Hot Identities and Vicious Abstractions in the (Homotopic) Foundations of Mathematics 21-29 October 2023

Workshop piloted by Gabriel Catren in the framework of the L.I.F.E. Project

“[…] as one gathers one of those magical, closed water lilies, which spring up all of a sudden, enveloping nothingness with their hollow whiteness […]”

S. Mallarmé

Something is shaking in the so-called “foundations” of mathematics.

One of the most basic mathematical notions, the notion of equality – symbolized by means of the equal sign ‘=’ – is currently undergoing a radical reconceptualization in the framework of a mathematical theory known as homotopy type theory. The mathematical (and political) notions of equality and identity have always taken on a mysterious allure: whereas a proposition a = b stating that two different entities are equal seems to be a contradiction, a proposition a = a stating that an entity is equal to itself – that is, the so-called principle of identity – seems to be nothing but a mere triviality. As we shall see, homotopy type theory allows us to get between contradictory equality and trivial self-identity by revisiting the sharp opposition between the notions of identity and difference from a fresh (topological) perspective. In particular, homotopy type theory shows that self-identity becomes non-trivial only when it envelops a hole, an “extimate” absence, a bounded chunk of nothingness.

A privileged operation by means of which we can introduce equalities is abstraction, that is, the capacity to abstract from certain differences that can be methodically considered irrelevant in a context-dependent manner. By “forgetting” such differences we “equalize” entities that are strictly speaking different. Whereas bracketing differences is a necessary operation in order to afford the otherwise overwhelming complexity of experience, abstraction might also entail forms of violence exerted against all the differences that are abstracted away. If abstract concepts result from an act by means of which we methodically “forget” certain differences, the risk concomitant with abstraction is that of forgetting that we have forgotten these differences.

In this introductory workshop addressed to non-scientists, we shall explore the possibilities of higher-dimensional thinking, we shall carnivalistically train our capacities to stretch strict identities, we shall gather some translucent holes in a bamboo forest, and we shall learn how to cope with the pitfalls of “vicious abstractionism” (W. James). By capitalizing on the topic of this year, we shall continue to experiment around “politics of equality” associated with linguistic differences, gender differences, and cognitive diversity. As in the previous editions, we shall explore strategies to deepen the rootedness of science in the material, political, aesthetical, emotional, and spiritual dimensions of the concrete life that both makes science possible and provides the ultimate horizon for its deployment. All in all, we shall not give up on the commitment that orients this overall project: “to serve science only to the extent that science serves life” (F. Nietzsche).

Practical Information

The lectures will be given by Gabriel Catren with the assistance of Manuel Eguía and Jonathan Lorand.

The workshop will take place at the Performing Arts Forum (Saint-Erme, France) and will start on October 21st (arrival day) and end on October 29th (departure day). The workshop consists of four working days (October 22nd, 24th, 26th, and 28th) interspersed by three “free” days (October 23rd, 25th, and 27th) during which the participants will have the time to rest, to work through the new material, and to participate in the related activities that we shall propose.

The costs including meal expenses, accommodation for 8 nights, PAF membership and the tuition fee add up to 316€ for the whole workshop per person (see below for details). In order to allow people with scarce financial resources to attend the meeting, we shall continue to use a differential pricing or sliding scale. So, we shall (un)fix the cost of the whole event to an amount between a baseline of 316€ and 400€ for eight nights per person. Even if you can pay 20€ more than the baseline, it will help someone else. If you would like to participate and you do not have the financial conditions to do so, please let us know and we shall do our best to make it possible for you.

Dates: Arrival on October 21st; Workshop from October 22nd to 28th; Departure on October 29th, 2023.

Location: Performing Arts Forum (PAF), Saint-Erme, France.

Cost (sliding scale): 316€ - 400€ per person. The baseline of 316€ corresponds to meal expenses (15€ per day including breakfast, lunch, dinner, and coffee), accommodation for 8 nights (18€ per night), PAF membership (12€ per year), and the tuition fee (40€).

Accessibility: We are in the process of improving accessibility at PAF. In terms of mobility, kitchens, workspaces, and a couple of bedrooms with bathrooms are available on the ground floor (we can provide pictures and measurements if needed). If you have other accessibility requirements, please let us know and we shall do our best to make it possible for you to participate.

Linguistic accessibility: the lectures will mainly take place in English. Nonetheless, we are strongly committed to hold space for linguistic pluralism and to explore strategies to overcome prevailing forms of Anglocentrism. So, languages other than English will be very welcome as alternative channels of interaction during the whole event.

Academic requirements: Kindergarten degree.

Reservations are mandatory by writing to The maximum number of participants is 40 persons and we ask participants to commit for the full length of the workshop.

Important note: Since reservations followed by cancellations entail a significant amount of additional work for the different persons involved in the organization (e.g. answering emails, waiting list, etc.), we kindly ask you to be as sure as possible that you will participate before booking.

For any questions that you might have, you can write to Gabriel Catren (