Showing posts with label structuralism. Show all posts
Showing posts with label structuralism. Show all posts

Friday, October 16, 2015

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.

Saturday, September 15, 2012

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.

Monday, February 20, 2012

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,...,An, 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:

  1. Necessarily, for any structure that satisfies the axioms A1,...,An, 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.