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. (shrink)

Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of category (...) theory' specifically in 'its alleged role as providing a foundation' (p. 130). I will also focus on the foundational question. Page numbers in parentheses are page references to Hellman [2003]. (shrink)

The standard view is that the Lagrangian and Hamiltonian formulations of classical mechanics are theoretically equivalent. Jill North, however, argues that they are not. In particular, she argues that the state-space of Hamiltonian mechanics has less structure than the state-space of Lagrangian mechanics. I will isolate two arguments that North puts forward for this conclusion and argue that neither yet succeeds. 1 Introduction2 Hamiltonian State-space Has less Structure than Lagrangian State-space2.1 Lagrangian state-space is metrical2.2 Hamiltonian state-space is symplectic2.3 Metric > (...) symplectic3 Hamiltonian State-space Does Not Have Less Structure than Lagrangian State-space3.1 Lagrangian state-space has less than metric structure3.1.1 A potential worry3.1.2 General Lagrangians3.2 Hamiltonian state-space has more than symplectic structure3.2.1 A dual structure on the Hamiltonian state-space3.2.2 Simple Hamiltonians3.3 Comparing Lagrangian and Hamiltonian structures3.3.1 Counting mathematical structure3.3.2 Symplectic and metric structure are incomparable4 An Alternative Argument for LS4.1 The argument for P3*4.2 Symplectic manifold vs. tangent bundle structure4.3 Trying to patch up the argument for P3*5 Interpreting Mathematical Structure5.1 State-space realism5.2 Model isomorphism and theoretical equivalence6 Conclusion. (shrink)

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. (shrink)

An affirmative answer is given to the question quoted in the title.

The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...) system must satisfy if it is to be regarded as a “structuralist foundation.” I will then explain why both set-theoretic foundations like ZFC and category-theoretic foundations like ETCS satisfy this criterion only to a very limited extent. Then I will argue that UF is better-able to live up to the proposed criterion for a structuralist foundation than any currently available foundational proposal. First, by showing that most criteria of identity in the practice of mathematics can be formalized in terms of the preferred criterion of identity between the basic objects of UF. Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics. (shrink)

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive `equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories provides a sense in which they are isomorphic, and mutually and canonically definable through an analog of `common definitional extension'.

This book, Foundations of Constructive Analysis, founded the field of constructive analysis because it proved most of the important theorems in real analysis by constructive methods. The author, Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego.

The article looks briefly at Fefermans own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

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 mathematics, at least according (...) to some speculative research programs. (shrink)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

Feferman argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro, we can be structuralists all the way down ; (...) we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up. (shrink)

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", (...) or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had (...) been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. (shrink)

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

A 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 ...

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...) real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. (shrink)

iinstein oered the prin™iple of gener—l ™ov—ri—n™e —s the fund—ment—l physi™—l prin™iple of his gener—l theory of rel—tivityD —nd —s responsi˜le for extending the prin™iple of rel—tivity to —™™eler—ted motionF „his view w—s disputed —lmost immedi—tely with the ™ounterE™l—im th—t the prin™iple w—s no rel—tivity prin™iple —nd w—s physi™—lly v—™uousF „he dis—greeE ment persists tod—yF „his —rti™le reviews the development of iinstein9s thought on gener—l ™ov—ri—n™eD its rel—tion to the found—tions of gener—l rel—tivity —nd the evolution of the ™ontinuing de˜—te (...) over his viewpointF.. (shrink)

In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study.

One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, (...) to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. (shrink)

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired (...) by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature. This paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in [20]. (shrink)

We discuss ways in which category theory might be useful in philosophy of science, in particular for articulating the structure of scientific theories. We argue, moreover, that a categorical approach transcends the syntax-semantics dichotomy in 20th century analytic philosophy of science.

of type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is also employed extensively as a framework for the development of high-level programming languages, in virtue of its combination of expressive strength and desirable proof-theoretic properties [NPS90, Str91]. In addition to simple types A, B, . . . and their terms x : A b(x) : B, the theory also has dependent types x : (...) A B(x), which are regarded as indexed families of types. There are simple type forming operations A × B and A → B, as well as operations on dependent types, including in particular the sum x:A B(x) and product x:A B(x) types (see the appendix for details). The Curry-Howard interpretation of the operations A × B and A → B is as propositional conjunction and.. (shrink)

The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory in the standard sense of the word. However these theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert constructive. I show that the formal axiomatic (...) method always requires a support of some more basic constructive method. (shrink)

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using (...) regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic. (shrink)