Should a naturalist be a realist or not?
Unsurprisingly, the answer to the question in the title of this post depends on the details of what one takes 'naturalism' about science to mean. The shared conception of naturalism is something like 'There is no first philosophy' (I think Penny Maddy explicitly calls this her version of naturalism) -- that is, philosophy does not stand above or outside the sciences. "As in science, so in philosophy" is (one of) Bas van Fraassen's formulations.
Each of the following two quotes comes from a naturalist, but the first appeals to naturalism to justify realism (about mathematics), while the second appeals to naturalism in support of anti-realism (about science).
In his review of Charles Chihara's A Structuralist Account of Mathematics in Philosophia Mathematica 13 (2005), John Burgess writes:
"If you can't think how we could come justifiably to believe anything implyingCompare van Fraassen, in The Empirical Stance (2002):
(1) There are numbers.
then 'Don't think, look!' Look at how mathematicians come to accept
(2) There are numbers greater than 10^10 that are prime.
That's how one can come justifiably to believe something implying (1)." (p.87)
But [empiricism's] admiring attitude [towards science] is not directed so much to the content of the sciences as to their forms and practices of inquiry. Science is a paradigm of rational inquiry. ... But one may take it so while showing little deference to the content of any science per se.(p.63)Both Burgess and van Fraassen are naturalists about their respective disciplines (mathematics and empirical sciences) -- but they disagree on what the properly scientific reaction to questions like "Are there numbers?" and "Does science aim at truth or merely empirical adequacy?" is.
The mathematician deals in proof. And proof is (at least a large part of) the source of mathematics' epistemic force. The number theorist (e.g.) assumes the existence of the integers and proves things about them; that's what she does qua mathematician. People with the proclivities of Burgess and van Fraassen would agree thus far, I think. But they part ways when we reach the question "Are there integers?" A Burgessite (if not John B. himslef) could say "If you're really going to defer to number theorists and their practice, they clearly take for granted the existence of the integers." A van-Fraassen-ite could instead say: "What gives mathematics its epistemic force and evidential weight is proof, and the number theorist has no proof of the existence of integers (or the set theorist of sets, etc.). Since there is no proof of the integers' existence forthcoming, asserting the existence of the integers (in some sense) goes beyond the evidential force of mathematics. Thus, a naturalist about mathematics should remain agnostic about the existence of numbers (unless there are other arguments forthcoming, not directly based on naturalism)."
Is there any way to decide between these forms of naturalism -- one which defers (for the most part) to the form and content of the sciences, and the other which defers only to the form? (Note: van Fraassen's Empirical Stance takes up this question, but this post is too long already to dig into his suggestions.)