Wikipedia:Reference desk/Archives/Mathematics/2022 November 7

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November 7

Is mathematics a mostly finished field?

And do we say mathematics is a mostly finished field? The thrust is mostly finished with? All the new stuff is just little proofs here and there. Or do we say we haven't even scratched the surface of math? 67.165.185.178 (talk) 00:26, 7 November 2022 (UTC).

The surface hasn't even been scratched yet. But it is getting harder to have a wide grasp of it, I doubt anyone has had a really strong grasp of most of it since the end of the ninteenth century. The stuff taught in schools and colleges doesn't go beyond the seventeenth century, at least in physics people have heard of quantum mechanics and relativity. NadVolum (talk) 12:55, 7 November 2022 (UTC)

There have been several relatively recent incredible breakthroughs in mathematics. The most publicized one is probably the proof by Andrew Wiles of Fermat's Last Theorem, in 1993–94. The problem had been open for more than three-and-a-half centuries, and many excellent mathematicians had struggled in vain to find a proof. Wiles's proof of Fermat's Last Theorem built on other recent important advances by several mathematicians, such as Ribet's theorem, proved by Ken Ribet in 1986, basing this in return on the properties of the Frey curve noted in 1982 by Gerhard Frey. And in 2002–03, in an equally stunning breakthrough, Grigori Perelman, building on the work by Richard S. Hamilton, proved the Poincaré conjecture, which had been open for a century. For some of the mathematical breakthroughs just in the last year, 2021, read this article: "The Year in Math and Computer Science". One has to be somewhat of an accomplished mathematician to even understand what these breakthroughs really are about, which is the reason they do not get much attention in the media. Fermat's Last Theorem was the exception, but in this case no more than elementary high-school algebra is needed to understand the problem. Most people who know Fermat's Last Theorem was proved, probably do not know that this was just a corollary of the proof of a much more important conjecture, the Taniyama–Shimura–Weil conjecture. While Wiles only proved it for a special class, his techniques were extended by Brian Conrad, Fred Diamond, Richard Taylor and Christophe Breuil to produce a proof of the full theorem in 2001. At the moment there is no indication that this flow of new results will stop; in fact, there are reasons to think it will only increase as mathematicians harness the potential of AI support.^[1][2][3]  --Lambiam 15:55, 7 November 2022 (UTC)

There is TONS of really important math that has eluded us till now, and that's just the known unknowns. When we get to the stuff that we don't even know that we don't know, it feels boundless. The Millennium Prize Problems are some of the most famous "known unknowns" out there, if I had to hazard a guess the most important single thing we don't know yet is likely the Riemann hypothesis, because the implications of it being true really are so huge, large aspects of modern mathematics basically hinge on it. Modern cryptography is built on the notion that, for very large prime numbers, there is no method better than brute force to find them. If true, the same tools used to verify the Riemann hypothesis would also give us ways to essentially break modern cryptography, rendering essentially all digital security vulnerable, [4]. Similar things could be said about the P versus NP problem, which is closely related to Riemann and to cryptography. There's also fun little bits of recreational mathematics that are still unsolved, which have far less import, but still are unknown problems, such as the Collatz conjecture. --Jayron32 16:37, 7 November 2022 (UTC)

Finding huge prime numbers is easy. Factoring the product of two such numbers absent operational quantum computers is – as far as we know – computationally intractable. But as of today there is no proof that the problem is not in complexity class P.  --Lambiam 18:46, 7 November 2022 (UTC)

Math is like mining gold. Sure, there are some well picked over areas where the nuggets are few and hard to find, but people still find new veins well worth mining. Besides, we know math isn't finished because the Collatz conjecture still hasn't been resolved. --RDBury (talk) 22:54, 7 November 2022 (UTC)

Yes, I agree that Calculus I and II, and possibly III, are all 1600s-level math. But what about someone who finished a bachelor's degree in math? Is that late 1800s? 67.165.185.178 (talk) 23:27, 7 November 2022 (UTC).

It isn't up to any particular time. They'd learn a good introduction to set theory and topology and statistics and some combinatorics which is mainly 20th century, but very often they'd not even be introduced to the calculus of variations which dates back to Newton and the Bernoulli's. And the coverage of number theory and groups and fields is rather basic. And somebody from the 19th century would run rings around young graduates nowadays in various forms of geometry or appplied maths and solving differential equations by hand though of course they might learn about manifolds in a course on relativity. And things change, who knows if they'll be taught div and grad and Green's theorm using differential forms for instance even though that was first introduced in the 19th century. NadVolum (talk) 00:20, 8 November 2022 (UTC)

The standard undergraduate curriculum (through abstract algebra and real analysis) goes broadly as far as state-of-the-art 19th century mathematics. The main exception is that it's common for mathematics students to also take some courses in discrete math or computer science, and some of that content is 20th century. --100.36.106.199 (talk) 15:40, 13 November 2022 (UTC)

• In 1900, someone made a well-informed speech along the lines of "there’s only two things left do solve in physics and after that we’re good". Those two things were only the sticking ends of very long threads, that we are still untangling today. (One could argue a posteriori that hydrodynamics had a long way to go from its 1900 state, too.)

I would bet against any "end of mathematics" prediction too. Tigraan^{Click here for my talk page ("private" contact)} 09:58, 8 November 2022 (UTC)

It's also important to note that the notion that research "for the love of it" basically has not, does not, and probably will not happen, even in "pure mathematics", at any practical scale. Advances in math are largely motivated by the needs of fields that are using that math. In the early 20th century, the hot fields of mathematics were in things like non-Euclidean geometry, topology, set theory, group theory, etc. Much of this was motivated by the needs of the advances in physics being made at the time. Things like quantum theory and special relativity and general relativity needed new mathematics to work out the details in the same way that classical physics ala Newton needed Calculus to do what it needed to do. When we think about the mathematical work of people from 100ish years ago like David Hilbert and Emmy Noether and Hermann Minkowski, they were often motivated by what physicists were needing to do their work. A few decades later, mathematicians like John von Neumann and Alan Turing were doing new and exciting mathematics motivated by the early growth of computers, having to create the mathematical language for the new field. Today, much of the work is indeed motivated by cryptography; everything from quantum computing to information security and the like is tied very closely to number theory, especially that of prime numbers, and many of today's top mathematical minds (Terence Tao, Tamar Ziegler, James Maynard, Yitang Zhang, to name but a few) are working in that and allied fields, largely because that's where the money for research is coming from, so that's where the research is focused. By no means is this 100% of what mathematicians are doing, but it is one example of how they do not work in a vacuum, and the needs of many other fields often are the driver for the direction that mathematical research goes in. Even goofy bits of math like 3n+1 (Collatz) are in number theory. --Jayron32 15:17, 8 November 2022 (UTC)

Most useful thing discovered by Ramanujan

Also, question for anyone that really specializes in Ramanujan's works. What would be the most useful thing he discovered? 67.165.185.178 (talk) 05:14, 7 November 2022 (UTC).

For breakthrough results in which a discovery by Ramanujan was pivotal, read this article: "Mathematicians Chase Moonshine’s Shadow".^[5] Whether this is useful depends on one's definition of usefulness. Ramanujan's protector, the English mathematician G. H. Hardy, famously wrote, Pure mathematics is on the whole distinctly more useful than applied. Seemingly contradicting himself, he also wrote, I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. He may have been mistaken, since the Hardy–Weinberg principle, in large part due to his work, is a widely used tool applied in population dynamics.  --Lambiam 07:48, 7 November 2022 (UTC)

"Give him threepence, since he must make a gain out of what he learns." - Euclid. NadVolum (talk) 12:32, 7 November 2022 (UTC)

It's hard to nail down one specific thing as most important, especially because his work really was so varied. As the lead of the article Srinivasa Ramanujan notes "his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Hardy (who tended to lionize his ward to the point of hagiography) thought him at least as important to mathematics as Carl Friedrich Gauss and Leonhard Euler. --Jayron32 17:00, 7 November 2022 (UTC)