Towards a Categorical Foundation of Mathematics

Makkai, Michael

doi:10.1017/9781316716830.014

Cited by 13 publications

(12 citation statements)

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“…The study of such a theory, only hinted at in [4], was originally explored by the second author in an unpublished note, and developed further by his PhD student Joao Belo. This dependently-sorted logic can be compared with the the formulation of First-Order Logic with Dependent Sorts (FOLDS) [25,26,27]. The motivation for FOLDS is quite different to ours, and leads to a different technical focus.…”

Section: Discussionmentioning

confidence: 99%

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The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions

Aczel¹

1986

Studies in Logic and the Foundations of Mathematics

125

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Abstract. We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

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Section: Discussionmentioning

confidence: 99%

“…Also, by (26), p ≤ (∃y ∈ β)τ (jy). As (∀y ∈ β)(y ≤ q) and j is monotone we get that (∀y ∈ β)(jy ≤ jq) and hence p ≤ jq.…”

Section: Reinterpreting Logicmentioning

confidence: 99%

The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions

Aczel¹

1986

Studies in Logic and the Foundations of Mathematics

125

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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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“…A = X B := ∀P : P(X). P (A) ⇒ P (B), 3 The importance of this principle has also been emphasized by Makkai [8].…”

Section: Identitymentioning

confidence: 95%

Structuralism, Invariance, and Univalence

Awodey¹

2013

Philosophia Mathematica

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The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy type theory. It also gives the new system of foundations a distinctly structural character.Recent advances in foundations of mathematics have led to some developments that are significant for the philosophy of mathematics, particularly structuralism. Specifically, the discovery of an interpretation of Martin-Löf's constructive type theory into abstract homotopy theory [3] suggests a new approach to the foundations of mathematics, one with both intrinsic geometric content and a computational implementation [5]. Leading homotopy theorist Vladimir Voevodsky has proposed an ambitious new program of foundations on this basis, including a new axiom with both geometric and logical significance: the Univalence Axiom [4]. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to the framework of homotopical type theory.

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Towards a Categorical Foundation of Mathematics

Cited by 13 publications

References 29 publications

The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions

The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions

Axiomatic Method and Category Theory

Structuralism, Invariance, and Univalence

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