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I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X.

My special interest is in the area of mathematics. I am a research mathematician working in applicable, if not applied, mathematics and I also have some academic background in philosophy, though no research experience.

I guess there are some connections or interdisciplinary areas between philosophy and mathematics - presumably in logic and maybe elsewhere. I am not interested in these as such. Rather, I am looking for work in philosophy of mathematics (as defined in e.g. both answers at What is the Philosophy of Mathematics?) that is used by practicing mathematicians when doing mathematics, or used to argue the importance of this or that mathematical theory, or that is formalized in the language of mathematics.

A concrete sign of this would be a work of philosophy cited by mathematics papers or textbooks. Mathematics papers and textbooks can be defined, for the purposes of this question, as those where proving theorems is a significant preoccupation.

I am interested in recent developments; say, philosophy within during this millenium. If there is a thriving research community in this direction, than some overview of that or a major research question or two would be nice to know. If not, examples, or a persuasive argument that this does not happen, would be good answers.

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  • Very limited interaction. We can consider Intuitionism and its impact on Constructive Mathematics. At the same time, we have to consider that the "best effort" towards constructive mathematics : Bishop's one was developed without explicit ref to Intuitionsitic Phil.
    – Mauro ALLEGRANZA
    Mar 28, 2019 at 13:49
  • It used to a lot more than now, for example Godel's work in the 1930-s (axiom of choice, continuum hypothesis, incompleteness) was directly motivated by philosophical concerns about the nature of mathematics he picked up from Hilbert and Carnap. But even then it was largely confined to mathematical logic and abstract set theory. Recent developments include "philosophy of mathematical practice" and the study of relations between formal and informal proofs, see Mancosu volume here. But all of that has little effect on mathematics.
    – Conifold
    Mar 28, 2019 at 19:19
  • I have never seen any examples of direct influence of philosophy on maths. I'm sure it exists, but my feeling is that it's extremely limited to obscure corners of the field.
    – Ben W
    Mar 30, 2019 at 19:41

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The answer to that in a short sentence would be: it depends.

Now for the long one: Quine, Carnap, Putnam, Kripke and other Philosophers of Mathematics have had profound influence on the way Mathematics is done. For most of them, like Quine, set theory is just that, philosophy (Refer to Quine's On Carnap's View On Ontology and Word and Object). Obviously, now, this was not one sided. His philosophy at times aided in his logical endeavours. Furthermore, philosophy of logic, and logic in-general is a subset of philosophy of mathematics; this tradition has continued since the days of logicism. Although now we consider logic a separate and independent subject, originally it was not and I believe it is still not distinct since logic has and will have direct metaphysical implications (Refer to non-classical Paraconsistent Logic).

Now for useless Philosophy of Mathematics

Don't get me wrong. I don't think it is useless in general, but useless in terms of real mathematical implication.

So which part of Philosophy of Mathematics is useless, you wonder? Well, that would be Ontology of Mathematics. That is, the branch that preoccupies itself with existence (be it abstract or concrete) and non-existence of mathematical object. This branch of Philosophy has absolutely no bearing on the way mathematics is conducted just like the platonic numbers have no bearing on reality. Hehe. Don't get me wrong, I am not saying neoplatonism is the only useless tradition, but I am saying the whole Ontology of Mathematics is useless (including nominalism, fictionalism, any form of realism, and the unnamed subsets of anti-realism).

Summary

There are branches within Philosophy of Mathematics which definitely effect Mathematics, but there are also branches stemming out of it (like Ontology of Mathematics) which have absolutely no effect on the way mathematics is conducted.

I hope I answered your question, feel free to ask for any clarification.

Some Contemporary Works Signifying Interplay Between Philosophy of Mathematics and Mathematics

Menzel, Christopher. "WIDE SETS, ZFCU, AND THE ITERATIVE CONCEPTION." Journal of Philosophy111.2 (2014): 57-83. Web.

Brauer, Ethan. "Second-order Logic and the Power Set." Journal of Philosophical Logic 47.1 (2018): 123-42. Web.

Horsten, L. (2015). The Influence of the Polish School in Logic on Mathematical Philosophy. European Review, 23(1), 150-158. doi:10.1017/S1062798714000635

Incurvati, Luca. "The Graph Conception of Set." Journal of Philosophical Logic 43.1 (2014): 181-208. Web.

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    @TommiBrander Brauer, Ethan. "Second-order Logic and the Power Set." Journal of Philosophical Logic 47.1 (2018): 123-42. Web.
    – Bertrand Wittgenstein's Ghost
    Mar 29, 2019 at 17:33
  • @TommiBrander There is active interplay between philosophy of mathematics and mathematics, especially within the sub-disciplines I mentioned in my main answer. Regards.
    – Bertrand Wittgenstein's Ghost
    Mar 29, 2019 at 18:02
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    @TommiBrander Definitely! Regards.
    – Bertrand Wittgenstein's Ghost
    Mar 29, 2019 at 18:52
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    @TommiBrander I added them, and also linked some more contemporary articles. Regards.
    – Bertrand Wittgenstein's Ghost
    Mar 29, 2019 at 19:51
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Philosophers of mathematics and Mathematicians seem to ignore one another. The Philosophers come up with theories that don’t seem to have any impact on what the Mathematicians do or think. The mathematicians might listen to what the philosophers say once a fortnight, and then they go and get on with their work just as they did before. You can’t tell from somebody’s mathematics if they are a fictionalist, a rationalist, a platonist, a realist, an operationalist, a logicist, a formalist, structuralist, nominalist, intuitionist.

Worse, they can argue themselves to be one thing on one day, the opposite the next day, and yet still carry on doing the same mathematics. Does this mean that mathematicians are deluded? Or that philosophy is irrelevant?

[...]

The trouble is that from a mathematician’s point of view, philosophers ask questions that have no impact on mathematical practice.

[...]

Mathematicians generally think that they are studying things that are true. Unless interrogated by a philosopher, they don’t spend their working time worrying about exactly what is the nature of that truth.

Cheng, E. (2004). Mathematics, morally.

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  • This should be the accepted answer. Even in logic you won't find hardly any publication by a mathematician that cites a logician employed at a philosophy department.
    – Jishin Noben
    Apr 16, 2019 at 10:30
  • @JishinNoben: That's completely false. George Boolos did a PhD in philosophy and was a professor of philosophy at MIT.
    – user21820
    Feb 25, 2021 at 6:42
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Adjacent to the question, there's a lot of Philosophy of Science (Physics, often) done in a vacuum (hehe), with complete disregard and proud ignorance of actual scientific practice. To the extent that you have societies explicitly countering this attitude, e.g. the Soc for Phil of Sci in Practice.

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    Although I find the link interesting, I wonder if this actually fits as an answer to this question of philosophy of mathematics. Perhaps you could expand your answer with other references. Is there a specific research question? Is this philosophy of physics cited in physics papers and textbooks? Welcome.
    – Frank Hubeny
    Jul 23, 2019 at 10:32

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