Philosophical Trials interview: Joel David Hamkins on Infinity, Gödel’s Theorems and Set Theory

I was interviewed by Theodor Nenu as the first installment of his Philosophical Trials interview series with philosophers, mathematicians and physicists.

 
Theodor provided the following outline of the conversation:
 
  • 00:00 Podcast Introduction
  • 00:50 MathOverflow and books in progress
  • 04:08 Mathphobia
  • 05:58 What is mathematics and what sets it apart?
  • 08:06 Is mathematics invented or discovered (more at 54:28)
  • 09:24 How is it the case that Mathematics can be applied so successfully to the physical world?
  • 12:37 Infinity in Mathematics
  • 16:58 Cantor’s Theorem: the real numbers cannot be enumerated
  • 24:22 Russell’s Paradox and the Cumulative Hierarchy of Sets
  • 29:20 Hilbert’s Program and Godel’s Results
  • 35:05 The First Incompleteness Theorem, formal and informal proofs and the connection between mathematical truths and mathematical proofs
  • 40:50 Computer Assisted Proofs and mathematical insight
  • 44:11 Do automated proofs kill the artistic side of Mathematics?
  • 48:50 Infinite Time Turing Machines can settle Goldbach’s Conjecture or the Riemann Hypothesis
  • 54:28 Nonstandard models of arithmetic: different conceptions of the natural numbers
  • 1:00:02 The Continuum Hypothesis and related undecidable questions, the Set-Theoretic Multiverse and the quest for new axioms
  • 1:10:31 Minds and computers: Sir Roger Penrose’s argument concerning consciousness