[asdflkj]

Mathematics is a product of the mind; the mind is a product of evolution; evolution is a product of natural laws. If you want answers to these questions, it helps to look at what we are, exactly, and how we came to be.

In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of arithmetic, or basic plane geometry. Mathematical proofs are not formal proofs--they are instructions for our brains. Evolution made the relevant parts of our brains the same, so same instructions lead to same results. That's why there's never any argument over whether a proof is correct, once a few people got to study it in detail. This also explains Hamming's observation that when proofs turn out to be "wrong" after math has evolved a bit, theorems are still usually correct. We find a new, better route to the same place in our brain, and recognize the hazards of the old route, now deprecated.

Okay, here is the key bit: if evolution made the relevant parts of our brains the same, that means it has arrived at a maximum, or at least a local maximum. What is the nature of this maximum? Physiologically, there are constraints on the amount of brain circuity our body can maintain. Brains consume a lot of energy, take up space, etc. So naturally, evolution ended up with a design where the same circuity can serve the greatest possible number of functions.

Of course, evolution only concerns itself with those functions relevant to our survival and reproduction. But there is nothing niche about those goals. If some general pattern occurs often in our quest for survival, then it likely occurs often in other quests that evolution never knew about--like building airplanes.

[calcnerd256]

Mathematics is not a product of the mind any more than physics is. All theorems are true (or, more precisely, all theorems follow from their axioms), even the ones we haven't discovered yet. That a mind can choose an axiomatic system to explore does not mean the relationships between those axioms and their theorems are created by that mind.

[asdflkj]

Mathematics and physics are products of the mind, obviously. Mathematicians and physicists do their work by using their minds. They don't channel some divine truth--they merely filter what their mind makes through certain criteria. I don't understand this common tendency, exemplified by your comment, to shift attention away from how mind makes things, to the criteria according to which we filter them before we call them "science" or "mathematics". I've studied mathematical logic, and it has been of little use to me in mathematics. Philosophy of mathematics has been of no use at all. The practice of mathematics can get by perfectly well without that stuff. So maybe it's time to put aside the mysticism, and start looking at how our brains actually make what they make. Especially since we are just now starting to understand what brains are and how they've evolved.

[calcnerd256]

I'm not talking about how we build them. I'm talking about what they are, and what they are is as they would be whether they were built by humans or computers or nature. That the theorems follow from their axioms is not a human invention, nor could it be. There is a difference between discovering something and inventing it. Man could not invent mathematics any more than man could invent electricity. When I say physics exists, I mean that the physical world exists and follows rules. If we discover those rules, it does not mean we have invented them.

[asdflkj]

You're "not talking about how we build them", but you were responding to my post where I am talking about how we build them. I'm gonna say it again: I don't understand this tendency to shift attention away from the "how"--rather insistently, in your case.

[calcnerd256]

Either you're not just talking about how we build them or I misunderstand you, because the first thing you said was, "Mathematics is a product of the mind."

[calibraxis]

On philosophy of mathematics, have you checked out Reuben Hersh? Or Lakatos? They're much more interesting to me than the usual platonism/formalism.

I'm guessing platonism/formalism were popular in arguing against other ways of understanding the world, like folk science, authoritarianism and mysticism. (I'm not equating the last three.) Maybe also as a foundation myth for professional mathematics.

[asdflkj]

Thanks--I'll check them out. I hope it's the kind of stuff that is only called philosophy, but actually falls under the purview of science, which doesn't yet have a framework to deal with it. That's the only kind of "philosophy" I've ever found of value.