Type Theory and Homotopy

Awodey, Steve

doi:10.1007/978-94-007-4435-6_9

Cited by 42 publications

(63 citation statements)

References 43 publications

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“…The homotopy type theory has been not yet exposed in textbooks but there are very clear expository papers [9] and [7].…”

Section: Vladimir Arnoldmentioning

confidence: 99%

“…The logical relations which hold with regard to judgments, concepts, 7 Beware that this interpretation of algebra is anachronistic. Arnauld in his New Elements of Geometry [5] suggests that algebraic operations are operations with general mathematical magnitudes; his notion of magnitude generalizes upon geometrical magnitudes and arithmetical magnitudes aka numbers.…”

Section: Foundations Of 1899: Logicality and Logicismmentioning

confidence: 99%

“…Here is the precise definition 7 . In addition to functors f, g adjunction g f comprises natural transformations α : A → f g and β : gf → B such that (gα)(βf ) = g and (αf )(gβ) = f .…”

Section: Conceptual Theories and Their Presentationsmentioning

confidence: 99%

“…While Venn thinks of multiple universes of discourse as certain independent domains of application of his logic, Lawvere and Rosebrugh claim that "thinking actually involves relationships 7 The concept of adjunction has been first introduced in 1958 by Kahn [128] in the context of algebraic geometry.…”

Section: Conceptual Theories and Their Presentationsmentioning

confidence: 99%

“…In a category, two objects can be "the same in a way" while still being different (p. 7) so in category theory the notion of identity is relative in Geach's sense. But from the logical point of view the notion of relative identity remains highly controversial.…”

Section: According To Baez and Dolan [11]mentioning

confidence: 99%

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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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“…The homotopy type theory has been not yet exposed in textbooks but there are very clear expository papers [9] and [7].…”

Section: Vladimir Arnoldmentioning

confidence: 99%

Section: Foundations Of 1899: Logicality and Logicismmentioning

confidence: 99%

Section: Conceptual Theories and Their Presentationsmentioning

confidence: 99%

Section: Conceptual Theories and Their Presentationsmentioning

confidence: 99%

Section: According To Baez and Dolan [11]mentioning

confidence: 99%

See 3 more Smart Citations

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

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Constructive Type Theory

Ranta¹

2015

The Handbook of Contemporary Semantic Theory

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A New Foundational Crisis in Mathematics, Is It Really Happening?

Džamonja

2019

Synthese Library

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The article reconsiders the position of the foundations of mathematics after the discovery of the homotopy type theory HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.

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Type Theory and Homotopy

Cited by 42 publications

References 43 publications

Axiomatic Method and Category Theory

Axiomatic Method and Category Theory

Constructive Type Theory

A New Foundational Crisis in Mathematics, Is It Really Happening?

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