17 ms·

Computer proof ‘blows up’ centuries-old fluid equations

196 points by nsoonhui 4y ago

paulpauper 4y ago
How is it even possible to learn this much math or organize such complex ideas. Undergrad comes nowhere even close to this. I guess the leap occurs in grad school. But even though, this is way more advanced than 99.75% of math papers I have read.
- d2049 4y ago
  In my limited few years of doing undergraduate / graduate research in this kind of field, and thus having to read research papers like this, I can say that this is BY FAR the longest paper I have ever encountered. Several times over. Forget writing a 177-page paper, how do you even begin to read this? Yes there is a helpful outline on page 6, but sheesh!
Agingcoder 4y ago
I remember my math professor at university telling me that truth in mathematics was a social construct, and that nothing was true until a social consensus had been reached between mathematicians. This struck me at the time as a very powerful statement, yet unexpected, since very much not what most people expect from mathematics. After all, it's supposed to be a field where there is such a thing as a (most of the time) reachable truth!
- torotonnato 4y ago
  I would have asked how he could assert the truth of the proposition "Truth in mathematics is a social construct", since its truthfulness has to be a social construct too. (I assume that mathematics encompasses formal logic too)
- rapsey 4y ago
  Huh? Mathemathical proof is not a social construct. This makes no sense.
  - hugh-avherald 4y ago
    It's certainly a social construct, but it is not merely a social construct.
  - karmakurtisaani 4y ago
    I suppose they mean what's commonly accepted as true and can be referred to as truths. No one can read all the proofs, so they have to trust others who have. There was that one example where a mathematician "proved" something terribly complicated using his own methods and terminology developed over several years. The truth value of that kind of proof is very much a social construct.
    - Agingcoder 4y ago
      Yes, mochizuki and the abc conjecture. It was an interesting conundrum : he was a really good mathematician, not a crank so his funky proof couldn't be dismissed. However, people were wary of approaching the proof since it was risky career wise (it takes time, etc). You end up with a weird situation where something is probably true, but you won't know that until a trusted group of mathematicians have read it and said so.
  - uKVZe85V 4y ago
    The mathematical truths are the fruit of the work. The social consensus is an implementation "detail". Yet it's the implementation we have. Does this make better sense now?
  - kingkawn 4y ago
    And yet nonetheless you feel the need to object to this formulation publicly and have it considered by others
  - mpweiher 4y ago
    The proof is not a social construct. The truth is. The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is. There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true? Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans. So for example, if the proof doesn't convince other mathematicians, then it's not a a proof. Or it might convince other mathematicians and later turn out to be wrong after all. For more on the practical aspects of math, I highly recommend The Mathematical Experience. https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... I read it in German: https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis/dp/3764329963 https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...
- planck01 4y ago
  There are such reachable truths, but every mathematical system has a non empty set of axioms - or assumptions- which are 'given'.
  - jojobas 4y ago
    These axioms are not given in course of socialization and generally are observations of nature rather than human.
    - roywiggins 4y ago
      There are multiple set theories using different axioms. Is the Axiom of Choice based on an observation of nature or do mathematicians keep it around because it's useful? It's a statement about infinities that absolutely have no physical reality. You can do mathematics without it, and the question of whether to do math relying on it is a matter of opinion. (Yes, proofs relying on AC are arguably true even if you don't accept AC, but as a social reality some sets of axioms are considered valid bases for work and some aren't, you can keep adding stronger axioms to ZFC to prove more things more easily, but how far you go with that before it stops being interesting is a matter of opinion)
    - pencilguin 4y ago
      ... of what people can and also choose to observe about nature.
- mpweiher 4y ago
  A proof is a rhetorical device to convince others of the truth of a proposition.
  - deleted 4y ago
    [deleted]
  - pencilguin 4y ago
    If you can't get anybody to read your proof, does it demonstrate anything? Fred Moxley has (what seems to me like, but what do I know?) a nice proof of the Riemann conjecture that he got by quantizing the problem. But nobody will read it, because mathematicians don't like that method. It might be right or not, but it anyway doesn't tell you anything surprising about prime numbers, so nobody can be bothered.
- faraaz98 4y ago
  Is it because maths is "incomplete" ala godel incompleteness theorem?
- flanked-evergl 4y ago
  > and that nothing was true until a social consensus had been reached between mathematicians. So how do calculators and computers work then?
  - yazaddaruvala 4y ago
    1+1=2 is a social construct. It’s simple, repeatable and therefore programable - but still a social construct.
    - jojobas 4y ago
      Well if "1+1=2" is a social construct then "there are planets, stars, and, in general, objects" is also a social construct. No kind of society can make 1+1=3, and also there is mathematical proof that 1+1=2 (built of course on some sort of axioms that somebody might also consider social constructs).
      - prmph 4y ago
        Imagine a universe, where, when you combine two apples (or really anything), you always, through some weird physical process, end up with three items. Would that invalidate (or not), the statement that 1 + 1 = 2?
        eesmith 4y ago
        1 cloud combined with 1 cloud gives 1 cloud.
        feanaro 4y ago
        That's just some additional structure pertaining to, specifically, clouds. One cloud standing close to another cloud still equals two clouds, which is what 1 + 1 = 2 is really encoding.
      - eesmith 4y ago
        We know "planet" is a social construct because the astronomers decided, some years back, that Pluto wasn't a planet. And the rules they created to make that definition only apply in the Solar System, not to exoplanets around other stars nor to rogue/free-floating planets. And Ceres, Pallas, Juno, and Vesta were considered planets for over half a century.
- ThereIsNoWorry 4y ago
  I don't know why you're being downvoted, but that is a perfectly valid statement. Everyone thinks proofs are this holy grail and totally rigorous, and they are on a certain level. But the idea is floating around that Mathematicians are infallible when in fact lots of proofs in highly complex areas of mathematics are NOT 100% perfectly rigorous. They contain a lot of skipping, because "it's trivial" and consensus. This approach may work very often, but there is a danger that sometimes it doesn't work and things get overlooked. Since mathematics is done in a bottom-up approach, at some point some fundament may or may not turn out to be wrong, which endangers parts built on top of it. The whole movement of rigorous automated proof systems is to prove mathematics from the very bottom to the very top in a 100% rigorous and verifiable way. Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation.
  - zozbot234 4y ago
    > because the inherently subjective "elegance" and "beauty" gets lost in translation. That's a very subjective POV, and perhaps one that varies by area of math. Many computer proof developments are more cleanly refactored/abstracted than the manual equivalent, because it's so easy to refactor a computer proof without worrying that the new proof might fail to prove the same statement.
  - q-big 4y ago
    > Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation. Couldn't we also interpret this fact that computerized proofs are currently often very unelegant as strong evidence that not a lot is understood about this topic and thus doing such "ugly" computerized proofs is the best we can (in most cases) currently do? Science at the boundary of human knowledge is often quite ugly; as our understanding of it grows, it often becomes more beautiful and elegant.
  - 77pt77 4y ago
    You should read "Proofs and Refutations by Imre Lakatos" if you haven't already.
- mattigames 4y ago
  "Yes profesor, truth in any academic field is result of agreement between the people working on that field. Just wondering... its everything Ok at home?"
- auggierose 4y ago
  It certainly is a social construct, because what tools are at my disposal to convince someone who disagrees otherwise? In that sense everything is a social construct. Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. It also is much less cumbersome than it used to be, and will continue to improve quickly. I can sit down and prove something using a tool like Isabelle, and I will be as sure of its "truth" as I can possibly be, and it really doesn't matter what other people, mathematicians or not, think about it. That's the beauty of it. Of course, you could say my belief in Isabelle is also a social construct. Except it is not, I know exactly how Isabelle works. There could be issues with Isabelle, but these issues adding up to make my proof wrong are very unlikely, especially in addition to my independent understanding of the proof. But of course, it is much nicer if others can see the same truth that I do, and for this, computer-assisted proof is actually great, because it allows to understand and trust in the high-level structure of a proof without having to verify every little gritty low-level detail.
  - prmph 4y ago
    > Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. I think you are mistaken. The idea that math proofs are a social construct relates to, in my view, much deeper ideas than you seem to think [1]. It is not just that convincing other mathematicians that a proof is correct is a social process, but also that the reasoning on which any proof relies, even if it seems unassailable, even if built into an automated checker, is still a product of the human mind. Usually there is a level of logic that can challenge even what seems so basic as to be fool-proof. Take the proof that the square root of 2 is irrational. The proof relies on a contradiction that arises if one assumes the root is rational, but one can imagine a logic system where such a contradiction does not imply that the original assumption is false. How possible, you say? It's all math, where one is allowed any starting assumptions, and works out the implications of those. But, there is something deeply satisfying about thinking that contradictions are (or should be) impossible in our universe, and so this "proof" seems solid. 1. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/
- krcz 4y ago
  It's a social construct in the the same sense anything not directly verifiable using senses is. Is there an Eiffel tower in Paris? Most people haven't seen it, so they can only accept the social consensus that it is there. If one can afford it, they can travel to Paris and check themselves. The same with mathematical truth: if one has means (time, intelligence, access to training), they can check the proof themselves. Otherwise they need to trust the consensus. So again, is the truth in mathematics just a social construct? In some sense, I guess, but probably not the one some people might assume hearing such a statement.
  - tigerlily 4y ago
    To illustrate the point further, once you get to Paris how can you be sure it's an Eiffel tower? I guess you have to ask the man in the street. See the truth of it is a social construct. And whether you accept this as truth is a social construct, and so on. QED.
    - quonn 4y ago
      > I guess you have to ask the man in the street. How about checking with a GPS? A social construct has nothing to do with simple facts about the universe. And whether the Eiffel tower exists as an object at a particular spot as indicated on maps is such a fact. And if there were maps that would place it elsewhere, those maps would be a lie. Even if the every single map ever made and every other person would deny that there is such a tower at that position one could still go there and check for oneself. Maybe you are talking about the name? The fact that we call it the Eiffel tower? Well, that tower has a history and again one could lie about the history, who built it, how it was historically called as a matter of fact etc. But an observer would have seen who actually built this tower. It's a fact.
- jules 4y ago
  There are (short) computer programs where you input a mathematical proposition and a proof in a kind of proof programming language, and the program will then check if it's a valid proof. Saying that mathematical truth is a social construct is technically true but misses the point entirely.
  - e12e 4y ago
    How does one decide which axioms to build on?
    - threatofrain 4y ago
      You can choose whatever you want, and mathematicians do sometimes choose different axioms. The question is whether the consequences of some axioms are up for social negotiation.
    - deleted 4y ago
      [deleted]
- aaron695 4y ago
  > that truth in mathematics was a social construct. This is garbage. Everything under this definition is a social construct and as such why would you only relate it to mathematics? The rock I'm holding is a social construct. Deep. If they want to get stoned and talk about the meaning of life cool, but it's beneath a math professor (Who's not at home getting stoned) Following it logically you quickly find murder, rape, genocide being bad are just social constructs. And why exactly should we follow social constructs? Lets all go and start the next FTX because everything is just a social construct so who cares? And we've just rediscovered nihilism like the other 120 billion teens did.
  - dist1ll 4y ago
    Just because something is a social construct doesn't mean it's worthless, impure or required to be rejected. Going from self-reflection to nihilism is a pretty big overreaction.
- bheadmaster 4y ago
  The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus. If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks. I hate this sociologist view that everything depends on the social consensus. Going extreme with it is how you end up in a 1984-esque society: Anything could be true. The so-called laws of Nature were nonsense. The law of gravity was nonsense. ’If I wished,’ O’Brien had said, ’I could float off this floor like a soap bubble.’ Winston worked it out. ’If he thinks he floats off the floor, and if I simultaneously think I see him do it, then the thing happens.
  - jfengel 4y ago
    That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks. That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry. Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented. That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.
    - deleted 4y ago
      [deleted]
    - bheadmaster 4y ago
      > But the hard part is convincing other mathematicians to care. My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world. Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection. Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.
- 77pt77 4y ago
  What people accept as truth is a social construct. That's a different thing. Your teacher was just a sophist.
- IIAOPSW 4y ago
  I'm going to relay a story. For a short period in my life, I had a roommate whom I wasn't sure if he was real for the first two weeks I knew him. At first it was just we agreed unreasonably well about our view on the world. Like I could not think of a single thing we differed on. But then it started to get uncanny. He had this way of knowing all the same trivia as I did. And also of not being able to recall the same bits of trivia I was struggling with. I'm talking really obscure sorts of things, not the sort of stuff you could dismiss as "20% random hn person recognizes it." Then there was the scavenging. Practically any time I mentioned off hand an idea for something we might have a use for, he would randomly find that or a similar item thrown out on the side of the curb (this was in NY). He wasn't buying these items, it was all just "lucky coincidence". Then there was the absurd situational similarity. We had started out as guests in an airbnb, permanent temporaries, but now we were both effectively bartering for rent making improvements on this guys apartment in exchange for free board. Its the sort of weird niche situation few people ever find themselves in, and we were both doing it. At some point the thought occurs to me. Which is more likely, there's a guy who knows all the same stuff I know, is in the same awkward work situation I am in, happens to find the exact things I am looking for, OR I am having a psychotic break, this guy is my delusion, and all those things he does is actually just me doing it? After thinking this I started to realize, I had never really seen this guy outside the apartment. No one else I knew from before had ever seen or knew of this guys existence. One day I'm idly humming a tune that got stuck in my head. You might recognize it as "Battle hymn of the Republic." But, there's actually 4 prominent songs in American history with this exact same tune. The others are "John Browns body", "Blood on the Risers" and "Solidarity Forever". My new roommate walks in and starts singing the words. But how did he know which one I was humming? It wasn't the obvious well known one! No, surely I am going mad. My roommate had mentioned that he lived in Russia until he was 8 and could speak basic Russian. I do not know Russian. I ask him to teach me about Russian grammar. He agrees but then changes the topic. I push the issue again latter that day. He once again agrees to teach me some Russian and then proceeds to divert attention elsewhere again. I ask him to teach me some Russian. He pushes it off yet again. Whereas before the thought was idle, the evidence keeps on growing. I'm having a psychotic break. This guy can't be real. We are sitting around one day. My roommate points out that all of us sitting in the room have hazel eyes, and that this is the rarest of the eye colors. They then proceed to pull up the statistics and crudely calculate the probability of this happening (pretending our genetic demographic is unrelated to the circumstances that led us all to this room). The result was some outrageously small number, less than a tenth of a percent. At this point I'm pretty sure my own delusion is taking the piss out of me, actively shoving the implausibility of his own existence in my face as a joke. It turns out all of this really was amazingly coincidental. As weeks went by, guests at the airbnb would come in go, we would meet each others friends, and eventually there were enough people who also acknowledged his existence that I am now convinced he is real. So I pose it to you. Was my roommate real, was everyone involved in this story a figment of a madman's imagination, or am I completely making up this roommate story to make a point? The answer is, reality is shared consensus. If you all are also convinced that this person existed and these events transpired, then we share a common set of facts. If there is no shared consensus, then he only exists for me. Perhaps there is some underlying truth beyond the shared consensus, but shared consensus is the instrument we use to measure realness. At some point, there is no difference between "every multimeter says this battery is 9 volts" and the battery actually being 9 volts. I'm going to relay another story. Neils Bohr used to keep a horse shoe nailed to his door. When asked, he would say its for good luck. One day someone asked "do you really believe that?" He responds "No, but they say it works even if you don't believe in it." Why believe quantum mechanics over lucky horse shoes? If everyone chooses lucky horse shoe theory, does that become reality? If powerful interests in government start forcing everyone to adapt horse shoe theory, does that make it real? Thus I arrive at a truly bothersome set of contradictions. Reality is shared consensus, but reality is also the set of things not subject to popularity. There is no truth only power, but also the essence of science and math is that truth does derive from authority. One day I will reconcile these. One day.
- eternalban 4y ago
  "2 + 2 = 4" "the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2." Most of us conflate arithmetic with mathematics. In arithmetic, things start getting 'conceptual' as soon as we no longer can map certain measures and operations to a realizable physical construct. At that precise juncture, math becomes a semantic system and is therefore subject to social consensus. For example, consider introducing infinity, or even zero, into 'shopkeepers' sense of numbers. Before, you could never add something to a number and end up with the same number, but now 0 + 0 = 0, and a + \infty = \infty . And to the shopkeepers' surprise, some mathematicians may even argue over it.
- samatman 4y ago
  Mathematical truth is socially constructed, but using rules, and it is the rules, rather than the process of social construction, which give this process its power. An interesting meditation here on mathematics itself, which is also simply certain rules, and not others. Merely invoking social construction ignores this difference, which is the essential difference, between mathematics and, say, hide and go seek.
- kenjackson 4y ago
  Truth in math is not a social construct. But belief in math is a social construct.
- robertlagrant 4y ago
  Social constructionalism is to my understanding least surprisingly found in universities. I think the issue is if you call every type of thought and communication "social construction" then you don't end up with anything useful.
  - robertlagrant 4y ago
    Sorry: constructionism*.
- kzz102 4y ago
  I think this is not the right way to look at it. You can think of mathematical proofs as computer program that is compiled by the mathematician by hand. There is a lot of room for error, but with practice and peer review, it's relatively easy to avoid the common errors. This human compiler also brings the benefit of error correcting, which commonly correct two types of errors: sometimes the proof makes syntactical mistakes that the human compiler fixes automatically, sometimes the proof claims something that's not fully justified (similar to calling a function that is not implemented), but the human compiler just fill in the detail themselves. The social part of mathematics is really about how much error the reviewer is willing to accept, because the reviewer can also be wrong with how they correct the proof.
thaumasiotes 4y ago
> Hou and Luo’s work was suggestive, but not a true proof. That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it — meaning that the solution might be very accurate, but it’s still an approximation. I feel certain that if you run a process that approaches infinity using ordinary floating-point numbers, you will actually reach infinity. This is a case ("can a calculation yield an infinite result?") where computers have less of a problem than people do. You'd have to deal with the question of whether the infinite value accurately reflected an infinite limit of the process or whether it was spurious. But there's no difficulty in calculating infinite values.
- blurbleblurble 4y ago
  Double precision floats have a maximum value of 1.7976931348623158 E + 308
  - onion2k 4y ago
    Aka not infinity
    - PartiallyTyped 4y ago
      They meant maximum finite value. 0 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ 7FF0 0000 0000 0000 (base 16) ≙ +∞ (positive infinity) 1 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ FFF0 0000 0000 0000 (base 16) ≙ −∞ (negative infinity)
  - thaumasiotes 4y ago
    That is not correct; they have a maximum value of positive infinity. See what you get when you square 1.7e+307.
    - mjburgess 4y ago
      A computer is always just a function from {0,1}^N -> {0,1}^M The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. And, in general, almost every function isnt representable on the discrete domain above. floats are a hacky interpretation of discrete bit patterns
somat 4y ago
"That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it" Why not? Is it impossible to calculate infinite values in general? I suspect not, My understanding is that a lot of calculus is in fact on how to calculate infinite values. And a computer is a universal machine, this means that while it can not calculate everything, it can calculate anything that is calculable.
- raverbashing 4y ago
  Because infinity is a mathematical trick, to say where something is going but it is not calculable (it is not a number in itself)
  - agumonkey 4y ago
    well you end up in bounded axiomatic ranges
- mjburgess 4y ago
  Almost all functions aren't computable, as they aren't discrete. A "computer" is just a function from {0,1}^N -> {0,1}^M
  - jojobas 4y ago
    Symbolic engines are way above and beyond number crunching in many respects.
    - mjburgess 4y ago
      Mathematics is constrained by properties of abstract objects that the symbols are about. Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario. Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario, which the machine is unable to represent.
wwalker3 4y ago
The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes. For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device! Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do. 1) https://arxiv.org/pdf/2210.07191.pdf https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.
- nuclearnice1 4y ago
  What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
  - isoprophlex 4y ago
    I think (not a physicist), simply put, an infinity or NaN value. As these are step-wise methods, having such values show up anywhere will seriously mess up subsequent calculation steps.
  - prolyxis 4y ago
photochemsyn 4y ago
> "Yet much remains unknown about the Euler equations — including whether they’re always an accurate model of ideal fluid flow." In general the word 'model' implies a mathematical construct that mimics the behavior of a real-world experimental or observational system, i.e the experimental or observational data can be generated by the model. Where's the real-world ideal fluid flow? No such systems exist. > "In principle, if you know the location and velocity of each particle in a fluid, the Euler equations should be able to predict how the fluid will evolve for all time. But mathematicians want to know if that’s actually the case." At some scale in a real fluid, quantum effects become important, and the notion that it's possible to know both position and momentum falls apart. To quote Peter Atkins: > "The trouble is when you're dealing with operators, is it turns out you can't always extract explicit information about all of them simultaneously, and this leads to Heisenberg's uncertainty principle, which most people think of as a great confuser of the world and denier of information. I like to think of it as a great clarifier because old-fashioned people, like Newton and Einstein, and Lagrange, and all the people who developed classical mechanics, took it as certain that to specify the state of a particle, you had to specify where it was and how fast it was going." That's why there are no perfect ideal fluids. Atkins expands: > "What quantum theory did, through the uncertainty principle, was to clarify - what it said, was discuss the world if you like in terms of positions, or if you prefer, discuss the world in terms of linear momenta, don't try both at once. You get very simple descriptions in terms of positions, you get very simple descriptions in terms of linear momenta, it's only when you - like trying to start a sentence in English and ending in Latin or something, trying to mix the two together, that you get into confusion. So think of the Heisenberg principle as a clarifier - try to think one way, or try to think the other way, but don't try to think like Newton thought."
- Ferrotin 4y ago
  No, atoms are why there aren’t perfect ideal fluids, and quantum effects have nothing to do with it.
kzz102 4y ago
Computer assisted proof is not a new idea. The most famous example is the four color theorem, which (as far as I know) does not have a humanly understandable proof. For this particular proof, it looks like the computer assisted part would be similar to writing down hundreds of pages of checkable inequalities. One way to do this is to use interval arithmetic, which always guarantees the answer is within the a certain given interval. Mathematicians have split opinions about computer assisted proof. On one hand, there doesn't seem to be a real difference between one page of checkable inequalities vs 500 pages of checkable inequalities. On other other hand, computer assisted proof do not help humans gain clarity on the logical process. The fear is that the computer result will just be an "one and done" result, which people believe is true, but cannot build upon because they don't fully understand it.
- lambdatronics 4y ago
  > people believe is true, but cannot build upon because they don't fully understand it. OK, maybe it doesn't help people build on the proof, but does it help computers build on the proof? Why does it have to come back to human understanding? Maybe it's not possible right now for computers to generalize proof strategies, but I wouldn't bet against it becoming possible down the road.

196 points by nsoonhui 4y ago

How is it even possible to learn this much math or organize such complex ideas. Undergrad comes nowhere even close to this. I guess the leap occurs in grad school. But even though, this is way more advanced than 99.75% of math papers I have read.

d2049 4y ago
In my limited few years of doing undergraduate / graduate research in this kind of field, and thus having to read research papers like this, I can say that this is BY FAR the longest paper I have ever encountered. Several times over. Forget writing a 177-page paper, how do you even begin to read this? Yes there is a helpful outline on page 6, but sheesh!

Agingcoder 4y ago

I remember my math professor at university telling me that truth in mathematics was a social construct, and that nothing was true until a social consensus had been reached between mathematicians. This struck me at the time as a very powerful statement, yet unexpected, since very much not what most people expect from mathematics. After all, it's supposed to be a field where there is such a thing as a (most of the time) reachable truth!

torotonnato 4y ago
I would have asked how he could assert the truth of the proposition "Truth in mathematics is a social construct", since its truthfulness has to be a social construct too. (I assume that mathematics encompasses formal logic too)
rapsey 4y ago
Huh? Mathemathical proof is not a social construct. This makes no sense.
- hugh-avherald 4y ago
  It's certainly a social construct, but it is not merely a social construct.
- karmakurtisaani 4y ago
  I suppose they mean what's commonly accepted as true and can be referred to as truths. No one can read all the proofs, so they have to trust others who have. There was that one example where a mathematician "proved" something terribly complicated using his own methods and terminology developed over several years. The truth value of that kind of proof is very much a social construct.
  - Agingcoder 4y ago
    Yes, mochizuki and the abc conjecture. It was an interesting conundrum : he was a really good mathematician, not a crank so his funky proof couldn't be dismissed. However, people were wary of approaching the proof since it was risky career wise (it takes time, etc). You end up with a weird situation where something is probably true, but you won't know that until a trusted group of mathematicians have read it and said so.
- uKVZe85V 4y ago
  The mathematical truths are the fruit of the work. The social consensus is an implementation "detail". Yet it's the implementation we have. Does this make better sense now?
- kingkawn 4y ago
  And yet nonetheless you feel the need to object to this formulation publicly and have it considered by others
- mpweiher 4y ago
  The proof is not a social construct. The truth is. The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is. There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true? Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans. So for example, if the proof doesn't convince other mathematicians, then it's not a a proof. Or it might convince other mathematicians and later turn out to be wrong after all. For more on the practical aspects of math, I highly recommend The Mathematical Experience. https://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687 https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... I read it in German: https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis/dp/3764329963 https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...
planck01 4y ago
There are such reachable truths, but every mathematical system has a non empty set of axioms - or assumptions- which are 'given'.
- jojobas 4y ago
  These axioms are not given in course of socialization and generally are observations of nature rather than human.
  - roywiggins 4y ago
    There are multiple set theories using different axioms. Is the Axiom of Choice based on an observation of nature or do mathematicians keep it around because it's useful? It's a statement about infinities that absolutely have no physical reality. You can do mathematics without it, and the question of whether to do math relying on it is a matter of opinion. (Yes, proofs relying on AC are arguably true even if you don't accept AC, but as a social reality some sets of axioms are considered valid bases for work and some aren't, you can keep adding stronger axioms to ZFC to prove more things more easily, but how far you go with that before it stops being interesting is a matter of opinion)
  - pencilguin 4y ago
    ... of what people can and also choose to observe about nature.
mpweiher 4y ago
A proof is a rhetorical device to convince others of the truth of a proposition.
- deleted 4y ago
  [deleted]
- pencilguin 4y ago
  If you can't get anybody to read your proof, does it demonstrate anything? Fred Moxley has (what seems to me like, but what do I know?) a nice proof of the Riemann conjecture that he got by quantizing the problem. But nobody will read it, because mathematicians don't like that method. It might be right or not, but it anyway doesn't tell you anything surprising about prime numbers, so nobody can be bothered.
faraaz98 4y ago
Is it because maths is "incomplete" ala godel incompleteness theorem?
flanked-evergl 4y ago
> and that nothing was true until a social consensus had been reached between mathematicians. So how do calculators and computers work then?
- yazaddaruvala 4y ago
  1+1=2 is a social construct. It’s simple, repeatable and therefore programable - but still a social construct.
  - jojobas 4y ago
    Well if "1+1=2" is a social construct then "there are planets, stars, and, in general, objects" is also a social construct. No kind of society can make 1+1=3, and also there is mathematical proof that 1+1=2 (built of course on some sort of axioms that somebody might also consider social constructs).
    - prmph 4y ago
      Imagine a universe, where, when you combine two apples (or really anything), you always, through some weird physical process, end up with three items. Would that invalidate (or not), the statement that 1 + 1 = 2?
      - eesmith 4y ago
        1 cloud combined with 1 cloud gives 1 cloud.
        feanaro 4y ago
        That's just some additional structure pertaining to, specifically, clouds. One cloud standing close to another cloud still equals two clouds, which is what 1 + 1 = 2 is really encoding.
    - eesmith 4y ago
      We know "planet" is a social construct because the astronomers decided, some years back, that Pluto wasn't a planet. And the rules they created to make that definition only apply in the Solar System, not to exoplanets around other stars nor to rogue/free-floating planets. And Ceres, Pallas, Juno, and Vesta were considered planets for over half a century.
ThereIsNoWorry 4y ago
I don't know why you're being downvoted, but that is a perfectly valid statement. Everyone thinks proofs are this holy grail and totally rigorous, and they are on a certain level. But the idea is floating around that Mathematicians are infallible when in fact lots of proofs in highly complex areas of mathematics are NOT 100% perfectly rigorous. They contain a lot of skipping, because "it's trivial" and consensus. This approach may work very often, but there is a danger that sometimes it doesn't work and things get overlooked. Since mathematics is done in a bottom-up approach, at some point some fundament may or may not turn out to be wrong, which endangers parts built on top of it. The whole movement of rigorous automated proof systems is to prove mathematics from the very bottom to the very top in a 100% rigorous and verifiable way. Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation.
- zozbot234 4y ago
  > because the inherently subjective "elegance" and "beauty" gets lost in translation. That's a very subjective POV, and perhaps one that varies by area of math. Many computer proof developments are more cleanly refactored/abstracted than the manual equivalent, because it's so easy to refactor a computer proof without worrying that the new proof might fail to prove the same statement.
- q-big 4y ago
  > Doing actual rigorous proofs a computer can verify is enormously tedious and many Mathematicians dislike it for that reason, because the inherently subjective "elegance" and "beauty" gets lost in translation. Couldn't we also interpret this fact that computerized proofs are currently often very unelegant as strong evidence that not a lot is understood about this topic and thus doing such "ugly" computerized proofs is the best we can (in most cases) currently do? Science at the boundary of human knowledge is often quite ugly; as our understanding of it grows, it often becomes more beautiful and elegant.
- 77pt77 4y ago
  You should read "Proofs and Refutations by Imre Lakatos" if you haven't already.
mattigames 4y ago
"Yes profesor, truth in any academic field is result of agreement between the people working on that field. Just wondering... its everything Ok at home?"
auggierose 4y ago
It certainly is a social construct, because what tools are at my disposal to convince someone who disagrees otherwise? In that sense everything is a social construct. Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. It also is much less cumbersome than it used to be, and will continue to improve quickly. I can sit down and prove something using a tool like Isabelle, and I will be as sure of its "truth" as I can possibly be, and it really doesn't matter what other people, mathematicians or not, think about it. That's the beauty of it. Of course, you could say my belief in Isabelle is also a social construct. Except it is not, I know exactly how Isabelle works. There could be issues with Isabelle, but these issues adding up to make my proof wrong are very unlikely, especially in addition to my independent understanding of the proof. But of course, it is much nicer if others can see the same truth that I do, and for this, computer-assisted proof is actually great, because it allows to understand and trust in the high-level structure of a proof without having to verify every little gritty low-level detail.
- prmph 4y ago
  > Apart from that, with the help of computers, it can be made absolutely precise and clear which statements follow from which axioms, and in that sense it is not a social construct at all. I think you are mistaken. The idea that math proofs are a social construct relates to, in my view, much deeper ideas than you seem to think [1]. It is not just that convincing other mathematicians that a proof is correct is a social process, but also that the reasoning on which any proof relies, even if it seems unassailable, even if built into an automated checker, is still a product of the human mind. Usually there is a level of logic that can challenge even what seems so basic as to be fool-proof. Take the proof that the square root of 2 is irrational. The proof relies on a contradiction that arises if one assumes the root is rational, but one can imagine a logic system where such a contradiction does not imply that the original assumption is false. How possible, you say? It's all math, where one is allowed any starting assumptions, and works out the implications of those. But, there is something deeply satisfying about thinking that contradictions are (or should be) impossible in our universe, and so this "proof" seems solid. 1. https://plato.stanford.edu/entries/intuitionism/ https://plato.stanford.edu/entries/intuitionism/
krcz 4y ago
It's a social construct in the the same sense anything not directly verifiable using senses is. Is there an Eiffel tower in Paris? Most people haven't seen it, so they can only accept the social consensus that it is there. If one can afford it, they can travel to Paris and check themselves. The same with mathematical truth: if one has means (time, intelligence, access to training), they can check the proof themselves. Otherwise they need to trust the consensus. So again, is the truth in mathematics just a social construct? In some sense, I guess, but probably not the one some people might assume hearing such a statement.
- tigerlily 4y ago
  To illustrate the point further, once you get to Paris how can you be sure it's an Eiffel tower? I guess you have to ask the man in the street. See the truth of it is a social construct. And whether you accept this as truth is a social construct, and so on. QED.
  - quonn 4y ago
    > I guess you have to ask the man in the street. How about checking with a GPS? A social construct has nothing to do with simple facts about the universe. And whether the Eiffel tower exists as an object at a particular spot as indicated on maps is such a fact. And if there were maps that would place it elsewhere, those maps would be a lie. Even if the every single map ever made and every other person would deny that there is such a tower at that position one could still go there and check for oneself. Maybe you are talking about the name? The fact that we call it the Eiffel tower? Well, that tower has a history and again one could lie about the history, who built it, how it was historically called as a matter of fact etc. But an observer would have seen who actually built this tower. It's a fact.
jules 4y ago
There are (short) computer programs where you input a mathematical proposition and a proof in a kind of proof programming language, and the program will then check if it's a valid proof. Saying that mathematical truth is a social construct is technically true but misses the point entirely.
- e12e 4y ago
  How does one decide which axioms to build on?
  - threatofrain 4y ago
    You can choose whatever you want, and mathematicians do sometimes choose different axioms. The question is whether the consequences of some axioms are up for social negotiation.
  - deleted 4y ago
    [deleted]
aaron695 4y ago
> that truth in mathematics was a social construct. This is garbage. Everything under this definition is a social construct and as such why would you only relate it to mathematics? The rock I'm holding is a social construct. Deep. If they want to get stoned and talk about the meaning of life cool, but it's beneath a math professor (Who's not at home getting stoned) Following it logically you quickly find murder, rape, genocide being bad are just social constructs. And why exactly should we follow social constructs? Lets all go and start the next FTX because everything is just a social construct so who cares? And we've just rediscovered nihilism like the other 120 billion teens did.
- dist1ll 4y ago
  Just because something is a social construct doesn't mean it's worthless, impure or required to be rejected. Going from self-reflection to nihilism is a pretty big overreaction.
bheadmaster 4y ago
The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus. If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks. I hate this sociologist view that everything depends on the social consensus. Going extreme with it is how you end up in a 1984-esque society: Anything could be true. The so-called laws of Nature were nonsense. The law of gravity was nonsense. ’If I wished,’ O’Brien had said, ’I could float off this floor like a soap bubble.’ Winston worked it out. ’If he thinks he floats off the floor, and if I simultaneously think I see him do it, then the thing happens.
- jfengel 4y ago
  That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks. That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry. Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented. That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.
  - deleted 4y ago
    [deleted]
  - bheadmaster 4y ago
    > But the hard part is convincing other mathematicians to care. My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world. Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection. Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.
77pt77 4y ago
What people accept as truth is a social construct. That's a different thing. Your teacher was just a sophist.
IIAOPSW 4y ago
I'm going to relay a story. For a short period in my life, I had a roommate whom I wasn't sure if he was real for the first two weeks I knew him. At first it was just we agreed unreasonably well about our view on the world. Like I could not think of a single thing we differed on. But then it started to get uncanny. He had this way of knowing all the same trivia as I did. And also of not being able to recall the same bits of trivia I was struggling with. I'm talking really obscure sorts of things, not the sort of stuff you could dismiss as "20% random hn person recognizes it." Then there was the scavenging. Practically any time I mentioned off hand an idea for something we might have a use for, he would randomly find that or a similar item thrown out on the side of the curb (this was in NY). He wasn't buying these items, it was all just "lucky coincidence". Then there was the absurd situational similarity. We had started out as guests in an airbnb, permanent temporaries, but now we were both effectively bartering for rent making improvements on this guys apartment in exchange for free board. Its the sort of weird niche situation few people ever find themselves in, and we were both doing it. At some point the thought occurs to me. Which is more likely, there's a guy who knows all the same stuff I know, is in the same awkward work situation I am in, happens to find the exact things I am looking for, OR I am having a psychotic break, this guy is my delusion, and all those things he does is actually just me doing it? After thinking this I started to realize, I had never really seen this guy outside the apartment. No one else I knew from before had ever seen or knew of this guys existence. One day I'm idly humming a tune that got stuck in my head. You might recognize it as "Battle hymn of the Republic." But, there's actually 4 prominent songs in American history with this exact same tune. The others are "John Browns body", "Blood on the Risers" and "Solidarity Forever". My new roommate walks in and starts singing the words. But how did he know which one I was humming? It wasn't the obvious well known one! No, surely I am going mad. My roommate had mentioned that he lived in Russia until he was 8 and could speak basic Russian. I do not know Russian. I ask him to teach me about Russian grammar. He agrees but then changes the topic. I push the issue again latter that day. He once again agrees to teach me some Russian and then proceeds to divert attention elsewhere again. I ask him to teach me some Russian. He pushes it off yet again. Whereas before the thought was idle, the evidence keeps on growing. I'm having a psychotic break. This guy can't be real. We are sitting around one day. My roommate points out that all of us sitting in the room have hazel eyes, and that this is the rarest of the eye colors. They then proceed to pull up the statistics and crudely calculate the probability of this happening (pretending our genetic demographic is unrelated to the circumstances that led us all to this room). The result was some outrageously small number, less than a tenth of a percent. At this point I'm pretty sure my own delusion is taking the piss out of me, actively shoving the implausibility of his own existence in my face as a joke. It turns out all of this really was amazingly coincidental. As weeks went by, guests at the airbnb would come in go, we would meet each others friends, and eventually there were enough people who also acknowledged his existence that I am now convinced he is real. So I pose it to you. Was my roommate real, was everyone involved in this story a figment of a madman's imagination, or am I completely making up this roommate story to make a point? The answer is, reality is shared consensus. If you all are also convinced that this person existed and these events transpired, then we share a common set of facts. If there is no shared consensus, then he only exists for me. Perhaps there is some underlying truth beyond the shared consensus, but shared consensus is the instrument we use to measure realness. At some point, there is no difference between "every multimeter says this battery is 9 volts" and the battery actually being 9 volts. I'm going to relay another story. Neils Bohr used to keep a horse shoe nailed to his door. When asked, he would say its for good luck. One day someone asked "do you really believe that?" He responds "No, but they say it works even if you don't believe in it." Why believe quantum mechanics over lucky horse shoes? If everyone chooses lucky horse shoe theory, does that become reality? If powerful interests in government start forcing everyone to adapt horse shoe theory, does that make it real? Thus I arrive at a truly bothersome set of contradictions. Reality is shared consensus, but reality is also the set of things not subject to popularity. There is no truth only power, but also the essence of science and math is that truth does derive from authority. One day I will reconcile these. One day.
eternalban 4y ago
"2 + 2 = 4" "the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2." Most of us conflate arithmetic with mathematics. In arithmetic, things start getting 'conceptual' as soon as we no longer can map certain measures and operations to a realizable physical construct. At that precise juncture, math becomes a semantic system and is therefore subject to social consensus. For example, consider introducing infinity, or even zero, into 'shopkeepers' sense of numbers. Before, you could never add something to a number and end up with the same number, but now 0 + 0 = 0, and a + \infty = \infty . And to the shopkeepers' surprise, some mathematicians may even argue over it.
samatman 4y ago
Mathematical truth is socially constructed, but using rules, and it is the rules, rather than the process of social construction, which give this process its power. An interesting meditation here on mathematics itself, which is also simply certain rules, and not others. Merely invoking social construction ignores this difference, which is the essential difference, between mathematics and, say, hide and go seek.
kenjackson 4y ago
Truth in math is not a social construct. But belief in math is a social construct.
robertlagrant 4y ago
Social constructionalism is to my understanding least surprisingly found in universities. I think the issue is if you call every type of thought and communication "social construction" then you don't end up with anything useful.
- robertlagrant 4y ago
  Sorry: constructionism*.
kzz102 4y ago
I think this is not the right way to look at it. You can think of mathematical proofs as computer program that is compiled by the mathematician by hand. There is a lot of room for error, but with practice and peer review, it's relatively easy to avoid the common errors. This human compiler also brings the benefit of error correcting, which commonly correct two types of errors: sometimes the proof makes syntactical mistakes that the human compiler fixes automatically, sometimes the proof claims something that's not fully justified (similar to calling a function that is not implemented), but the human compiler just fill in the detail themselves. The social part of mathematics is really about how much error the reviewer is willing to accept, because the reviewer can also be wrong with how they correct the proof.

thaumasiotes 4y ago

> Hou and Luo’s work was suggestive, but not a true proof. That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it — meaning that the solution might be very accurate, but it’s still an approximation. I feel certain that if you run a process that approaches infinity using ordinary floating-point numbers, you will actually reach infinity. This is a case ("can a calculation yield an infinite result?") where computers have less of a problem than people do. You'd have to deal with the question of whether the infinite value accurately reflected an infinite limit of the process or whether it was spurious. But there's no difficulty in calculating infinite values.

blurbleblurble 4y ago
Double precision floats have a maximum value of 1.7976931348623158 E + 308
- onion2k 4y ago
  Aka not infinity
  - PartiallyTyped 4y ago
    They meant maximum finite value. 0 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ 7FF0 0000 0000 0000 (base 16) ≙ +∞ (positive infinity) 1 11111111111 0000000000000000000000000000000000000000000000000000 (base 2) ≙ FFF0 0000 0000 0000 (base 16) ≙ −∞ (negative infinity)
- thaumasiotes 4y ago
  That is not correct; they have a maximum value of positive infinity. See what you get when you square 1.7e+307.
  - mjburgess 4y ago
    A computer is always just a function from {0,1}^N -> {0,1}^M The "Inf" interpretation, of, eg., 11111111111111111 isnt infinity. And, in general, almost every function isnt representable on the discrete domain above. floats are a hacky interpretation of discrete bit patterns

somat 4y ago

"That’s because it’s impossible for a computer to calculate infinite values. It can get very close to seeing a singularity, but it can’t actually reach it" Why not? Is it impossible to calculate infinite values in general? I suspect not, My understanding is that a lot of calculus is in fact on how to calculate infinite values. And a computer is a universal machine, this means that while it can not calculate everything, it can calculate anything that is calculable.

raverbashing 4y ago
Because infinity is a mathematical trick, to say where something is going but it is not calculable (it is not a number in itself)
- agumonkey 4y ago
  well you end up in bounded axiomatic ranges
mjburgess 4y ago
Almost all functions aren't computable, as they aren't discrete. A "computer" is just a function from {0,1}^N -> {0,1}^M
- jojobas 4y ago
  Symbolic engines are way above and beyond number crunching in many respects.
  - mjburgess 4y ago
    Mathematics is constrained by properties of abstract objects that the symbols are about. Here, for example, the mathematician has to imagine a scenario to describe with mathematics (two couter-flow fluids, etc.). The notation gains its meaning from this imagined scenario. Rules for manipulating symbols are therefore insufficient. The proof has to follow from the scenario, which the machine is unable to represent.

wwalker3 4y ago

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes. For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device! Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do. 1) https://arxiv.org/pdf/2210.07191.pdf https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.

nuclearnice1 4y ago
What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
- isoprophlex 4y ago
  I think (not a physicist), simply put, an infinity or NaN value. As these are step-wise methods, having such values show up anywhere will seriously mess up subsequent calculation steps.
- prolyxis 4y ago

photochemsyn 4y ago

> "Yet much remains unknown about the Euler equations — including whether they’re always an accurate model of ideal fluid flow." In general the word 'model' implies a mathematical construct that mimics the behavior of a real-world experimental or observational system, i.e the experimental or observational data can be generated by the model. Where's the real-world ideal fluid flow? No such systems exist. > "In principle, if you know the location and velocity of each particle in a fluid, the Euler equations should be able to predict how the fluid will evolve for all time. But mathematicians want to know if that’s actually the case." At some scale in a real fluid, quantum effects become important, and the notion that it's possible to know both position and momentum falls apart. To quote Peter Atkins: > "The trouble is when you're dealing with operators, is it turns out you can't always extract explicit information about all of them simultaneously, and this leads to Heisenberg's uncertainty principle, which most people think of as a great confuser of the world and denier of information. I like to think of it as a great clarifier because old-fashioned people, like Newton and Einstein, and Lagrange, and all the people who developed classical mechanics, took it as certain that to specify the state of a particle, you had to specify where it was and how fast it was going." That's why there are no perfect ideal fluids. Atkins expands: > "What quantum theory did, through the uncertainty principle, was to clarify - what it said, was discuss the world if you like in terms of positions, or if you prefer, discuss the world in terms of linear momenta, don't try both at once. You get very simple descriptions in terms of positions, you get very simple descriptions in terms of linear momenta, it's only when you - like trying to start a sentence in English and ending in Latin or something, trying to mix the two together, that you get into confusion. So think of the Heisenberg principle as a clarifier - try to think one way, or try to think the other way, but don't try to think like Newton thought."

Ferrotin 4y ago
No, atoms are why there aren’t perfect ideal fluids, and quantum effects have nothing to do with it.

kzz102 4y ago

Computer assisted proof is not a new idea. The most famous example is the four color theorem, which (as far as I know) does not have a humanly understandable proof. For this particular proof, it looks like the computer assisted part would be similar to writing down hundreds of pages of checkable inequalities. One way to do this is to use interval arithmetic, which always guarantees the answer is within the a certain given interval. Mathematicians have split opinions about computer assisted proof. On one hand, there doesn't seem to be a real difference between one page of checkable inequalities vs 500 pages of checkable inequalities. On other other hand, computer assisted proof do not help humans gain clarity on the logical process. The fear is that the computer result will just be an "one and done" result, which people believe is true, but cannot build upon because they don't fully understand it.

lambdatronics 4y ago
> people believe is true, but cannot build upon because they don't fully understand it. OK, maybe it doesn't help people build on the proof, but does it help computers build on the proof? Why does it have to come back to human understanding? Maybe it's not possible right now for computers to generalize proof strategies, but I wouldn't bet against it becoming possible down the road.