In the Univalent Foundations of mathematics spatial notions like “point” and “path” are primitive, rather than derived, and all of mathematics is encoded in terms of them. A Homotopy Type Theory is any formal system which realizes this idea. In this paper I will focus on the question of whether a Homotopy Type Theory can be justified intuitively as a theory of shapes in the same way that ZFC can be justified intuitively as a theory of collections. I first clarify (...) what such an “intuitive justification” should be by distinguishing between formal and pre-formal “meaning explanations” in the vein of Martin-Löf. I then go on to develop a pre-formal meaning explanation for HoTT in terms of primitive spatial notions like “shape”, “path” etc. (shrink)

The starting point of this paper is a claim defended most famously by Graham Priest: that given certain observed similarities between the set-theoretic and the semantic paradoxes, we should be looking for a ‘uniform solution’ to the members of both families. Despite its indisputable surface attractiveness, I argue that this claim hinges on a problematic reasoning move. This is seen most clearly, I suggest, when the claim and its underlying assumptions are examined by the lights of a novel, quite general (...) and, I contend, promising take on inter-theoretic analogy. The ensuing discussion is intended to serve as both a possible case study and a first step towards the broader aim of the paper: namely, to initiate a wider conversation on the methodology of paradox-solving on the one hand, and the use of inter-theoretic analogies on the other. (shrink)

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.

Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.

Homotopy Type Theory 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, 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. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

This article explores the connection between Boolean-valued class models of set theory and the theory of arbitrary objects in roughly Kit Fine’s sense of the word. In particular, it explores the hypothesis that the set-theoretic universe as a whole can be seen as an arbitrary entity. According to this view, the set-theoretic universe can be in many different states. These states are structurally like Boolean-valued models, and they contain sets conceived of as variable or arbitrary objects.

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)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

LandryElaine, * ed. Categories for the Working Philosopher. Oxford University Press, 2017. ISBN 978-0-19-874899-1 ; 978-0-19-106582-8. Pp. xiv + 471.

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well-known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo-logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that (...) if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research. (shrink)

The claim that there can be relations without relata, submitted by the radical ontic structural realist, mounts a serious challenge to her: on the one hand, the world is constituted, according to this sort of realism, just by structures and relations, and on the other hand, relations depend, mathematics says, on individual objects as relata. To resolve the problem, Steven French has argued that while the dependency of relations on relata is conceivable concerning the structure associated with the source of (...) representation, radical ontic structural realism assumes the existence of relations devoid of relata regarding the target of representation. In this article, I will argue this solution considered together with his semantics for the sentences that contain terms referring apparently to individual objects seems incoherent. (shrink)

This paper proposes an approach to the philosophy of mathematics, deductive pluralism, that is designed to satisfy the criteria of inclusiveness of and consistency with mathematical practice. Deductive pluralism views mathematical statements as assertions that a result follows from logical and mathematical foundations and that there are a variety of incompatible foundations such as standard foundations, constructive foundations, or univalent foundations. The advantages of this philosophy include the elimination of ontological problems, epistemological clarity, and objectivity. Possible objections and relations with (...) some other philosophies of mathematics are also considered. (shrink)