Mathematical Existence

2 05 2007

Last night, I mentioned to my roommate (who’s a mathematician) the fact that some realist philosophers of mathematics have (perhaps in a tongue-in-cheek kind of way) used the infinitude of primes to argue for the existence of mathematical entities. After all, if there are infinitely many primes, then there are primes, and primes are numbers, so there are numbers.

As might be expected, he was unimpressed, and suggested there was some sort of ambiguity in these existence claims. He wanted to push the idea that mathematically speaking, all that it means for a statement to be true (in particular, an existence statement) is that it be provable from the relevant axioms. I was eventually able to argue him out of this position, but it was surprisingly harder than expected, and made major use of the Gödel and Löwenheim-Skolem-Tarski theorems.

The picture that is appealing to the mathematician is that we’re just talking about any old structure that satisfies the axioms, just as is the case when we talk about groups, graphs, or topological spaces. However, I pointed out that the relevant axioms for the natural numbers can’t just be Peano Arithmetic, because we’ll all also accept the arithmetical sentence Con(PA), once we’ve figured out what it’s saying. This generalizes to whatever set of axioms we accept for the natural numbers. As a result, Gödel’s theorem says that we don’t have any particular recursive set of axioms in mind.

To make things worse for the person who wants to think of the natural numbers as just given by some set of axioms, there’s also the fact that we only mean to count the members of the smallest model of PA, not any model. By the LST theorem, any first-order set of axioms that could possibly characterize the natural numbers will also have uncountable models. In order to pick out the ones we really mean, we either have to accept some mathematical class of pre-existing entities (the models of PA, so we can pick out the smallest one), or accept a second-order axiomatization, or give up any distinction between standard and non-standard natural numbers.

Of course, the same argument can be run for set theory, and at this point, accepting a second-order axiomatization causes trouble. Some philosophers might argue a way around this using plural quantification, but at least on its face, it seems that second-order quantification requires a prior commitment to sets, so it can’t underlie set theory. And we really don’t want to admit non-standard models of set theory either, because they give rise to non-standard models of arithmetic, which include certain infinite-sized members as “natural numbers”.

So it seems, for number theory and set theory at least, the mathematician has to admit that something other than just axioms underlies what they’re talking about. I’ve argued that the existence of and agreement on these axioms means that mathematicians don’t often have to think about what’s going on there. And from what I can see so far, the mathematician could take a platonist, fictionalist, or some other standpoint without any trouble. And in fact, most of the reasoning is going on at a high enough level that even if there’s some problem with the axioms (say, ZFC turns out to be inconsistent) this will have almost no effect on most other areas of mathematics, just as a discovery that quantum mechanics doesn’t correctly describe the behavior of subatomic particles will have relatively little effect on biology. But the mathematician can’t insist that existence in mathematics just means provability from the axioms – she can insist on some distinction between mathematical existence and ordinary physical existence, but it will be something like the minor distinction that the platonist recognizes, or possibly something like the distinction between real and fictional existence.




22 responses

2 05 2007
Noah Snyder

There’s something very unintuitive to me in the argument that “you can’t possibly mean X, because X applies to way more situations than you had in mind.” Much of the point of mathematics is to distill things to their essence and thus be able to apply them to many different cases. The “right” statement of Euclid’s version of the infinitude of primes in my mind is (something more like): “If D is an ordered integral domain with no elements between 0 and 1, then D has infinitely many primes.” Whether you think that theorem applies to the version of the integers you have in your head is between you and your integers and is none of my business.

2 05 2007

It’s true that the infinitely many primes result is just fine even if we’re dealing with non-standard models. But it’s less good if we think about finiteness. We can establish a numerical coding of sentences, and sequences of sentences of our language (following Gödel and others), and then exhibit an arithmetically definable property that a number has only if it is the code number of a valid proof, or a theorem. If we’re just talking about the standard natural numbers, then a number is the code of a valid proof iff it has this property. However, if non-standard naturals are allowed, then this will result in codes of “infinitely long sequences of formulas”, and also give us finite standard natural numbers that satisfy the arithmetic property of being a theorem. However, they won’t in fact be theorems, because there is no actual proof of them, since actual proofs are finite.

You might just say, so much the worse for the arithmetical representability of meta-mathematical concepts like valid proof. But I think you’ll admit some notion of valid proof then that isn’t captured by the arithmetic coding. Either that notion of valid proof is itself mathematical (in which case I can run the same argument as from the post) or not, in which case the mathematics rests on something non-mathematical.

4 05 2007

Can you please elaborate, why Con(PA) is necessary to be accepted once PA axioms are accepted?

Existence of the “real world” with objects that can be counted if not enough, as we can only believe that PA axiomatizes these objects faithfully (a belief, in my view, quite similar to the belief that Euclidean geometry is the geometry of the real world—to which some time ago almost all mathematicians were subscribed, but which is at best irrelevant nowadays).

Assuming Con(PA) on the basis of “our work is useless otherwise” is like assuming theorem T as an additional axiom, while trying to prove it.

4 05 2007

If we accept PA, then we believe that it is consistent. The only way to believe that it is consistent without accepting Con(PA) is to allow that the natural numbers don’t properly characterize the notion of finiteness. That’s ok if we’re just interested in studying models of PA (because some of those certainly don’t characterize finiteness and don’t satisfy Con(PA)). But if we’re interested in the natural numbers themselves then we care about more than just PA.

As for your point that we only believe that PA axiomatizes objects faithfully, there is a result of Gottlob Frege, which has since been vindicated, showing that PA follows from the association of numbers with finite sets.

5 05 2007

If we accept PA, then we believe that it is consistent.
I am not sure. As any platonist (read: average mathematician) would tell you: we believe that natural numbers exist, and any system of axioms is but a tool to look at the properties of these ideal objects. We use PA to formalize our understanding of numbers, and if it turns out to be inconsistent, then what? Our precious number-theoretic theorems would be an no larger risk than mathematics was when paradoxes of set-theory were found (i.e., zero).

And as a formalist would tell you, we only shuffle symbols according to rules, if rules turn out to be inconsistent then… game is not interesting anymore. Though luck.

Thank you for the reference to the article about Frege, but I don’t see how reducing arithmetics to sets helps here: numbers are much more simple and direct notion that sets.

In addition, one might argue that Presburger arithmetics is sufficient to capture behavior of “real objects”, because in reality we never multiple some set of objects by some set of objects, we only ever multiple some set by some number (i.e., repeat it certain number of times), which is what Presburger arithmetics allows.

5 05 2007

I don’t know what you’re saying, nikita. If “accept”ing a theory is taking it to be a collection of true statements, then in “accept”ing PA, you are saying that the statements of PA are all true, which implies that PA is consistent. Moving from there to the truth of the arithmetic sentence Con(PA) requires, as Kenny noted, taking Con(PA) to actually capture the statement “PA is consistent” (i.e., it requires taking the natural numbers discussed in the language of PA to be the genuine arbiters of finiteness, so that the finite proofs are precisely the ones coded by such natural numbers), but that doesn’t seem to be your objection. It seems you have some different view of what it means to “accept” a theory.

I’d say the quick response to a naive formalism of saying that a mathematical claim’s truth is no more than its provability from some theory is the one found in Kenny’s first response: if “X is true” only has meaning relative to some theory T, it then being just a shorthand for “T proves X”, then what of the claim “T proves X” itself? This would only have meaning relative to some theory T2, it then being a shorthand for “T2 proves ‘T proves X’ “. And so on, of course; there would be an infinite regress. I suppose you could hold fast and say it is indeed turtles all the way down, but that seems fairly awkward.

6 05 2007

It seems you have some different view of what it means to “accept” a theory.
Indeed, that seems to be the core of misunderstanding. To me „to accept“ a theory means to „to agree to use it“, and my point is that for neither platonist nor formalist such form of acceptance necessarily implies acceptance of Con(PA).

10 05 2007

“However, I pointed out that the relevant axioms for the natural numbers can’t just be Peano Arithmetic, because we’ll all also accept the arithmetical sentence Con(PA), once we’ve figured out what it’s saying. This generalizes to whatever set of axioms we accept for the natural numbers. As a result, Gödel’s theorem says that we don’t have any particular recursive set of axioms in mind.”

I suppose I don’t understand Godel’s theorem well enough, but this paragraph is mysterious to me. Why does accepting Con(PA) imply “we don’t have any particular recursive set of axioms in mind”? What is meant by a “recursive set of axioms” anyway?

I just discovered this blog by the way, it’s quite interesting…

11 05 2007

You’re right that Con(PA) doesn’t imply those things itself. But I was just skipping some analogous steps in reasoning to get there.

To say that a set of sentences is recursive, intuitively means that there is a procedure, such that given any sentence, the procedure will tell you in some finite amount of time whether or not that sentence is a member of the set. So ideally, we would certainly like our set of axioms to be recursive, because if we can’t even tell whether or not a sentence is an axiom, then we would seem to have no hope of telling whether it’s a theorem. (More formally, you use Gödel’s system for coding sentences by numbers, and then say that a set of sentences is recursive iff there is a Turing machine that always terminates, with one ending state if the number is the code of a sentence in the set, and another state if the number is not.)

Gödel’s theorem more fully states that for any recursively axiomatized theory T extending PA (in fact, we can weaken this to Robinson’s arithmetic, standardly called Q), there is a sentence we can formalize as Con(T) that intuitively says that T is consistent, and that T does not prove Con(T).

Since any finite extension of a recursive set of axioms is itself recursive, we can’t just stop at PA+Con(PA), or PA+Con(PA)+Con(PA+Con(PA)), for exactly the same reason that we can’t stop at PA. There is no recursive system of axioms that we can recognize to characterize our complete understanding of the natural numbers.

Anyway, thanks for the interest!

12 05 2007

Your view that accepting axiomatic system means that you just agree to use it (and your comments on though luck) is kind of depressing. It makes it sound as if it was luck that we could do so much math in PA. There were deep reasons to believe that PA was the right axiomatic system and its success just supports the view that it is indeed the right one (modulo the equivalent axiomatizations). For me it is also hard to see how can one have a reasonable view of mathematics, but not work under the assumption that CON (PA) is true. If you think of math as a tool for other sciences then you better give some kind warranty that your tools are consistent. If you are doing math because you believe that you are adding something to human knowledge then again you better have some way of ensuring that what you say is indeed true. The only view of math that I can think of which seems to be consistent with not taking CON (PA) as a true statement is the one that basically says that mathematicians produce true statements of the form if P then Q. These statements become true only because of the logical form they have. Again this is a quite depressing view of math which basically suggests that the usefulness of math in other areas of science is just luck. When we, mathematicians, make the assumption that P is true we almost always have a reason to believe that it is at least probable that P is true. It seems to me that no mathematician will ever investigate the consequences of some completely accidental and unnatural axiomatic system P.

12 05 2007
Russell O'Connor

Since I believe that (some) mathematical objects exist in as much as software exists, I wonder, do philosophers debate the existence of software as well?

12 05 2007

Grigor – I think I largely agree with you, except perhaps with regards to the role that axioms are playing. I think there’s a lot of good reasons for these axioms, but that these axioms don’t exhaust the content of what we care about.

Russell – I don’t know of anyone who’s talked about that specifically. But I think most people working in this area would treat software in about the same way as math. Though I could imagine someone making a distinction, because there is some sort of obvious physical representation or realization associated with software, while there generally isn’t as much for math. (Of course, that’s only a prima facie claim, and it’ll affect platonists, fictionalists, constructivists, and others in different ways.)

13 05 2007

Kenny, I didn’t mean to suggest that we should always work in PA+CON (PA). In fact to understand PA well enough we actually need to allow the possibility of not CON(PA). Moreover, not CON(PA) is a very useful mathematical tool. Anybody who has seen the effective version of Perfect Set Property would agree. There the nonstandard models of PA is used to show that Cantor-Bendixon analysis if halts then halts before omega_1^CK. This is just one very pretty use of non-standard models. Let’s not get into technicalities, but I even would argue that the first instance of large-cardinal- like-properties is in fact the overspill in PA.

14 05 2007

Correction: not nonstandard models of PA, but nonstandard models of KP. This is what one needs for the effective perfect set property.

15 05 2007
Jean-Luc Delatre

in which case the mathematics rests on something non-mathematical.

Didn’t you notice that you use language to make such statements?
AFAIK language does not lend itself to mathematical definition and therefore “mathematics rests on something non-mathematical”.

P.S. I am not a mathematician.

21 05 2007
Brian Johnson, Ph.D.

Dear Sir or Madam:

I have, for quite some time now, been interested in the generalized infinitude of primes, with regard to an ordered domain. Here is my version of this result, for which I can present the proof if you would like:
(1) Let D be an ordered PID which has only units between 0 and 1, and which is not a field. Suppose C is a choice subset of positive irreducibles, one from each associate class of irreducibles. Then C is infinite.
(2) For each whole number n, D[x] has infinitely many irreducible classes of every possible degree.

I am an assistant professor of mathematics at Northern Virginia Community College, and I teach the Basic Abstract Algebra course at this institution. I look forwared to further communicating with you about this. Please send an e-mail to the address above whenever you can.

Brian Johnson, Ph.D.

23 05 2008

It makes sense and is true to say “Prime numbers exist in the set of all numbers”. It makes no sense to say “prime numbers exist”, you need to specify your domain of discourse. It’s like saying “for breakfast I am going to have a”. You need to say what cereal you are going to have, etc. Of course, the former is grammatically well formed and the latter is not, but that is no criteria for being meaningful.

Anywho, you mentioned that your friend thought that all it meant for a mathematical statement to be true, was that it was derivable formally, from a specific list of axioms. This is of course not the case. However, you don’t need LST for this, Godel will suffice, provided one agrees that there is a fact of the matter about statements of first order arithmetic (or even questions of solvability of Diophantine equations).

23 05 2008

Is there a reason you have to specify your domain of discourse? Most existential claims don’t seem to need such a specification – when I say “tigers exist” or “black holes exist” those seem to be meaningful claims. Maybe mathematical claims are different, but I don’t see a good argument for this claim.

24 05 2008

It’s a basic requirement. When we say “tigers exist”, technically that does not make sense. Of course, usually, there is the implicit spatiotemporal/physical domain of discourse of our space-time continuum, or maybe just our galaxy. If you don’t allow implicit meanings, you really would have to write “tigers exist in our space-time continuum” or something like that. However, humans have come to, in everyday discourse, drop the domain and mean implictly “the world” or “spacetime” or whatever is appropriate. This is fine and convenient, but one should not forget entirely about the domain of discourse just because it has been dropped in everyday language.

For mathematical objects there is a similar state of affairs, in fact pretty much the same. For example, “prime numbers exist” makes no sense. If you add to the sentence a domain of discourse, the set of all numbers, then it is true. However if it is the set of all Carmichael numbers, then the statement is false.

24 05 2008

I understand the claim – you say that existence is not a bare predicate, but instead requires a domain in order to make sense. For some sorts of claims, the domain can be filled in contextually, but for mathematical claims we need to be explicit.

However, I still don’t see a positive reason to believe that this is the right theory of existence, rather than the one that looks more plausible on its face, which just says that there is a unique notion of existence. I don’t have any arguments against the understanding you mention yet, but I don’t see any arguments in its favor either. The one consideration I can imagine is that in model theory, we represent the semantics of first-order sentences involving an existential quantifier by giving a model with a domain. In model theory, sentences aren’t true or false, but merely true in an interpretation or false in an interpretation. But I don’t see any reason to impose this particular restriction on the word “exists”, any more than we impose it on the word “tigers” – on this model-theoretic account, when I say “tigers exist” I don’t need to just specify the domain of quantification, but also which objects (if any) in that domain count as tigers.

Additionally, this thesis looks like a claim about the meanings of natural language words like “exist”, but one might be able to argue that what’s really at stake here is a technical notion of existence from metaphysics, which doesn’t obviously have to be the same as the notion expressed in ordinary language.

25 05 2008

I still don’t see a positive reason to believe that this is the right theory of existence, rather than the one that looks more plausible on its face, which just says that there is a unique notion of existence.

This is only because you are some sort or “realist”, for you tigers exist “in reality” or “in the world” but it’s only because you are oblivious of the HUGE implicit model you have of reality (like anyone) and this “model” is no different in its kind than a mathematical model, only fuzzier and much much larger.

when I say “tigers exist” I don’t need to just specify the domain of quantification, but also which objects (if any) in that domain count as tigers.

Yes you should, but you cannot (nor anyone else) because of the size and murkiness of the underlying model, just try do dig out all the “premisses” you would need to formally ascertain that “there is a tiger here”.

25 05 2008

“However, I still don’t see a positive reason to believe that this is the right theory of existence, rather than the one that looks more plausible on its face, which just says that there is a unique notion of existence. I don’t have any arguments against the understanding you mention yet, but I don’t see any arguments in its favor either.”

That’s why I said it was basic, because I do not need an argument for it. If someone said to you “For breakfast I will have a.” or “Today I am going to.” you would probably not be very satisfied. That is the same feeling I have when I look at a sentence like “pi exists”. If someone was insisting to me that the sentence “for breakfast I will have a.” was perfectly literally meaningful, I would feel and respond in the same way as if someone insisted that the sentence “pi exists” really made literal sense.

You cannot give an argument for everything. The english language has only finitely many letters, and thus there are only finitely many collections of sentences that can be compiled in less than 10 billion years, say. Some things have to be basic. It is obvious to me that the sentences “pi exists” or “super middleweight boxers exist”, without an implicit domain of discourse, make no sense. You might say that I am just saying that, and not justifying it, and yes, indeed I am.

Obviously a lot of metaphysical, non-humanitarian philosophy goes out of the window when one realises this. The platonism/antiplatonism debate is gibberish. Anything which concerns itself with the so-called “ontological commitment” is meaningless. However I expect people will continue to waste their time with them.

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