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1468The History of Categorical Logic: 1963-1977In Dov Gabbay, Akihiro Kanamori & John Woods (eds.), Handbook of the history of logic, Elsevier. 2011.
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1395Categorical foundations of mathematics or how to provide foundations for abstract mathematicsReview of Symbolic Logic 6 (1): 51-75. 2013.Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.
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1170Forms of Structuralism: Bourbaki and the PhilosophersStructures Meres, Semantics, Mathematics, and Cognitive Science. 2020.In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
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979Canonical MapsIn Elaine Landry (ed.), Categories for the Working Philosophers, . pp. 90-112. 2018.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 a…Read more
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968The Structuralist Mathematical Style: Bourbaki as a case studyIn Stefano Boscolo Claudio Ternullo Gianluigi Oliveri (ed.), Boston Studies in the Philosophy and the History of Science. pp. 199-231. 2022.In this paper, we look at Bourbaki’s work as a case study for the notion of mathematical style. We argue that indeed Bourbaki exemplifies a mathematical style, namely the structuralist style.
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902Vérité partielle et réalisme scientifique: une approche bungéenneMεtascience: Discours Général Scientifique 1 293-314. 2020.Le réalisme scientifique occupe une place centrale dans le système philosophique de Mario Bunge. Au cœur de cette thèse, on trouve l’affirmation selon laquelle nous pouvons connaître le monde partiellement. Il s’ensuit que les théories scientifiques ne sont pas totalement vraies ou totalement fausses, mais plutôt partiellement vraies et partiellement fausses. Ces énoncés sur la connaissance scientifique, à première vue plausible pour quiconque est familier avec la pratique scientifique, demanden…Read more
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861A path to the epistemology of mathematics: homotopy theoryIn Jeremy Gray & Jose Ferreiros (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy, Oxford University Press. pp. 239--260. 2006.
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768Category theory and the foundations of mathematics: Philosophical excavationsSynthese 103 (3). 1995.The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 sections. We first show that already in th…Read more
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754Mathematical Abstraction, Conceptual Variation and IdentityIn Peter Schroeder-Heister, Gerhard Heinzmann, Wilfred Hodges & Pierre Edouard Bour (eds.), Logic, Methodology and Philosophy of Science, Proceedings of the 14th International Congress, . pp. 299-322. 2014.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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692Stairway to Heaven: the abstract method and levels of abstraction in mathematicsThe Mathematical Intelligencer 38 (3): 41-51. 2016.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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573Bunge’s Mathematical Structuralism Is Not a FictionIn Michael Robert Matthews (ed.), Mario Bunge: A Centenary Festschrift, Springer Verlag. pp. 587-608. 2019.In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involv…Read more
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462Abstract logical structuralismPhilosophical Problems in Science 69 67-110. 2020.Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latte…Read more
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397Abstract mathematical tools and machines for mathematicsPhilosophia Mathematica 5 (3): 250-272. 1997.In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. W…Read more
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375Mathematical Models of Abstract Systems: Knowing abstract geometric formsAnnales de la Faculté des Sciences de Toulouse 22 (5): 969-1016. 2013.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 model…Read more
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352Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional MathematicsSynthese 190 (12): 2141-2164. 2013.In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathemat…Read more
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240Categories in context: Historical, foundational, and philosophicalPhilosophia Mathematica 13 (1): 1-43. 2005.The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism.…Read more
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231Categories, sets and the nature of mathematical entitiesIn Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 181--192. 2006.
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196Unfolding FOLDS: A Foundational Framework for Abstract Mathematical ConceptsIn Landry Elaine (ed.), Category for the Working Philosophers, Oxford University Press. pp. 136-162. 2018.
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113Mathematical Conceptware: Category Theory: Critical Studies/Book ReviewsPhilosophia Mathematica 18 (2): 235-246. 2010.(No abstract is available for this citation)
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110John L. BELL. The continuous and the infinitesimal in mathematics and philosophy. Monza: Polimetrica, 2005. Pp. 349. ISBN 88-7699-015- (review)Philosophia Mathematica 14 (3): 394-400. 2006.Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no lon…Read more
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94A Study of the History and Philosophy of Category Theory Jean-Pierre Marquis. to say that objects are dispensable in geometry. What is claimed is that the specific nature of the objects used is irrelevant. To use the terminology already ...
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70Mario Bunge: A Centenary Festschrift (edited book)Springer Verlag. 2019.This volume has 41 chapters written to honor the 100th birthday of Mario Bunge. It celebrates the work of this influential Argentine/Canadian physicist and philosopher. Contributions show the value of Bunge’s science-informed philosophy and his systematic approach to philosophical problems. The chapters explore the exceptionally wide spectrum of Bunge’s contributions to: metaphysics, methodology and philosophy of science, philosophy of mathematics, philosophy of physics, philosophy of psychology…Read more
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62Book Review: Colin McLarty. Elementary Categories, Elementary Toposes (review)Notre Dame Journal of Formal Logic 39 (3): 436-445. 1998.
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61Menger and Nöbeling on Pointless TopologyLogic and Logical Philosophy 22 (2): 145-165. 2013.This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was sup…Read more
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55Mathematical engineering and mathematical changeInternational Studies in the Philosophy of Science 13 (3). 1999.In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathe…Read more
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52Approximations and truth spacesJournal of Philosophical Logic 20 (4). 1991.Approximations form an essential part of scientific activity and they come in different forms: conceptual approximations (simplifications in models), mathematical approximations of various types (e.g. linear equations instead of non-linear ones, computational approximations), experimental approximations due to limitations of the instruments and so on and so forth. In this paper, we will consider one type of approximation, namely numerical approximations involved in the comparison of two results,…Read more
Montreal, Quebec, Canada
Areas of Interest
Metaphysics |
Philosophy of Physical Science |
Science, Logic, and Mathematics |