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

quantamagazine.org

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

#71

Post by ducttapecrown »

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 existence of a set with infinite cardinality is one of the axioms of ZFC.

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

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

#72

Post by cdelsolar »

taeric wrote: post #11

Fun story. And it makes me want to play with plots showing these circles, now. :D Makes me think it would be neat to have a list of old conjectures that have been proven/disproven in the past year or so. Surely such a thing already exists?

there's a super fun Soddy circles Project Euler problem. When I had an appendectomy like 15 years ago I had nothing to do but lie in bed and eventually solved that problem in a Vicodin-induced haze.

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

#73

Post by PaulDavisThe1st »

constantcrying wrote: post #57
gowld wrote: post #25

What 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?

They were undergrads. Nobody expects undergrads to solve decade old research problems in a few months. These seminars are about deep dives into particular mathematical questions. Maybe including some recent "doable" unsolved problems.

She was a grad student already.

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

#74

Post by rthnbgrredf »

anon291 wrote: post #47
Jeff_Brown wrote: post #31

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

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.

Any statement which is not logically valid (read: always true) is unprovable. The statement ∃x∃y(x>y) is not provable from the theory of linear orders, since it is false in the singleton order. On the other hand, it is not disprovable since any other order type would satisfy it.

The statement ∃x(x2−2=0) is not provable from the axioms of the field, since Q thinks this is false, and C thinks it is true.

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

#75

Post by taeric »

cdelsolar wrote: post #72
taeric wrote: post #11

Fun story. And it makes me want to play with plots showing these circles, now. :D Makes me think it would be neat to have a list of old conjectures that have been proven/disproven in the past year or so. Surely such a thing already exists?

there's a super fun Soddy circles Project Euler problem. When I had an appendectomy like 15 years ago I had nothing to do but lie in bed and eventually solved that problem in a Vicodin-induced haze.

I should bring myself to do more of those. Did the first hundred ish, and I think I'm a better programmer now. So, I should be able to do more. Sounds fun!

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

#76

Post by kdmccormick »

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? :)

Sure, I'll take a swing!

If we define "knowing" some fact F as "we have proven that F follows from prerequisite facts A1..AN", then we can imagine all knowledge forms a graph where prerequsities point to the facts that they prove. Either this graph is cyclic, or there are nodes with no inbound edges. Therefore, there are facts which are either non-demonstrable, or only demonstrable via cyclic logic.

I think this works, unless of course you have another definiton of "knowing" :)

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

#77

Post by alasdair_ »

kdmccormick wrote: post #76
alasdair_ wrote: post #52

Earlier quoted context omitted.

ALL knowledge does? Can you prove that? :)

Sure, I'll take a swing! If we define "knowing" some fact F as "we have proven that F follows from prerequisite facts A1..AN", then we can imagine all knowledge forms a graph where prerequsities point to the facts that they prove. Either this graph is cyclic, or there are nodes with no inbound edges. Therefore, there are facts which are either non-demonstrable, or only demonstrable via cyclic logic. I think this work…

I see no reason why knowledge needs to come from prerequisite facts.

Showing my ignorance here, but isn't this the whole point of "cogito ergo sum"? The "fact" that we are thinking is a first principle that comes from nothing else without a prerequisite axiom.

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

#78

Post by NeoTar »

Spivak wrote: post #50
quchen wrote: post #19

Earlier quoted context omitted.

Unless it's not. I'm sure lots of mathematicians have open bets about that. I still remember my supervisor hoping for no Higgs "because then physics will be boring for the foreseeable future".

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 such a big number it's hard to put in context - I've been trying to make a physical analogy, but I think its bigger than the number of Planck-length cubes that could fit in the visible universe.

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

#79

Post by dilippkumar »

Extigy wrote: post #58
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.

It can happen, the Pólya conjecture is the usual example which holds until n = 906150257. Another fun one I just found is the statement “n^17 + 9 and (n + 1)^17 + 9 are relatively prime”. The first counterexample is at n=8424432925592889329288197322308900672459420460792433.

How does one even find something like this? Let alone prove that this is the first counterexample. That number looks to be in the order of the age of the universe in millionths of a quectosecond!!

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

#80

Post by prox »

alasdair_ wrote: post #77
kdmccormick wrote: post #76

Earlier quoted context omitted.

Sure, I'll take a swing! If we define "knowing" some fact F as "we have proven that F follows from prerequisite facts A1..AN", then we can imagine all knowledge forms a graph where prerequsities point to the facts that they prove. Either this graph is cyclic, or there are nodes with no inbound edges. Therefore, there are facts which are either non-demonstrable, or only demonstrable via cyclic logic. I think this work…

I see no reason why knowledge needs to come from prerequisite facts. Showing my ignorance here, but isn't this the whole point of "cogito ergo sum"? The "fact" that we are thinking is a first principle that comes from nothing else without a prerequisite axiom.

That is still a presupposition you are taking, you are making a statement “that thinking is the first principle that comes from nothing.”

There are host of ways to debate that philosophically.

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