For those who understand French, here is an inspired explanation right to the point by Alain Connes about the work of mathematicians

The two following videos by André Lichnerowicz also contain thoughts about the nature of mathematics. I had the chance to meet Pr. Lichnerowicz when I was about 15 since he was a friend of my grandmother. I was already curious about mathematics at that age and he took a lot of time to discuss with me about the philosophy of mathematics and thus played a decisive role in my vocation.

A. Connes and A. Lichnerowicz confront their different opinions about the nature of mathematics in the beautiful book Triangle de Pensées.