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  1. A general framework for a Second Philosophy analysis of set-theoretic methodology.Carolin Antos & Deborah Kant - manuscript
    Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and (...)
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  2. Mathematicians against the myth of genius: beyond the envy interpretation.Terence Rajivan Edward - manuscript
    This paper examines Timothy Gowers’ attempt to counter a mythology of genius in mathematics: that to be a mathematician one has to be a mathematical genius. Someone might take such attacks on the myth of genius as expressions of envy, but I propose that there is another reason for cautioning against placing a high value on genius, by turning to research in the humanities.
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  3. Can AI Abstract the Architecture of Mathematics?Posina Rayudu - manuscript
    The irrational exuberance associated with contemporary artificial intelligence (AI) reminds me of Charles Dickens: "it was the age of foolishness, it was the epoch of belief" (cf. Nature Editorial, 2016; to get a feel for the vanity fair that is AI, see Mitchell and Krakauer, 2023; Stilgoe, 2023). It is particularly distressing—feels like yet another rerun of Seinfeld, which is all about nothing (pun intended); we have seen it in the 60s and again in the 90s. AI might have had (...)
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  4. The Epistemological Subject(s) of Mathematics.Silvia De Toffoli - 2023 - In B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Switzerland: Springer Nature. pp. 1-27.
    Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...)
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  5. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
    Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
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  6. Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2023 - Global Philosophy 33 (38):1-29.
    If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...)
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  7. Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups.Joshua Habgood-Coote & Fenner Stanley Tanswell - 2023 - Episteme 20 (2):281-307.
    In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little (...)
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  8. The hidden use of new axioms.Deborah Kant - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory. Palgrave Macmillan.
    This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic practitioners with (...)
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  9. Rethinking inconsistent mathematics.Franci Mangraviti - 2023 - Dissertation, Ruhr University Bochum
    This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in (...)
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  10. Szemerédi’s theorem: An exploration of impurity, explanation, and content.Patrick J. Ryan - 2023 - Review of Symbolic Logic 16 (3):700-739.
    In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...)
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  11. Signs as a Theme in the Philosophy of Mathematical Practice.David Waszek - 2023 - In Handbook of the History and Philosophy of Mathematical Practice. Springer.
    Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...)
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  12. Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza (eds.), Précis de philosophie de la logique et des mathématiques. Volume 2, philosophie des mathématiques. Paris, France: pp. 387-425.
    Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.
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  13. Naive cubical type theory.Bruno Bentzen - 2022 - Mathematical Structures in Computer Science:1-27.
    This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation (...)
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  14. What is Mathematical Rigor?John Burgess & Silvia De Toffoli - 2022 - Aphex 25:1-17.
    Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.
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  15. Predicativity and constructive mathematics.Laura Crosilla - 2022 - In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo (eds.), Objects, Structures and Logics. Springer Cham.
    In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive (...)
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  16. Unrealistic Models in Mathematics.William D'Alessandro - 2022 - Philosophers’ Imprint.
    Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...)
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  17. What are mathematical diagrams?Silvia De Toffoli - 2022 - Synthese 200 (2):1-29.
    Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a (...)
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  18. Objectivity and Rigor in Classical Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2022 - Noesis 38:195-212.
    The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...)
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  19. Symmetry and Reformulation: On Intellectual Progress in Science and Mathematics.Josh Hunt - 2022 - Dissertation, University of Michigan
    Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...)
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  20. A Hippocratic Oath for mathematicians? Mapping the landscape of ethics in mathematics.Dennis Müller, Maurice Chiodo & James Franklin - 2022 - Science and Engineering Ethics 28 (5):1-30.
    While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is (...)
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  21. Virtue theory of mathematical practices: an introduction.Andrew Aberdein, Colin Jakob Rittberg & Fenner Stanley Tanswell - 2021 - Synthese 199 (3-4):10167-10180.
    Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...)
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  22. Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - 2021 - Review of Symbolic Logic:1-55.
    Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...)
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  23. Groundwork for a Fallibilist Account of Mathematics.Silvia De Toffoli - 2021 - Philosophical Quarterly 7 (4):823-844.
    According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then (...)
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  24. Giving the Value of a Variable.Richard Lawrence - 2021 - Kriterion - Journal of Philosophy 35 (2):135-150.
    What does it mean to ‘give’ the value of a variable in an algebraic context, and how does giving the value of a variable differ from merely describing it? I argue that to answer this question, we need to examine the role that giving the value of a variable plays in problem-solving practice. I argue that four different features are required for a statement to count as giving the value of a variable in the context of solving an elementary algebra (...)
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  25. Review Of Joseph C. Pitt, Heraclitus Redux: Technological Infrastructures and Scientific Change. [REVIEW]Andrew Aberdein - 2020 - Social Epistemology Review and Reply Collective 9 (7):18–22.
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  26. Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.
    Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...)
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  27. Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth.William D’Alessandro - 2020 - Synthese (9):1-44.
    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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  28. 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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  29. Cognitive processing of spatial relations in Euclidean diagrams.Yacin Hamami, Milan N. A. van der Kuil, Ineke J. M. van der Ham & John Mumma - 2020 - Acta Psychologica 205:1--10.
    The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, we (...)
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  30. Mathematizing as a virtuous practice: different narratives and their consequences for mathematics education and society.Deborah Kant & Deniz Sarikaya - 2020 - Synthese 199 (1-2):3405-3429.
    There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is (...)
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  31. Proof, Explanation, and Justification in Mathematical Practice.Moti Mizrahi - 2020 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 51 (4):551-568.
    In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of (...)
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  32. Mathematical cognition and enculturation: introduction to the Synthese special issue.Markus Pantsar - 2020 - Synthese 197 (9):3647-3655.
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  33. Mathematical Monsters.Andrew Aberdein - 2019 - In Diego Compagna & Stefanie Steinhart (eds.), Monsters, Monstrosities, and the Monstrous in Culture and Society. Wilmington, DE, USA: pp. 391-412.
    Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...)
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  34. 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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  35. 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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  36. Teaching and Learning Guide for: Explanation in Mathematics: Proofs and Practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11):e12629.
    This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".
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  37. Explanation in mathematics: Proofs and practice.William D'Alessandro - 2019 - Philosophy Compass 14 (11):e12629.
    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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  38. 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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  39. Redefining revolutions.Andrew Aberdein - 2018 - In Moti Mizrahi (ed.), The Kuhnian image of science: Time for a decisive transformation? London: Rowman & Littlefield. pp. 133–154.
    In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that inglorious revolutions are (...)
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  40. 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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  41. 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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  42. 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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  43. 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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  44. 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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  45. Mohan Ganesalingam. The Language of Mathematics: A Linguistic and Philosophical Investigation. FoLLI Publications on Logic, Language and Information. [REVIEW]Andrew Aberdein - 2017 - Philosophia Mathematica 25 (1):143–147.
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  46. 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. [REVIEW]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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  47. 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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  48. 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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  49. 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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  50. 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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