UnAxiMa is an interdisciplinary project integrating mathematics, computer science, and philosophy to address one of the most fundamental questions in science:

What should the rules of mathematics be?

This question was investigated already 100 years ago by the Vienna Circle, a group of leading intellectuals from mathematics, philosophy, and the natural sciences. Their work culminated in Gödel’s incompleteness theorems, one of the most significant achievements in the foundations of science. According to Gödel’s theorems, there will always be mathematical questions which cannot be answered using the rules of mathematics. Such questions have appeared in algebra, analysis, combinatorics, topology, information theory, game theory, machine learning, and other fields. This ever-growing list compels us to seek more powerful rules which extend mathematics beyond its current reach. In UnAxiMa, we aim to identify new rules which could change forever the way that mathematics is done, taught, and applied, as well as our very conception of what mathematics is.

Adequate rules for mathematics must reflect current scientific understanding, which has changed dramatically since Gödel’s time: mathematical disciplines have become increasingly specialized and the world has seen radical technological advances. For the first time, UnAxiMa addresses both phenomena by uniting six disciplines towards distilling new rules which are theoretically sound and practically viable. The fields of Reverse Mathematics, Complexity Theory, Forcing, and Inner Model Theory evolved in the second half of the 20th century, and nourish our insight into the axioms of mathematics and their limits. Automated Theorem Proving, as part of artificial intelligence, is now seeing accelerated progress, thereby challenging our rules by the practical need of efficiency. Philosophy of Mathematics serves as the guiding light in our quest for rules that reflect our understanding of truth.

UnAxiMa isolates core questions on the relationship between axioms and computation, incompleteness, and necessity. The answers to these questions require interdisciplinary collaboration and have the potential to lead to a paradigm shift in the foundations of mathematics.

UnAxiMa gathers, in a single geographic location, world-leading experts from each of the six fields necessary to realize its collaborative vision. The project emerges as the culmination of 100 years of research, which have created an unprecedented scientific landscape where the momentum to rethink our scientific foundations is centered in Vienna once again. In a historic effort, UnAxiMa shall uncover the axioms of mathematics.