How would you account for suboptimal mathematical systems such as that used by the Piraha tribe? Their numerical system is limited to one, two, and many (3 or more). Research suggests that they can't match sets of objects to numbers past two or three. If math reflects axioms present in nature, those axioms should be the same globally.
My argument isn't that math per se reflects axioms in nature, but rather that we may be using mathematical axioms to approximate a set of axioms that exist a priori in nature. Obviously for thousands of years in the west we thought Euclidean geometry was the only geometrical system. Then we discovered that non-Euclidean geometries existed and could make accurate predictions about space that Euclid's system couldn't. This was a disaster for the Kantian view of mathematics, in which it was one rigid, accurate system.
So basically I think it's possible that there are two kinds of math-natural mathematics which we discover, and synthetic mathematics which we invent. Both have the status of abstract objects, but synthetic mathematics can be explained as a product of human minds (therefore subsisting on the physical, not violating physicalist ontology), whereas it's not clear that all math is just imagined axioms-it certainly seems like nature respects certain mathematical principles of some kind, and it's hard on a purely naturalist account to explain why. I don't think it's impossible, but it's safe to say I'm not smart enough for that question, so platonism remains a genuine possibility to me.
Why not abandon physicalism but keep naturalism? There's few enough non-physical entities with any explanatory power (math, maybe normativity or consciousness) that you wouldn't have to open the floodgates to some supernatural rogue's gallery.
This could be a terminological error on my part, but I think physicalism does allow for non-physical things so long as they supervene on the physical and depend on it for existence, in the style of Aristotle’s metaphysics. The way I’ve written about the term naturalism encompasses a lot more than that, but in this respect I think the two terms are synonymous.
To be clear, I’m not admitting the existence of supernatural mathematics, only allowing that it *is* a distinct possibility for which no refutation presently exists, so far as I’m aware. I don’t think it’s unfalsifiable-if theoretical physics turns up some explanation for all the math we keep finding everywhere, platonism can be dismissed. As it is, though, I do think it remains a plausible theory.
To be clear, by this I mean specifically mathematical platonism (the convention now is to capitalize the P when talking about Plato’s actual ontology, and drop it when discussing the belief in a priori abstract mathematical entities), but it is a good question whether one can do that without allowing robust Platonism in with it. Quite possibly not.
How would you account for suboptimal mathematical systems such as that used by the Piraha tribe? Their numerical system is limited to one, two, and many (3 or more). Research suggests that they can't match sets of objects to numbers past two or three. If math reflects axioms present in nature, those axioms should be the same globally.
Edit: Piraha, not Pirahi
My argument isn't that math per se reflects axioms in nature, but rather that we may be using mathematical axioms to approximate a set of axioms that exist a priori in nature. Obviously for thousands of years in the west we thought Euclidean geometry was the only geometrical system. Then we discovered that non-Euclidean geometries existed and could make accurate predictions about space that Euclid's system couldn't. This was a disaster for the Kantian view of mathematics, in which it was one rigid, accurate system.
So basically I think it's possible that there are two kinds of math-natural mathematics which we discover, and synthetic mathematics which we invent. Both have the status of abstract objects, but synthetic mathematics can be explained as a product of human minds (therefore subsisting on the physical, not violating physicalist ontology), whereas it's not clear that all math is just imagined axioms-it certainly seems like nature respects certain mathematical principles of some kind, and it's hard on a purely naturalist account to explain why. I don't think it's impossible, but it's safe to say I'm not smart enough for that question, so platonism remains a genuine possibility to me.
Why not abandon physicalism but keep naturalism? There's few enough non-physical entities with any explanatory power (math, maybe normativity or consciousness) that you wouldn't have to open the floodgates to some supernatural rogue's gallery.
This could be a terminological error on my part, but I think physicalism does allow for non-physical things so long as they supervene on the physical and depend on it for existence, in the style of Aristotle’s metaphysics. The way I’ve written about the term naturalism encompasses a lot more than that, but in this respect I think the two terms are synonymous.
To be clear, I’m not admitting the existence of supernatural mathematics, only allowing that it *is* a distinct possibility for which no refutation presently exists, so far as I’m aware. I don’t think it’s unfalsifiable-if theoretical physics turns up some explanation for all the math we keep finding everywhere, platonism can be dismissed. As it is, though, I do think it remains a plausible theory.
To be clear, by this I mean specifically mathematical platonism (the convention now is to capitalize the P when talking about Plato’s actual ontology, and drop it when discussing the belief in a priori abstract mathematical entities), but it is a good question whether one can do that without allowing robust Platonism in with it. Quite possibly not.