Sunday, May 1, 2011

Daily affirmation

All of us must live in constant contradiction. Philosophers wage constant war against it.

Induction and the Axiom of Choice

In this short but strange paper, Alexander George gives a mathematical "proof" justifying induction. The problem of (Humean) induction basically is:

Now as concerns the argument, its conclusion is that in induction (causal inference) experience does not produce the idea of an effect from an impression of its cause by means of the understanding or reason, but by the imagination, by “a certain association and relation of perceptions.” The center of the argument is a dilemma: If inductive conclusions were produced by the understanding, inductive reasoning would be based upon the premise that nature is uniform; “that instances of which we have had no experience, must resemble those of which we have had experience, and that the course of nature continues always uniformly the same.” (Hume THN, 89) And were this premise to be established by reasoning, that reasoning would be either deductive or probabilistic (i.e. causal). The principle can't be proved deductively, for whatever can be proved deductively is a necessary truth, and the principle is not necessary; its antecedent is consistent with the denial of its consequent. Nor can the principle be proved by causal reasoning, for it is presupposed by all such reasoning and any such proof would be a petitio principii.

George's proof uses the axiom of choice and a recent result called the Hardin-Taylor theorem. George's proof requires the axiom of choice. However, it seems to me that this would not make induction any less problematic because the grounds for skepticism on which the problem of induction is formed is analogous, indeed, very similar, to the grounds for doubting the truth of the axiom of choice. Consider the axiom of choice which basically states:

For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection.

This axiom presupposes the existence of a choice function for any collection of non empty sets, even infinite collections and even if we cannot specify the details of that function (for example by specifying a selection rule or the distinguishing property of the elements to be picked out). The function is assumed to exist under that axiom. Likewise the reason the problem of induction does not bother scientists, common folks and philosophers alike (who actually doubts the sun will rise tomorrow?) is because we assume that nature has principles of uniformity such as laws, etc even if we cannot directly observe such laws or discover them by deductive reasoning reason as Hume insisted. Hume suggest that we arrive at the principle of uniformity for any series of past events to continue to hold into the future by acts of "imagination" and inference through that imagining. To claim that we can discover the principles through observation is to beg the question, or worse, to get into a vicious circle (what would justify the principle?). To claim that we can discover them through deductive reasoning is to commit a category error.

Similarly, assuming the existence of choice functions is assuming that there is some regularity in any collection of items from a collection of non empty sets even when we cannot specify what that regularity is. We cannot prove it from the axioms (it is an axiom after all). In the case of induction we are not bothered by the "problem" of induction because we assume that our imagination in conjoining repeated contiguous events have some basis in a natural uniformity that is existing independently outside of our imagination (such as some natural law). We assume the existence of some law e.g., which we cannot directly observe but must infer the existence.

So "proving" induction from the axiom of choice seems to do no anti-skeptical work at all. It throws the problem in the lap of mathematics when the foundations of such reasoning can be put under a very similar skeptical lens as the foundations of our everyday inductive reasoning. This is made especially salient when we consider the reasons why mathematicians have accepted the axiom of choice. First, let's consider what George says about his proof:

But might there be some feature of the proof that compromises its ability to offer the kind of justification we are after? One aspect of the proof that suggests itself is its reliance on the Axiom of Choice....
But then he goes on:
Where a condition on conceptual coherence is consistency with ZFC, Zermelo-Fraenkel set theory plus the Axiom of Choice, a theory most mathematicians believe to be true and indispensable for the formal development of mathematics.

If this is the justification then it would render the whole proof implausible because as Penelope Maddy has correctly pointed out, mathematicians have "accepted" the axiom of choice largely because of pragmatic reasons (for ease of proving some theorems like Zorn's lemma). They haven't accepted it because of its philosophically sound obviousness (in fact, many of them and logicians and philosophers have skeptically questioned its truth much as the later have inductive reasoning). Given these pragmatic historical considerations, they seem to mirror the pragmatic considerations of why people accept inductive reasoning as generally true (how will we live without assuming natural regularity and uniformity?).

Many mathematicians are equally adept and comfortable working with axioms that are inconsistent with choice (such as the axiom of determinacy). Maddy's experience with mathematicians on the philosophical aspects of their discipline is similar to my own experience in asking them about foundational issues. They are, almost unanimously in my experience, far more pragmatic and anti-platonist than philosophers would believe and are in general less likely to have platonist intuitions than philosophers of math in my experience. When asked if they think the axiom of choice (or the Continuum Hypothesis) is true they will usually say something like "well, it's true under some models but not others." This is not them being flippant but stems from their general pragmatic outlook. So George's appeal to the authority of the mathematician in his reliance on that axiom for his proof of induction seems odd and rather unconvincing.