I'm a practical person. What's the point of spending "time" (brain cycles) on such problems? Is it just a mathematician's version of brain teaser?
What useful development comes out of such?
What useful development comes out of such?
Aleksandr Lyapunov's theory of stability for dynamical systems, which forms the bedrock for much of control theory, was first proposed in 1892 and went unnoticed until the 1930's [2]. How practical was it for Lyapunov to study that topic?
It's virtually impossible to predict which mathematical tools will be "practical" or "useful" in the future. The point is that if someone needs the tool down the road, they'll have access to the solution due to the efforts of these mathematicians.
[1]: https://en.wikipedia.org/wiki/Neural_network#History
[2]: https://en.wikipedia.org/wiki/Lyapunov_stability#History
Mendel's discoveries weren't appreciated until after he died. When Narinder Singh Kapany invented fiber optic cable in the 50s he had no idea it would lead to revolutions in computer networking, gastroenterology, and other fields.
On a long enough timeline, you're right! But if we limit ourselves to, say, the next 5,000 years, it gets a lot easier to accomplish. ;)
Fast Fourier transforms, complex numbers, complexity classes, etc. are clearly useful. It's much harder to rigorously claim that a result will _never_ be practical or useful.
Just to be clear, I think it's fair to say that a result isn't _immediately_ useful. But who can say whether or not it will have a more practical application down the road? Only time will tell.
Math is like code: technical debt makes moving forward much harder.
Some think mathematics is something like a physical law, or property of the universe. So figuring out such things may be like fundamental physics research. Hard to quantify the value, but there is value. Seemingly unrelated developments in mathematics have had a habit of helping to solve problems in physics in the past, at least. I don't think anyone really expected the exciting new mathematics of statistics in the 18th century, to one day be relevant to physics. But it would be.
my kid asked me if math was invented or discovered. i said "invented", i hope i'm right considering he views me as some all knowing oracle of information. just a little bit of pressure there.
There's no such thing as "one" in nature: there's not "one" apple. An apple is a monstrously complex collection of cells and dynamic chemical processes, it's not "one".
The concept of numbers are a tool that humans use to categorize, summarize and model the world around us.
The "truths" we discover in math are only true in the mathematical realm, which is an inexact model of the physical realm. Useful, but undeniably a human construct.
So yes, math is invented.
Though we constantly discover new ways to use it in the physical world.
The fact that the physical space we live in has three dimensions also isn’t invented. These numbers, three and five here, are not a human invention. They correspond to facts we discovered, and they represent truths that are independent of human existence or inventions.
Maths is exactly about that, discovering the fundamental truths that, based on logic, we find couldn’t have been any other way.
The concept of "three orthogonal dimensions" is a pretty good model of the physical reality that is space.
There's no such thing as "three", other than a categorization that we, as humans, apply to model the physical phenomenon.
We've developed language and symbols over time to more precisely model the things we observe, but it's important to remember that our symbols are different from the things they are invented to represent.
It's really hard to separate this abstract concept from our observed reality when they are so tightly coupled - the idea of "three dimensional space" is a model we've constructed to categorize and represent, in a convenient way, observed phenomenon that is too difficult to talk about in other terms.
We didn't "discover" that there are three dimensions, for example. We developed language and a mental framework for categorizing space in such a way that we could more easily reason about it and communicate our reasoning effectively with others.
We didn't "discover" Ohm's law. We created a set of units, names and concepts that would allow us to model the relationship in an analytic way, between other concepts such as current, voltage and resistance. Ohm's law isn't not a fundamental law of nature. Above all else, it's a series of struggles with language to describe things we observe. "Ohm" as a unit of resistance, "Voltage", "Amperage" - all of this was invented in order to be able to reason about things we were seeing, and it seems to be a good mental model and tool for getting things done, but it is very different than the actual reality.
We didn't "discover" the number three - we created the concept of numbers and refined their meaning over time. For a very long time, in most civilizations there wasn't a concept of "zero" in number symbols. We needed a way to communicate that concept though, so invented a symbol to represent it.
Number systems and the symbols that represent them have varied wildly throughout human history, and there appears to be a neurological attachment in the human brain that associates quantity with physicality (specifically fingers) [1] that makes grokking this concept especially difficult. When we become sophisticated enough that using our fingers wasn't sufficient to reason and communicate, we started using bone tally marks as a quantity representation - but all of these things, fingers, tally marks, glyphs - are just symbols that humans have created over time to imperfectly model physical phenomena.
It's no different than any other sort of naming. We look at differences between birds, and attach a symbol of "hawk" to one, and "eagle" to another. There's no such thing as "hawk" or "eagle" in nature, these are human invented symbols that help us categorize and communicate in a short-hand way, differences between species we observe. Numbers are the same, but we've also invented a bunch of rules for manipulating those symbols to communicate more advanced concepts: addition, subtraction, etc...
Now it's true, of course, that we discover new things all the time. Frequently we need to invent new ways of describing it. We discovered, by observing the stars, that planets follow an elliptical orbit. We couldn't describe that well using the tools we had, so invented Calculus.
[1] https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys...
To take an example from computer science, we discovered that comparison sorting algorithms have a time complexity of Ω(n log n). This discovery is independent from the notation we use to express it, and independent from the programming languages we use and independent of the choice of sorting algorithm. It is a fundamental fact of the matter, and there is nothing invented about it. Any alien intelligence would end up with exactly the same insight about sorting.
Or to give a more hands-on example: If you have two, three, or four balls of the same size, you can arrange them such that each ball touches all of the other balls (a tetrahedron shape in the case of four balls). But you can’t do that with five or more balls. Convincing yourself that that’s true is an essential example of doing math. And it is true regardless of what “names” you use. You can also use apples and oranges if they are approximately spherical and of the same size (or, alternatively, “look the same from all angles”).
For these reasons, when chasd00 tells their kid that math is invented, not discovered, they are misrepresenting what math is really about. The above example with balls could be one way to illustrate to a kid how math isn’t invented. (I’m sure there are much better examples, this is just from the top of my head.)
If you don't understand how this is a human construct for human cognition and information processing, I don't have a way to explain it to you. I would recommend reading more about number theory and philosophy.
Reading about number theory helped me understand these concepts, and I recall having a similar resistance to the ideas at the time. Specifically set theory and "zero" as a proxy for the empty set, and "one" being the set that contains the empty set, etc...
Math is an imperfect mechanism for representing the physical realm, invented by humans for that purpose. It has truths that are only true in the mathematical realm and have no bearing or representation in the physical realm.
As an example, I suspect you'd have a difficult time pointing to a physical "real" example of the square root of negative one. It's called the imaginary plane for a reason.
> gives the impression that it is building fantasy castles in the sky
That's exactly what it is, as most mathematicians will tell you.
That doesn't imply however, that it's not useful.
Imaginary numbers (to continue the example) are quite handy for dealing with signal processing, among other things. Which doesn't change the fact that they are a human invention - one that stumped societies for hundreds, if not thousands of years (the Greeks struggled mightily with square roots for this reason, for them math and philosophy were not separate things). We only really got past it by inventing the symbol i and tossing all the gnarly stuff in it like a waste bin, because most of the time for practical applications it's factored out.
> absolute and objective truths
These are most appropriately the realm of religion, not of physical sciences.
This is not a fact and is the oldest debate in mathematics. All of it is just as likely (if not more) to be the way the brain models things than some objective truth of “reality.”
Mathematics have nothing to do with the real world.
The set of all true statements provable with a set of axioms is determined only by the axioms. All the true statements are already true whether we know they are or not; in that sense math is like exploring the unknown land that exists outside of the physical realm. It's discovered.
Math was invented and refined over time in order to find ways to model, communicate and reason about our observed physical phenomenon.
I understand though, that there is a ton of subject matter within the field that is completely divorced from the "real" world.
Even in the supposedly simple idea of counting apples.
To continue to (ab)use the apple example, tell me how many apples this is [1].
Sure, there's a problem with the "apple" category/symbol (to your point) but similar exceptions and confounding examples are abundant throughout messy physical reality.
So I think my point stands, "one" is an imperfect model of reality. Useful, but inexact.
Given that "one" may be the most basic fundamental axiom of mathematics, as more abstractions are built upon it the accuracy with which those concepts reflect the physical world declines.
Hence the invention of fractions, etc...
[1] https://www.dailymail.co.uk/news/article-2433619/Siamese-app...
> The concept of numbers are a tool that humans use to categorize, summarize and model the world around us.
By the same argument, aren't apples invented too?
Ask him back, if it's discovered, where was it "stored" all this time.
If it's invented, does it mean that other potential sentient species will have a different math than ours ?
In the process of creating them I had a lot of fun, honed my existing skills and picked up new ones, broke out of various boxes I had been stuffed into in terms of how to architect systems, and just generally worked out my creative muscles.
I've also written a lot of software that is useful and used, and in a practical sense makes (lots of) money.
It's meaningless to break out the time and cycles spent on one Vs the other. The necessary skills required were enabled by one another. Moreover it would have been to a large extent impossible to determine a priori which would be which.
Am I an impractical person because I've spent time and cycles on the former?
What's the point of anything?
> Is it just a mathematician's version of brain teaser?
All of mathematics can be characterised as "brain teasers". Make of that what you will.
> What useful development comes out of such?
I'm not qualified for this particular example. However, it seems quite number-theoretic, and number-theory has given us such "practical" things as asymmetric encryption and universal computation (Turing's famous paper from 1936 was titled "On Computable Numbers", after all).
Not that I disagree with you, but some things have an impact on people's quality of life: nutrition, healthcare, and psychological well-being.
I would perhaps classify that as "immediate" impact, not "lack of impact".
Number theory has driven computer science, calculus, statistics, and so on.
CS, calculus, stats... they all drive (for example) "nutrition". For example, how do you know what a "vitamin" is? How do you measure it? How do you quantify it's effects on the body (and mind)? A whackload of analysis DEPENDS on number theory for us to understand almost everything about the natural world!
Perhaps not that many people in the world understand the formulation of the problem, but then not many people "get" abstract art, either.
:-)
Only a fool pretends the ground floor of a house is worthless because they live on the second floor.
Irrigation is very important. Now, is Irrigation Maths more important to us as a species than Terrence Taos work on spontaneously combusting water? Probably! Would it be more useful to have PhDs be funded in optimizing food routes between countries after climate change over game theory applications for blackjack? Probably!
What someone might call "a silly little brain teaser" today could actually result in a breakthrough paper weeks (or centuries) from now in a different subfield because someone far smarter than us realized that part of the problem they were working was actually analogous to a number theory related problem that was simplified, even a tiny bit, by this solution. (Hell, Nash built his entire career on spotting those kind of links and then telling other mathematicians to focus on working out the individual pieces)
Maths plays out over "we don't even know how long or short" time scales. What's the use? We don't know, it's probably completely useless. Until someone suddenly realizes that it's not.
I imagine you would be equally annoyed at Euler in 1736 when he was wasting his time with bridge brain teasers (and invented graph theory in the process) instead of solving bubonic plague or optimizing irrigation. Science just doesn't (in general) work the way you propose.
People pointing out that maths is full of advancements that had no immediately identifiable use at the time, but that came to be useful later, is correct. Yet it doesn't even begin to answer the OP's question.
I doubt very much that many people choose to pour their lives into endeavours that they don't particularly enjoy just because some hypothetical person at some hypothetical future point in time might hypothetically find a hypothetical use (hypothetically ;P).
The answer is that "value" presupposes the question "valuable to who and why?"
Newton invented Calculus because he had an immediate use for it. Other mathematicians pour themselves into solving problems because they enjoy it and find a lot of reward in the prospect of solving a previous unsolved problem. Both are "valuable", just to different people for different reasons.
The pragmatic, practical perspective here is that funding the egg heads has had incredible outcomes (and its so cheap too), so dont let the simpletons shake the golden goose down just because they don't understand anything they cant fuck, fight, or eat.
That's the problem with math from a "so what is this good for?" perspective: we don't know yet, but we sure have a litany of instances where seemingly useless proofs had a profound impact anywhere from weeks to centuries later.
Unless you can make a prediction that water will combust in low energy conditions, in which you can use this combusting water to generate excess power. Then use that power to compress nitrogen into ammonia, and then use that product as fertilizer.
The British show Connections went over how completely different things sometimes 'connect' and bring fruitful new ideas.
The problem space of reality has emergent behaviors that are not (easily?) predictable. Sometimes you have to iterate large portions of it.
I wish they went into more detail about how this problem was relevant to the other problem.
[1] https://drops.dagstuhl.de/opus/volltexte/2021/14276/pdf/LIPI...
Why do people play board games, or video games, or sports? Why do people read fiction? Why do people have hobbies which strictly distract from productive work time? The answer to any of these questions is the same as the answer for "Why do people do theoretical research?": it's fun.
It's very hard to know how these things end up being useful. Even if a specific piece of knowledge isn't useful by itself, it can lead to things that are useful down the line. It's both very hard to know which ones will be, or which ones do lead to actually useful results; for example, think of work on propositional logic which ends up being the bedrock of computation, work on elliptic curves (cryptography), work on PDEs (physics), quantum physics (transistors), photonics (communications), etc.
Some fields have pretty "fast-to-application" times, but it's not always clear which ones will _not_ yield useful results. Plus, and here's the real, honest answer: all of this shit is very fun! Why not do it?
I'm also a (somewhat) practical person, and that guides many of the problems I try to solve, but sometimes there's just such a tantalizing puzzle that it's hard not to resist! It's pretty damn hard to "only" work on "useful" stuff and not be just an ok academic/programmer/etc., you have to let yourself get nerd-sniped into doing random things. With very high probability, these little "side-quests" end up being very useful down the line, for one reason or another.
I can't speak for others but for me math is kind of like a brain teaser on crack. It is immensely satisfying to think about and I would imagine most mathematicians get a similar kick out of it. Not a mathematician myself mind you.
Solid state physics problems of that sort are used in the process of engineering new materials (e.g. a more efficient dielectric for capacitors, a magnetic material that can store magnetic memory data at higher temperatures without destablizing, a higher temperature superconductor, etc). Idk if this particular result has such an application, but I wouldn't be surprised if it did.
In addition to solid state physics, "special functions" show up in many areas of applied math and physics, including the design, calibration, and data analysis of/from MRI machines and antennas. Many special functions including the hypergeometric functions can be approximated more efficiently using fancy number theory algorithms.
There is also a broad category of physics and chemistry problems that are best solved by Monte Carlo simulations, and sometimes Monte Carlo approximations can be made using many fewer iterations by choosing a set of points within the sample space that satisfy number theoretic constraints.
Also, physics models often inspire machine learning models so it's not impossible that some niche ML algorithm could eventually benefit from a faster algorithm that uses this result or a derivative of it.
None of these directly answer your question about a definitive application for this particular number theory result, but I hope it gives you an idea of what it could potentially be used for. The closest analogy I know if is that certain multi-dimensional Monte Carlo integral approximations can be done orders of magnitude more quickly if you choose lattice points whose coordinates satisfy some similar-ish constraints to the ones described for this problem.
Take cryptography for instance. Why large prime numbers? Why elliptic curves? If you have proven the fundamental mathematics behind factoring large primes, then you have a basis for thinking that a cryptographic system based on large primes will be reasonably secure.
Many of the developments you use everyday are based on the understanding of just such "brain teasers" :-).
Humanity already tried thinking it was better than the selfless pursuit of knowledge, that path leads nowhere good.
And it's math all the way down in computers. So anything that makes math easier or quicker has the potential to affect anything a computer can process.
People didn't believe there was non-Euclidean geometry and not long after the taboo was broken, we had General Relativity. Einstein was a genius, but he did not invent the math for GR. Mathematicians like Riemann, Minkowski, Hilbert, Ricci, Levi-Civita and others created the mathematics for it first.
Finding the area under a curve or the tangent line of an arbitrary curve were considered two unconnected brain teasers until Calculus was invented by Newton and Leibniz and applied to gravity by Newton.
The "Math First" approach has done more than enough to prove its continued viability and without it you wouldn't have been able to type that here.
Mathematicians aren't concerned with that at all. If something they do ends up being useful to science and engineering, then wonderful. But it's not the point.
I can already see it being useful for verifying entropy and for verifying anonymization of statistical data.
The ways I've seen combinatorics (and graph theory!) interact with software engineering is typically showing that a problem is too expensive to solve at scale.
Considering that way vaster amounts of brain cycles are devoted to making people watch one more TikTok video, then, I think this kind of brain-scratching is more useful than much of what most of us do.
If you can't see why this is useful, then great: that's literally what it means to be a layman. But that also means that you don't know enough about the subject field for the real answer (the one that explains what other maths this affects) to satisfy your question because you won't know why _those_ things would matter.
So you're going to get the layman's answer: "because this shows that any work already performed in the assumption of its truth is now known to be valid", which is critically important because a lot of what maths is used for in real life is based on assumptions about the properties of numbers. Now others can stop wasting their time trying to determine whether this aspect of number theory can be relied upon, and under which circumstances, and trying to find all expressions of it in both software and hardware in order to determine whether those might have problems on a case-by-case basis. And at the same time, this result allows others still to move forward with work that is known to rely (in part!) on this property either being true or false, for which it was previously uncertain whether it would even be worth the effort because it was unknown whether the results would turn out to be useless because no one knew whether the basis it relied on was flawed.
As for what useful development comes out of things like this:
points at the computer you're using to comment on HN threads, and every single application on it that you've ever used
That.