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  1. H. Leitgeb A. Hieke (ed.) (2009). Reduction – Abstraction – Analysis. Ludwig Wittgenstein Society.
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  2. Michael A. Arbib (1990). A Piagetian Perspective on Mathematical Construction. Synthese 84 (1):43 - 58.
    In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The axiom set does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view truth not as something to be defined within (...)
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  3. Conrad Asmus (2009). Jody Azzouni. Tracking Reason: Proof, Consequence and Truth. Philosophia Mathematica 17 (3):369-377.
    (No abstract is available for this citation).
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  4. Steve Awodey & A. W. Carus, The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism From Tractatus on Logical Syllogism.
    Steve Awodey and A. W. Carus. The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus on Logical Syllogism.
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  5. Carla Bagnoli (2004). Introduction. Croatian Journal of Philosophy 4 (3):311-316.
    This volume collects articles in realism, anti-realism, and constructivism.
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  6. Alan Baker (2007). Is There a Problem of Induction for Mathematics? In M. Potter (ed.), Mathematical Knowledge. Oxford University Press 57-71.
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  7. Jeffrey A. Barrett (2013). On the Coevolution of Basic Arithmetic Language and Knowledge. Erkenntnis 78 (5):1025-1036.
    Skyrms-Lewis sender-receiver games with invention allow one to model how a simple mathematical language might be invented and become meaningful as its use coevolves with the basic arithmetic competence of primitive mathematical inquirers. Such models provide sufficient conditions for the invention and evolution of a very basic sort of arithmetic language and practice, and, in doing so, provide insight into the nature of a correspondingly basic sort of mathematical knowledge in an evolutionary context. Given traditional philosophical reflections concerning the nature (...)
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  8. André Bazzoni (2015). Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts. Journal of Philosophical Logic 44 (5):507-516.
    The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact (...)
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  9. Claus Beisbart (2008). Review of M. Wille, Mathematics and the Synthetic A Priori: Epistemological Investigations Into the Status of Mathematical Axioms. [REVIEW] Philosophia Mathematica 16 (1):130-132.
    Kant famously thought that mathematics contains synthetic a priori truths. In his book, Wille defends a version of the Kantian thesis on not-so-Kantian grounds. Wille calls his account neo-Kantian, because it makes sense of Kantian tenets by using a methodology that takes the linguistic and pragmatic turns seriously.Wille's work forms part of a larger project in which the statuses of mathematics and proof theory are investigated. The official purpose of the present book is to answer the question: what is mathematics. (...)
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  10. David Bell & W. D. Hart (1979). The Epistemology of Abstract Objects: Access and Inference. Proceedings of the Aristotelian Society 53:153-165.
  11. Kajsa Bråting & Johanna Pejlare (2008). Visualizations in Mathematics. Erkenntnis 68 (3):345 - 358.
    In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.
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  12. Otávio Bueno (2008). Truth and Proof. Manuscrito 31 (1):419-440.
    Current versions of nominalism in the philosophy of mathematics face a significant problem to understand mathematical knowledge. They are unable to characterize mathematical knowledge as knowledge of the objects mathematical theories are taken to be about. Oswaldo Chateaubriand’s insightful reformulation of Platonism (Chateaubriand 2005) avoids this problem by advancing a broader conception of knowledge as justified truth beyond a reasonable doubt, and by introducing a suitable characterization of logical form in which the relevant mathematical facts play an important role in (...)
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  13. Stefan Buijsman (forthcoming). Accessibility of Reformulated Mathematical Content. Synthese:1-18.
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
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  14. Stefan Buijsman (2016). Philosophy of Mathematics for the Masses : Extending the Scope of the Philosophy of Mathematics. Dissertation, Stockholm University
    One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...)
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  15. Paola Cantù & De Zan Mauro (2009). Life and Works of Giovanni Vailati. In Arrighi Claudia, Cantù Paola, De Zan Mauro & Suppes Patrick (eds.), Life and Works of Giovanni Vailati. CSLI Publications
    The paper introduces Vailati’s life and works, investigating Vailati’s education, the relation to Peano and his school, and the interest for pragmatism and modernism. A detailed analysis of Vailati’s scientific and didactic activities, shows that he held, like Peano, a a strong interest for the history of science and a pluralist, anti-dogmatic and anti-foundationalist conception of definitions in mathematics, logic and philosophy of language. Vailati’s understanding of mathematical logic as a form of pragmatism is not a faithful interpretation of Peano’s (...)
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  16. Carlo Cellucci (2000). The Growth of Mathematical Knowledge: An Open World View. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge, pp. 153-176. Kluwer 153--176.
    In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (Poincaré (...)
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  17. Clarke-Doane, Ethics and Mathematics: The Reliability Challenge.
    It is sometimes alleged that “the reliability challenge” to moral realism is equally compelling against mathematical realism. This allegation is of interest. The reliability challenge to moral realism is increasingly taken to be the most serious challenge to moral realism. However, the specific considerations that are said to motivate it – such as considerations of rational dubitability and evolutionary influence – are widely held not to motivate an analogous challenge to mathematical realism. If it turned out that, in fact, they (...)
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  18. Justin Clarke-Doane (forthcoming). Debunking Arguments: Mathematics, Logic, and Modal Security. 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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  19. Julian C. Cole (2013). Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics. Philosophia Mathematica 21 (1):9-36.
    I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.
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  20. Chris Daly & David Liggins (2014). Nominalism, Trivialist Platonism and Benacerraf's Dilemma. Analysis 74 (2):224-231.
    In his stimulating new book The Construction of Logical Space , Agustín Rayo offers a new account of mathematics, which he calls ‘Trivialist Platonism’. In this article, we take issue with Rayo’s case for Trivialist Platonism and his claim that the view overcomes Benacerraf’s dilemma. Our conclusion is that Rayo has not shown that Trivialist Platonism has any advantage over nominalism.
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  21. Helen De Cruz & Johan De Smedt (2013). Mathematical Symbols as Epistemic Actions. Synthese 190 (1):3-19.
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...)
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  22. Michael Detlefsen (2008). 7.1 Purity as an Ideal of Proof. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. OUP Oxford 179.
    This is a paper on a type of purity of proof I call topical purity. This is purity which, practically speaking, enforces a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content. -/- For some, this has been regarded as an epistemic ideal concerning the type of knowledge that proof (...)
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  23. Michael Detlefsen (2008). Proof: Its Nature and Significance. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America 1.
    I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have a (...)
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  24. Michael Detlefsen (2001). What Does Gödel's Second Theorem Say. Philosophia Mathematica 9 (1):37-71.
    We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We (...)
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  25. Michael Detlefsen (1990). Brouwerian Intuitionism. Mind 99 (396):501-534.
    The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The (...)
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  26. Michael Detlefsen & Andrew Arana (2011). Purity of Methods. Philosophers' Imprint 11 (2).
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  27. Francisco Antonio Doria (2007). Informal Versus Formal Mathematics. Synthese 154 (3):401 - 415.
    We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.
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  28. Jacques Dubucs (1986). Réalisme et antimécanisme chez K. Gödel. Dialectica 40 (4):297-308.
    RésuméSelon K. Gödel, les restrictions méthodologiques des constructivistes sont aberrantes, et le point de vue réaliste est beaucoup plus fécond: il est hi‐même parvenu à ses résultats logiques fondarnentaux en donnant une place à la notion de vérite, qui n'est pas de type combinatoire. Le premier théorème d'incomplétude est hi‐même un argument décisif en faveur du réalisme, si l'on accepte le principe d'«accessibilité» en vertu duquel nous pouvons décider de toutes les propriétés de nos constructions intellectuelles. Par ailleurs, il existe (...)
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  29. Kenny Easwaran (2009). Probabilistic Proofs and Transferability. Philosophia Mathematica 17 (3):341-362.
    In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...)
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  30. Philip A. Ebert & Marcus Rossberg (2009). Ed Zalta's Version of Neo-Logicism: A Friendly Letter of Complaint. In Hannes Leitgeb & Alexander Hieke (eds.), Reduction – Abstraction – Analysis. Ontos 11--305.
    In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
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  31. Editor (1973). Is Mathematics an “Anomaly” in the Theory of “Scientific Revolutions” ? Philosophia Mathematica (1):92-101.
  32. S. Feferman (2006). Are There Absolutely Unsolvable Problems? Godel's Dichotomy. Philosophia Mathematica 14 (2):134-152.
    This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...)
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  33. J. Ferreiros (1999). Matemáticas y Platonismo(S). Gaceta de la Real Sociedad Matemática Española 2 (446):473.
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  34. Jose Ferreiros (2010). Mathematical Knowledge and the Interplay of Practices. In Mauricio Suárez, M. Dorato & M. Rédei (eds.), EPSA Philosophical Issues in the Sciences · Launch of the European Philosophy of Science Association. Springer 55--64.
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  35. Curtis Franks (2009). The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited. Cambridge University Press.
    Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles.
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  36. S. Friederich (2011). Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms. Philosophia Mathematica 19 (1):1-19.
    The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on (...)
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  37. Haim Gaifman, Some Thoughts and a Proposal in the Philosophy of Mathematics.
    The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathematical material. The paper’s (...)
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  38. Mihai Ganea (2008). Epistemic Optimism. Philosophia Mathematica 16 (3):333-353.
    Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism. I would like to thank an anonymous referee for helpful comments. CiteULike Connotea Del.icio.us What's this?
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  39. Ludovico Geymonat (2008). Storia E Filosofia Dell'analisi Infinitesimale (1945-1949). Bollati Boringhieri.
  40. Ludovico Geymonat (1980). Traduzione di Bertrand Russell, I principi della matematica. Longanesi.
  41. Ludovico Geymonat (1942). Traduzione di Friedrich Waismann, Introduzione al pensiero matematico. Einaudi.
  42. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...)
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  43. V. Giardino (2007). Gabriele Lolli. Fenomenologia Della Dimostrazione. Turin: Il Mulino, 2005. ISBN 88-339-1588-3. Pp. 182. Philosophia Mathematica 15 (1):132-134.
    In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of T generally involves (...)
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  44. Valeria Giardino (2010). Intuition and Visualization in Mathematical Problem Solving. Topoi 29 (1):29-39.
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I (...)
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  45. Eduard Glas (1989). Testing the Philosophy of Mathematics in the History of Mathematics. Studies in History and Philosophy of Science Part A 20 (2):157-174.
    Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...)
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  46. Eduard Glas (1989). Testing the Philosophy of Mathematics in the History of Mathematics. Studies in History and Philosophy of Science Part A 20 (1):115-131.
    Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn (...)
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  47. Emily Grosholz & Herbert Breger (eds.) (2000). The Growth of Mathematical Knowledge. Kluwer Academic Publishers.
    This book draws its inspiration from Hilbert, Wittgenstein, Cavaillès and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question (...)
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  48. Jacques Hadamard (2008). An Essay on the Psychology of Invention in the Mathematical Field. Read Books.
    We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
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  49. Jacques Hadamard (1980). In Particular and in Retrospect: “The Psychology of Invention in the Mathematical Field” (a Sum-Up). Philosophia Mathematica (1):29-38.
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  50. Jacques Hadamard (1949). The Psychology of Invention in the Mathematical Field. Philosophy and Phenomenological Research 10 (2):288-289.
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