How do we make sense of true, but unprovable statements in math?

We will look at the video https://youtu.be/JJLBZ4C1OGw?list=TLPQMDQxMTIwMjXIcaGWA5yj3A

We will watch clips of it, look at the arguments, discuss it, and repeat with another clip until finished.

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• The No Coincidence Principle The talk is inspired by mathematician Tim Gowers' question about why mathematicians believe unproved statements. Gowers suggested the "no coincidence principle": the more surprising a mathematical truth is, the more likely there is a good reason for it to be true. [03:03]
• Proof as Explanation A mathematical proof should not just certify a truth, but it must explain and provide reasons why theorems are true. [04:35]
• The Computational Limit of the World The limits of our computational resources define the limits of our world, making the understanding of mental effort and resource allocation a key focus for 21st-century science. [08:25]
• NP (Easy to Check) vs. Co-NP (Hard to Check) The computational class NP describes problems where a solution is hard to find but easy to check (e.g., finding the murderer in a detective story). The class Co-NP involves problems where proving a solution is unique may require an exponentially long, hard-to-understand proof. [14:29]
• Unreasonable Truths "Hard proofs" that scale exponentially in length are called unreasonable truths—provable facts for which there is no good reason or humanly comprehensible explanation (e.g., a hypothetical, extremely large counterexample to the Goldbach conjecture found by exhaustive computer search). [17:35]
• AI and the Future of Proofs Artificial intelligence is likely to find mathematically true results in the "light blue zone" (the region of truths with hard/unreasonable proofs) using non-intuitive methods, leading to lemmas we trust but cannot fully comprehend. [26:02]
• Non-Contingency and Coincidence She notes that we cannot apply tools used in empirical science, like probability distributions, to rule out mathematical coincidences because mathematical truths are non-contingent (they cannot be otherwise). [34:03]
• The Tractability Puzzle The question of how mathematicians are so good at selecting problems they can solve is a significant puzzle, as there is no algorithm to know in advance if a provable statement is tractable (has a short proof). [36:54]
• Long Proofs Can Still Reveal Reasons Dr. Grosholz argues against the strict identification of long proofs with "no good reason." She cites the proof of Fermat's Last Theorem, which is very long but impressed people because of what it revealed about the underlying structure of number theory and the connections between different areas of mathematics. [39:13]
• Knowledge Outside of Proof She questions whether the lack of an efficient proof (or any proof within a system, as in Gödel's theorems) means there is no knowledge or reason for a truth. She gives the example of the consistency of arithmetic, which we believe to be true for good reasons even though it cannot be proved within arithmetic. [40:50]