Wikipedia:Reference desk/Mathematics#A Second Opinon

[https://www.youtube.com/watch?v=sGCmu7YKgPA This video] talks about some of the new mathematical results from Google Deepmind. Before anyone changes their major, it seems worth pointing out that these are all examples of a certain type of optimization problem and that Deepmind only improved on the best known result, it didn't prove or even claim they were the best possible, only that they were an improvement on what human intelligence was able to find in those particular cases. So I wouldn't count on the Collatz conjecture being solved tomorrow. But it does seem to indicate that the way math is done in the future will be significantly different than the way it was done in the 20th century. Thoughts? RDBury (talk) 02:07, 18 May 2025 (UTC)

:The main takeaway of the video is that Matt Parker should take a vacation more often.

:Constructing formal proofs is not an optimization problem, but in a sense it is a combinatorial puzzle, and I expect more artificial help there in the future. Formalizing published proofs (possibly uncovering seemingly obvious but unjustifiable skips) could be a nice start. Another area where an indefatigable agent can prove its worth may be finding interesting conjectures connecting different fields, similar to the discovery of the monstrous moonshine, which started as an observation of a curious coincidence.

:I do not expect the Collatz or Goldbach conjectures to be solved ever, also not with the help of a superintelligent agent. We cannot, with the knowledge we have, exclude the possibility that although true they are not members of the theory of ZFC. (Consider that a simple generalization of the Collatz problem is undecidable, so it is (non-constructively) certain that some Collatz variants are true but don't hold a ZFC club membership card.)  ​‑‑Lambiam 09:32, 18 May 2025 (UTC)

::Formalizing proofs and developing proof verifiers is a program which has been underway since well before the recent AI boom.--Antendren (talk) 03:45, 20 May 2025 (UTC)

:::The use of generative AI in constructing formal proofs is still largely uncharted territory. The Flyspeck project to produce a formal proof of the Kepler conjecture took seven calendar years and an order of magnitude more person years. Where are the formal proofs of Fermat's Last Theorem and the Poincaré conjecture, to mention just two? I know a project for FLT has been under way for about a year now,^{[https://github.com/ImperialCollegeLondon/FLT/blob/main/GENERAL.md]} but (to the best of my knowledge) current AI systems are not yet helpful for this task.  ​‑‑Lambiam 06:22, 20 May 2025 (UTC)

= May 25 =

If a point is x decibels cause 1,000 randomly distributed mouths in radius r then how loud is 10r (10⁵ mouths)? What if it's 3D (10⁶ mouths)? What if the inner ⅓r & 3⅓r respectively have no mouths? Sagittarian Milky Way (talk) 00:21, 25 May 2025 (UTC)

:First, note that a reading of $x$ decibels is equal to a sound pressure of $p\_0\; 10^\{x/20\}$ pascals or a sound intensity of $I\_0\; 10^\{x/10\}$ watts per square meter. I'll call these $p\_x$ and $I\_x$ for short.

:Second, note that standard wave laws mean that the sound intensity given a spherical source will be

:$I(r)=\backslash frac\{P\}\{4\backslash pi\; r^2\}$

:so multiplying the distance by $10$ gives an intensity $1/100$ the original intensity for a single mouth. Sound pressure is proportional to the square root of sound intensity and thus to $1/r$.

:However, we're considering a sphere with the same density, so $100$ times as many mouths. To combine different sound sources, we assume the different mouths are not coordinated enough in their frequency and phase to give strongly constructive or destructive interference, in which case the intensity will be additive and the pressure will be square-root-of-summed-square. (This relates to the additivity of variance in the central limit theorem.) This means that with a constant density of mouths on the surface of a sphere, we multiply the intensity at the reader by $r^2$, and get a constant that does not depend on the radius; this will carry back to the decibel reading, which will continue to be $x$.

:For the case where the mouths are evenly distributed in space, one can consider it as an integral/sum of spherical shells, and will get a result proportional to $r$ for intensity, $\backslash sqrt\{r\}$ for pressure, and a reading of $x\; +\; 10\backslash ,\; \backslash log\_\{10\}(r\_2/r\_1)$. Sesquilinear (talk) 18:45, 25 May 2025 (UTC)

= May 26 =