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Is there any mathematical statement or theorem or theory, which was used to be TRUE in the past, but then found out FALSE later? In short, my question is: everything in math that we have found and proved to be TRUE so far (theorem, theory, etc), will it remain true forever?

More generally, natural science in general, may be true in the past, but then may turn out to be false later. Is math an exception?

SiXUlm
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    Sounds like you're asking us to predict the future. BTW, apart from the future-prediction request, I think that it's a very interesting question, though more philosophical than mathematical, so I doubt you'd get any concrete answer here (or anywhere else for that matter). – barak manos Jan 15 '16 at 12:16
  • not really, I'm asking for something which we already found out up to now, not what we will or may find out in the future :D – SiXUlm Jan 15 '16 at 12:20
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    You're asking if something that we have found TRUE in the past will become FALSE in the future. That pretty much goes under the definition of predicting the future IMO. – barak manos Jan 15 '16 at 12:22
  • Ooops, pardon my capitals on TRUE and FALSE, was in the middle of trying to draw some stuff with Win32 API... – barak manos Jan 15 '16 at 12:23
  • I would not even call math a science whatsoever. In my view it is more an art. The only thing you need is a human brain that constructs his own theories and dives deeper into it. In that sense math differs from natural science wich needs external fenomena to be studied. If anything has been proved to be consistent with its surrounding theory then that will not change in the future. – drhab Jan 15 '16 at 12:25
  • Yes, it will be true forever. Regarding whether math is an exception... define "true". This is off topic here, I believe. – Stefan Perko Jan 15 '16 at 12:33
  • Parallel postulate (https://en.wikipedia.org/wiki/Parallel_postulate) is close to what you're asking. It is true for Euclidean geometry but doesn't hold in not Euclidean geometry. – mathcounterexamples.net Jan 15 '16 at 12:36
  • @mathcounterexamples.net And that's commonly taken to be an axiom, and even if it was an honest theorem, it is still a non-example. Just because you can consider non-Euclidian geometry, does not make Euclidian geometry "wrong". Reality does not matter for mathematical truth. – Stefan Perko Jan 15 '16 at 12:42
  • @StefanPerko I'm not considering reality in my comment. Something that was TRUE and becomes FALSE can be either because the axioms changed, because the deduction rules changed or because there was an error in a proof. What I meant is that parallel postulate was considered as an axiom and that now we're doing geometry with or without this axiom. Changing the axioms imply changes in what is TRUE or not. – mathcounterexamples.net Jan 15 '16 at 12:48
  • @mathcounterexamples.net That is a matter of convention, but I suppose since the question is philosophical in nature this kind of discussion ought to be expected. What I meant is, if you change axioms or deduction rules you are simply considering another logical system. Otherwise it's like saying "not all elements in a group are invertible" because somebody made up the definition of a monoid and called it a group. – Stefan Perko Jan 15 '16 at 13:01
  • Mathematics will be true as long as humans will exist.After that the human kind will be gone who will check up our proofs ? – Mr. Y Jan 15 '16 at 13:25
  • @drhab: Your comment suggests that you do not know how much modern technology depends on the real-world truth to some extent of some mathematical facts, such as Fermat's little theorem in HTTPS, and real analysis in modeling engineering constructions. In no way are these mere mental constructs of humans but attempts to understand the real world. There are of course areas of mathematics that do not share this status, but not all are like that. – user21820 Jan 17 '16 at 01:59
  • @StefanPerko: Reality does not matter when considering the mathematical definition of truth (over classical first-order logic and in a meta-system). But reality does matter when asking whether there is a reasonable interpretation of mathematical statements in some formal system such that all statement that it proves would be interpreted to assert something true about the real world. The latter is probably what the asker had in mind, not realizing that within mathematics itself truth can never be defined (as Tarski noticed long time ago). – user21820 Jan 17 '16 at 02:04

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I assume you're talking about mistakes, rather than changing truth (maybe we'll find that our axiom system is inconsistent, we can't prove it otherwise) which would be most unfortunate. Check out https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry for an example of established theory which turned out to be rubbish.

Sean D
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Truth cannot be defined precisely. Mathematically, you can only define provability, which is whether a statement in some precise language can be derived in a sequence of steps from axioms according to some fixed deductive rules. If you decide to use a different language or axioms or rules, you would of course get a different collection of provable statements. The language, axioms and rules together form a formal system.

Whether one formal system is more 'true' than another leaves the realm of mathematics and goes into science and philosophy. Science, because we could define 'truth' as what is true in the real world, and some of these truths can possibly be empirically verifiable to some extent. Philosophy, because one has to make some initial assumptions anyway, such as that there is actually a real world...

But at least for this definition of truth based on the real world, it is not impossible that in the future someone might empirically verify some real world fact that contradicts the standard interpretation of some mathematical theorem. In that case, we would know that our mathematical model of the world is flawed, and examine to see what other theorems would be invalidated as well.

Historically, there was naive set theory that was proven to be inconsistent, which implies that every statement of naive set theory can be proven, both it and its negation, and so technically one could say that 'half' of all provable theorems would be false, regardless of whatever interpretation we choose. That was 'fixed' by modifying the axioms of set theory, so now we have ZF[C]. As of today mathematicians have proven a lot of statements in ZFC, so it is conceivable that if ZFC turns out to be inconsistent, then we might find that two of the published theorems contradict one another, in which case we would have to decide which one to reject. However, most mathematicians believe that ZFC is consistent.

user21820
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Don't forget that Math and science are two different things. In science, truth is rarely if ever absolute. We just say things are true when there's so much empirical evidence for them. However some very basic statements can be known. For example "something exists other than nothing." If that were not true then we could not be even pondering the question.

In contrast, in math we work in purely formal systems and we say things are true when we can logically derive them from a set of axioms - where axioms are a set of statements we all agree on as a starting point - since you have to start somewhere. The problem is, for most sets of axioms we cannot prove they are consistent in the sense that no contradictory statement could ever be derived from them. Therefore ultimately we can never know for sure that mathematical results are even sensible. However I think most people believe that statements such as "There are infinitely many primes in the natural numbers." must be true in some absolute sense, even if humans can't prove a set of axioms of arithmetic is consistent. I mean most people believe the axioms are consistent, even if we can't prove it. If they aren't consistent, then that would really be mind blowing.

However, all that being said, there are undoubtedly proofs in the literature that are wrong and will some day be discovered as such. But that's not to say those things will go from true to false, it just means they were never true to begin with. If you believe in absolute truth, then things are either true or false and nothing can really go from one state to the other just because human beings change their minds about what to believe.

Gregory Grant
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  • In what absolute sense do you think "infinitude of primes" is true? What if the universe is discrete and finite in spacetime and energy, and hence there are not even infinitely many natural numbers in any sense? – user21820 Jan 15 '16 at 13:11
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    @user21820 I meant true in the abstract domain, not true in the sense of there being a physical manifestation of them. – Gregory Grant Jan 15 '16 at 13:23
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    But what is the abstract domain? I am well aware of all the usual ways to explain induction, but all of them without exception rely on some physical intuition that may well break down at very large numbers. – user21820 Jan 15 '16 at 13:31
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    @user21820 No, I'd disagree. The infinite structure of natural numbers, and the induction principle which is true of it, are independent of whether there are $0, 17, 10^54, \aleph_0, $ or more particles in the physical universe. Induction is basically the statement that $\Bbb N$ is well-ordered: every nonempty set of integers has a least element. I don't know of any would-be explanation of induction which appeals to physical intuition. You yourself may translate the term "least" into some spatial relation, but then it's you doing that, not the explanation. – BrianO Jan 15 '16 at 14:30
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    The notion that induction might break down at very large numbers sounds like a form of ultrafinitism, a defensible point of view but hardly mainstream among mathematicians. // Erratum: in my previous comment, "$10^54$" should be "$10^{54}$". – BrianO Jan 15 '16 at 14:36
  • @BrianO: Then you're just assuming the truth of induction or well-ordering, and cannot possibly justify it. That's my point. – user21820 Jan 16 '16 at 02:48
  • @BrianO: Perhaps you do not understand proof theory. PA is usually defined with the signature $(+,\times,0,1)$, and the standard natural numbers can be observed (from within a stronger meta-system) to be representable by all the strings of the form "$1+1+\cdots+1$". This is the actual concrete representation that underlies our understanding of counting, and the intuitive properties of counting numbers were later abstracted out into the axioms of PA. Induction captures the idea that only strings of that form correspond to natural numbers. Now what if strings can't go beyond some length? – user21820 Jan 16 '16 at 03:04
  • @BrianO: By the way, I'm well aware of ultrafinitism. They assume that induction will break down at large numbers. I don't, but rather I'm objecting to with the claim that most people believe in some kind of abstract truth with no physical intuition behind it. Everybody who claims to understand induction are relying on some very physical intuition, even if they don't realize it. Since I also do not assume that the universe is infinite, I therefore stated that it may well be that our intuitive understanding of natural numbers does not apply for large numbers. – user21820 Jan 16 '16 at 03:09
  • @user21820 I don't understand what you mean by "what if strings can't go beyond some length?" What would be the reason for such a limitation? I mean sure in the physical world there may be a maximum number of things you can put in a string. But in our minds in the abstract domain we can always add one to any natural number so they cannot be finite. – Gregory Grant Jan 16 '16 at 03:09
  • Gregory, your abstract domain is based on what you can do in the real world. As I said already, the idea of being able to add one item to a list is the key assumption in proof theory. If you cannot in the real world add items to a list, you would never have thought that you can also "abstractly" do so. Furthermore, you assume that adding one does not cause it to become a previous natural number. How do you know that? Well again it's because you rely on the intuitive understanding of lists, where increasing its length can never make it the same as a previous list... – user21820 Jan 16 '16 at 03:12
  • @user21820 I respect your view, but I actually have the opposite belief. I believe in that math is the only thing that really exists and the physical universe is just a complex mathematical system so complex that it implies consciousness. The universe only becomes concrete in our minds, and it's tangibility is just an illusion. This is called the Mathematical Universe Hypothesis (MUH) popularized by Max Tegmark. The nice thing about this is that it explains how we got everything from nothing. Because math exists a priori. See here: https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis – Gregory Grant Jan 16 '16 at 03:16
  • I have read about that viewpoint before, but I'm afraid it's not tenable. What kind of mathematical universe exists a priori? Is it a set-theoretic universe? Is it ZFC? Don't forget Godel's incompleteness theorem, which shows that within any formal system that has arithmetic on natural numbers, there are statements that are neither provable nor disprovable, so there are two incompatible extensions of the formal system obtained by adding any such statement or its negation. Which then is true in your mathematical universe? [continued] – user21820 Jan 16 '16 at 03:32
  • [continued] If you say all consistent universes exist, then you still need some overarching universe to contain them all. But again, the same holds for this universe, unless you stipulate that there is no formal system that captures this overarching universe. But if so, then it is not called mathematics anymore unless you subscribe to Brouwer-style intuitionism. If there is a formal system for the overarching universe, then either it does not have arithmetic on natural numbers or it is again incomplete. And all this is just concerning formal systems extending PA or ZFC. What about others? – user21820 Jan 16 '16 at 03:36
  • If there's any part of my comments that you'd like me to explain in detail, I'd be glad to do so, but perhaps we should create a dedicated chatroom for that. =) – user21820 Jan 16 '16 at 03:40
  • @user21820 Yeah I don't think we're supposed to discuss this at length here. Can you start an appropriate discussion somewhere and point me to it, because I do want to reply. You think mathematical truths have to happen somewhere, which means you have to already have a "place" for them to happen, which makes MUH sound circular. But MUH rejects the contention that math has to have a place to exist. The fact that you cannot create a place where there is counting but there are only finitely many primes, means the infinitude of primes somehow supersedes all physical reality. – Gregory Grant Jan 16 '16 at 12:34
  • Let us continue this discussion in chat. – user21820 Jan 16 '16 at 13:07
  • Are you able to respond in the chat-room I created? Anyway let me just make one more remark here. You are not at all justified in claiming that there cannot be a 'place' where there is counting but only finitely many primes. Just because you cannot think of such a 'place' does not mean it does not exist. Better still, such actually does 'exist'. ( ZF + not Inf ) is bi-interpretable with PA, but there is no infinite set! Also, if counting is impossible beyond a certain size, which I mentioned right at the start, then the question of having counting but finitely many primes is simply irrelevant. – user21820 Jan 17 '16 at 01:51
  • Although I believe that PA (with induction) is consistent up to any physically possible length of proof, it does not mean that it is true. Consistency means very little actually, since ( PA + not Con(PA) ) is consistent too. – user21820 Jan 17 '16 at 01:54
  • @GregoryGrant: The chat-room was frozen because of inactivity, so if you're interested feel free to start another one and respond to my question there, and I'll reply when I see it. – user21820 Feb 21 '16 at 14:17
  • @user21820 Oh man I didn't know they froze chats, that seems wrong. I still intended to take up this discussion it's just never seem to find the time because I know it's going to get involved and time-consuming. I'll start a new chat when I get up for air. Thanks, G – Gregory Grant Feb 21 '16 at 14:33
  • @GregoryGrant: Sure sure. No hurry. I gladly look forward to our discussion. =) – user21820 Feb 21 '16 at 14:45