Does mathematics contain a lot of theorems that rely on unproven assumptions?
Students’ insight proves that the local-global conjecture doesn’t hold
31–40 of 107 posts
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#32Does mathematics contain a lot of theorems that rely on unproven assumptions?
Absolutely.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#33What would happen if the conjecture happened to be true, and they did their data collection/computation, and got a bunch of data (called "0%" in mathematics) consistent with the conjecture? Would their summer research be an utter failure? Would they be unimpressive mathematicatians? How much of math success is being lucky enough to stumble upon a tractable problem?
"Luck favors the prepared mind"
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#34Does mathematics contain a lot of theorems that rely on unproven assumptions?
Technically, all of them do, that's what axioms are.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#35A lot of recent math discoveries seem to be geometrical: circle packing, infinitely tiling patterns, neighbor coloring theorem, ... I don't seem to see as much discoveries in other areas like number theory, calculus, algebra etc... Is this a bias of what gets covered on HN, or are this type of geometrical problems the currently most active field of mathematics?
I think other fields it's gotten much harder to explain the results. For example, it was a big deal in number theory a few years ago when 10 authors proved a weak form of the Shimura-Taniyama conjecture for number fields with complex multiplication. (The simplest case of Shimura-Taniyama implies Fermat's last theorem.) It would take an entire course to explain just the statement of the theorem.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#36Does mathematics contain a lot of theorems that rely on unproven assumptions?
All knowledge ultimately relies on some self-evident first principals which are not demonstrable.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#37Earlier quoted context omitted.
Counterpoint: am American and I read it exactly as intentioned the first time. It didn't even occur to me to read it in the context you did until you mentioned it. I don't watch network news though.
I don’t watch network news either and had the same reaction as OP reading “two students shoot down” A better headline could be “Widely Believed Math Conjecture shot down by two students.”
Yeah the title of this article is a very poor choice of words
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#38A bit surprising that no one seriously attempted to find a counter example before.
I don't think it is unexpected if the counter examples are really really big. It haven't been so long we have gotten lot of processing power and memory. If the counter examples only appear when you are in territory that is unfeasible by hand it is not so surprising.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#39What would happen if the conjecture happened to be true, and they did their data collection/computation, and got a bunch of data (called "0%" in mathematics) consistent with the conjecture? Would their summer research be an utter failure? Would they be unimpressive mathematicatians? How much of math success is being lucky enough to stumble upon a tractable problem?
Well, same as anything. It sounds like they weren't seriously looking for counterexamples, they were looking for a starter project to "implement" the hypothesis and generate a nice chart to illustrate the property. It wouldn't have been a failure if that's all they'd produced, they would have moved on to the next thing.
Re: Students’ insight proves that the local-global conjecture doesn’t hold
#40tl;dr: the conjecture is that if you start with three touching circles, each of which has an integer curvature (i.e. 1/r is a whole number) and you then draw a circumscribing circle around those, then starting filling in the gaps between the circles with ever smaller circles, every one of those smaller circle will also have an integer curvature.
Turns out, they don't, but no one actually sat down to do the grunt work necessary, because as it turns out you need a lot of circles before you see the pattern breaking down.