Originally posted by Lanny
In general I lean nominalist and while that doesn't totally commit one to believing mathematics is invented rather than discovered it lends itself to that conclusion and it is my general inclination. I watched a walk by Philip Wadler and he made this argument about how three or four mathematicians seemed to independently discover the decidability problem and that that was an argument for a discovery model of mathematics. Wadler is a really smart guy but I thought that was kind of a disingenuous move, it fails to acknowledge the distinction between axioms and theorems, the decidability problem is the product of systems in which it emerges, yes we may "discover" new properties of a formal system as in this case but that discovery is predicated on the invention of the system to start with. I'll claim that nothing which is discovered can be fully reduced to invented parts (I recently discovered my phone has a "do not disturb" feature, that's not an argument that my phone or that feature are discovered properties of the universe rather than something invented).
One might naturally make some argument from isomorphisms between our mathematics and the world as being evidence that mathematical systems are founded in natural, discovered elements of the world. But this fails too, to model the world post-hoc is not to be discovered. Consider natural languages model the world yet no one would claim that English was "discovered". Likewise the relationship of mathematics to the world is an intentional product of its genesis, it contains similar structures not because it is similar to the world but because it exists to model it.
I agree up to a certain point with you. I'm more of an intuitionist than a platonist myself, and I think that axioms are invented and theorems are discovered within the structural limits drawn by the design of the axioms, which is basically what undecidability is all about (I guess that the three mathematicians you were talking about are godel, emil post and turing). So, for instance, working in number theory using the peano axioms rather than the presburger arithmetic is a whole different thing, and that's what I think has been the finishing blow to platonism: the fact that there are undecidable statements within a system implies that there's not such a thing as a perfect bijective function between the mind and the so called perfect platonic realm. Actually, I don't see much difference between religion and platonism since the last one asks for a blind faith in the aforementioned hyperuranion and it makes sense from an historic perspective since platonism stems from the pythagorean mysticism.
However I find your critics of the analogy with the "do not disturb" phone feature kinda fallacious. The designers of a phone will know (virtually) every property of the phone they're developing, so there will be nothing to discover. But mathematics is on a whole different level of complexity. When cantor single-handedly created set theory he could have not predicted that the continuum hypothesis is undecidable, so this is something that has been "discovered" by paul cohen, but of course it was already a sort of "built-in feature" of axiomatic space they were working on. I dunno, man, it's fucked up and I'm getting confused, but I'd go as far as to say that the very fact that when you invent a system you cannot virtually know every property means that then everything else from that point forth is discovered.
About isomorphism between math and the real world, those are discovered too and since I think that the so called "pure mathematics" is an hoax and doesn't exist (fuck that faggot of harold hardy), I infer that every mathematical model will eventually be linked to something in the real world (that doesn't necessarily imply the opposite): for instance, category theory, which is known as the most abstract field of math, has been proved useful in many real-world fields (computational linguistics, sorting algorithms, AI and so on): ah, with real-world applications I talk about things that are discovered (such as new physics shit) and things that are invented (such as computer science stuff, artificial intelligence, automata theory). So math could be invented and sometimes only used for other things that have been invented and not discovered. Sounds very autistic.
Originally posted by thelittlestnigger
Does the mind have no mathematical properties? Is there any way we could define it as a mathematical object? If I was a smarter man I would say perhaps but all I can say now is I don't know.
Yeah I think we could define the mind as a mathematical object and eventually we will in a much more rigorous way. But we do now by now is that the mind is not a turing machine because we do not crash by bumping into the halting problem when we reason around undecidable statements (at least we can prove that they undecidable, a computer cannot)