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OK, maybe the title is exaggerated, but is it true that the rest of math is just "good enough", or a good approximation of absolute truth - like Newtonian physics compared to general relativity? How do we know that our "approximation" is the right one? Another analogy: Fundamental physics is also not well-fundamented (where is the Higgs boson?) but most of the rest of the physics is on top of it, and it does its job well (it's a good-enough approximation).

Summary: According to the responses, math is indeed an imperfect domain, but can be seen as perfect for all practical purposes. In this case I wonder if math is indeed pure and identical across all possible universes. Maybe another universe comes up with a different set of axioms, more or less consistent than what we have now.

Adrian
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    There is no reason to think that general relativity is absolute truth. The universe is the universe, and general relativity is a good mathematical model of some of its features. – André Nicolas Aug 01 '11 at 16:57
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    I think you do not actually understand what Godel's (in)completeness theorems are about (though, admittedly, your presentation seems to be similar to those in a lot of popular books in mathematics). Also, the concept of "absolute truth" is philosophical, not really mathematical. – Willie Wong Aug 01 '11 at 16:58
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    Why bother? I, for one, have lots of fun doing math... – Mariano Suárez-Álvarez Aug 01 '11 at 17:00
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    The way relativity showed that newtonian physics is incorrect, but just a (mostly acceptable) approximation, the same way Gödel's Incompleteness Theorem showed that math (as in Principia Mathematica) is incorrect if stretched to its limits. But it still did not change the rest of "general math". I find this a bit unsettling. – Adrian Aug 01 '11 at 17:05
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    Incompleteness has nothing to do with approximation. Gödel emphatically did not show that math is incorrect. Stay calm and let someone explain, all will be well. – anon Aug 01 '11 at 17:05
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    Why bother determining the position of a car about to hit you if Heisenberg's uncertainty principle is true? – Qiaochu Yuan Aug 01 '11 at 17:07
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    A not-so-well-kept secret of mathematics is that mathematicians are not that bothered about rigour. Mathematicians have been working with things not well-defined since the dawn of time and will continue to do so. What changes is the standard of ‘well-definedness’, and from time to time there will be a crisis over the lack of foundations. People have been doing arithmetic for aeons before Peano wrote down his axioms, and for the most part, no one worried about it. Perhaps a better question to ask is, what is mathematics? – Zhen Lin Aug 01 '11 at 17:10
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    Why bother with history, if we will never know the number of hairs Julius Caesar had on his head? – Levon Haykazyan Aug 01 '11 at 17:12
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    A much more productive question for this site is: What does Godel's (in)completeness theorem tell us? I am sure that'll at least clear some of your misconceptions. – Srivatsan Aug 01 '11 at 17:24
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    Ask again if you have proven all you can prove. – Raphael Aug 01 '11 at 18:05
  • From the Wiki: "Gödel's incompleteness theorems show that in certain cases it is not possible to obtain an effectively generated, complete, consistent theory." Yeah, math is wrong, right :\ – Aleksei Averchenko Aug 01 '11 at 18:17
  • @Adrian Why bother leaving your house now? You could get hit by a falling brick from a building and die! So never leave your house. –  Aug 01 '11 at 21:54
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    @D Lim: The problem is, if you are really gonna doubt and be skeptic until the very end, you must also concede that the brick might hit you while you are in your house. Better not to live at all, it kills all the insecurity for sure. – Raskolnikov Aug 02 '11 at 08:57
  • Agree with @anon, incompleteness certainly has nothing to do with mathematics being an "approximation". – Josh Aug 02 '11 at 09:55
  • @Adrian, re your edit: One does not need to think about other universes having different mathematical axioms. Right here on Earth there are mathematicians who study the possibility of mathematics under different axiom systems. Perhaps if history turned out differently we would have the axiom of determinacy instead of the axiom of choice. – Zhen Lin Aug 03 '11 at 03:00
  • @Zhen Lin Agree. Going out on a limb, I would say that the axioms are likely influenced by human psychology, as it evolved in this planet, or it may evolve on another planet... – Adrian Aug 03 '11 at 04:22
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    These arguments are funny. Godel himself wrote a letter to Von Neumann in which he basically said "If P = NP, then nothing will be left for human mathematicians apart from the postulation of axioms." (He used different language because P v NP wasn't really known at the time). See this paper and my answer to another Godel question for more. – ely Apr 14 '12 at 02:05
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    ... and concerning the Higgs: it was found at the LHC ... – draks ... Aug 01 '12 at 20:06

6 Answers6

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The first incompleteness theorem says that, under appropriate conditions, and for appropriate definitions of "true" and "provable," there exist statements that are true but not provable. This is interesting, but not a big deal. It is more or less a consequence of the existence of nonstandard models in first-order logic, which is an interesting feature of first-order logic, but mathematics is more than first-order logic.

The second incompleteness theorem says that, again under appropriate conditions, a sufficiently strong formal system can't prove its own consistency. Okay, this is kind of a big deal, but in practice it doesn't matter as much as it sounds like it does. Mathematicians don't actually do everything in a fixed formal system. There is a whole subject called reverse mathematics dedicated to finding the weakest formal systems that are capable of proving various things, which vary widely.

There is a possibility that ZFC, the formal system that sounds like it's what mathematicians work in (but isn't really), could be inconsistent. So what? If that ever happened we would just find another set of axioms to use. ZFC is a ridiculously strong formal system and in practice we never use its full strength, so it wouldn't really matter if it were inconsistent.

Mathematics is not the study of what formal statements can be proven in ZFC (although there are mathematicians who study this kind of thing). Proof: mathematics is thousands of years old, and ZFC is not.

Qiaochu Yuan
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    One could add that there is such a thing as experimental mathematics... Wolfram has tried to explore this... sometimes a computer crashes into the gate of the marriage between geometry and number theory... apologies... – Condor Aug 01 '11 at 18:25
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    In first-order Peano arithmetic, the statement that there is no natural number greater than $1$, $1+1$, $1+1+1$,$\dots$ is not a statement. – André Nicolas Aug 02 '11 at 05:09
  • @André: oops. I could add a constant and use a sequence of statements, but I feel like that looks less impressive. I was hoping to avoid a more algebraic example... – Qiaochu Yuan Aug 02 '11 at 13:35
  • Could you elaborate (in a comment is fine, if it fits!) on the statement that "ZFC ... sounds like it's what mathematicians work in (but isn't really)" ? (I understand that it could be inconsistent, but not that we aren't actually working in it!) – The Chaz 2.0 Aug 02 '11 at 21:21
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    @The Chaz: ask any practicing non-logician whether they can list the axioms of ZFC off the top of their head. They probably can't. That's because we don't do mathematics by checking if all of our constructions make sense in ZFC. They probably do, but that's of secondary importance. Like I said, people did mathematics just fine for thousands of years before ZFC existed. – Qiaochu Yuan Aug 02 '11 at 21:30
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    @Qiaochu: Your portrayal of mainstream mathematicians as being happily unconcerned (if not ignorant) of such foundational integers rings true to me (not to mention about me; I certainly don't care about these things professionally, although from time to time I get idly interested). But I think you may be underplaying the significance of some of these issues. 1) Both of Godel's theorems are a "big deal"; they are probably among the most important theorems in math.... – Pete L. Clark Aug 03 '11 at 02:39
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    ...2) I would be flabbergasted if ZFC were proved inconsistent because all of its axioms express what I believe to be true facts about sets. I think this is a more or less mainstream position. What's not such a big downer is that you can't prove the consistency of ZFC within ZFC: since, as you say, I couldn't recite the axioms of ZFC to you anyway, this is nothing that I have ever tried or would want to try to do. If, as Hilbert hoped, this had actually panned out, it would be a pretty neat thing. But the lack of it has no impact on my mathematical life. – Pete L. Clark Aug 03 '11 at 02:41
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    @Pete: yes, I admit I'm downplaying things. But when the popular conception is "Godel's theorems prove that math is broken" it seems like a good direction to go is downplaying. Feel free to restore the balance a little in your own answer if you choose. I am a little suspicious of the fact that you say both that the axioms of ZFC express what you believe to be true facts about sets and that you can't recite the axioms to me! – Qiaochu Yuan Aug 03 '11 at 02:52
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    @Qiaochu: I couldn't recite the "Second Isomorphism Theorem" for groups to you either, but I saw once that it was one of many facts about groups that I know to be true. It's similar with ZFC: I look at the list of axioms and I say, "Yes, that's right, sets satisfy these properties." But then I forget exactly what's on the list. – Pete L. Clark Aug 03 '11 at 03:51
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    Also, it's no big deal, but I don't really understand why you're giving an answer which is different from what you actually think in order for the weighted average of the answers to be closer to the truth. If the question is "How many sporadic simple groups are there?" and four other people have already said $25$, I presume you would say $26$, not $30$. – Pete L. Clark Aug 03 '11 at 04:00
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    @Pete: fair enough. As for your other question, I guess I don't feel qualified to discuss the ways in which the incompleteness theorems are important, and I was hoping another answer would do this instead. This is an important issue and certainly not as clear-cut as the number of sporadic simple groups. – Qiaochu Yuan Aug 03 '11 at 04:07
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    @Pete: Maybe they are saying 25 because someone slayed the monster! – Michael Joyce Apr 14 '12 at 04:07
  • I can't accept "thousands of years" as a proof of something like consistency. Everything goes around earth was also the major view for "thousands of years"... – Zeta.Investigator Sep 05 '12 at 09:44
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    @Pooya: you're misinterpreting what I'm saying. I'm not claiming that anything is consistent. All I'm observing is that mathematics predates ZFC by thousands of years. This is not contestable. – Qiaochu Yuan Sep 05 '12 at 16:36
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Your writing demonstrates some pop-math-esque misconceptions surrounding Gödel's Incompleteness Theorems. The analogy between math and physics is inaccurate, for one, because physics is specific to the contingent governing rules of our universe, whereas math is transcendent in that it looks toward truths that must hold in every possible world, i.e. are logically necessary. If I draw a curve on a piece of paper and ask you to model it, just because you only find yourself capable of approximating the curve doesn't change the reality that there are true facts about curves in general that can be investigated and discovered. Just because you don't know everything about the user "anon" on Math.SE doesn't mean you're incapable of knowing things about human beings in general. Just because we can't yet for certain nail down the exact form of the universe doesn't mean we can't figure out the logic of space and time and combination at all.

The key to understanding this is: Gödel did not demonstrate any of mathematics was incorrect or inaccurate in any way. I'm not sure how you even came to that misinterpretation. The theorems show, in a nutshell, that any formal mathematical theory with a given set of axioms (starting assumptions) and a given set of inference rules (ways of deducing things), which is capable of expressing basic arithmetic, is only self-consistent if it is incomplete (there exists some true proposition that the theory is capable of expressing but not proving) and if it is incapable of proving its own consistency.

The consistency of mathematics isn't really a problem; we can be confident of all of our theorems exactly as much as we can be confident in all of our axioms taken as a whole. The only real contentious one I'm aware of is the Axiom of Choice, but it's instructive to know that we have yet to have ever generated a contradiction or falsehood of any sort from our standard axioms. So why not do mathematics, as a society (I leave out personal reasons for doing math, as that is essentially another discussion entirely), if it has a pristine, 100% perfect track record of getting everything right? If absolute certainty is the standard for the worth of human endeavor as you tacitly posit, then that would make mathematics literally the most respectable endeavor humans have ever achieved.

The incompleteness of mathematics is likewise not really a problem; all of the ideas we've discovered so far that are undecidable, are either so extremely far-removed from reality and our lives that they are effectively insignificant or meaningless to us, or they are still far-removed but capable of being proven within a stronger system. Incompleteness means we will never fully have all of truth, but in theory it also allows for the possibility that every truth has the potential to be found by us in ever stronger systems of math. (I say in theory because, technically, the human brain is finite so there is an automatic physical limit to what we can know.) In a way, instead of being unsettling, incompleteness should almost be reassuring and reinvigorating to mathematicians, because it means the adventure is never-ending.

anon
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    The axiom of choice is contentious (to some), but not because of consistency issues. It has been shown by Godel and Cohen that we are free to add either the axiom of choice or its negation without fear of adding any new contradictions which weren't already present. – Jason DeVito - on hiatus Aug 01 '11 at 22:00
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Okay. They used a differential equation in establishing the engineering specifications used for building a certain bridge near here. Now that the troublemaker Gödel has had his say, should I conclude that differential equations are no good, and therefore avoid using that bridge?

Seriously: we can still do mathematics. It is just "Hilbert's Program" that should be (and was) abandoned.

GEdgar
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    I am sure you can ignore both Godel and Einstein while building a bridge. It's just that math is supposed to be more "pure" than anything else, and I thought that mathematicians are bothered by Godel more than a structural engineer is bothered by Einstein. – Adrian Aug 01 '11 at 17:53
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    I don't think engineers can ignore Einstein as easily as you think. I hear GPSes use general relativity! (This was a random link I got by searching for "GPS general relativity" http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html .) – Srivatsan Aug 01 '11 at 18:28
  • GEdgar, The previous comment was, of course, addressed to @Adrian. I forgot to add his username, and was too late to edit the comment. – Srivatsan Aug 01 '11 at 18:35
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I think this thread could also benefit from my answer to another Godel question. But more importantly, it could benefit from looking at this paper written last year by Scott Aaronson.

The OP question is absolutely a legitimate one. I hate to see people saying that "you don't understand Godel's theorems" if someone asks this... because by that logic then, if you are a reductionist, you must also believe that Roger Penrose doesn't understand Godel's theorems.

Community
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ely
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    I have no opinion, but it's worth knowing that people do indeed argue that the Penrose argument misunderstands the incompleteness theorems. For example, see http://www.theassc.org/files/assc/2327.pdf – Carl Mummert Apr 14 '12 at 11:43
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    About your second paragraph: And yet, some people do not understand Gödel's theorems... Hence why should one object to other, more expert, people saying so? Especially when said experts then embark on some precise debunking of said misconception? In the case at hand, it seems (to me) most valuable to explain why the good approximation of absolute truth stuff in the question is just wrong. – Did Apr 15 '12 at 09:34
  • @Didier, I agree with you 100%. That component of this question should be isolated from question that the title of the post asks. Godel himself asked the question that is in the title and the only answered that seemed to satisfy him was "well, let's hope $P\neq{NP}$". But the stuff about approximating absolute truth, etc., is not appropriate for Godel's theorems. That's a whole different philosophical beast and the analogy with Newtonian-to-Quantum physics isn't a good one given the state of the art in the philosophy of truth. – ely Apr 15 '12 at 17:33
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Godel's theorem says what we should expect anyways, namely that one cannot simply write down some simple rules, and mechanically derive the deepest mysteries of our Universe.

Godel's theorem is only a limitation of what mechanical non-thinking beeings can figure out about math, truth and the Universe, it does not represent an ending of reasoning.

TROLLHUNTER
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Why do maths at all? It's fun, it's interesting, it tells us about the world we live in, and a bunch of other stuff. Gödel's incompleteness theorem doesn't tell us we're not really getting anywhere - it's a statement about the confines of pure logic, not a statement about how maths relates to the world. But even if it did tell us we were going to hit a wall somewhere, we've got pretty far along in our quest for knowledge so far - wouldn't you be interested in seeing how far we could get?

Billy
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  • Just to stretch it a bit, Astrology could be fun and interesting too. – Adrian Aug 01 '11 at 17:11
  • @anon I honestly believe that the question is valid, and it was not properly addressed here or elsewhere. Of course I don't expect a definite answer in the near future either. – Adrian Aug 01 '11 at 17:19
  • If you don't believe my answer was an answer, the most helpful thing for you to do is tell me why, not make cryptic comments. – Billy Aug 01 '11 at 17:20
  • I believe that math is fun and interesting. But if somehow it may not ne well-fundamented, it may go to waste. Like Principia Mathematica did. OK, the astrology parallel was a bit rude, sorry. – Adrian Aug 01 '11 at 17:34
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    @Adrian Sure. It may have gone to waste if it wasn't well fundamented. But, importantly, it hasn't gone to waste. It has proven its worth. It has told us lots of fun and interesting and, most importantly, correct things about the world. As I said, this isn't surprising when you realise that Gödel's statement sits very firmly in the realms of logic. As an analogy, take a piece of paper, write "this statement is false" on it, and conclude that English is broken. If you want better than an analogy (which is all popular maths books give...), you need to start with a study of formal logic. – Billy Aug 01 '11 at 17:52
  • @Adrain: I agree the question is perfectly legitimate and worthwhile. The way you broached the issue with the community here, however, reminded me very strongly of some good times had getting my leg jerked over the internet. Nothing personal. – anon Aug 01 '11 at 18:21