Naturalized philosophy of mathematics

There are three types of naturalism in the philosophy of mathematics: metaphysical, epistemological and methodological. Metaphysical naturalists maintain that all entities are natural. One reading of this claim is that mathematical ontology is the ontology of natural science - which of course leads immediately to the question as to just what ontology is indispensably needed by the natural sciences. Another reading is that all mathematical entities are spatiotemporal. This view faces considerable difficulties, as it seems to go against the claims and methods of mathematics. Epistemological naturalists maintain that we can only know about entities spatiotemporally or causally connected to us. Though prima facie plausible, epistemological naturalism has encountered resistance on many fronts.

Methodological naturalism sees scientific standards, suitably understood, as authoritative. In its canonical version, science is construed as natural science, and thus the acceptability of mathematics is linked to its role in natural science. The most obvious argument for this form of methodological naturalism is the success argument: natural science is the most successful sphere of human inquiry and should consequently trump other disciplines. However, it turns out that the success argument is difficult to develop convincingly. Some philosophers also believe, controversially, that there is room for a naturalism that takes the authoritative standards in the philosophy of mathematics to be those of mathematics itself.