# 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.
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 Introductions Horsten 2008
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1. Zofia Adamowicz & Teresa Bigorajska (2001). Existentially Closed Structures and Gödel's Second Incompleteness Theorem. 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.

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2. J. Agassi (1981). Lakatos on proof and on mathematics. Logique Et Analyse 24 (95):437.
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4. Alice Ambrose (1982). Wittgenstein on Mathematical Proof. Mind 91 (362):264-272.

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5. 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 (...)
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6. Andrew Arana (2015). On the Depth of Szemerédi's Theorem. 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 (...)

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7. Sr Arthur H. Copeland (1966). Mathematical Proof and Experimental Proof. 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 (...)

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8. 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 (...)

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9. 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 (...)

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10. Jeremy Avigad (2010). Proof Theory. Gödel and the Metamathematical Tradition. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic
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11. Jeremy Avigad (2010). Understanding, Formal Verification, and the Philosophy of Mathematics. 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 (...)

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12. Jeremy Avigad (2009). Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. 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 (...)

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13. J. Azzouni (2013). The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof. 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.

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14. J. Azzouni (2013). That We See That Some Diagrammatic Proofs Are Perfectly Rigorous. 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 (...)

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15. 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 (...)

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17. 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 (...)

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18. H. P. Barendregt (1976). The Incompleteness Theorems. Rijksuniversiteit Utrecht, Mathematisch Instituut.
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19. O. Bradley Bassler (2006). The Surveyability of Mathematical Proof: A Historical Perspective. 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 (...)

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20. O. Bradley Bassler (2006). The Surveyability of Mathematical Proof: A Historical Perspective. 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 (...)

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22. Lev D. Beklemishev (2003). On the Induction Schema for Decidable Predicates. 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. (...)

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23. Paul Benacerraf (1967). God, the Devil, and Gödel. The Monist 51 (1):9-32.

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24. Donald C. Benson (1999). The Moment of Proof: Mathematical Epiphanies. 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. (...)

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27. 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 (...)

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28. 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.

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29. 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.

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30. Otávio Bueno & Jody Azzouni (2005). Review of D. Mac Kenzie, Mechanizing Proof: Computing, Risk, and Trust. Philosophia Mathematica 13 (3):319-325.

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32. Carlo Cellucci (1993). From Closed to Open Systems. 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 (...)

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33. 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 (18th century), Georg Cantor (19th century) and Srinivasa Ramanujan (20th century).

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34. C. Cieslinski (2002). Heterologicality and Incompleteness. Mathematical Logic Quarterly 48 (1):105-110.
We present a semantic proof of Gödel's second incompleteness theorem, employing Grelling's antinomy of heterological expressions. For a theory T containing ZF, we define the sentence HETT which says intuitively that the predicate “heterological” is itself heterological. We show that this sentence doesn't follow from T and is equivalent to the consistency of T. Finally we show how to construct a similar incompleteness proof for Peano Arithmetic.

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35. Cezary Cieśliński (2001). Arytmetyka i intensjonalność. 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 (...)
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36. Justin Clarke-Doane (2013). What is Absolute Undecidability?†. 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 (...)

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37. Edwin Coleman (2009). The Surveyability of Long Proofs. 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 (...)

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38. Gödel's incompleteness theorems establish the stunning result that mathematics cannot be fully formalized and, further, that any formal system containing a modicum of number or set theory cannot establish its own consistency. Wilfried Sieg and Clinton Field, in their paper Automated Search for Gödel's Proofs, presented automated proofs of Gödel's theorems at an abstract axiomatic level; they used an appropriate expansion of the strategic considerations that guide the search of the automated theorem prover AProS. The representability conditions that allow the (...)

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39. Sr Copeland (1966). Mathematical Proof and Experimental Proof. Philosophy of Science 33 (4):303-.
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 (...)

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40. Newton C. A. Da Costa (2012). Gödel's Incompleteness Theorems and Physics. Principia 15 (3):453-459.
This paper is a summary of a lecture in which I presented some remarks on Gödel’s incompleteness theorems and their meaning for the foundations of physics. The entire lecture will appear elsewhere. doi: http://dx.doi.org/ 10.5007 / 1808-1711.2011v15n3p453.

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41. We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

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42. Smoryński Craig (1977). The Incompleteness Theorems. In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co. 822--865.
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44. Newton da Costa & Francisco Doria (1994). Gödel Incompleteness in Analysis, with an Application to the Forecasting Proble.. Philosophia Naturalis 31:25-62.
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45. John W. Dawson Jr (2006). Why Do Mathematicians Re-Prove Theorems? Philosophia Mathematica 14 (3):269-286.
From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)

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46. John W. Dawson (1984). The Reception of Godel's Incompleteness Theorems. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:253 - 271.
According to several commentators, Kurt Godel's incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Godel's Nachlass.

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47. Walter Dean (2015). Arithmetical Reflection and the Provability of Soundness. Philosophia Mathematica 23 (1):31-64.
Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...)

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48. Józef Dębowski (2004). Pułapki komputacjonizmu. Filozofia Nauki 1.
The paper concerns basic restrictions (and also simplifications and misinterpretations) which happens when one tries to explain mind processes (especially cognitive ones) by an analogy to formal, algorithmical and anticipated computation processes. The paper puts together the most important reasons why these attempts come to grief. The essence of computative reduction is shown among other things on the basic theorems of modern metamathematics. Especially it gives prominence epistemological consequences Gödel's theorems and recent discovers in metamathematics made by Gregory J. Chaitin. (...)
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49. Michael J. Degnan (1994). Principles and Proofs. Review of Metaphysics 48 (1):154-156.