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  1. added 2020-02-14
    Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth.William D'Alessandro - forthcoming - Synthese.
    Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...)
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  2. added 2019-10-13
    Last Bastion of Reason. [REVIEW]James Franklin - 2000 - New Criterion 18 (9):74-78.
    Attacks the irrationalism of Lakatos's Proofs and Refutations and defends mathematics as a "last bastion" of reason against postmodernist and deconstructionist currents.
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  3. added 2019-09-28
    Tools of Reason: The Practice of Scientific Diagramming From Antiquity to the Present.Greg Priest, Silvia De Toffoli & Paula Findlen - 2018 - Endeavour 42 (2-3):49-59.
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  4. added 2019-09-23
    À Maneira de Um Colar de Pérolas?André Porto - 2017 - Revista Portuguesa de Filosofia 73 (3-4):1381-1404.
    This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.
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  5. added 2019-08-09
    Teaching and Learning Guide For: Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".
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  6. added 2019-08-06
    Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11).
    Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...)
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  7. added 2019-06-10
    Tales of Wonder: Ian Hacking: Why is There Philosophy of Mathematics at All? Cambridge University Press, 2014, 304pp, $80 HB.Brendan Larvor - 2015 - Metascience 24 (3):471-478.
    Why is there Philosophy of Mathematics at all? Ian Hacking. in Metascience (2015).
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  8. added 2019-06-08
    Category Theory is a Contentful Theory.Shay Logan - 2015 - Philosophia Mathematica 23 (1):110-115.
    Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.
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  9. added 2019-06-03
    Canonical Maps.Jean-Pierre Marquis - 2018 - In Elaine Landry (ed.), Categories for the Working Philosophers. Oxford, UK: pp. 90-112.
    Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...)
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  10. added 2019-06-03
    Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  11. added 2019-06-03
    The Exact (Up to Infinitesimals) Infinite Perimeter of the Koch Snowflake and its Finite Area.Yaroslav Sergeyev - 2016 - Communications in Nonlinear Science and Numerical Simulation 31 (1-3):21–29.
    The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and (...)
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  12. added 2019-06-03
    Mathematical Abstraction, Conceptual Variation and Identity.Jean-Pierre Marquis - 2014 - In Peter Schroeder-Heister, Gerhard Heinzmann, Wilfred Hodges & Pierre Edouard Bour (eds.), Logic, Methodology and Philosophy of Science, Proceedings of the 14th International Congress. London, UK: pp. 299-322.
    One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.
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  13. added 2019-04-28
    Mathematical Models of Abstract Systems: Knowing Abstract Geometric Forms.Jean-Pierre Marquis - 2013 - Annales de la Faculté des Sciences de Toulouse 22 (5):969-1016.
    Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. (...)
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  14. added 2019-04-26
    Introduction.Andrew Aberdein & Matthew Inglis - 2019 - In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics. Bloomsbury Academic. pp. 1-13.
    There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such (...)
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  15. added 2019-04-26
    Reconstruction in Philosophy of Mathematics.Davide Rizza - 2018 - Dewey Studies 2 (2):31-53.
    Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to the (...)
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  16. added 2019-04-26
    Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics.Jean Pierre Marquis & Jean-Pierre Marquis - 2016 - The Mathematical Intelligencer 38 (3):41-51.
    In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.
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  17. added 2019-03-30
    Evidence, Proofs, and Derivations.Andrew Aberdein - 2019 - ZDM 51 (5):825-834.
    The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, (...)
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  18. added 2019-03-23
    Imagination in Mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), Routledge Handbook on the Philosophy of Imagination. Routledge. pp. 463-477.
    This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.
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  19. added 2019-03-23
    On the Relationship Between Plane and Solid Geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
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  20. added 2019-03-23
    Review of D. Corfield's Toward A Philosophy Of Real Mathematics. [REVIEW]Andrew Arana - 2007 - Mathematical Intelligencer 29 (2).
    When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.
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  21. added 2019-03-04
    Tools for Thought: The Case of Mathematics.Valeria Giardino - 2018 - Endeavour 2 (42):172-179.
    The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced.
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  22. added 2018-12-17
    Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Yaroslav D. Sergeyev - 2019 - Foundations of Science 24 (1):153-170.
    This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...)
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  23. added 2018-12-17
    The Difficulty of Prime Factorization is a Consequence of the Positional Numeral System.Yaroslav Sergeyev - 2016 - International Journal of Unconventional Computing 12 (5-6):453–463.
    The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the prime factorization is (...)
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  24. added 2018-12-17
    The Olympic Medals Ranks, Lexicographic Ordering and Numerical Infinities.Yaroslav Sergeyev - 2015 - The Mathematical Intelligencer 37 (2):4-8.
    Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...)
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  25. added 2018-11-24
    Viewing-as Explanations and Ontic Dependence.William D’Alessandro - 2020 - Philosophical Studies 177 (3):769-792.
    According to a widespread view in metaphysics and philosophy of science, all explanations involve relations of ontic dependence between the items appearing in the explanandum and the items appearing in the explanans. I argue that a family of mathematical cases, which I call “viewing-as explanations”, are incompatible with the Dependence Thesis. These cases, I claim, feature genuine explanations that aren’t supported by ontic dependence relations. Hence the thesis isn’t true in general. The first part of the paper defends this claim (...)
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  26. added 2018-10-19
    The "Artificial Mathematician" Objection: Exploring the (Im)Possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Cham: Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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  27. added 2018-09-07
    O nouă filosofie a matematicii?Gabriel Târziu - 2012 - Symposion – A Journal of Humanities 10 (2):361-377.
    O tendinţă relativ nouă în filosofia contemporană a matematicii este reprezentată de nemulţumirea manifestată de un număr din ce în ce mai mare de filosofi faţă de viziunea tradiţională asupra matematicii ca având un statut special ce poate fi surprins doar cu ajutorul unei epistemologii speciale. Această nemulţumire i-a determinat pe mulţi să propună o nouă perspectivă asupra matematicii – una care ia în serios aspecte până acum neglijate de filosofia matematicii, precum latura sociologică, istorică şi empirică a cercetării matematice (...)
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  28. added 2017-12-25
    The Necessity of Mathematics.Juhani Yli‐Vakkuri & John Hawthorne - 2018 - Noûs 52.
    Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.
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  29. added 2017-08-23
    Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Zurich, Switzerland: Birkhäuser. pp. 25-50.
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...)
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  30. added 2017-08-23
    An Inquiry Into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: Springer International Publishing. pp. 315-336.
    The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used (...)
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  31. added 2017-06-14
    The Great Gibberish - Mathematics in Western Popular Culture.Markus Pantsar - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Springer International Publishing. pp. 409-437.
    In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics is ...
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  32. added 2017-05-22
    Proof Phenomenon as a Function of the Phenomenology of Proving.Inês Hipólito - 2015 - Progress in Biophysics and Molecular Biology 119:360-367.
    Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...)
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  33. added 2017-05-21
    Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...)
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  34. added 2017-02-17
    The Interplay Between Mathematical Practices and Results.Mélissa Arneton, Amirouche Moktefi & Catherine Allamel-Raffin - 2014 - In Léna Soler, Sjoerd Zwart, Michael Lynch & Vincent Israel-Jost (eds.), Science After the Practice Turn in the Philosophy, History, and Social Studies of Science. New York - London: Routledge. pp. 269-276.
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  35. added 2017-01-12
    The Changing Practices of Proof in Mathematics: Gilles Dowek: Computation, Proof, Machine. Cambridge: Cambridge University Press, 2015. Translation of Les Métamorphoses du Calcul, Paris: Le Pommier, 2007. Translation From the French by Pierre Guillot and Marion Roman, $124.00HB, $40.99PB.Andrew Arana - 2017 - Metascience 26 (1):131-135.
    Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.
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  36. added 2017-01-12
    On the Depth of Szemeredi'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 (...)
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  37. added 2017-01-12
    L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
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  38. added 2016-01-02
    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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  39. added 2015-09-10
    Mathematics as Language.Adam Morton - 1996 - In Adam Morton & Stephen P. Stich (eds.), Benacerraf and His Critics. Blackwell. pp. 213--227.
    I discuss ways in which the linguistic form of mathimatics helps us think mathematically.
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  40. added 2012-04-20
    Jeremy Gray. Plato's Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press, 2008. Isbn 978-0-69113610-3. Pp. VIII + 515. [REVIEW]A. Arana - 2012 - Philosophia Mathematica 20 (2):252-255.
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