Philosophy of Science Colloquium TALK: Adrien Champougny (IVC Fellow) | Reverse Mathematics: Why Should the Philosopher Care About It?


Reverse Mathematics: Why Should the Philosopher Care About It?

Philosophy of Science Colloquium
The Institute Vienna Circle holds a Philosophy of Science Colloquium with talks by our present fellows.

Date: 04/04/2024

Time: 16h45

Venue: New Institute Building (NIG), Universitätsstraße 7, 1010 Wien, HS 3F


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.

NIG, Universitätsstraße 7, 1010 Wien, HS 3F