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.
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".
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!!
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 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.
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.
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.
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.
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.
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.
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.
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.
Solving integrals (beyond simple cases) has never been all that useful, simply because most integrals don't even have a solution. You do it in calculus to get a feeling for the subject.
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.
I don't think the point is that nobody needs to understand the theory. The point is that 3-5 years of analysis might be an overkill in most of practical situations.
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.
I didn't say "too much math" :-) if anything, I'd like MORE math.
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.
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"
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.
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.
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!]
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.
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!
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.
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.
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.
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?
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?
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.
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.”
> 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.
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.
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" :)
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.
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.
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
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.
Axioms can just be thought of as baseline assumptions that are included by the terms in your hypothesis. If your theorem is about some vector space V, then you’re assuming that vector spaces exist, and also assuming all of the building blocks do as well (fields all the way down to natural numbers and set theory).
Existence is a very common term in mathematics. It doesn't mean existence like a table or a piece of paper. It means that a mathematical object could (in principle) be constructed without leading to a contradiction.
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.
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.
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.
> 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.
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.
> 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'.
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_...)
That proof assumes that "s(n)=n+1 is clearly an injection" - ie that every natural number has a unique successor, and uses it to prove something about the size of the set. But it doesn't prove there is "always a number following another number", it assumes it.
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.
Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that object to produce another object that is neither the original object nor any of the objects produced before by repeated applications of 'S' to some root object. Whether you simply define a number as 0 and S(x) where x is a natural number or you define natural numbers as nested sets in ZFC or as church-encoded lambda terms, etc, you assume the existence of such a transformation.
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.
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.
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 Ω(√N). In particular, the local-global conjecture is false for these packings.
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.
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!
106 comments
[ 0.22 ms ] story [ 183 ms ] threadNice and elegant.
https://en.wikipedia.org/wiki/Riemann_hypothesis
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.
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)”
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.
I don't watch network news though.
A better headline could be “Widely Believed Math Conjecture shot down by two students.”
https://web3isgoinggreat.com/?id=wyre-finally-shuts-down
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?
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.
Assuming you're referring to natural numbers or integers, that's not an assumption:
https://proofwiki.org/wiki/Natural_Numbers_are_Infinite
https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
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_...)
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.
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.
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!