Badiou and Philosophy of Mathematics

OP: “I understand he’s developed a fairly complicated ontology based on mathematics, but does this involve any new insights into mathematics as mathematics?”

egbertus_b: “I guess that strongly depends on what qualifies as ‘insights’. He has published writings which clearly try to address questions in the field of philosophy (and history) of mathematics, for example, Number and Numbers.

I work in philosophy of mathematics and I’m not really aware of his work being referenced anywhere in relevant academic publications in my field. It certainly hasn’t started a new school of thought and it hasn’t had a significant impact on ongoing discussions in academia. His reasoning is a bit wonky to be honest and while he’s definitely educated in math, there are some gaps in his knowledge. I think this is a fair review.

On balance, Number and Numbers is a highly creative interpretation, but I think Badiou has the roles of informal mathematical narrative and proof exactly reversed. He believes, like set theorists of old, in mathematical realism. […] Believe what you want. What matters are new systems, logics, heuristics, conjectures, counterexamples, theorems, proofs. However you explain these is fine, but don’t take mathematical metaphors too seriously, even as these are essential to understanding, communication, and teaching. In particular, the idea that ZF, or other set theories, provide ‘foundations’ is itself a metaphor, true in part, but today far from having the ultimate status envisioned by Frege, Russell, or Gödel.

And more than anything:

Anti-constructivist bias distorts Badiou’s history of mathematics, in which virtually nothing happens vis à vis number after around 1900. In particular, he ignores Alan Turing and related theories of symbolic processing (Post, Church, Chomsky, et al.), from which came modern computing and its world historical consequences. Of course, Badiou detests such ‘constructive’ thought, so he ignores its milestones, including equivalencies of many mathematical theories and computational procedures.

He seems to be completely unaware of the development in constructive mathematics and modern proof theory. Now, if you lack knowledge in the field of mathematics, you’ll run into problems trying to explain, justify or describe the discpline ‘mathematics’.

Now recall Badiou’s fundamental premise, namely the primacy of real numbers and their algebra for representing nature or other expressions of Being. It used to be assumed, as Badiou does, that to do much applied science, you needed the full real number line and its basic operations (integrals, derivatives, functional operators, etc.) as found in college physics. Hence it was thought that some infinitary set theory was needed to justify all that, in the spirit of Quine’s ontological commitment. But that turns out to be false. Hermann Weyl already saw in the 1920s that much abstract mathematics could be developed using weak set-theoretical methods, sometimes called ‘predicative,’ in contrast to ‘impredicative’, definitions implicit in many real number concepts (e.g. the ‘least upper bound’ of infinite sets of real numbers). The question then is whether modern science really can be codified using such weak assumptions, thus greatly reducing any supposed ontological burden implied by a Quine or Badiou. Solomon Feferman, the proof theorist most critical of much modern set theory, and editor-in-chief of Gödel’s Collected Works argues persuasively in In the Light of Logic (1998) that Weyl was essentially correct: very little set theory, and perhaps none of the ‘completed’ infinite, is needed for natural science. You can study the higher infinite all you like for aesthetic or intellectual reasons, but it can’t be justified by an ideology of natural scientific need. So unless Badiou has other grounds for his starting premise, the project is unstable from the start, built on the sand Weyl saw beneath the Cantorian infinite.

I mean, not only that, large parts of mathematics can arguably be recovered from a position of strict finitism, without acknowledging the existence of anything infinite in any meaningful way.

Some people on the internet have a boner for him and apparently, Žižek thinks Number and Numbers is ‘breathtaking’. I’d say he barely managed to demonstrate a sufficient understanding of mathematics as a discipline, let alone to contribute significant insights into the philosophy of math.”