Recreational Mathematics?

One of Penelope Maddy’s objections to the indispensability argument as the justification for mathematical practice is that it seems to make set theory a hostage to quantum gravity. That is, if set theorists are realists, and are justified in their position because their work is an indispensable part of fully specifying the mathematical theories that are an inherent part of physics, then they should be eagerly awaiting the results of physical research to find out just what things need to be included in order to support the physics.

A potential response is to notice that the form of this argument is just the same as the argument against Quine’s web of belief that points out that “2+2=4” is not going anywhere. I think Maddy is right to notice that higher set theory is at least more vulnerable to this sort of attack than basic number theory. I would be hard-pressed to imagine a version of science that doesn’t apply basic number theory, while the motivations for set theory stem from deep reasoning about the continuum and about various other transfinite objects.

Mark Colyvan responds by suggesting that in fact, some mathematics may not be applied. Such mathematics carries with it no ontological commitments, and he calls it “recreational mathematics”. (I don’t recall if Maddy used this term as well.) He suggests that set theorists may be free to investigate set theory however they want, because part of it will be applied, and part will be purely recreational.

However, this doesn’t seem right to me. Set theory seems to have a fairly unified methodology (ignoring the fact that Californians work on extending large cardinal axioms and sloving CH, while east coasters and Israelis do something else), and this applied/recreational divide would cut across this work in some totally unexpected way. It doesn’t seem plausible that this divide between the applied and the recreational could be important enough to base ontology on, but unimportant enough that practitioners don’t even notice it.

I think that it is far more likely that either all set theory falls on the applied side, or almost all of it falls on the recreational side. (I say “almost all”, because I could imagine the countable being seen as applied while the uncountable is recreational.) It seems that once one adopts ZFC, there are important reasons to adopt large cardinal axioms. Maddy gives these explanations quite well (see her “Believing the Axioms” parts I and II in the 1988 Journal of Symbolic Logic) and I think both she and Colyvan are unnecessarily worried that proper scientific methodology might only pick out some of this. The real worry about indispensability will come much lower down, and I think the argument may well be found wanting at that level. But rather than considering the rest of mathematics to be “recreational”, a Fieldian fictionalist position will be the right attitude to take. In practice, this should be no different from the realism that Colyvan supports, or the agnosticism that Maddy seems to endorse now.

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On What There Is (Or Might Be)

In Quine’s essay “On What There Is”, he suggests that the sorts of entities we are ontologically committed to are all and only the ones that our best scientific theories of the world quantify over existentially, when spelled out in the fullest and most rigorous terms. So if our best theories say that water exists (and I’m pretty sure that no revolution is going to overturn that statement), then we are committed to the existence of water. And if they go on to say that all water contains hydrogen atoms, and all hydrogen atoms contain protons, then we are also committed to the existence of protons (though those two steps are both more conceivable revisable). And if they don’t mention the little green men that push the electrons around the nuclei of the atoms, then we should be committed to the non-existence of those little green men. Anyway, on this view, in developing our best scientific theories, we end up with a large set of axioms and theorems in some formal language. Our ontological commitment is just the sum total of the existential claims in that set, and nothing else.

But this doesn’t fully settle the question of what there is. A theory like Peano arithmetic states the existence of zero, a successor to zero, a successor to the successor to zero, and many other things. But it doesn’t fully specify what there is. As Gödel made painfully clear (though it was already clear before), PA has many different models. The standard model just has the quantifiers range over the natural numbers. In non-standard models, there are extra elements that are strictly greater than any standard natural number, and these elements come in Z-chains (ie, chains whose order type is the same as that of the integers, since each of these extra elements will have to have both an immediate successor and an immediate predecessor). By the Löwenheim-Skolem theorem, there are models of each infinite cardinality. Even if we take the complete theory of the natural numbers (which Gödel showed we can’t access in any computable manner), we have many non-isomorphic models. Each one of them says that different things exist. So what are we committed to?

In this case, it seems fairly obvious. We are committed to the existence of the standard natural numbers, because the standard model embeds elementarily into every other model. That is, not only is every model isomorphic to an extension of the standard model, but also any formula with free variables is true of some elements in the standard model just in case it’s true of their counterparts in any of these other models, so that it’s not just a submodel that happens to satisfy all the same sentences. (An example of the latter is the model of PA where the symbol “0” is interpreted as referring to the number one, and every other numeral is interpreted as referring to the successor of the number it standardly refers to. This model including only the positive integers is a submodel of the standard one, and is isomorphic to it, but the object referred to by “0” in the submodel has the property of being the least element there, but not in the standard model.)

But although we’re only committed to the elements of the standard model here, we should be open to the possibility of the existence of elements from any of the other models. But we shouldn’t be open to the existence of anything beyond these elements. That is, we should be committed to the existence of the standard natural numbers, agnostic about the possibility of further Z-chains, and committed to the non-existence of anything between the standard natural numbers, or less than zero, or otherwise not satisfying the axioms.

To generalize to other theories, I’ll point out that the property I relied on for the standard model of PA here was not merely the fact that it was in fact the standard model, but rather the fact that it embeds elementarily into any other model of PA. In general, for any theory T, if there is a model of T that embeds elementarily into all other models of T, then we call this model a prime model. So I propose an amendment to Quine’s criterion stating that we should be committed to the existence of all the elements of the prime model for our theory (assuming such a model exists), and only the elements that appear in some model for our theory. Thus, on my proposed amendment, the “all” and “only” parts of Quine’s thesis can come apart. Specifying a complete theory doesn’t necessarily specify a unique model, so we can still have some uncertainty in our ontological commitments, though I agree with him that this is the way to find them.

Of course, many theories don’t have prime models, like for instance, any incomplete theory, like PA, as opposed to complete number theory. In such cases I’m less sure what we should be committed to, because there is no “minimum” set of commitments to hold. If there’s some way to determine which objects exist in all the models, then we could be committed to those, even though they won’t themselves form a model of the theory (or else that model would be prime). But of course, there’s not in general any natural way to identify elements of distinct models, unless one model has a canonical elementary embedding in the other. But even if there is no prime model, we can at least look at the domains of all these distinct models and recognize that these objects are the ones that we should be open to, and not anything else.

Quine seems to have been worried only about which ontological categories there are, and these tend not to vary from one model to another of a given theory. But which particular objects exist does, and I think we can use the notion of a prime model to make Quine’s thesis more specific.