Answer by Mikhail Katz for Has philosophy ever clarified mathematics?
Bernhard Riemann's famous 1854 Habilitation lecture "On the hypotheses…" owed more to the influence of the anti-Kantian philosopher Johann Friedrich Herbart than to Kant himself. Here one could mention...
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Answer by Peter Heinig for Has philosophy ever clarified mathematics?
Kant has featured in Geog Cantor's argument that time is irrelevant for describing the continuum.This is a partly a (tangentially) relevant answer to the question, and partly a relevant comment on a...
View ArticleAnswer by Thomas Benjamin for Has philosophy ever clarified mathematics?
Can the works of Penelope Maddy ("Believing the Axioms", I and II), Georg Kreisel (his papers on "Informal Rigour"), and Benjamin Rin's paper "Transfinite recursion and the iterative concept of set"...
View ArticleAnswer by Samuel David Bravo for Has philosophy ever clarified mathematics?
Yes, philosophy has clarified mathematics.In 'Naming Infinity', by Loren Graham and Jean-Michel Kantor, they argue that Russian mathematicians in the early 1900s were able to introduce new concepts to...
View ArticleAnswer by Korbonits for Has philosophy ever clarified mathematics?
There's a great book called the "Synthetic Philosophy of Contemporary Mathematics" by Prof. Fernando Zalamea that came out a few years ago. I've read about half and would highly recommend.It's...
View ArticleAnswer by PMar for Has philosophy ever clarified mathematics?
I'm surprised no one has yet mentioned Cantor. His work leading to Transfinite Arithmetic, and from there to general Set Theory, overturned millenia of received opinion on the concept of the infinite...
View ArticleAnswer by Mikhail Katz for Has philosophy ever clarified mathematics?
After Newton and Leibniz developed infinitesimal calculus, a number of critics emerged to criticize the new technique. These included Berkeley, Cantor, and others. One of the few schools to battle the...
View ArticleAnswer by AlainD for Has philosophy ever clarified mathematics?
Logic (and formal systems) is a contribution of philosophy.The point is that up to recent times (~1800), mathematics was a discipline of philosophy, just as music was a discipline of mathematics at the...
View ArticleAnswer by Thomas Klimpel for Has philosophy ever clarified mathematics?
Just like René Descartes'La Géométrie from Discours de la méthode (1637) might have had a profound influence on mathematics, also Immanuel Kant's Transzendentale Ästhetik from Kritik der reinen...
View ArticleAnswer by Noam Zeilberger for Has philosophy ever clarified mathematics?
Per Martin-Löf has done work spanning mathematics, logic, and philosophy. I am not sure how much his early work on randomness was influenced by philosophical concerns, but his work on type theory in...
View ArticleAnswer by Sean for Has philosophy ever clarified mathematics?
I'm not sure that some people would agree... but Popper's work on provability has clarified how things should work in "classical" statistics, that we can only prove something is wrong not that it is...
View ArticleAnswer by Colin for Has philosophy ever clarified mathematics?
Lawvere has evidently applied a philosophical outlook to some effect in his mathematics.This answer by Urs Schreiber on philosophy stack exchange gets at this and...
View ArticleAnswer by Ben McKay for Has philosophy ever clarified mathematics?
Ludwig Wittgenstein invented the truth table, in trying to clarify the structure of propositional logic. Of course, this is now a standard tool in mathematics and computer science. Wittgenstein was...
View ArticleAnswer by Marcin for Has philosophy ever clarified mathematics?
Would you expect to learn much about how to practice science (or any sub-branch thereof) from the philosophy of science? I hope the answer would be no; that falls within the study of science.The...
View ArticleAnswer by user58926 for Has philosophy ever clarified mathematics?
To add to the discussion: the very dichotomy Philosophy x Mathematics didn't make sense to most thinkers of the past. That still holds true for many of today's philosophers - Philosophy is not a...
View ArticleAnswer by Lennart Meier for Has philosophy ever clarified mathematics?
As already remarked by others: If one tries a narrow interpretation of your question, you are asking a lot. You want someone whose specialty is not mathematics to elucidate a mathematical argument in a...
View ArticleAnswer by Gerhard Paseman for Has philosophy ever clarified mathematics?
I paraphrase with italics from the original post to give my perspective:My naïve expectation would be that metamathematicians * might take a difficult construction or proof, and clarify it by isolating...
View ArticleAnswer by Avshalom for Has philosophy ever clarified mathematics?
Perhaps it has been cited elsewhere in this context, but Hilary Putnam's 1980 short essay "Models and reality" brings together mathematics and philosophy in its discussion of $V = L$ and the...
View ArticleAnswer by R Hahn for Has philosophy ever clarified mathematics?
Turing has already been mentioned in previous excellent answers as someone whose ideas sit at the boundary of philosophy and mathematics, conventionally understood. I want to mention Ludwig...
View ArticleAnswer by Timothy Chow for Has philosophy ever clarified mathematics?
In order to address this question, I think it is important to first take a step back and examine with a critical eye something that we normally take for granted, namely the professionalization and...
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