On October 16, 2019, Colin Rittberg and myself will give a joint talk about our upcoming paper on intended models in mathematical practice at the Work In Progress seminar of the CEFISES at UCLouvain, Louvain-la-Neuve.

The seminar is open and all interested are invited to join.

Venue and time

15:00 - 17:00
Salle Ladrière (a.124 - 1st floor), Collège Mercier
Place du Cardinal Mercier, Louvain-la-Neuve, Belgium

Can intended models of mathematical theories change over time?

The work of philosophers of mathematical practice has challenged our understanding of various concepts in mathematics. For example, observing how mathematical proofs develop over time and how they relate to our understanding of mathematical objects and phenomena, led Hanna (1989, 2002); Mancosu (2008); Macbeth (2012); Van Bendegem (2014) and others, to a conclusion that the adequate description of the notion of mathematical proof relies on processes that are much more complex than syntactic constructs and manipulations. In our paper we challenge another concept which has been highly idealised in classical philosophy of mathematics, namely the concept of an intended model. We will contest the widely accepted understanding of an intended model as simply one guaranteed to be found among all the models of a theory.

We will discuss the notion of the intended model in the context of the distinction between what we call content-driven practices and content-creating practices, borrowing from Lakatos’ notion of the informal (inhaltliche) mathematics (cf. Krause and Arenhart (2016, p. 63)). Content-driven mathematical practices aim to explicate in terms of a mathematical theory some content or structure, or the so-called intended model, which is assumed to be given in advance: content first, theory later. We intentionally use the term mathematical theory instead of axiomatic theory, as we aim highlight the order in which mathematical concepts emerge. For example, number theory had featured an intended model of natural numbers long before the development of PA. Another case is geometry trying to capture the realities of space.

In content-creating mathematical practices no such content is given in advance. Instead, an axiomatic theory defines a class of structures, its own object of study. The order is reversed compared to content-creating practices, that is we obtain theory first and content later. Classic examples of content-creating practices include algebra and graph theory, where the axiomatic theory is not meant to explicate any intended structure, but rather serves as a starting point of the process of discovering a class of structures which are presumed to be of mathematical interest.

Assuming that some mathematical practices are content-driven, where does their content come from? A plausible answer to this question has been proposed by Ferreirós, namely the content of mathematical theories arises from the historical development of “technical practices” (Ferreirós, 2016, p.40). We will supplement Ferreirós’ case with an example from contemporary set theory, where in the current axiom selection debate, the theory has caught up to the content in ways that require (and have produced) new ways of understanding the intended model of set theory. Our principal case study to show this is H. Woodin’s Ultimate L argument (Woodin, 2017). We will also touch upon other proposals by Steel and Magidor.

We will further argue that the understanding of an intended model for a given discipline of mathematics has the potential to change over time and even erode, so that the practice no longer aims at explicating an intended model at all. In this, within the same mathematical theory we can observe a shift from content-driven to content-creating practice. Starting with von Neumann, we will introduce the notion of epistemic pluralism and touch upon Väänänen’s work to formalise the notion. Epistemic pluralism already erodes, but does not discard, the idea of an intended model for a content-driven practice. We then report on how Hamkins shifts pluralism from the epistemic to metaphysical. We argue that with this shift, Hamkins places theory before content in the sense of our paper, and thereby suggests to transform set theory from a content-driven to a content-creating practice.