| > I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid" I think you're misinterpreting what I was saying. Of course, a formalist will say that some rules are "more valid" in the sense that they produce more interesting or useful theorems. My point was, to a formalist, there is nothing more to be said about the validity of axioms than the value of the theorems they produce. Whereas, from certain other perspectives in the philosophy of mathematics, that is not the only grounds on which axioms can be judged. > This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional. You sound like a logical positivist. And that's the issue – if your philosophical assumptions are positivist, then non-positivist philosophies of mathematics (and of anything else) simply aren't going to be intelligible to you. They can only make sense if you are at least willing to doubt for a moment your positivist assumptions. > Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about? Conceptualism claims that mathematics exists in the mind–but it doesn't claim necessarily only human minds. If animals have minds too, then mathematics can exist in animal minds as well, even if in a much more rudimentary form. I doubt any conceptualist would say, that if intelligent extraterrestrial life were discovered to exist, that their minds wouldn't contain mathematics simply because they are a different species from homo sapiens. > So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it. There are different types of constructivists: (a) those who have a philosophical commitment to constructivism; (b) those who are interested in constructivism for practical reasons (related to computer science); (c) those who are just interested in it as an interesting mathematical system to explore. You can be (b) or (c) without needing any philosophical commitments at all, and they are completely compatible with a formalist philosophy of mathematics. And, quite possibly, the majority working in constructive mathematics today are (b) or (c) not (a). But, historically, the founders of constructive mathematics (e.g. Brouwer) were very much (a) not (b) or (c). > There simply is no "non-formalist" mathematics. I think you are conflating mathematics with the philosophy of mathematics – they are two distinct disciplines. Disagreements about the philosophy of mathematics make no direct difference to mathematics itself; at the margins, they can influence judgements about which problems are interesting – although, even there, a person can find ultrafinitist mathematics interesting without needing any philosophical commitment to an ultrafinitist philosophy of mathematics. |