extensively in Simpson’s encyclopaedic Subsystems of Second Order Arithmetic. And it often seems that the minimum requirements for a system just to express the Incompleteness Theorems are sufficient to prove them. However, it will be (...)
32. 1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
33. Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
34. Review of D. Mac Kenzie, Mechanizing Proof: Computing, Risk, and Trust.Otávio Bueno & Jody Azzouni - 2005 - Philosophia Mathematica 13 (3):319-325.
35. Slides for the second tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
36. Slides for the third tutorial on Gödel's incompleteness theorems, held at UniLog 5 Summer School, Istanbul, June 24, 2015.
37. From Closed to Open Systems.Carlo Cellucci - 1993 - In J. Czermak (ed.), Philosophy of Mathematics, pp. 206-220. Hölder-Pichler-Tempsky.
While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system - as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's concept of (...)
38. Less Proof, More Truth.Gregory Chaitin - manuscript
MATHEMATICS is a wonderful, mad subject, full of imagination, fantasy and creativity that is not limited by the petty details of the physical world, but only by the strength of our inner light. Does this sound familiar? Probably not from the mathematics classes you may have attended. But consider the work of three famous earlier mathematicians: Leonhard Euler, Georg Cantor and Srinivasa Ramanujan.
39. Edgar Morin's Paradigm of Complexity and Gödel's Incompleteness Theorem.Yi-Zhuang Chen - 2004 - World Futures 60 (5 & 6):421 – 431.
This article shows that in two respects, Gödel's incompleteness theorem strongly supports the arguments of Edgar Morin's complexity paradigm. First, from the viewpoint of the content of Gödel's theorem, the latter justifies the basic view of complexity paradigm according to which knowledge is a dynamic, unfinished process, and develops by way of self-criticism and self-transcendence. Second, from the viewpoint of the proof procedure of Gödel's theorem, the latter confirms the complexity paradigm's circular line of inference through which is formed the (...)
40. Consistency, Optimality, and Incompleteness.Yijia Chen, Jörg Flum & Moritz Müller - 2013 - Annals of Pure and Applied Logic 164 (12):1224-1235.
Assume that the problem P0 is not solvable in polynomial time. Let T be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{ConT} as the minimal extension of T proving for some algorithm that it decides P0 as fast as any algorithm B with the property that T proves that B decides P0. Here, ConT claims the consistency of T. As a byproduct, we obtain a version of Gödelʼs Second Incompleteness Theorem. Moreover, we characterize problems (...)
41. From Consistency to Incompleteness: A Philosophical Study of Hilbert's Program and Goedel's Incompleteness Theorem.Byoung-il Choi - 1997 - Dissertation, University of California, Berkeley
The main objective of this thesis is a philosophical study of Hilbert's Program and Godel's Incompleteness Theorem. For this purpose we pursue historical, metamathematical, and conceptual investigations of them. ;By tracing the historical origins and conceptual developments of Hilbert's Program and Godel's Incompleteness Theorem, we will argue that both have inherently philosophical motivations. Also, by considering the relevant metamathematical developments such as Reverse Mathematics, the Paris-Harrington Incompleteness Theorem and related materials, we will argue that Hilbert's Program and Godel's Incompleteness Theorem (...)
42. Arytmetyka i intensjonalność.Cezary Cieśliński - 2001 - Filozofia Nauki 4.
The paper consists of two pats. The first part contains a critical review of "Gödel theorems, possible worlds and intensionality" by W. Krysztofiak. Krysztofiak argues that Gödel's incompleteness theorem and, in particular, the technique of aritmetization of syntax, gives rise to intensionality and intentionality in arithmetic. The author tries to show that these claims are mistaken and based on a simple misunderstanding of the incompleteness theorem and its proof. In the second part the author explains the traditional use (in the (...)
43. What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
44. The Surveyability of Long Proofs.Edwin Coleman - 2009 - Foundations of Science 14 (1-2):27-43.
The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because (...)
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