[odyssey7]

Yes, it’s difficult to predict where such an understanding might lead. If it reframes and redefines all of number theory, then we might call it one component of the foundational theory of number theory.

Analogously, if someone proves that P = NP, then that will be great, but the significance of lambda calculus and Turing completeness will remain. If the proof is constructive and practical, we’ll just have to reprioritize and update the list of algorithms we teach to undergrads, issue performance-enhancement updates to some software libraries, and patch any security vulnerabilities. Otherwise, we’ll only need to change a chapter or two in the Theory of Computation courses that universities are increasingly deprioritizing.

[guhbkji]

> if someone proves that P = NP

> we’ll just have to reprioritize and update the list of algorithms we teach to undergrads, issue performance-enhancement updates to some software libraries, and patch any security vulnerabilities.

Wow, your optimism sure is something.

What are you patching and with what?

How do you “patch any security vulnerabilities” when said vulnerability is “all of our security research, except one time pad, is now so obsolete we are unsure if security based on computational complexity is at all practical anymore”?

[odyssey7]

Gosh it was early in the morning and somehow I was thinking in terms of factoring prime numbers when I added the security point. But consider cryptography as an application of number theory and compatibility theory.

Interestingly, if there are cryptography alternatives that can still be relied upon if factoring primes is easy, but the same does not hold if P = NP in a practical sense, then that’s further support for the primary point that learning more about prime numbers would not reset the foundation of number theory.