a brief exchange on mathematical ontology
I was asked to clarify my ontology of mathematics, specifically, following upon an example involving types and literal strings, to say
what ontological status [I] would grant entities like A and B or the series ABABABA etc. Are they abstractions from actuality, and so logically significant but ontologically derivative, or are they real entities that inform the ordering structure of concrete reality, or both, or neither?
I took a stab at replying thus:
1. I think the most helpful way to begin to respond might be to distinguish my Platonism from what usually goes by the name.
2. Ordinarily, in the philosophy of mathematics, one is expected to have one understanding of what it is for something to be, to exist, to be an abstraction from something else, etc. and to give a uniform account of pretty much all “mathematical entities” according to this scheme.
3. In my attempt to think through the contemporary form of a Platonic dialectic (and I’m following Lautman and Badiou among others here) this would amount to a massive misunderstanding of the problem. I want to take a different hint from Plato, viz. that differences that we first articulate as being differences among forms (and that would therefore seem to demand a prior and univocal understanding of “[a] form”) – say the difference/s between isomorphism and adjunction, just to give an example – actually make an ontological difference. So the work of an ontology of mathematics (understood as a philosophical discourse developing in time and not just a static, explicit or dispositional, opinion held by someone doing mathematics) is not to defend and apply an account of the difference between “forms” and “things” or “abstracta” and “concrete reality” but to enter into an investigation of the relation between ontological concepts of form (etc.) and the form of ontological concepts. This reversibility – of form and concept, of philosophy and mathematics, and frankly of the states (of mind or brain) involved in the doing of these – is roughly what counts as dialectical for me.
[Notes: nonseparable & nonidentical, not ontologically neutral so much as metaontologically critical – conventional ontological positions (or a certain picture of what it is to have an ontology) are all excluded]