Some issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

Musings on mathematics, logical implication and metaphysical entailment

I intuitively find the following picture very plausible. On the one hand, there are mathematical claims, like the Banach-Tarski Theorem or Euclid's Theorem on the Infinitude of the Primes. These are mysterious (especially the former!), and tempt one to some sort of non-realism. On the other hand, there are purely logical claims, like the claim that the ZFC axioms logically entails the Banach-Tarski Claim or that the Peano Axioms logically entail the Infinitude of the Primes. Pushed further, this intuition leads to something like logicism, which we all know has been refuted by Goedel. But I want to note that the whole picture is misleading. What does it mean to say that p logically entails q? Well, there are two stories. One is that every model of p is a model of q. That's a story about models, which are mathematical entities (sets or classes). Claims about models are mathematical claims in their own right, claims in principle just as tied to set-theoretic axioms as the Banach-Tarski Theorem. The other reading is that there is a proof from p to q. But proofs are sequences of symbols, and sequences of symbols are mathematical objects, and facts about the existence or non-existence of proofs are once again mathematical facts, tied to axioms and subject to the foundational worries that other mathematical facts are. So the idea that there is some radical difference between first-order mathematical claims and claims about what logically entails what, such that the latter is innocent of deep philosophy of mathematics issues (like Platonism), is untenable.

Interestingly, however, what I said is no longer true if we replace logical entailment with metaphysical entailment. The claim that the ZFC axioms metaphysically entail the Banach-Tarski Claim is not a claim of mathematics per se. So one could make a distinction between the mysterious claims of mathematics and the unmysterious claims of metaphysical entailment--if the latter are unmysterious. (They are unmysterious if one accepts the causal theory of them.)

This line of thought suggests an interesting thing: the philosophy of mathematics may require metaphysical entailment.

Deflation of the foundations of probability

I don't really want to commit to the following, but it has some attraction.

Question 1: What is probability?

Answer: Any assignment of values that satisfies the Kolmogorov axioms or an appropriate analogue of them (say, a propositional one).

Question 2: Are probabilities to be interpreted along frequentist, propensity or epistemic/logical lines?

Answer: Frequency-based, propensity-based and epistemically-based assignments of weights are all probabilities when the assignments satisfy the axioms or an appropriate analogue of them. In particular, improved frequentist probabilities are genuine probabilities when they can be defined, but so are propensity-based objective probabilities if they satisfy the axioms, and likewise logical probabilities. Each of these may have a place in the world.

Question 3: But what about the big metaphysical and epistemological questions, say about the grounds of objective tendencies and epistemic probabilities?

Answer: Those questions are intact. But they are not questions about the interpretation of probability as such. They are questions about the grounds of objective propensity or about the grounds of epistemic assignments. Thus, the former question belongs to the philosophy of science and the metaphysics of causation and the latter to epistemology.

Question 4: But surely one of the interpretations of probability is fundamental.

Answer: Maybe, but do we need to think so? Take the axioms of group theory. There are many kinds of structures that satisfy these axioms. Why think one kind of structure satisfying the axioms of group theory is fundamental?

Question 5: Still, couldn't there be connections, such as that logical probabilities ultimately derive from propensities via some version of the Principal Principle, or the other way around?

Answer: Maybe. But even if so, that doesn't affect the deflationary theory. There are plenty more structures that satisfy the probability calculus that do not derive from propensities.

Question 6: But shouldn't we think there is a focal Aristotelian sense of probability from which the others derive?

Answer: Maybe, but unlikely given the wide variety of things that instantiate the axioms. Maybe instead of an Aristotelian pros hen analogy, all we have is structural resemblance.

Gentler structuralisms about mathematics

According to some standard structuralist accounts, a mathematical claim like that there are infinitely many primes, is equivalent to a claim like:

1. Necessarily, for any physical structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

There are two main motivations for structuralism. The first motivation is anti-Platonic animus. The second is worries about uniqueness: if there are abstract objects, there are many candidates for, say, the natural numbers, and it would be arbitrary if our mathematical language were to succeed in picking out on particular family of candidates.

The difficulty with this sort of structuralism is that while it may be fine for a good deal of "ordinary mathematics", such as real analysis, finite-dimensional geometry, dealing with prime numbers, etc., it is not clear that there are enough possible physical structures to model the axioms of such systems as transfinite arithmetic. And if there aren't, then antecedents in claims like (1) will be false, and hence the necessary conditional will hold trivially. One could bring in counterpossibles but that would be explaining the obscure with the obscurer.

I want to drop the requirement that the structures we're talking about are physical structures. Thus, instead of (1), we should say:

2. Necessarily, for any structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

If we do this, we no longer have a physicalist reduction. But that's fine if our motive for structuralism is worries about arbitrariness rather than worries about abstracta.

Next, restrict the theory to being about what modern mathematics typically means by its mathematical claims. If we do this, the claim becomes logically compatible with Platonism about numbers. Let us suppose that there really are numbers, and our ordinary language gets at them. Nonetheless, I submit, when a modern number theorist is saying that there are infinitely many primes, she is likely not making a claim specifically about them. Rather, she is making a claim about every system that satisfies the said axioms. If the natural numbers satisfy the axioms, then her claims have a bearing on the natural numbers, too.

Here is one reason to think that she's saying that. Mathematical practice is centered on getting what generality you can. What mathematician would want to limit a claim to being about the natural numbers, when she could, at no additional cost, be making a claim about every system that satisfies the Peano axioms?

Now, if we go for this gentler structuralism, and allow abstract entities, we can easily generate structures that satisfy all sorts of axioms. For instance, consider plural existential propositions. These are propositions of the form of the proposition that the Fs exist, where "the Fs" directly plurally refers to a particular plurality. We can define a membership relation: x is a member of p if and only if x is said by p to exist. Add an "empty proposition", which can be any other proposition (say, that cats hate dogs) and say that nothing is its member. Then plural existential propositions, plus the empty proposition, with this membership relations should satisfy the axioms of a plausible set theory with ur-elements. If all one wants is Peano axioms, we can take them to be satisfied by the sequence of propositions that there are no cats, that there is a unique cat, that there are distinct cats x and y and every cat is x or is y, that there are distinct cats x and y and z and every cat is x or is y or is z, and so on.

I am not completely convinced that this sociological thesis about modern mathematics is correct. Maybe I can retreat to the claim that this is what modern mathematics ought to claim.