Mathematicians don't care about foundations

Summary

Many people seem to believe mathematicians work in non-constructive, non-structural, battered foundations because they love their Platonic realm and have a kink for AC and LEM. The reality is most mathematicians don’t have a clue about foundations, they don’t care, and happily work informally for all their lives. Case in point, mathematical foundations are a pretty recent thing (19th century if we are being generous) but their establishment didn’t deprecate previous mathematics, which continued to be studied and used just as well. Even during the so-called ‘crisis in foundations’ at the start of the 20th century most mathematicians didn’t blink an eye. Only a few pages of math had to be rewritten, and they were about foundations themselves. I’m being intentionally provocative in calling out foundations here so, let me throw a bucket of water on this fire already. Foundations are not useless to study at all! On the contrary, mathematicians are thankful someone figured out foundations for them, so that they just need to know some TL;DR about which logical maneuvers they are allowed to perform and which objects they are allowed to claim the existence of. Such ‘irrelevance’ witnesses a robustness in mathematics, betraying a deeper nature behind it’s facade of rigour. Mathematics is irreducibly informal (even foundations), i.e. relying on some unspoken mutual understanding on how to interpret signs, concepts, and norms. The difference among mathematicians is how deep they have to shell such conventions before being satisfied. Thus math is not a castle built on a bedrock of unshakeable foundations. Math is rather a collective codification of intuitions squeezed into formal frames in the best way possible. This is why the ‘crisis in foundations’ didn’t really matter for most mathematics: what broke was the frame, not the ideas. This is also why we get new and improved mathematical theories every now and then. Saying ‘space’ today doesn’t evoke the same suggestions it used ...