As mentioned in my previous post, Jody Azzouni thinks that ultrathin posits (the kind that have no epistemic burdens at all) aren’t taken to exist. He suggests that mathematical posits are ultrathin, and thus we shouldn’t take them to exist, so we get nominalism the easy way.
[Ultrathin posits] can be found – although not exclusively – in pure mathematics where sheer postulation reigns: A mathematical subject with its accompanying posits can be created ex nihilo by simply writing down a set of axioms; notice that both individuals and collections of individuals can be posited in this way.
Sheer postulation (in practice) is restricted by one other factor: Mathematicians must find the resulting mathematics “interesting”. But nothing else seems required of posits as they arise in pure mathematics; they’re not even required to pay their way via applications to already established mathematical theories or to one or another branch of empirical science. (pgs. 127-8)
In a footnote to this passage, he suggests that he argues for this claim more thoroughly in his 1994 book Metaphysical myths, mathematical practice: the ontology and epistemology of the exact sciences. I haven’t looked at that book yet (though I plan to in the not-too-distant future), but the claim here just doesn’t seem convincing at all.
It’s true that mathematical posits aren’t required to have any applications, even to other areas of mathematics. After all, who thought of the applications of non-Euclidean geometries, aperiodic tilings, or measurable cardinals? Generally not the people that came up with them.
But as any set theorist knows from experience, mathematicians don’t just go along with any postulation that one makes. They require it to be consistent, and often even want it to be known to be consistent – so many mathematicians still object to various large cardinal axioms that postulate the existence of structures that actually even have a lot of the unnecessary virtues, like mathematical interest and application. But mathematicians sometimes seem not to want set theorists to just postulate these things willy-nilly. And on a more down-to-earth level, we can’t just postulate solutions to differential equations, or even algebraic equations – we have to prove existence theorems. In algebra, fortunately, people building on the work of Galois were able to show that for any polynomial with coefficients in some field K, there is some field K’ containing K in which that polynomial has a solution (this required a lot of tricks of algebra and knowledge of how to actually construct fields from scratch). Kurt Gödel actually helped this cause of proving existence theorems greatly – thanks to his completeness theorem, we (those who accept ZFC) now know that any consistent set of axioms defines a non-empty class of mathematical structures (this leads to much shorter, more powerful proofs of the existence of things like algebraically closed fields). Thus, thanks to Gödel (and our antecedent acceptance of at least a sufficient fragment of ZFC), Azzouni is almost right about postulating the existence of structures. But we still need to prove them consistent, or otherwise argue for their existence in the cases where we can’t prove them consistent.
Thus, mathematical posits don’t live down to his ultrathin expectations for them. It’s not clear to me if they are really “thin” in his sense either (and they’re almost certainly not “thick”), so it’s not immediately clear whether his criterion should say they exist or not.