Students’ insight proves that the local-global conjecture doesn’t hold
quantamagazine.org
quantamagazine.org
Nice and elegant.
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.
Which is to say, many of these ideas turn out to work after some modifications.
Fun quote (at least for me) from that page (https://en.wikipedia.org/wiki/Riemann_hypothesis#Littlewood'...:
“It has been computed that π(x) < li(x) for all x ≤ 10²⁵ (see this table)”
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".
Generally patterns like this get more regular as the numbers get bigger not less.
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.
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.
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?
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!]
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.
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.
Or unless by calculus he means analysis, which is really active. Especially things like PDEs.
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.
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.
If you want to calculate a numerical solution to a novel analytic problem you absolutely need a good understanding of analysis. Being able to plug things into Matlab is only useful if Matlab implements a good solver for that problem.
>It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths.
Indeed. We need 5+ years of analysis now.
And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
What point are you trying to make?
Yep, what you want is a good and predictable approximation.
> What point are you trying to make?
In practice most Matlab solutions you are describing are awful, in terms of performance and in terms of accuracy due to ignorance about underlying theory.
What if you want to make a numerical method for something you can't look up the recipe for?
Numerical methods are quite well-studied by now. If you need a new method or a variation then you probably specialise in these things, and that's a different question.
> The point is that 3-5 years of analysis might be an overkill
Rarely have I met an engineer and said, wow that guy knows TOO much about math. What do you think they should substitute?
But the amount of time available is fundamentally limited and it has to be balanced.
E.g., in my personal circumstances probability theory, statistics, deeper proof understanding would all be useful. I had all of these at varying depths.
But it's only analysis that we were getting for years and years and years...
And I see why and how this is such an important subject historically. It's just not that big anymore relative to other things.
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.
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.
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 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.”
These seminars are about deep dives into particular mathematical questions. Maybe including some recent "doable" unsolved problems.
> 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.
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?
Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.
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" :)
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.
There are host of ways to debate that philosophically.
Technically, we can derive one fact — there is something rather than nothing:
Asking the question is itself proof.
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.
Assuming you're referring to natural numbers or integers, that's not an assumption:
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.
Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the universe that could be used to write them down on or with.
So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. In what way is such a number distinguishable from any of the other infinite numbers that are supposedly greater than it? Since none can be written and all we can really say is that it's greater than whatever other number we have, one again questions whether or not the statement 'every natural number has a next' is truly well defined.
Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.
Depending on your philosophy of mathematics, there is good reason to believe that some real numbers do not exist. In particular, pi is a computable number, but many reals are not. Thus, we end up with a place where we conjecture that certain things exist yet simultaneously say there's (1) no way to write it down and (2) moreover, there's no systematic way to describe it. Given that with pi, there are many programs that given an N, can compute pi to N many digits, I do think it's reasonable to say that pi can be identified. But, there are infinitely many numbers that cannot. In fact, the vast majority of real numbers that supposedly exist cannot be computed to any arbitrary precision with a turing machine. Thus, they cannot be identified.
The general term for this philosophical approach towards mathematics is mathematical nominalism. What I'm seeing in this thread though is an implicit assumption that nominalism is false, despite being unaware of this assumption. I believe these sorts of hidden biases are dangerous. While I don't necessarily subscribe to nominalism, I think it's worth consideration, and I do think it brings up several interesting questions that cannot simply be ignored because 'well I believe it exists'.
References: https://plato.stanford.edu/entries/nominalism-mathematics/
More interesting reading:
https://philosophy.stackexchange.com/questions/81414/if-most...
https://math.stackexchange.com/questions/4322297/in-what-sen...
Many automated theorem provers can only prove things by construction, thus computability is the requirement for 'existence' in these systems (calculus of constructions via Coq, LEAN, etc). In other words, they follow a constructivist approach to mathematics, which is rather interesting as such approaches require us to elide a lot of 'obvious' axioms we take for granted (such as the law of excluded middle). Several things shake out of this approach such as the conclusion that all functions are continuous. (https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...)
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
imho it's part of the definition of what the set of natural numbers even is. Peano arithmetic just has S(n) always exists and is injective as axioms.
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.
> 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 Ω(√N). In particular, the local-global conjecture is false for these packings.
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.
I’ve ran really esoteric and dense research papers through GPT 4 and the original author confirmed that the summary was spot on!