Alexandre – that’s definitely a very interesting idea, that doesn’t seem to be very much pursued yet.

]]>But not the fact that there are substantives disputes *over the acceptability of particular axioms*, which is what I explicitly mentioned in my reply. On the face of it, that fact doesn’t look at all like it’s just the fact that there is controversy of the kind that might be bracketable by the mutual acceptance of some particular set of axioms for the theory. For this is by description substantive disagreement precisely over whether a particular axiom is acceptable.

]]>I believe that, for better understanding of the nature of mathematics, we should pay more attention to ephemera of everyday mathematical practice.

]]>Noah – that’s a consistent position to take, but I’m not entirely sure in what sense you’re going to mean that. Certainly the successor axiom has to be there, right? And we don’t want to count a structure that satisfies PA but also ~Con(PA) as the naturals – so we’re going to need to go beyond any recursive axiom system. But once we restrict to an isomorphism type, it certainly does seem plausible that there’s nothing in particular of that type that we mean.

Tanasije – that’s right. Mainly, I’m suspicious of lots of metaphysical claims, but that doesn’t mean that I don’t think they’re important and meaningful.

]]>I know the subtitle of the blog says “A general distrust of strong metaphysical claims in mathematics and philosophy.”, but you seem to be pointing to the difference between math as part of (more general) metaphysics, and math as formalism. ]]>

I don’t know whether this is constistant or sensical, but that’s what my intuition is.

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