I'm surprised Wittgenstein hasn't been mentioned here. He thought and wrote extensively about the foundation of mathematics. In fact, the write-ups of a series

I'm surprised Wittgenstein hasn't been mentioned here. He thought and wrote extensively about the foundation of mathematics. In fact, the write-ups of a series of his lectures features discussions between him and Turing (and other luminaries) about what maths is about (and W tends to come over stronger on the subject): https://www.amazon.co.uk/Wittgensteins-Lectures-Foundations-Mathematics-Cambridge/dp/0226904261 https://www.amazon.co.uk/Wittgensteins-Lectures-Foundations-... It's great fun to read.