[HN Gopher] Students' insight proves that the local-global conje...
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Students' insight proves that the local-global conjecture doesn't
hold
Author : Tomte
Score : 216 points
Date : 2023-08-10 14:54 UTC (8 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| gumby wrote:
| > "Once you see it, you just say 'Aha! Of course!'"
|
| These are my favorite kinds of things to learn. Once seen, they
| can't be unseen and can upend how I think of the world in some
| domain D.
| bonoetmalo wrote:
| [flagged]
| Ekaros wrote:
| Depends, what they shot down. Certain aeroplanes might be good
| or bad...
| phoe-krk wrote:
| Assume that something is true, write code to generate cases, run
| it, plot the generated data, expect that the plots show that
| everything matches the assertion, notice it doesn't, disprove the
| thing previously considered true.
|
| Nice and elegant.
| lcnPylGDnU4H9OF wrote:
| I'm reminded of a famous experiment in physics history: https:/
| /en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper....
|
| Similarly, the experiment was done with the intention of
| proving something that is widely believed to be true (I mean,
| _obviously_ light travels through _something_ ) only to
| undeniably disprove it.
| zmgsabst wrote:
| We call it LIGO these days.
|
| Which is to say, many of these ideas turn out to work after
| some modifications.
| amelius wrote:
| Now try that on this problem:
|
| https://en.wikipedia.org/wiki/Riemann_hypothesis
| Someone wrote:
| Mathematicians have been working on that for over 160 years.
| See https://en.wikipedia.org/wiki/Riemann_hypothesis#Numerica
| l_c....
|
| Fun quote (at least for me) from that page (https://en.wikipe
| dia.org/wiki/Riemann_hypothesis#Littlewood'...:
|
| _"It has been computed that p(x) < li(x) for all x <= 1025
| (see this table)"_
| ducttapecrown wrote:
| This one is true though.
| klyrs wrote:
| Prove it
| quchen wrote:
| 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".
| Spivak wrote:
| 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.
| [deleted]
| kongolongo wrote:
| Here's a hypothesis: no positive number is evenly
| divisible by 3 trillion one. True up to 3 trillion then
| false at 3 trillion 1.
| Extigy wrote:
| It can happen, the Polya 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.
| dilippkumar wrote:
| 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!!
| NeoTar wrote:
| 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: 1.397162x10^316
|
| 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.
| Analemma_ wrote:
| I also think the RH is true but it's unwise to be glib
| about about it. A whole lot of theorems about prime numbers
| involve growth rates like O(log log log log n / log log n),
| and one of those could derail the RH at some extreme value.
| jvanderbot wrote:
| It's so nice to hear these stories. But it makes the return to
| reality all the more bitter. Back to having zero effect on
| anything at my day job!
| Jeff_Brown wrote:
| Does mathematics contain a lot of theorems that rely on unproven
| assumptions?
| alanbernstein wrote:
| Probably, but we call them conjectures to indicate that we know
| that, and only call them theorems when we have tricked
| ourselves into believing otherwise
| chongli wrote:
| 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.
| LanceH wrote:
| Absolutely.
| roywiggins wrote:
| Technically, all of them do, that's what axioms are.
| agensaequivocum wrote:
| All knowledge ultimately relies on some self-evident first
| principals which are not demonstrable.
| alasdair_ wrote:
| ALL knowledge does? Can you prove that? :)
| kdmccormick wrote:
| 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" :)
| alasdair_ wrote:
| 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.
| prox wrote:
| 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.
| T-A wrote:
| https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness
| _...
| ducttapecrown wrote:
| Well, all knowledge capable of modeling the natural numbers
| is incapable of proving every true statement, so in a sense
| there can be no universal axiom set.
| zmgsabst wrote:
| > Dubito ergo cogito ergo sum.
|
| Technically, we can derive one fact -- there is something
| rather than nothing:
|
| Asking the question is itself proof.
| TheRealPomax wrote:
| Not just unproven, there are a decent number that are known to
| be unprovable.
| anon291 wrote:
| 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.
| denton-scratch wrote:
| > 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
| ducttapecrown wrote:
| 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...
| rthnbgrredf wrote:
| 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.
| Aardwolf wrote:
| 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?
| dboreham wrote:
| Aren't all those things the same ultimately?
| [deleted]
| travisjungroth wrote:
| All of those new discoveries involve running code. More people
| are learning to code, more libraries are being written, compute
| costs are going down. It's hard to break ground on a "sit and
| think" problem, especially one that is approachable enough that
| it's also understandable news when you solve it.
| syndicatedjelly wrote:
| Here's your answer as to why these kinds of discoveries make
| it to the front page of HN.
| vkazanov wrote:
| 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.
| constantcrying wrote:
| Nothing is more important for numerics than a veey good grasp
| of analysis.
| QuesnayJr wrote:
| 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.
| constantcrying wrote:
| It is probably mostly a reporting bias. Almost all new
| mathematical discoveries don't have simple pictures to draw and
| are about highly abstract concepts which are difficult to write
| about for a general audience.
|
| Analysis is an extremely active field, PDEs have almost endless
| amounts of open research questions. Usually they are not very
| flashy and can be very non-geometrical.
|
| Algebra is extremely hard to write about for a general
| audience. Trying to communicate any result which does not have
| a simple visual interpretation seems like a nightmare.
|
| Numerics are often things where advances are hard to relay to
| an audience, as it usually is about incremental improvements
| instead of breakthroughs.
| gowld wrote:
| Langlands Program, one if many examples, illustrates (ha!) that
| nearly all math is geometrical. Math doesn't care about silly
| human distinctions like "Algebra", "Analysis", and "Geometry"
| svat wrote:
| > _other areas like number theory_ ...
|
| In case you didn't notice: the article is about a discovery in
| number theory. The primary mathematician in this story (search
| the article for "Stange is a number theorist..."), the ones
| referenced and quoted ("Elena Fuchs, a number theorist", James
| Rickards, Peter Sarnak, Alex Kontorovich, Jeffrey C. Lagarias,
| ...) are all number theorists, and the paper itself
| (https://arxiv.org/abs/2307.02749) was posted under math.NT.
|
| Apollonian circles are geometry, but the conjecture is about
| the integers that show up as the curvatures of packings, and
| specifically about the "certain numerical buckets" they happen
| to fall into. Of course mathematics is ultimately a connected
| whole; e.g. Jean Bourgain mentioned in the article would not be
| considered primarily a number theorist.
|
| [And of course there's a bias in what gets covered: researchers
| work in all areas and it's far from true that "this type of
| geometrical problems the currently most active field of
| mathematics", but the ones that can be turned into a good story
| (and geometry is easier to explain / show) are more likely to
| get picked up by media like Quanta; and some of them are more
| likely to be posted to HN and to be upvoted. And some of them
| are likely to be interpreted as about geometry anyway!]
| klyrs wrote:
| Algebra and Calculus are more or less "solved." Unless by
| algebra, you mean abstract algebra... but the open questions
| there tend to be quite esoteric. That said, we did recently see
| a novel approach to solving quadratic equations (
| https://www.sciencealert.com/math-genius-finally-discovers-e...
| ).
|
| I'd say that HN posts a lot of quanta articles, and quanta has
| a "bias" towards results that can be explained to a semi-lay
| audience. You really don't want to know enough about modular
| elliptic curves to understand Wiles' proof of Fermat's
| conjecture. But sometimes number theory proofs come up here
| too.
| constantcrying wrote:
| >Algebra and Calculus are more or less "solved." Unless by
| algebra, you mean abstract algebra
|
| Or unless by calculus he means analysis, which is really
| active. Especially things like PDEs.
| sorokod wrote:
| A bit surprising that no one seriously attempted to find a
| counter example before.
| Ekaros wrote:
| 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.
| tgv wrote:
| OTOH, people have been using computer-assisted "configuration
| generation" for more than 50 years now. The 4-color problem
| was done on hardware that was already obsolete 30 years ago.
| I also would have expected mathematicians to take their
| hypotheses a bit more seriously. Especially since this does
| toppled at a rather modest problem size.
| tomaaron wrote:
| [flagged]
| itsmemattchung wrote:
| [flagged]
| arcbyte wrote:
| 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.
| wyre wrote:
| 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."
| popcalc wrote:
| Any relation to the defunct crypto on-ramp?
|
| https://web3isgoinggreat.com/?id=wyre-finally-shuts-down
| mcpackieh wrote:
| Airplanes get "shot down" but people just get shot (not
| down.) When a person not in a plane is said to be "shot
| down", it's always in a figurative sense.
| isuckatcoding wrote:
| Yeah the title of this article is a very poor choice of
| words
| kittensmittens wrote:
| [flagged]
| chrislan815 wrote:
| [flagged]
| taeric wrote:
| 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?
| cdelsolar wrote:
| 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.
| taeric wrote:
| 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!
| gowld wrote:
| 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?
| Sxubas wrote:
| From the article:
|
| > Stange added that none of this would have happened without
| the low-stakes summer project. "Serendipity and an attitude of
| playful exploration both have such a huge role in discovery,"
| she said.
| tedunangst wrote:
| Like what if there was a summer camp for 20 students, and each
| was randomly assigned a conjecture to programmatically test,
| and only one found something surprising?
| constantcrying wrote:
| 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.
| PaulDavisThe1st wrote:
| She was a grad student already.
| Laakeri wrote:
| "Luck favors the prepared mind"
| roywiggins wrote:
| 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.
| jagged-chisel wrote:
| If their approach is reproducible and correct, they have
| successfully disproved the conjecture. If errors are found,
| corrections will be made.
|
| They had started attempting to prove the conjecture. This
| result could be considered a failure of that proof. Showing
| they're capable of performing the research makes them good
| mathematicians.
|
| I believe the answer to your last question is "lots."
| TheRealPomax wrote:
| tl;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.
| svat wrote:
| That the curvatures are integers is proved (noticed by Soddy in
| 1936 as "a fairly straightforward consequence of Descartes'
| equation", as the article mentions), and a lot more is also
| proved about the integers' distribution; the conjecture
| specifically is about whether every sufficiently large
| "admissible (passing local obstructions) integer is the
| curvature of some circle in the gasket" -- see the second page
| of https://arxiv.org/abs/1205.4416, or indeed the second page
| of the paper the article is about
| (https://arxiv.org/abs/2307.02749), which puts it even more
| concretely:
|
| > Conjecture 1.1 ([GLM+03, FS11]). Let A be a primitive
| Apollonian circle packing containing curvatures equivalent to r
| (mod 24). The set of positive integers x [?] r (mod 24) not
| occurring in A is finite.
|
| which they disprove with:
|
| > Theorem 1.3. There exist infinitely many primitive Apollonian
| circle packings for which the number of missing curvatures up
| to N is ([?]N). In particular, the local-global conjecture is
| false for these packings.
| TheRealPomax wrote:
| Cheers. Can't edit my comment anymore, but that's a good
| tl;dr.
| claar wrote:
| 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 an integer curvature, all
| subsequent circles in the packing will also have integer
| curvatures.
|
| - Mathematicians later focused on identifying which integers
| emerge as the circles shrink and the curvatures grow.
|
| Key Developments:
|
| 1. Local-Global Conjecture: Elena Fuchs proved in 2010 that
| curvatures conform to a certain relationship. This led to the
| belief known as the local-global conjecture, which claims that
| all possible numbers within each category must appear in the
| circle packings.
|
| 2. Testing the Conjecture: James Rickards created software to
| examine any desired arrangement of circle packings. When
| researchers Summer Haag and Clyde Kertzer started using the
| software, they anticipated observing the regular patterns of the
| local-global rule.
|
| 3. A Surprise Discovery: After conducting extensive plotting,
| Haag observed patterns that didn't align with the local-global
| conjecture. This suggested that the conjecture may not hold
| universally.
|
| 4. Disproving the Conjecture: Upon further analysis, it was
| determined that the observed patterns indicated that the local-
| global conjecture was false. The team developed a rigorous proof,
| utilizing the principle of quadratic reciprocity, which explained
| why certain curvatures can't be tangent to each other.
|
| Implications:
|
| - The discovery was met with significant interest and surprise in
| the mathematical community.
|
| - The work questions the validity of other conjectures in number
| theory that have been largely assumed to be true.
|
| Conclusion:
|
| The study of Apollonian circle packings led to the challenge and
| ultimate disproval of the previously accepted local-global
| conjecture. This outcome underscores the importance of testing
| long-held beliefs in mathematics and the potential surprises that
| can emerge from seemingly simple problems.
| jiggawatts wrote:
| 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!
| squokko wrote:
| It is so amazing that I can read articles like this for free
| Sxubas wrote:
| I feel the same gratefulness. While I was reading I was
| thinking that whoever wrote this did a great job to make it
| understandable. Without their input I believe I would have
| never been told this story.
|
| I can't get enough of these mathematical stories
| proving/disproving conjectures. I think they show a more human
| part of mathematics, which I rarely got to see in my college
| courses.
| squokko wrote:
| I hope Quanta continues when Simons dies. There's no way the
| mathematicians in this story could have written anything like
| this.
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