Epistemology of Mathematics

Edited by Alan Baker (Swarthmore College)
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  1. The Epistemology of the Question of Authenticity, in Place of Strategic Essentialism.Emily S. Lee - 2011 - Hypatia 26 (2):258--279.
    The question of authenticity centers in the lives of women of color to invite and restrict their representative roles. For this reason, Gayatri Chakravorty Spivak and Uma Narayan advocate responding with strategic essentialism. This paper argues against such a strategy and proposes an epistemic understanding of the question of authentic- ity. The question stems from a kernel of truth—the connection between experience and knowledge. But a coherence theory of knowledge better captures the sociality and the holism of experience and knowledge.
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  2. The Hardness of the Iconic Must: Can Peirce's Existential Graphs Assist Modal Epistemology?C. Legg - 2012 - Philosophia Mathematica 20 (1):1-24.
    Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely to individual (...)
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  3. Is There a Good Epistemological Argument Against Platonism?David Liggins - 2006 - Analysis 66 (2):135–141.
    Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns of their dilemma.
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  4. Mathematical Metaphors in Natorp’s Neo-Kantian Epistemology and Philosophy of Science.Thomas Mormann - 2005 - In Falk Seeger, Johannes Lenard & Michael H. G. Hoffmann (eds.), Activity and Sign. Grounding Mathematical Education. Springer.
    A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that Natorp's (...)
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  5. Logicism, Interpretability, and Knowledge of Arithmetic.Sean Walsh - 2014 - Review of Symbolic Logic 7 (1):84-119.
    A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...)
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Apriority in Mathematics
  1. Justified Concepts and the Limits of the Conceptual Approach to the A Priori.Darren Bradley - 2011 - Croatian Journal of Philosophy 11 (3):267-274.
    Carrie Jenkins (2005, 2008) has developed a theory of the a priori that she claims solves the problem of how justification regarding our concepts can give us justification regarding the world. She claims that concepts themselves can be justified, and that beliefs formed by examining such concepts can be justified a priori. I object that we can have a priori justified beliefs with unjustified concepts if those beliefs have no existential import. I then argue that only beliefs without existential import (...)
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  2. Bolzano Versus Kant: Mathematics as a Scientia Universalis.Paola Cantù - 2011 - Philosophical Papers Dedicated to Kevin Mulligan.
    The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...)
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  3. Grassmann’s Epistemology: Multiplication and Constructivism.Paola Cantù - 2010 - In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context.
    The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...)
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  4. What is the Benacerraf Problem?Justin Clarke-Doane - 2017 - In Fabrice Pataut (ed.), New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity.
    In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...)
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  5. Closure of A Priori Knowability Under A Priori Knowable Material Implication.Jan Heylen - 2015 - Erkenntnis 80 (2):359-380.
    The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...)
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  6. Descriptions and Unknowability.Jan Heylen - 2010 - Analysis 70 (1):50-52.
    In a recent paper Horsten embarked on a journey along the limits of the domain of the unknowable. Rather than knowability simpliciter, he considered a priori knowability, and by the latter he meant absolute provability, i.e. provability that is not relativized to a formal system. He presented an argument for the conclusion that it is not absolutely provable that there is a natural number of which it is true but absolutely unprovable that it has a certain property. The argument depends (...)
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  7. Commentary: The Timing of Brain Events.Benjamin Libet - 2006 - Consciusness and Cognition 15:540--547.
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  8. Review of Space, Time, and Number in the Brain. [REVIEW]Carlos Montemayor & Rasmus Grønfeldt Winther - 2015 - Mathematical Intelligencer 37 (2):93-98.
    Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...)
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  9. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...)
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  10. Against Mathematical Convenientism.Seungbae Park - 2016 - Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...)
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  11. Review of The Art of the Infinite by R. Kaplan, E. Kaplan 324p(2003).Michael Starks - 2016 - In Philosophy, Human Nature and the Collapse of Civilization Articles and Reviews 2006-2016 by Michael Starks 662p (2016). Michael Starks. pp. 619.
    This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...)
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  12. Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge[REVIEW]Neil Tennant - 2010 - Philosophia Mathematica 18 (3):360-367.
    This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...)
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Mathematics and the Causal Theory of Knowledge
  1. Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design.Edward G. Belaga - manuscript
    Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability (...)
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  2. Debunking Arguments: Mathematics, Logic, and Modal Security.Justin Clarke-Doane - forthcoming - In Robert Richards and Michael Ruse (ed.), Cambridge Handbook of Evolutionary Ethics. Cambridge University Press.
    I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...)
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  3. What is the Benacerraf Problem?Justin Clarke-Doane - 2017 - In Fabrice Pataut (ed.), New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity.
    In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...)
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  4. Debunking and Dispensability.Justin Clarke-Doane - 2016 - In Uri D. Leibowitz & Neil Sinclair (eds.), Explanation in Ethics and Mathematics: Debunking and Dispensability. Oxford University Press.
    In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...)
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  5. Justification and Explanation in Mathematics and Morality.Justin Clarke-Doane - 2015 - In Russ Shafer-Landau (ed.), Oxford Studies in Metaethics: Volume 1. Oxford University Press.
    In an influential book, Gilbert Harman writes, "In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles [1977, 9 – 10]." What is the epistemological relevance of this contrast, if genuine? In this article, I argue that ethicists and philosophers of mathematics have misunderstood it. They have confused what I will call the justificatory challenge for realism about an (...)
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  6. Moral Epistemology: The Mathematics Analogy.Justin Clarke-Doane - 2014 - Noûs 48 (2):238-255.
    There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...)
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  7. The Epistemology of Geometry I: The Problem of Exactness.Anne Newstead & Franklin James - 2010 - Proceedings of the Australasian Society for Cognitive Science 2009.
    We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...)
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  8. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...)
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  9. Rejecting Mathematical Realism While Accepting Interactive Realism.Seungbae Park - 2018 - Analysis and Metaphysics 17:7-21.
    Indispensablists contend that accepting scientific realism while rejecting mathematical realism involves a double standard. I refute this contention by developing an enhanced version of scientific realism, which I call interactive realism. It holds that interactively successful theories are typically approximately true, and that the interactive unobservable entities posited by them are likely to exist. It is immune to the pessimistic induction while mathematical realism is susceptible to it.
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Mathematical Intuition
  1. On The Rabbinical Exegesis of an Enhanced Biblical Value of Pi.Edward G. Belaga - 1991 - In Hardi Grant, Israel Kleiner & Abe Shenitzer (eds.), Proc. of the 17th Congress of the Canadian Society of History and Philosophy of Mathematics. Kingston.
    We present here a biblical exegesis of the value of Pi, PI_{Hebrew} = 3.141509 ..., from the well known verse 1 Kings 7:23. This verse is then compared to 2 Chronicles 4:2; the comparison provides independent supporting evidence for the exegesis.
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  2. Nominalism and Mathematical Intuition.Otávio Bueno - 2008 - ProtoSociology 25:89-107.
    As part of the development of an epistemology for mathematics, some Platonists have defended the view that we have intuition that certain mathematical principles hold, and intuition of the properties of some mathematical objects. In this paper, I discuss some difficulties that this view faces to accommodate some salient features of mathematical practice. I then offer an alternative, agnostic nominalist proposal in which, despite the role played by mathematical intuition, these difficulties do not emerge.
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  3. Intuition in Mathematics.Elijah Chudnoff - 2014 - In Barbara Held & Lisa Osbeck (eds.), Rational Intuition. Cambridge University Press.
    The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem facing (...)
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  4. Awareness of Abstract Objects.Elijah Chudnoff - 2013 - Noûs 47 (4):706-726.
    Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding (...)
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  5. Intuitive Knowledge.Elijah Chudnoff - 2013 - Philosophical Studies 162 (2):359-378.
    In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? (...)
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  6. Intuição e Conceito: A Transformação do Pensamento Matemático de Kant a Bolzano.Humberto de Assis Clímaco - 2014 - Dissertation, Universidade Federal de Goiás, Brazil
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  7. Numerical Cognition and Mathematical Realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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  8. Forms and Roles of Diagrams in Knot Theory.Silvia De Toffoli & Valeria Giardino - 2014 - Erkenntnis 79 (4):829-842.
    The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...)
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  9. Mathematics and Conceptual Analysis.Antony Eagle - 2008 - Synthese 161 (1):67–88.
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...)
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  10. Kurt Gödels mathematische Anschauung und John P. Burgess’ mathematische Intuition.Eva-Maria Engelen - 2014 - XXIII Deutscher Kongress Für Philosophie Münster 2014, Konferenzveröffentlichung.
    John P. Burgess kritisiert Kurt Gödels Begriff der mathematischen oder rationalen Anschauung und erläutert, warum heuristische Intuition dasselbe leistet wie rationale Anschauung, aber ganz ohne ontologisch überflüssige Vorannahmen auskommt. Laut Burgess müsste Gödel einen Unterschied zwischen rationaler Anschauung und so etwas wie mathematischer Ahnung, aufzeigen können, die auf unbewusster Induktion oder Analogie beruht und eine heuristische Funktion bei der Rechtfertigung mathematischer Aussagen einnimmt. Nur, wozu benötigen wir eine solche Annahme? Reicht es nicht, wenn die mathematische Intuition als Heuristik funktioniert? Für (...)
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  11. Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).
    In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
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  12. The Epistemology of Geometry I: The Problem of Exactness.Anne Newstead & Franklin James - 2010 - Proceedings of the Australasian Society for Cognitive Science 2009.
    We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...)
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  13. The Eternal Unprovability Filter – Part I.Kiran Pai - 2016 - Dissertation, Thinkstrike
    I prove both the mathematical conjectures P ≠ NP and the Continuum Hypothesis are eternally unprovable using the same fundamental idea. Starting with the Saunders Maclane idea that a proof is eternal or it is not a proof, I use the indeterminacy of human biological capabilities in the eternal future to show that since both conjectures are independent of Axioms and have definitions connected with human biological capabilities, it would be impossible to prove them eternally without the creation and widespread (...)
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  14. Set Size and the Part-Whole Principle.Matthew W. Parker - 2013 - Review of Symbolic Logic (4):1-24.
    Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...)
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  15. Ortega y Gasset on Georg Cantor's Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos. Journal of Philosophy and Science (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  16. Review of The Art of the Infinite by R. Kaplan, E. Kaplan 324p(2003).Michael Starks - 2016 - In Philosophy, Human Nature and the Collapse of Civilization Articles and Reviews 2006-2016 by Michael Starks 662p (2016). Michael Starks. pp. 619.
    This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...)
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  17. Toward a Theoretical Account of Strategy Use and Sense-Making in Mathematics Problem Solving.H. J. M. Tabachneck, K. R. Koedinger & M. J. Nathan - 1994 - In Ashwin Ram & Kurt Eiselt (eds.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. Erlbaum.
    Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account of (...)
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  18. Hilbert's Finitism: Historical, Philosophical, and Metamathematical Perspectives.Richard Zach - 2001 - Dissertation, University of California, Berkeley
    In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...)
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Mathematical Proof
  1. 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 (...)
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  2. Corcoran Reviews Boute’s 2013 Paper “How to Calculate Proofs”.John Corcoran - 2014 - MATHEMATICAL REVIEWS 14:444-555.
    Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that term’s quotes (...)
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  3. Automated Theorem Proving and Its Prospects. [REVIEW]Desmond Fearnley-Sander - 1995 - PSYCHE: An Interdisciplinary Journal of Research On Consciousness 2.
    REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.
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  4. Teaching Proving by Coordinating Aspects of Proofs with Students' Abilities.Annie Selden & John Selden - 2009 - In Despina A. Stylianou, Maria L. Blanton & Eric J. Knuth (eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective. New York, USA: Routledge. pp. 339--354.
    In this chapter we introduce concepts for analyzing proofs, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We discuss how coordination of these two analyses can be used to improve students’ ability to construct proofs. -/- For this purpose, we need a richer framework for keeping track of students’ progress than the everyday one used by mathematicians. We need to know more than that a particular student can, or cannot, prove theorems by induction or contradiction or can, (...)
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  5. Does Gödel's Incompleteness Theorem Prove That Truth Transcends Proof?Joseph Vidal-Rosset - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 51--73.
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Godel's Theorem
  1. How to Say Things with Formalisms.David Auerbach - 1992 - In Michael Detlefsen (ed.), Proof, Logic, and Formalization. Routledge. pp. 77--93.
    Recent attention to "self-consistent" (Rosser-style) systems raises anew the question of the proper interpretation of the Gödel Second Incompleteness Theorem and its effect on Hilbert's Program. The traditional rendering and consequence is defended with new arguments justifying the intensional correctness of the derivability conditions.
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