I spent a couple of hours today in an interesting discussion with another PhD student which basically started (at least the part I am referring to here) because we could not agree on whether or not the question of how exactly do we know that we understand something makes sense or not (above all formulated in such general form; at opposition to more restricted --highly context dependent-- forms).
I wanted us to work on actually trying to answer the question, but could not get him to even agree on the fact that the question even makes sense to start with. This is sometimes how a career in philosophy starts, but in this case, this has lead (is leading) to a relatively deep change in the way I perceive mathematics.
Having been brought up as a logician (in France -- which probably made the entire thing worse), I have always had a relatively strict perception of mathematics; as a relatively formal activity taking place against axiomatic systems whose end objective is to establish that some statements are mathematically correct according to the cannon of mathematics reasoning: writing proofs. The other part of the mental activity of the mathematician has, to me, always belonged to the realm of meta-mathematics or just general human intuition (even though specialised against what we call mathematical objects). Given this mindset it is not surprising that I have always held Euclid as my favourite mathematican.
I have to mention actually how that all discussion started. I wanted to find the reason why mathematicians use the word "space" (as in "topological space", or "vector space") against some mathematical structures but not others. To me this was an almost linguistic question, whereas to him (even though he didn't formulate it explicitly at first, which contributed to the seemingly incompatible positions we had) this is very likely to be a mathematical question itself.
One thing leading to the other, I eventually understood, that mathematics goes much higher in his mindset than in mine, and that, to keep things simple, the split I have always felt between mathematics and mathematical intuition, simply results from my education rather than being a reflexion of what mathematics actually are, or at least, should I say, a reflexion of the various facets that mathematics come under (the classical, recognisable, form of what we write in books being just the somehow more trivial form).
So now, all this could appear as just a couple of educated people splitting hair over nothing, but the distinction has, among other things, a non trivial effect on the way one then refers to the actual contributions of another mathematician (which goes well beyond what the latter has written); and probably few other important things I need time to think about before being able to put words on them...
Thinking of it, this also explains why those people, yes those people, all swear by Alexander Grothendieck. I can see the appeal now :-)
Interestingly, we had Barry Simon from Caltech giving us a talk this afternoon called "Tales of Our Forefathers". A talk about mathematicians (even though not specifically about their mathematical contributions). I had never touched before somebody alive who has written almost 400 research papers...