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Comment by Eupraxias | Hacker News Reader

Neophyte speaking - I love it, but so much in this area feels deeply established. Who are the rebels, the provacateurs, and what are their arguments? If I wish to 'know thine enemy', who would I read?

For example:

- who questions the foundational axioms of mathematical structuralism these days?

- who argues from a comparative analysis of logic and field theory?

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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

> 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

It's not an answer to your question, but you may be interested in Alfred North Whitehead. He was initially trained as a mathematician and co-wrote some important works, but eventually shifted to religion and developed what is called "process philosophy."

> Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, and finally to metaphysics. He developed a comprehensive metaphysical system which radically departed from most of Western philosophy. Whitehead argued that reality consists of processes rather than material objects, and that processes are best defined by their relations with other processes, thus rejecting the theory that reality is fundamentally constructed by bits of matter that exist independently of one another. Today Whitehead's philosophical works – particularly Process and Reality – are regarded as the foundational texts of process philosophy.

https://en.wikipedia.org/wiki/Alfred_North_Whitehead

Alain Badiou also wrote some books on mathematics. He is definitely outside of the "official canon" in analytic philosophy of mathematics. His math-related work seems to have been controversial among more traditional mathematicians and philosophers:

https://en.wikipedia.org/wiki/Alain_Badiou#Being_and_Event

Badiou is a user, not a philosopher of mathematics (he himself would agree). Being and Event —neither volume of the now trilogy— is certainly not about mathematics, nor philosophy of mathematics. Rather, self-avowedly, it is a work in the service of radical politics.

Maybe not a direct answer to your question, but still something considered as rebellious (by some):

https://en.m.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle

Not sure though if the philosophy behind has ever been published in English.

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