Math and Ontology

This review of Ladyman and Ross is lengthy and gets a bit in the weeds of analytic metaphysics, but it’s well worth a read. This part of the review asks the right questions about any appeals to mathematical models (see Badiou and Meillassoux, who unlike Ladyman and Ross, are even farther out, since their models make no empirical claims—they are dealing with set theory, not general relativity). Basically, there’s been a lot of work in phil of mathematics that Badiou and Meillassoux jump past, but it has to with conflating math’s “representational” quality with its presentational quality—it is the thing itself. If you read Badiou or Meillassoux, you see that they claim the latter, but often use the language of the former, but these are contradictory claims. In any case, what is the language to use? Here’s the analysis:

One must earn the right to do so by describing a fundamental language within which no corresponding questions can be formulated. What might such a language look like? At some points, including the above quote about General Relativity, the authors seem to suggest taking as fundamental a language in which the world is described by reference to mathematical models. The idea that we should refuse to seek an account of what makes a given mathematical model an apt representation of the physical world is a recurring theme in the book. Another example:

What makes the structure physical and not mathematical? That is a question that we refuse to answer. In our view, there is nothing more to be said about this that doesn’t amount to empty words and venture beyond what the PNC allows. The ‘world-structure’ just is and exists independently of us and we represent it mathematico-physically via our theories. (158)

Perhaps, then we should attribute to them view (iii):

(iii) The fundamental language is some language adequate for pure mathematics, enriched with an additional predicate, ‘physically realized’, that attaches primitively to some mathematical entities and not to others.[7]

The problem with holding a view like this is that if one wants to accept principles of the form ‘if x is physically realized, and y bears such-and-such mathematical relation to x, then y is physically realized’, one must adopt them as axioms rather than deriving them as theorems from an account of what it is to be physically realized. And the authors clearly would want to accept many principles of this form — not only for isomorphism between models, but for other kinds of close relations, such as might obtain between a model for one language and a model for another language derived from the first model by some simple, invertible translation function. Some rather strong principles of this form had better be true; otherwise, the metaphysicians’ debates about the catalogue of fundamental relations will carry over unscathed into the context of view (iii), in the form of debates about the number and adicity of predicates interpreted by the physically realized model.[8]

It is tempting to think that one can get by with a very simple axiom: anything that stands as structure-preserving isomorphism to a physically realized entity is itself physically realized. The problem is that the standard (set-theoretic) way of doing pure mathematics doesn’t suggest any appropriately general meaning for ‘structure-preserving isomorphism’. Rather, this kind of talk is cashed out differently depending on the kind of mathematical entities we are talking about: groups, or vector spaces, or whatever. Interestingly, category theory seems to be different: there, the notion of a structure-preserving mapping seems to be one of the central primitives of the theory. If anyone can ever succeed in explaining category theory in a way that the larger philosophical community can understand, it will be interesting to see if it lets us carve out a version of (iii) out of which a principle capable of dissolving unwanted disputes about the catalogue of relations emerges naturally, rather than having to be written in by hand.

View (iii) is intriguing and radical. But the textual evidence for attributing it to the authors is not strong. Their talk about mathematical structures as representing the “real patterns” that constitute the physical world is hard to square with the idea that there is nothing more to reality than the mathematical realm. And the little they do say about mathematical ontology suggests that they are attracted by a view on which mathematical entities have the same non-fundamental status they assign to physical objects, as opposed to the fundamental status required by (iii).

If none of the languages I have considered is adequate for capturing the fundamental facts, what is left?