Sunday, 30 December 2012

Mathematical Explanation in Science, Conservativeness and Necessity

Mathematical explanation in science has become a hotly debated topic over the last ten years or so. There were, of course, discussions of this topic in various places, going back many decades and no doubt centuries! Physicists frequently commented on the issue (the most famous example being Eugene Wigner's "The Unreasonable Effectiveness of Mathematics": one can also find assorted comments in many places - for example in Feynman, Weinberg and Penrose). First book length discussion by a philosopher, at least that I am familiar with, is Mark Steiner's The Applicability of Mathematics as a Philosophical Problem. In that work, some very intriguing suggestions are made; in particular, the suggestion of a kind of pre-established, anthropocentric, harmony between mind and the mathematical nature of reality.

Here I'm more interested in how these debates have developed over the last decade or so. The question is:
do mathematical statements ever play an explanatory role in scientific theory?
Since I am some kind of metaphysical Pythagorean, I think reality is mathematical, my answer is: well, yes, obviously.

However, the usual setup for such discussions involves a presupposition that the world is non-mathematical, and the question arises whether mathematics plays a role in explaining the non-mathematical. Since I reject this nominalistic assumption, I don't feel any great urgency to answer that demand. For the physical quantities (mass, length, etc.; and field quantities with values in abstract spaces, etc.) that play such a crucial role in scientific theories are already mathematical entities - they are usually functions of some sort.

Still, suppose we simply grant the nominalist's basic assumption. That there is a way the world is that doesn't involve quantities, field quantities, symmetry groups, etc. There is a "nominalistic way the world is". (But, to repeat: I do not believe this is true.) The most perspicuous way of formalizing this, following the work of John Burgess, is to think of an interpreted 2-sorted theory language, $L$, with variables ranging over concreta and abstracta. The primitive predicates then fall into three kinds: those that express properties of, and relations amongst, concreta ("primary"); those that express properties of, and relations amongst, abstracta ("secondary"); and those that express relations between concreta and abstracta ("mixed").

The purely nominalistic statements, as it were, are simply those that involve no variables (and, a fortiori, no secondary or mixed predicates) for abstracta. Mathematical statements may be pure or mixed.

Then the question becomes: do either pure or mixed mathematical statements ever play a role in explaining nominalistic statements? Now one needs to be careful here, because the mixed mathematical statements come in two kinds. The usual laws of physics are mixed statements. (This is a quick formulation of one of the premises used in the Indispensability Arguments against nominalism.) For example, Maxwell's Law,
(1) $\nabla \cdot B = 0$
is, under analysis, a mixed statement. The same is true of Newton's Laws, the Laws of Quantum Theory, General Relativity, and so on. It is true of more mundane scientific laws, like the Gas Laws, and laws of optics and so on, of course.

However, there are mixed statements which are nothing like laws of physics. Two examples are:
(2) There are $n$ Fs if and only if the number of Fs = $n$.
(3) There is a set $X$ such that, for all $c$, $c \in X$ iff $\phi(c)$.
The primary difference, as it seems to me, between a mixed law of physics such as (1), and the mixed statements (2) and (3), is that the first is contingent, while the latter two are necessary. Of course, a nominalist will be inclined to claim that (2) and (3) are not necessary: rather, they are false! But the nominalist has a way of handling this, due to Field, that I come on to in a moment.

Statements such as (2) and (3) I shall call axioms of applicable mathematics. Other examples would include Hume's Principle. In fact, as far as I can tell, the primary such axioms are either Hume's Principle (from which (2) is derivable in the setting of second-order logic) or Comprehension Axioms (such as (3)).

In the setup favoured by nominalists, one imagines that mixed laws of successful physics, such as (1), have either been nominalized away by some kind of paraphrase or elimination, or can be downgraded to the status of "nominalistically adequate", as opposed to being "true" (or "approximately true"), as a realist requires. So, we can, at least temporarily, forget about (1) and other such laws of physics: contingent claims that some physical quantity has certain properties.

On the other hand, the status of applicable axioms---that is, mixed statements such as (2) and (3)---is that they are required to be conservative. It is this conservativeness property that gives them a deflated role. They are useful in some sense, but untrue. They are convenient scaffolding for formulating science, but, being conservative, they are insubstantial.

In fact, one can show that (2) and (3) are conservative in the required sense. In particular,
if the comprehension principle (3) implies a nominalistic statement $S$, then $S$ is a logical truth.
Let me slim this down a bit,
If $\exists X \forall c(c \in X \leftrightarrow \phi(c)) \vdash S$, then $\vdash S$.
Such results correspond to Hartry Field's Principle C, in his Science Without Numbers.

How does this relate to the contemporary debate about mathematical explanation in science? I think it points to a very important feature:
The applicable mathematical axioms are necessities. If they imply a purely nominalistic statement, then that statement is a necessity too. Consequently, contingent nominalistic statements cannot be explained by applicable mathematical axioms.
The examples that have been given in the literature have tended to obscure this point. This is not to deny that there examples of mathematical explanation! On the contrary, I think there are. It is simply that if applicable mathematical axioms (such as comprehension axioms for the existence of sets of concreta) imply a purely nominalistic statement, then that statement must be a necessity (a logical truth, in fact).

Even so, such necessities may themselves be quite interesting, particularly if they are conditionals. Suppose we abbreviate the applicable mathematical axioms that we are assuming, in the background as it were, as $AM$. We may have,
$AM \vdash \phi \to \theta$
where $\phi$ and $\theta$ are purely nominalistic claims. So, with the mathematical axioms, we can deduce that: if the concreta do this, then they do that. This conditional $\phi \to \theta$ is, in the end, a complicated logical truth: that is,
$\vdash \phi \to \theta$
So, there is a derivation $\Gamma$ of $\phi \to \theta$ using some sound and complete deductive system for FOL. It may also happen that the shortest such $\Gamma$ is huge (e.g., contains at least $10^{10^{10}}$ symbols, say), and thus one cannot deduce this conclusion in practice without using mathematics, for speed-up reasons.

If this is right, then the only output we can get from applicable mathematical axioms at the purely concrete level must be complicated logical truths. Does it make sense to talk of explaining a logical truth? I'm not sure, but actually, I think it might; but that's a topic for another M-Phi post.

(Postscript: I've included no references or links, so if I have time to update this, I'll try to add some references and links.)

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