Recent Debate over Categorical Foundations

Structuralism is the view that mathematics is the science of structure. It has been noted that category theory expresses mathematical objects exactly along their structural properties. This has led to a programme of categorical structuralism, integrating structuralist philosophy with insights from category theory for new views on the foundations of mathematics. In this thesis, we begin by by investigating structuralism to note important properties of mathematical structures. An overview of categorical structuralism is given, as well as the associated views on the foundations of mathematics. We analyse the different purposes of mathematical foundations, separating different kinds of foundations, be they ontological, epistemological, or pragmatic in nature. This allows us to respond to both the categorical structuralists and their critics from a neutral perspective. We find that common criticisms with regards to categorical foundations are based on an unnecessary interpretation of mathematical statements. With all this in hand, we can describe “schematic mathematics”, or mathematics from a structuralist perspective informed by the categorical structuralists, employing only certain kinds of foundations.

1990

First we introduce some basic theoretical issues that set the stage for subsequent accounts. Secondly, we touch upon some important issues: Set-theory vs. category theory, various conceptions of sets, the problem of universals, combining set-theory and category theory, structuralism, and ¯nally category theory as an application tool. We argue that in the present time, the categorical holistic way of structuring is much needed. Also we would like to see a kind of uni¯cation, not only to mathematics but to structuralism as well.

The two works made available here for the first time in either English or French – Topos, or Logics of Onto-Logy: An Introduction for Philosophers and Being-There: Mathematics of the Transcendental – represent Badiou’s early forays into category theory (and in particular the sub-branch of topos theory) from two vastly different philosophical angles. While the former comes hot on the heels of the publication of his first ‘great book’, Being and Event, the latter captures in painstaking detail the conceptual development of what would ultimately become the belated ‘sequel’ to this foundational work, Logics of Worlds. As such, they catalogue Badiou’s philosophico-mathematical trajectory over a period of nearly two decades – a period of astonishing philosophical and artistic productivity on Badiou’s part – from his enthusiastic embrace of set theory as the theory of pure multiplicity and consequent assertion that ‘mathematics is ontology’, up to his parallel claim that ‘logic is appearing’ together with his difficult excursions through the intricacies of topos theory.

Maddy, Penelope - Naturalism in Mathematics (PDF), 2002

Rivista Italiana Di Filosofia Analitica Junior, 2011

Philosophia Mathematica, 2005

The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning.

Synthese, 2008

This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe 'unchangeable properties of unchangeable objects'. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary 'structural properties'. Although I grant that mathematical statements could be considered as necessarily true in this sense, I question whether this justifies the claim that mathematics as a whole is necessary.