Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Students’ insight proves that the local-global conjecture doesn’t hold

quantamagazine.org

Post reply on HN
Print view View original

81–90 of 107 posts

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#81

Post by jiggawatts »

claar wrote: post #69

ChatGPT v4 Technical Article Version of this long-form post: Apollonian Circle Packings and the Local-Global Conjecture Background: - Apollonian circle packings is the study of how circles can fit into a larger circle. - Rather than using diameter to measure these circles, mathematicians employ curvature — the inverse of the radius. The smaller the circle, the larger its curvature. - When the first four circles have…

This type of task is what LLMs are ideally suited for. It makes me laugh when people try to use them for something we already have optimal tools and then walk away disappointed.

I’ve ran really esoteric and dense research papers through GPT 4 and the original author confirmed that the summary was spot on!

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#82

Post by T-A »

alasdair_ wrote: post #52
agensaequivocum wrote: post #36

Earlier quoted context omitted.

All knowledge ultimately relies on some self-evident first principals which are not demonstrable.

ALL knowledge does? Can you prove that? :)

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#83

Post by wyre »

popcalc wrote: post #23
wyre wrote: post #9

Earlier quoted context omitted.

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.”

Any relation to the defunct crypto on-ramp? https://web3isgoinggreat.com/?id=wyre-finally-shuts-down

Nope

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#84

Post by amenhotep »

NeoTar wrote: post #78
Spivak wrote: post #50

Earlier quoted context omitted.

I think anyone that's betting on it being false is nuts at this point. It would be astounding if it held for the first three trillion then broke down. Generally patterns like this get more regular as the numbers get bigger not less.

https://en.m.wikipedia.org/wiki/Skewes%27s_number In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which the prime-counting function is greater than the logarithmic integral function. The current best estimate we have for when this happens is: 1.397162×10^316 To put that in context... it's suc…

Way bigger. The universe is a mere 10^185 Planck volumes.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#85

Post by anon291 »

denton-scratch wrote: post #64
anon291 wrote: post #47

Earlier quoted context omitted.

Godel's incompleteness theorems say that all mathematics that is complicated enough to encode basic arithmetic must rely on unproven assumptions. Unproven assumptions are the basis of all mathematics. For example, even numbers rely on very unproven assumptions. We assume, for example, that there is always a number following another number, but it is not at all obvious what that means.

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite

The statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real.

Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that eventually you arrive at a set that, due to known physical constraints of the universe, is impossible to identify and you must reasonably ask yourself if such a number exists.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#86

Post by rcme »

chongli wrote: post #45
Jeff_Brown wrote: post #31

Does mathematics contain a lot of theorems that rely on unproven assumptions?

A theorem is just a statement which has been proved. Proofs are logical deductions that begin with a set of assumptions (called the hypothesis of the theorem) and follow a sequence of valid steps to reach a result (called the conclusion of the theorem). Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.

However, in order to be a theorem, it's generally required that the hypothesis be consistent with the axiomatic system, which is not possible if the hypothesis can be proven false.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#87

Post by agensaequivocum »

alasdair_ wrote: post #52
agensaequivocum wrote: post #36

Earlier quoted context omitted.

All knowledge ultimately relies on some self-evident first principals which are not demonstrable.

ALL knowledge does? Can you prove that? :)

If first principals were demonstrated by definition they rely on some other first principles (see syllogisms). Hence you have an infinite regress which is unknowable. Thus no knowledge could be acquired. Therefore the first principals in syllogisms must be self-evident.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#88

Post by vkazanov »

constantcrying wrote: post #55
vkazanov wrote: post #24

Earlier quoted context omitted.

Well, algebra and calculus have been around for a while. And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.

Nothing is more important for numerics than a veey good grasp of analysis.

Yup.

But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers.

Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations.

It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths... I mean, this used to be THE subject in most engineering programs round the world. These days it's more like bootstrapping the intuition.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#89

Post by denton-scratch »

anon291 wrote: post #85
denton-scratch wrote: post #64

Earlier quoted context omitted.

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite

The statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real. Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that e…

Real numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.

Re: Students’ insight proves that the local-global conjecture doesn’t hold

#90

Post by commonlisp94 »

vkazanov wrote: post #24
Aardwolf wrote: post #6

A 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?

Well, algebra and calculus have been around for a while. And then, Calculus took about 3 years for me to begin to scratch the surface... and realise I will never need it as numeric methods took over completely with the advent of cheap compute.

> I will never need it as numeric methods took over completely

What if you want to make a numerical method for something you can't look up the recipe for?

Post reply on HN