I see, so as a low-dimensional topologist I'm not a mathematician, thanks!* But more seriously, if you haven't read it already you might take a look at Thursto

I see, so as a low-dimensional topologist I'm not a mathematician, thanks!* But more seriously, if you haven't read it already you might take a look at Thurston's "On Proof and Progress in Mathematics." He had a stunningly accurate intuition, and putting what was in his mind into a form that other experts could understand and consider to be a proof gave many mathematicians a job for quite a long time. So long as you had him around, you could query him for any amount of detail for why his claims were true, so in that sense he had real proofs. Symbolic proofs are sort of a lowest-common-denominator, serializing what's in one's mind to paper and reducing things to machine-checkable rules. It's also a rather modern idea that this is what math is. I think it's very much worth considering math to be the study of what can be made perfectly obvious, which is closer to what Thurston seemed to be doing vs formalism. * This is sort of a joke. At least I'm able to convince people it's math because in principle it can be written symbolically, and formalism is rather compelling as a foundational philosophy, if a bit nihilistic.