Exploring Categorical Structuralism

McLarty, Colin

doi:10.1093/philmat/12.1.37

Cited by 61 publications

(30 citation statements)

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“…I shall argue that the relationships between the category theory and the mathematical structuralism are more involved: although the category theory helps the mathematical structuralism to win over some of its traditional rivals (including all varieties of mathematical substantialism) it also transforms the traditional structuralism into something very different. Although many people in this context prefer to talk about a new form of structuralism [178], [93], [94], [6], [186] 1 I find this terminological choice rather unfortunate because it only points to the continuity of the conceptual change but does not help one to describe this change itself. I wholly recognize the fact that the category theory grew up on the basis of the structuralist mathematics, and that structuralist motivations played a major role in the historical development of this theory.…”

Section: Chapter 8 Categories Versus Structuresmentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

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PrefaceI first learned about category theory about 20 years ago from Yuri I. Manin's course on algebraic geometry [180] when I was preparing my dissertation on Euclid's Elements and was focused on studying Greek mathematics and classical Greek philosophy. Then I convinced myself that the mathematical category theory is philosophically relevant not only because of its name but also because of its content and because of its special role in the contemporary mathematics, which I privately compared to the role of the notion of figure in Euclid's geometry. Today I have more to say about these matters. The broad historical and philosophical context, in which I studied category theory, is made explicit throughout the present book. My interest to the Axiomatic Method stems from my work on Euclid and extends through Hilbert and axiomatic set theories to Lawvere's axiomatic topos theory to the Univalent Foundations of mathematics recently proposed by Vladimir Voevodsky. This explains what the two subjects appearing in the title of this book share in common.The next crucial biographical episode took place in 1999 when I was a young scholar visiting Columbia University on the Fulbright grant working on ontology of events under the supervision of Achille Varzi. As a part of my Fulbright program I had to make a presentation in a different American university, and I decided to use this opportunity for talking about the philosophical significance of category theory (I cannot now remember how exactly I married then this subject with the event ontology). Achille Varzi kindly arranged for me the invitation from Barry Smith to give a talk at his seminar on formal ontology in the SUNY in Buffalo. When I sent to Barry Smith my abstract he replied that nobody except probably Bill Lawvere will be able to understand my paper, and suggested to make the paper more accessible to the general audience.By that time I had already read some of Lawvere's papers but was wholly unaware about the fact that Lawvere worked in the same university and could attend my planned talk. So I took Smith's words for a joke. When I realized that this was not a joke I was very excited and, as Vladimir ArnoldThe main motivation of writing this book is to develop the view on mathematics described in the above epigraphs. Some 200 years ago this view used to be by far more common and easier to justify than today. It is sufficient to say that it made part of Kant's view on mathematics, and that Kant's view on mathematics remained extremely influential until the very end of the 19th century. When Cassirer defended this Kantian view in the beginning of the 20th century it was already polemical. When Arnold defended it in the end of the 20th century and in the beginning of this current century it already sounded as an intellectual provocation, and so his words sound today. Kant, Cassirer and Arnold do not speak about the same mathematics: each speaks about mathematics of his own time. So the growing polemical attitude to their shared view reflects not only a change of...

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Section: Chapter 8 Categories Versus Structuresmentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

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“…On the other hand, we propose category-theoretic properties which correspond to set-theoretic axioms such as separation, replacement, and collection. In particular, we characterize toposes that are the category of sets in a model of ZFC, as did Osius, Lawvere, and also McLarty [McL04]; but our result generalizes to the intuitionistic case.…”

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confidence: 78%

Comparing material and structural set theories

Shulman

2019

Annals of Pure and Applied Logic

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We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and predicative theories of pretoposes, and we also propose category-theoretic versions of stronger axioms such as unbounded separation, replacement, and collection. Finally, we compare all of these theories formally to traditional membership-based or "material" set theories, using a version of the classical construction based on internal well-founded relations.

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“…On the other hand, the categorical perspective on foundations has been criticized as inadequate for recovering all of set theory, as well as lacking autonomy insofar as it relies on hidden set-theoretic presuppositions. Colin McLarty has already answered objections of this sort in McLarty (1990) and McLarty (2003).…”

Section: Category Theory Foundations and Philosophy Of Mathematicsmentioning

confidence: 99%

The Meaning of Category Theory for 21st Century Philosophy

Peruzzi¹

2006

Axiomathes

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Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its ''received view''. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a ''non-standard'' approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which ''ontological'' interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax-semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century's philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational ''crisis'', as if algebraic geometry and synthetic differential geometry -not to mention algebraic topology -were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as 'the' philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties * Text of a talk given at the

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Exploring Categorical Structuralism

Cited by 61 publications

References 22 publications

Axiomatic Method and Category Theory

Axiomatic Method and Category Theory

Comparing material and structural set theories

The Meaning of Category Theory for 21st Century Philosophy

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