[behel]

There are people like the late Edward Nelson who represent radical formalism: https://web.math.princeton.edu/~nelson/papers/rome.pdf

and question the consistency of the commonly used axiomatic systems like peano arithmetic: https://www.lesswrong.com/posts/gsvQSpeDHKXxjXwuM/edward-nel... (Although his inconsistency proof of PA turned out to be flawed.)

One should also mention Wittgensteins remarks: https://plato.stanford.edu/entries/wittgenstein-mathematics/

And in general thoughts going into the direction of ultrafinitism are quite provocative.

See for example D. Zeilbergs opinions: https://sites.math.rutgers.edu/~zeilberg/OPINIONS.html

[lordgrenville]

> A funny story about someone who was an ultrafinitist:

> -Do you believe in 1?

> -Yes, he responded immediately

> -Do you believe in 2?

> -Yes, he responded after a brief pause

> -Do you believe in 3?

> -Yes, he responded after a slightly longer pause

> -Do you believe in 4?

> -Yes, after several seconds

> It soon become clear that he would take twice as long to answer the next question as the previous one. (I believe Alexander Esesin-Volpin was the person.)

https://math.stackexchange.com/a/532/687466