Foundations of Mathematics: a new proposal based on Homotopy Type Theory (original) (raw)
Related papers
Does Homotopy Type Theory Provide a Foundation for Mathematics?
The British Journal for the Philosophy of Science, 2016
Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the 'HoTT Book'. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation), and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory.
Homotopy Type Theory and the Vertical Unity of Concepts in Mathematics
What is a Mathematical Concept?
The mathematician Alexander Borovik speaks of the importance of the 'vertical unity' of mathematics. By this he means to draw our attention to the fact that many sophisticated mathematical concepts, even those introduced at the cutting-edge of research, have their roots in our most basic conceptualisations of the world. If this is so, we might expect any truly fundamental mathematical language to detect such structural commonalities. It is reasonable to suppose then that the lack of philosophical interest in such vertical unity is related to the prominence given by philosophers to languages which do not express well such relations. In this chapter, I suggest that we look beyond set theory to the newly emerging homotopy type theory, which makes plain what there is in common between very simple aspects of logic, arithmetic and geometry and much more sophisticated concepts.
The vertical unity of concepts in mathematics through the lens of homotopy type theory
2016
The mathematician Alexander Borovik speaks of the importance of the 'vertical unity' of mathematics. By this he means to draw our attention to the fact that many sophisticated mathematical concepts, even those introduced at the cutting-edge of research, have their roots in our most basic conceptualisations of the world. If this is so, we might expect any truly fundamental mathematical language to detect such structural commonalities. It is reasonable to suppose then that the lack of philosophical interest in such vertical unity is related to the prominence given by philosophers to languages which do not express well such relations. In this chapter, I suggest that we look beyond set theory to the newly emerging homotopy type theory, which makes plain what there is in common between very simple aspects of logic, arithmetic and geometry and much more sophisticated concepts.
A Primer on Homotopy Type Theory Part 1: The Formal Type Theory
2014
This Primer is an introduction to Homotopy Type Theory (HoTT). The original source for the ideas presented here is the ``HoTT Book'' -- Homotopy Type Theory: Univalent Foundations of Mathematics published by The Univalent Foundations Program, Institute for Advanced Study, Princeton. In what follows we freely borrow and adapt definitions, arguments and proofs from the HoTT Book throughout without always giving a specific citation. However, whereas that book provides an introduction to the subject that rapidly involves the reader in advanced technical material, the exposition in this Primer is more gently paced for the beginner. We also do more to motivate, justify, and explain some aspects of the theory in greater detail, and we address foundational and philosophical issues that the HoTT Book does not. In the course of studying HoTT we developed our own approach to interpreting it as a foundation for mathematics that is independent of the homotopy interpretation of the HoTT B...
2022
This dissertation is concerned with the foundations of homotopy theory following the ideas of the manuscripts Les Dérivateurs and Pursuing Stacks of Grothendieck. In particular, we discuss how the formalism of derivators allows us to think about homotopy types intrinsically, or, even as a primitive concept for mathematics, for which sets are a particular case. We show how category theory is naturally extended to homotopical algebra, understood here as the formalism of derivators. Then, we proof in details a theorem of Heller and Cisinski, characterizing the category of homotopy types with a suitable universal property in the language of derivators, which extends the Yoneda universal property of the category of sets with respect to the cocomplete categories. From this result, we propose a synthetic re-definition of the category of homotopy types. This establishes a mathematical conceptual explanation for the the links between homotopy type theory, ∞-categories and homotopical algebra, and also for the recent program of re-foundations of mathematics via homotopy type theory envisioned by Voevodsky. In this sense, the research on foundations of homotopy theory reflects in a discussion about the re-foundations of mathematics. We also expose the theory of Grothendieck-Maltsiniotis ∞-groupoids and the famous Homotopy Hypothesis conjectured by Grothendieck, which affirms the (homotopical) equivalence between spaces and ∞-groupoids. This conjectured, if proved, provides a strictly algebraic picture of spaces.
Expressing ‘the structure of’ in homotopy type theory
Synthese, 2017
As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities to express ‘the structure of’ within homotopy type theory are explored, and it is shown there is little or no need for this expression.
Axiomathes, 2014
Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an active role in the development of computational science and core mathematics. It is worth exploring some of them in depth, particularly predicative Martin-Löf's intuitionistic type theory and impredicative Coquand's calculus of constructions. The logical and philosophical differences and similarities between them will be studied, showing the relationship between these type theories and other fields of logic.
Philosophia Mathematica, 2016
One of the main motivations for Homotopy Type Theory (HoTT) is the way it treats of identity and the rich ∞-groupoid structure of types and identifications to which this gives rise. This paper investigates the conceptual and philosophical status of identity in HoTT. We examine the formal and technical features of identity types in the theory, and how these relate to other features of the theory such as its intensionality, constructive logic, and the interpretation of types as propositions and concepts. We explore the possibility that identity types might be better understood as encoding the indiscernibility of two tokens. We argue that identity types are a primitive component of HoTT.