Metamathematics, often explored in Lex Fridman's podcast, delves into the space of all possible theorems and their proofs.

1. Concept of Metamathematical Space: Stephen Wolfram defines metamathematical space as the infrastructure of all possible theorems that can be proven. He explains that the theorem dependency graph grows and you can map out the relationships between axioms and theorems 1.

2. Mathematics vs Metamathematics: Wolfram likens the axiomatic level of mathematics to the molecular dynamics of a gas. Mathematics is the larger structure humans are familiar with, while metamathematics deals with the detailed proofs and underlying rules 2.

3. Automated Theorem Proving: Automated theorem provers are tools used to navigate through the multiway graphs in metamathematics. These graphs consist of paths that represent proofs, helping in understanding complex mathematical spaces and even aspects of quantum mechanics 3.

   

   1

   Metamathematical Space

   03:34:54

   

   2

   Mathematics and Metamathematics

   02:17:06

   

   3

   Mathematics and Theorem Proving

   03:46:42

   

   4

   Quantum Theory of Mathematics

   02:20:01

   

   5

   The Foundations of Mathematics

   02:14:40

4. Quantum Theory of Mathematics: Wolfram introduces the idea of a "quantum theory of mathematics," where multiple proofs can exist for a single result, potentially leading to phenomena like destructive interference in proof space, analogous to concepts in quantum mechanics 4.

5. Foundation of Mathematics: Wolfram discusses whether mathematics is merely a construct based on arbitrary axioms or if it holds some fundamental truth. He acknowledges that mathematics was historically formalized with the hope of building everything from symbolic axioms, a notion challenged by computational irreducibility 5.

These insights provide a deeper understanding of metamathematics and its role in comprehending and proving mathematical theorems.