Adrien Champougny MA

October 1, 2023 until March 31, 2024

Affiliation:  Université Paris 1 Panthéon-Sorbonne

Research for a study about:

Philosophical issues of reverse mathematics

My main area of research concerns the philosophical aspects of reverse mathematics which is a sub-field of mathematical logic. It is used to identify exactly what is needed to prove a given theorem t. The way this usually goes is that, in addition to t, we start with a hierarchy H of systems S1, …, Sn and with a proof of t in Sk and we then show that S1+t proves all the axioms of Sk, which implies, since H is strict, that no system weaker than Sk proves t.
One could maybe wonder, why one should care about this type of results from a philosophical perspective. The main response offered by the founding fathers of reverse mathematics (that is Steven Simpson and Harvey Friedman) is that reversals are about identifying what are the set existence principles necessary to do contemporary mathematics.
This answer is on the ontological side and seems to rest on the idea that we might be suspicious about the existence of this or that set, or alternatively, that we might think that we don’t have the same kind of access to all kinds of sets. In my PhD thesis, I try to develop another approach to this question that is more centered on the type of knowledge we can gain when something is proven in a weak system. This led me to take an interest in the question of the epistemological gains that one might expect when one goes through a constructive proof rather than a classical one.

Lecture

Reverse Mathematics: Why Should the Philosopher Care About It?

Philosophy of Science Colloquium

Date: cancelled 2024 January 25, postponed to 28/03/2024

Time: 3-4.30 pm CET

Venue: NIG, Universitätsstraße 7, 1010 Wien, SR 2H

Abstract:

Reverse mathematics is a sub-field of mathematical logic. It is used to, a certain theorem t being given, be able to identify exactly what is needed to prove t. The goal of this talk is to provide a brief introduction to reverse mathematics and to give a few insights on why it is an interesting subject from a philosophical point of view.

I will show how the founding fathers of reverse mathematics (that is Harvey Friedeman and Steven Simpson) offered a first philosophical reading of their work that was mainly ontological in character: according to their view, the goal of reverse mathematics is to identify “[…] which set existence axioms are needed to prove the known theorems of mathematics” [Simpson,2009].

 I will then present another way to see the philosophical interest of reverse mathematics that is more focused on the epistemological side. This reading rests on a simple idea: all other things being equal, one has a deeper epistemological control over a constructive proof than over an unconstructive one (I will try to make this concept precise in the course of the presentation). According to this reading, reverse mathematics can be seen as a way to evaluate what kind of knowledge we can hope to acquire concerning a particular mathematical theorem.

Finally, assuming that time permits, I will close my presentation by mentioning a new field of research started by Benedict Eastaugh and Walter Dean: reverse philosophy. The idea in this field is to find an argument in philosophy that somehow rests on a mathematical theorem and to show that this mathematical theorem necessitates some non-trivial mathematical resources to be proven.