Monday 5 January 2009

Consistency and existence, with some random ranting

The idea that the completeness theorem vindicates the very natural notion that in mathematics consistency implies existence, that mathematics is a free creation, subject only to the requirement of consistency, and so on, crops up now and then. Some time ago it did just that in sci.logic, but I failed to subject the poor poster bringing it up to my characteristic tedious pedantry and obnoxiousness. Happily, I can do that here in this blog.

The completeness theorem, proven in the usual Henkin style, establishes that if a theory is consistent it has a model, the domain of which is the set of naturals, constant symbols naming particular naturals, and the interpretations of the relation and function symbols being horribly complex random configurations of naturals. Consider now the claim that infinite sets exist if the story we tell about them, as formalised in e.g. ZFC, is consistent. As noted, this is a very natural idea. Alas, the completeness theorem does nothing to support it, because even though it does guarantee the existence of something for a consistent theory, this something is not what we'd like to have. For certainly the claim that infinite sets exist does not mean that there is some horribly complex configuration of naturals under which the axioms of ZFC come out true. It is also apparent that for example inaccessible cardinals do not exist if ZFI = ZFC + "there are inaccessible cardinals" is consistent but proves "ZFI is inconsistent". If we wish to take the natural idea seriously and flesh it out in some philosophically fruitful way, we must explain just what sort of consistency is at issue here, as it is apparent, at least in light of rather trivial observations such as the above, that it is not consistency of formal theories in the technical sense.

The confused idea at issue is a perfect illustration of a very real danger in philosophy of mathematics, of reading philosophical significance into technical results without special argument. Before we get very worked up over and excited about this or that piece of mathematics in a philosophical context, we must carefully figure out just how it is related to the notions under philosophical scrutiny. Usually, the answer is that, despite superficial similarities -- sometimes on terminological level! -- the technical result is just that, a technicality. Another aspect the confused idea nicely illustrates is a sort of intellectual equivocation very common when thinking about mathematics: we wish to view claims about sets and what not from a distance, instead of taking them at face value, as something requiring explanation, philosophical justification and so on, while simultaneously interpreting a technical mathematical result -- in this case, the completeness theorem -- in the usual straightforward manner. This leads to nothing but confusion and bad philosophy, but has its appeal for those of certain bent, in that it seemingly allows us to wax philosophical with mathematical precision.

This is not to say that the completeness theorem is devoid of philosophical significance, or that mathematical results never have philosophical relevance. Perhaps the most famous example of "informal rigour" (if we don't count Turing's analysis of mechanical computability) is Kreisel's proof (in the sense of compelling piece of reasoning, not in the mathematical sense -- hence the "informal" in "informal rigour") that logical validity in the "informal" sense is coextensive with validity in the technical sense. (There's a bit about this in my sci.logic post on formalisation.) In case of completeness and the idea that consistency implies existence what is missing is just such a proof, a piece of compelling reasoning, a careful explanation and analysis. What we have instead is simply a wonderful philosophical revelation, of mathematical clarity, that unfortunately evaporates when subjected to scrutiny. Beware of revelations! Most real insights turn out to be rather dull and not at all exciting, the way things we are intimately familiar and comfortable with are. I suggest that instead of seeking exciting revelations we seek dullness of this kind, that is, dullness that is the opposite of bewilderment; this without any suggestion that philosophy should be dull the way a phonebook is.

6 comments:

  1. Hi, Aatu!

    I don't agree with you completely about this. In many cases, abstract mathematical objects such as the collection of all naturals are plainly convenient for reasoning. It seems to me that there is no reason *not* to treat such collections as sets, unless for some unforeseen reason it is inconsistent to do so.

    I know that for some questions of existence, such as "Does there exist a counterexample to Goldbach's conjecture?", we don't want to assume that something exists just because it is consistent to assume it. But in other cases, such as "the set of all naturals" or "the square root of -1", it seems that we are free to invent them, as long as we are able to do it consistently.

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  2. Here's a sci-logic thread on this topic

    http://groups.google.com/group/sci.logic/browse_frm/thread/852870627c48590f/1d36cc46dd5ddd3c?hl=en&lnk=gst&q=consistency+and+existence#1d36cc46dd5ddd3c

    (If that doesn't work, search sci.logic for "consistency and existence".)

    Also, of course, the Completeness Theorem doesn't establish that the consistency of PA means the natural numbers exist, since the Completeness Theorem assumes that the natural numbers exist.

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  3. Hello Daryl.

    I'm not sure what you're disagreeing with. My purpose was not to argue against the very natural idea that in mathematics some things exist simply by stipulation, and that we are free to introduce anything we like so long as we are consistent, and clear enough to allow for mathematical reasoning, etc. My point was simply that this very natural idea is not at all supported by the completeness theorem, as sometimes suggested. That a particular argument for something is faulty is of course not grounds for rejecting that something, and indeed I am very sympathetic to the general thrust of the idea of mathematics as free creation.

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  4. "I am very sympathetic to the general thrust of the idea of mathematics as free creation."

    I think it is safe to say that a structure exists at least if you can in principle construct every object of the domain. Only when things are not constructible, the question comes more subtle. We can construct a nonstandard model for PA, but if we could also construct a non standard natural (I suppose we can`t, but I am not an expert.)like we can construct a non-euclidean geometry, I think we could say that there is no "true" integers like we can say that there is no "true" geometry.

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  5. Gc,

    The fact that there exists a nonstandard model for PA does not in any way call into question the existence of a "standard" model of PA.

    There is a unique description of the standard model of the naturals: a model M is standard if it has no submodels.

    In more detail: A structure for PA consists of a structure <A,S,P,T,z> where A is a set, S is a function from A to A, P and T are functions from AxA to A, and z is an element of A. We can say what it means for a statement of arithmetic to be true in such a structure. Then the structure is a model of PA if every theorem of PA is true in that structure.

    Then we can say <A,S,P,T,z> is a standard model of PA if it is a model, and for every proper subset A' of A, <A',S,P,T,z> is not a model of PA.

    In contrast, nonstandard models all have submodels.

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  6. "The fact that there exists a nonstandard model for PA does not in any way call into question the existence of a "standard" model of PA."

    You misunderstand me completely. The contex which I am referring to is not the existence in set theoretic sense. Only one structure exist as a true arithemetic, I pointed out that maybe if in those countable non-standard models every object would be constructible, it would be fair to say that no true arithmetic exists. We could still speak about standard model, but it would not have the philosophical meaning it has now.

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