Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a (...) solution to this difficulty, but his solution, I argue, evens the score between Shapiro’s structuralism and fictionalism. (shrink)

Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...) abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic principle related to (...) the search for new invariants. (shrink)

This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist criterion of identity can (...) be applied successfully in graph-theoretic proofs. Part B will turn to the philosophical interpretation and assessment of the theory. (shrink)

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)

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...) the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. (shrink)

The concept of process is ubiquitous in science, engineering and everyday life. Category theory, and monoidal categories in particular, provide an abstract framework for modelling processes of many kinds. In this paper, we concentrate on sequential and parallel decomposability of processes in the framework of monoidal categories: We will give a precise definition, what it means for processes to be decomposable. Moreover, through examples, we argue that viewing parallel processes as coupled in this framework can be seen as a category (...) mistake or a misinterpretation. We highlight the suitability of category theory for a structuralistic interpretation of mathematical modelling and argue that for appliers of mathematics, such as engineers, there is a pragmatic advantage from this. (shrink)

Einer weit verbreiteten Auffassung zufolge ist es eine zentrale Aufgabe der Philosophie der Mathematik, eine ontologische Grundlegung der Mathematik zu formulieren: eine philosophische Theorie darüber, ob mathematische Sätze wirklich wahr sind und ob mathematischen Gegenstände wirklich existieren. Der vorliegende Text entwickelt eine Sichtweise, der zufolge diese Auffassung auf einem Missverständnis beruht. Hierzu wird zunächst der Grundgedanke der Hilbert'schen axiomatischen Methode orgestellt, die Axiome als implizite Definitionen der in ihnen enthaltenen Begriffe zu behandeln. Anschließend wird in Anlehnung an einen Wittgenstein'schen Gedanken (...) zur normativen Rolle mathematischer Sprache argumentiert, dass im Kontext der Hilbert'schen Axiomatik mathematische Sätze als Standards für die Verwendung der in ihnen enthaltenen Begriffe dienen und dass dies die Idee einer ontologischen Grundlegung für die Mathematik untergräbt. (shrink)

The approach of structuralism came to philosophy from social science. It was also in social science where, in 1950–1970s, in the form of the French structuralism, the approach gained its widest recognition. Since then, however, the approach fell out of favour in social science. Recently, structuralism is gaining currency in the philosophy of mathematics. After ascertaining that the two structuralisms indeed share a common core, the question stands whether general structuralism could not find its way back into social science. The (...) nature of the major objections raised against French structuralism – concerning its alleged ahistoricism, methodological holism and universalism – are reconsidered. While admittedly grounded as far as French structuralism is concerned, these objections do not affect general structuralism as such. The fate of French structuralism thus does not seem to preclude the return of general structuralism into social science, rather, it provides some hints where the difficulties may lie. (shrink)