## The truth about mathematics.

I often read paraphrases of one or other of Gödel’s theorems that talk about true, unprovable statements. I’ve said before that I’m a formalist of sorts. Talk of undecidable statements in a system being true gives me headaches. And I’m an analyst so I work in ZFC. If I say “statement X is true” I’m telling you that there exists a proof of statement X in ZFC. If you ask me if I think the continuum hypothesis is true, I’ll explain to you that it’s known to be undecidable. If you tell me you know it’s undecidable but still want to know if I think it’s true, I’ll look at you as if you asked me what colour integrity is.

If it was clear that the question “Is X true?” didn’t mean “Can X be proven in ZFC?”, say the question was “Is the Axiom of Choice true?”, I guess I’d them take an explicitly formalist stance: “I don’t think that either AC or its negation is truer than the other; the term “true” doesn’t really mean anything in this context”. We analysts pretty much always work using the axiom of choice, but for reasons that don’t have anything to do with delcaring it to be “really, really true”. We use choice because it’s better for doing the maths in. Results would need expressing in tiresomely fussy ways if we couldn’t reach for Zorn’s lemma or the Hahn-Banach theorem.

So, “true” is a word I might use to talk about maths but “true” as I would ever use it isn’t itself maths; the meaning is a bit slippery as you would expect for a word in a natural language. Also any uses I would give it wouldn’t be allowed to slip outside a particular system. I don’t think this is a particularly unusually position for a mathematician to take. All the same writing aimed at non-mathematicians often goes like this by Chad Orzel:

…Kurt Gödel’s famous Incompleteness Theorem, which shows that any formal logical system complex enough to describe arithmetic must allow the formation of statements that are true, but cannot be proved to be true within that system.

Now I’m not saying that this formulation is unsupportable since (some) people who have looked at this very deeply accept it. Roger Penrose always states it like that and I understand Gödel himself saw it that way. Also, at a push, I can sort of see it; the Gödel sentence (which is proved to be undecidable) “says” that a certain number is not the Gödel number of a provable statement in the system. If such a statement existed so would its proof and that proof could be turned into a proof of the negation of the Gödel sentence (which of course we can have). Therefore it is true that the number was not the Gödel number of a provable statement in the system.

OK, but you see those quotation marks around “says”? What if we take a “glibly” formalist view of this; that mathematics is a game played by the formal manipulation of meaningless symbols? Then the formula doesn’t say anything at all; the interpretation may be handy for thinking about it (like thinking of Euclid’s axioms as being about space) but it doesn’t “really say” anything, especially if take them out of the system and look at them on there own. The theorem is (as was cautiously pointed on the In Our Time episode on the subject but is usually ignored in popular treatments) purely syntactic and not necessarily about any semantic interpretation.

Again I’m not saying the version with “true” isn’t true; I’m just saying it is a headache. If it is true, it’s true in a way that’s somewhat complicated. And, note, this is the version we see in things for a general audience.

Help me out here; is it really that laypeople can understand this with no trouble, but it makes my mathematician’s head explode? Or is it that the popularizations are actually close to incomprehensible but people just don’t notice because they are typically happy with intuition that statements are (“really”) either true or false?

### 13 Responses to “The truth about mathematics.”

1. G.D. Says:

You seem to going off in two different directions here.

If one adopts the kind of formalism you seem sympathetic to in general, the problem isn’t primarily with truth – the deep problem is that the languages of mathematics are meaningless. If mathematics are just a collection of uninterpreted languages, of course statements in them aren’t true or false and of course the Gödel theorems don’t show that truth in any ordinary sense and provability come apart (there is a sense in which they don’t show anything at all; they are just certain dots of ink on paper, structured after certain (aesthetic) rules which characterize them as “provable”). Of course this kind of formalism avoids certain ontological problems – but at quite a cost.

But notice this: I can define a predicate T with the following axiom schema (for some higher-order system L* incorporating such systems as ordinary arithmetics)

“S” is T in L* if and only if S

(of course, you’ll need the distinction between various orders, a hierarchy of (in the case of formalism, uninterpreted) systems, but I don’t see the problem about this, even given formalism). And with this schema (and the higher-order structure), I can show that provability isn’t coextensive with the application of the predicate T (i.e. with the Gödel sentences). This result is still formally interesting, but it doesn’t say anything about truth in any ordinary language sense. That is because NO mathematical statement is true or false in that sense given formalism. There is no special problem with Gödel’s theorems.

But your earlier considerations involving the sentence X provable in ZFC but not in ZF seems to go in a different direction – that one seems to indicate sympathy towards some kind of intuitionism, and intuitionism and formalism are incompatible approaches. Intuitionism is (among other things) a view about what mathematical truth amounts to. And some versions (not all) of intuitionism do indeed have trouble with the Gödel theorems. But they don’t have any trouble with truth in mathematics in general. Indeed, intuitionism is motivated, in part, by taking mathematical truth extremely seriously.

Anyway, your post touches on several deeply problematic issues and I don’t think I am in any position to solve them.

2. mattheath Says:

Thanks for you reply. My only really important point here was that talking about truth with regards Gödel’s theorem makes some complicated assumptions about the nature of mathematics which it isn’t useful to make in general audience writing. I think what you say about various positions problems with it supports that.

To clear up what else you had to say. I don’t think I’m defending contradictory positions as to the nature of mathematics. I’m pretty sure if I sound like a inuitionist anywhere it is only because I am not expressing myself sufficiently clearly in this post. As I understand it intuitionists reject the axiom of choice, while as I say I always work with it .

I’ve just realised that I put the wrong URL in my first link it should have gone here http://mattheath.wordpress.com/2008/12/12/reasonable_effectiveness/

It explains the extent to which I am “sort of” a formalist but not what is sometimes called a “glib” formalist. Briefly, I think that out of the vast array of possible formal games we could play, our predecessors picked ones that have some connection to our universe for non-arbitrary, human reasons, but all the same they a still “only” formal games without any metaphysical claim to truth..

My position on word “true” is only that is a word in ordinary English and like all words in natural languages it doesn’t have a completely consistent meaning that makes sense in all circumstances. My description of how I, as working mathematician, would use the word in relation to mathematical statements is designed to show that it is messy and really not the sort of thing you want in the statement of a theorem. I don’t think my usage is the “really real” meaning of truth in mathematics and I wouldn’t certainly say mathematics done in any other system is just as true.

3. Daryl McCullough Says:

Matt,

I agree that it is not completely clear what the question “Is the continuum hypothesis true?” means. However, I think it is clear enough what the question “Is it true that every even natural number is equal to the sum of two prime numbers?” In the latter case, it certainly does *not* mean “Is it provable in ZFC that …?” For questions about arithmetic to ask “Is it true that every X has property Y?” is to ask, simply “Does every X have property Y?” The Tarskian notion of truth is appropriate here. And this is the notion of “truth” that is relevant in Godel’s theorem.

4. TooMuchCoffeeMan Says:

Daryll,

How about the question “Does every Goodstein sequence terminate?” Perfectly arithmetical statement…

I’m not sure what you mean by your last two sentences. What reality are we ascribing to X, if any?

5. Daryl McCullough Says:

TooMuchCoffeeMan,

Yes, to me, the question “Is it true that every Goodstein sequence terminates?” means exactly the same as the question “Does every Goodstein sequence terminate?” The use of “Is it true that” is eliminable.

In Godel’s theorem, Godel constructs a sentence G such that (under some additional assumptions about the soundness of PA, Peano’s theory of arithmetic) G is true, but PA doesn’t prove G. The use of “is true” here is in principle eliminable, but in practice, eliminating it would make the claim humanly incomprehensible. Instead of saying

G is true but PA cannot prove G

one could spell out G in all its gory details and write

forall natural numbers x, blah blah blah,
but PA cannot prove forall natural numbers x, blah blah blah.

where blah blah blah is some enormously complicated statement involving the variable x. After expanding G to get an explicit sentence of arithmetic, the use of “true” is no longer necessary. True is only necessary if we are using a *name* for the sentence instead of explicitly quoting the sentence.

6. mattheath Says:

Daryll,

That’s how I’m taking “true” in the part where I say “I can sort of see it”. But as I say and (if I understood rightly) G.D. agreed this does rely on the statement having the semantic content about numbers and not just being a “meaningless”, uninterpreted formula (as I understand many respectable people would argue it was). It also relies on the formula still having this meaning when we look at it from “outside” as opposed to only meaning that relative to the system around it.

Let me repeat again, I’m not asserting that this is wrong. Just that if it is right it relies on deep subtleties of the philosophy of mathematics that are apparently not settled amongst those who specialise in this stuff (see of what G.D.. wrote about formalists and intuitionists different problems with it). Thus, I think the language of truth it is best avoided if you are writing about a theorem like this in, say, New Scientist. Writing “can neither be proved or disproved” doesn’t seem a great loss in readability. .

7. mattheath Says:

O my last comment was just in response to Daryll’s first, in case there is any confusion.

I think TMCM questions wasn’t “Is “Does every Goodstein sequence terminate?” the same as “Is it true that every Goodstein sequence terminates?”” . I think his question was “Does every Goodstein sequence terminate?”

I think it’s a fairly interesting question if you are asserting that arithmetic statements have absolute truth values (which I think you were).

8. Daryl McCullough Says:

matt,

My comment is agnostic about the philosophical question of whether statements of arithmetic have absolute truth values. I’m saying that you don’t need to solve that question to understand what “true but unprovable” means.

My complaint about rephrasing Godel’s result as saying that the Godel sentence G can neither be proved nor disproved is that, to me, the ontological status of “G cannot be proved” is *exactly* the same as the status of “G is true”. To say that G is true is to say that there does not exist a natural number satisfying such and such a property. To say that G cannot be proved is to say that there does not exist a proof satisfying such and such a property. I don’t see how the latter makes any less ontological commitment than the former.

9. Thony C. Says:

You are probably right that this isn’t the best forum to discuss this but no theorem about formal systems proves anything about the correct use of the English word “true”. What constitutes “truth in mathematics” is an open discussion amongst philosophers and I don’t think any of them mean things like a Tarski truth function.

Now most mathematicians are not set theorists or logicians. If we say “the closed graph theorem is true” we mean “in our fixed set theory (presumably ZFC) there is a proof of the closed graph theorem”. Most (IME) would hold that the question of whether the continuum hypothesis is true has no real meaning; “it’s undecidable” would be the whole answer. If asked specifically about different systems we would say “Hahn-Banach is true in ZFC” but not in “ZF (NOT C)”. “True” is not in this set up a formal concept in maths itself much less a formula within a theory, but why should it be?

Matt, my immediate facetious reaction to the first one of your quotes that I have copied is that both Gödel and Tarski wrote in German so the relevant phrase would be “ Deutsches Wort Wahrheit” however here you have the answer to your question in the second quote that I have copied.

When we talk about truth in a formal language we are talking about a concept of that formal language and not a concept in English or Deutsch. In formal languages the concepts that we use are defined exactly and specifically for that language and although we my use a concept in English that is homophone to the concept in our formal language we should not be deluded into thinking that they mean the same. As far as Gödel Theorem goes, it was thought before he published that provability and truth were equivalent concepts for formal languages, his theorem shows that they are not for any formal language “rich” (Reichhaltig) enough to contain Peano arithmetic. There are formal languages where it is the case that provability and truth are equivalent, First Order Predicate Logic for example, as was in fact proved by Gödel in his doctoral thesis. In the second quote of yours that I copied you are confusing talking about a formal language informally in a meta-language, in this case English, and using that formal language itself.

That at least is my reading of you problem! Thony C.

10. mattheath Says:

Matt, my immediate facetious reaction to the first one of your quotes that I have copied is that both Gödel and Tarski wrote in German so the relevant phrase would be “ Deutsches Wort Wahrheit

Strictly speaking, I think that would be “wahr” rather than “Wahrheit”. In any case I was referring to popular treatments in English.

When we talk about truth in a formal language we are talking about a concept of that formal language and not a concept in English or Deutsch. In formal languages the concepts that we use are defined exactly and specifically for that language and although we my use a concept in English that is homophone to the concept in our formal language we should not be deluded into thinking that they mean the same. …
…you are confusing talking about a formal language informally in a meta-language, in this case English, and using that formal language itself.

Well I knew I wasn’t using the formal language (I’m not a total moron). I do confess that I wasn’t totally aware that there was a technical usage of “true” in formal logic. But meh, using a technical term, which sounds like a common term in the language you are writing in, without defining it sounds to me an unforgivable sin.

11. mattheath Says:

@Darryl: I suspect you have a completely valid point in the second paragraph of your last comment. That pushes me back almost to thinking that people shouldn’t state the theorem at all except in settings where they can properly define every term used, but of course that’s silly; giving even a vague feeling of the meaning of a theorem to a general audience is always pretty cool..

Hmm if the same problems exist with any other sensible way of expressing it, I guess I don’t really have a good reason to be bothered by the version with “true”.

Harumph! It still causes an unconformable feeling in the back of my head.

I should get back to doing maths where I’m far enough from the foundations that I can just assume that everything works. If I don’t prove stuff about those bounded derivations… it doesn’t really matter.

12. gowers Says:

I agree that it is not completely clear what the question “Is the continuum hypothesis true?” means. However, I think it is clear enough what the question “Is it true that every even natural number is equal to the sum of two prime numbers?” In the latter case, it certainly does *not* mean “Is it provable in ZFC that …?” For questions about arithmetic to ask “Is it true that every X has property Y?” is to ask, simply “Does every X have property Y?”

In response to the above remarks of Daryl McCullough, let me say what my instincts are. In general they’re pretty similar to Matt’s: I am uneasy about any notion of mathematical truth that smacks of metaphysics. However, I have degrees of unease, and I’m not sure I can justify them. I can see that I would have a hard time defending the view that there is no fact of the matter as to whether every even natural number is the sum of two primes (jokes about 2 being a counterexample apart), but if I go one level deeper in the quantification then my attitude changes.

For example, is there a fact of the matter about whether the decimal expansion of $\pi$ contains infinitely many 0s? The difference here is that an imaginary thought experiment (in which, say, you are punished for your earthly sins by being put in a cell and told to watch as the digits of $\pi$ run past you for ever) never comes to an end, regardless of what you observe at any finite stage. The analogous Goldbach verification probably wouldn’t come to an end either, but one can at least entertain the possibility that an even integer might come along and spring a huge surprise.

One might argue as follows. If I concede that there is a fact of the matter as to whether $\pi$ contains at least one 0 after the nth digit, then surely I am now reduced to asking whether some factual statement holds (namely not having any more 0s after the nth digit) for at least one $n.$ And now I am back with a statement of the Goldbach type. But I don’t buy that, because in the Goldbach situation one can check statement $n$ in finite time and here one cannot: it seems to me to be genuinely worse.

If we deepen the quantification further, by asking whether the proportion of 0s in the decimal expansion of $\pi$ tends to 1/10, then all these concerns only increase. The imaginary thought experiment becomes wilder and wilder.

In fact, I have some sympathy with ultrafinitism: I’m tempted to say that there may be no fact of the matter as to whether the proportion of 0s in the first Ackermann(10,10) digits of $\pi$ is under 20 percent. Of course, if someone came up with a proof, then I would change my tune completely. But what if it was a statement that had no proof short enough to write down?

I do have one argument on the other side, which is that although we may not have a proof, we do at least have evidence that we can look at, such as how the digits of $\pi$ behave. The fact that we take this evidence seriously could be taken as showing that we do have some concept of the truth or falsity of statements about the long-term behaviour: it’s just that to justify our beliefs we use inductive rather than deductive methods. But this is to use the word “truth” in a somewhat different way.

13. Surbhi Says:

hey could u answer this that how is truth different in ethics and maths?
maths to is based on assumptions..which then hints that it is not absolute..but relative depnding on the assumption chosen