**Financial support through the following projects:**

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

Funding body: FCT – Fundação para a Ciência e Tecnologia

Project reference: PTDC/MHC-FIL/2583/2014

Starting date: June 1, 2016

Duration in months: 36

Principal contractor: NOVA.ID.FCT

Principal investigator: Reinhard Kahle (CMA)

Abstract: In 2000, a draft note of Hilbert was found concerning a *24th problem* for his Paris problem list. This problem concerns **simplicity of proofs**. The aim of the project to reassess Hilbert’s 24th problem as a philosophical challenge (rather than a purely formal exercise). We take a closer look at the specifically mathematical examples that concerned Hilbert himself, in particular syzygies (the modern understanding of which is subsumed by the notion of Gröbner bases). Syzygies are, however, just a starting point for looking at Hilbert’s problem in other suitable mathematical contexts. Special emphasis is put on the potential impact of Hilbert’s 24th problem on contemporary philosophy of mathematics.

Keywords: Proof Theory | Purity of Method | Explanation | Combinatorial Proofs

See the website of the project: http://hilbert.dm.fct.unl.pt/

**Members of the project**:

- Reinhard Kahle
- Alan James Cain
- António Malheiro
- Augusto José Franco de Oliveira
- Helena Rocha
- Inês Hipólito
- Isabel Oitavem
- João Araújo
- Josiano Nereu
- Manuel Silva
- Michael Kinyon
- Nuno Jerónimo
- Wilfried Sieg