As you may know, many mathematicians turned philosophers while trying to do work on the foundations of mathematics. It seems like logic was the gateway discipline. What we now know as the analytic turn in philosophy of early 20th century came from such as lineage before it devolved into philosophy of language. Frege, Russel, Whitehead, and Wittgenstein are well known in this line of thought. However, the history of philosophy and mathematics goes way back to pre-socratic philosophers like Pythagoras and centuries later Aquinas and then Descartes. The question posed is rather strange given that mathematical development has often been formed by philosophical thought. I guess by clarifying the author means providing solutions (since he/she mentions that "mathematical insight" is nowhere to be found in the literature)? Since philosophy is not the same as doing mathematics the only kinds of clarification that philosophy will provide is in terms of distinctions, definitions and criteria: what is a proof, etc. This is because 'philosophy of x' is always meta discipline.
Carlo Rovelli: "Why Physics needs Philosophy"<p><a href="https://www.youtube.com/watch?v=IJ0uPkG-pr4" rel="nofollow">https://www.youtube.com/watch?v=IJ0uPkG-pr4</a><p>A few notes from the above:<p>* the beginning of astronomy == plato's school<p>* the scientific method as falsification == Popper<p>* quantum theory / relativity / Heisenberg == positivism (If I don't see it (e.g. electron orbitals) it doesn't exist) (* e.g. complementarity)<p>* Einstein claimed that his reading of Schopenhauer was crucial to thinking about time, space, etc...<p>in essence, you are doing philosophy when you're re-evaluating your methodology and using a evolving reflective feedback loops to change your thinking.
Frege, Russell, Whitehead, Pierce--of course!<p>It doesn't help that many philosophers of mathematics are, for obvious reasons, either also logicians or mathematicians, so demarcating between advancements in philosophy of mathematics that clarify mathematics and advancements in mathematics that clarify mathematics can be a bit of a fool's errand.<p>Whatever the case, I dislike it when folks from the sciences or mathematics try to discredit or dismiss philosophy--funnier still, and luckily not as bad, is when they question the <i>value</i> of philosophy without realizing that question is in and of itself a highly philosophical question!<p>Philosophy has been around for a long time and isn't going anywhere in the perceivable future (though I suppose it depends on what metaphysics of time you subscribe to :) ).
On the flip side, here's an example of mathematicians clarifying philosophy.<p><a href="https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flaming_laser_sword" rel="nofollow">https://en.m.wikipedia.org/wiki/Mike_Alder#Newton.27s_flamin...</a>
Bill Lawvere was heavily influenced by philosophy in his [revolutionary] contributions to the development of Category Theory [1][2].<p>[1] <a href="https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPhilosophy" rel="nofollow">https://ncatlab.org/nlab/show/William%20Lawvere#RelationToPh...</a><p>[2] <a href="http://philosophy.stackexchange.com/questions/9768/have-professional-philosophers-contributed-to-other-fields-in-the-last-20-years/9814#9814" rel="nofollow">http://philosophy.stackexchange.com/questions/9768/have-prof...</a>
In some sense Grothendieck's investigations could be considered "philosophical"; in the early 20th century algebraic geometers studied objects called "varieties" and Grothendieck's coup resulted from asking the question "what is the general class of object with which we can do algebraic geometry?". Today scheme theory is entirely mathematical, but a scheme had to be conceived as a philosophical concept first. The link also mentions Turing's elucidation of the Turing machine as a similar process.<p>There is also the case of Frank Ramsey and Piero Sraffa, who were the only close friends of Ludwig Wittgenstein, and who went on to make major epistemological contributions to economics (and Ramsey was a philosopher in his own right): Ramsey was the first person to really clarify the concept of a subjective probability, and Sraffa was central in the capital aggregation controversy.
Isn't the crux of this back and forth proofs that rely upon the axiom of choice vs those that do not?<p>It is my understanding that philosophers have added a lot to our understanding of how the axiom of choice impacts logical reasoning about things which matter to humanity in concrete ways.
Bishop Berkeley's response to Newton might qualify. Newton had something useful but Berkeley showed he hadn't proved anything. It wasn't until Cauchy (?) proved things rigorously that calculus was on a solid footing.
Philosophy often revolves around questions of purpose in life, and any non-mathematician might wonder why someone would devote their life to the study of advanced math. The question of purpose becomes even more interesting when a mathematician spends decades solving theoretical math problems that have no immediate practical use in engineering or programming. Any mathematician who's engaged in such problems wouldn't have any trouble understanding his own motives; for him, the philosophy is so obvious that it goes without saying. But an outsider might see his actions as inexplicable - as inexplicable as an ascetic monk spending a week meditating on a frigid mountaintop.<p>Philosophy functions as an aid in that understanding. The mathematician might read it and recognize truth, if it is written well, and a non-mathematician might be able to learn the rationale for devoting one's life to math. It might have a strong enough effect to make someone gravitate towards that type of life.<p>Different works in the philosophy of math could deal with different aspects of the mathematical life, such as the inherent beauty of an elegant proof, and it could also go into the larger, more long-term purposes of math, whose effects might not be recognized until long after the mathematician is dead.<p>I would think that the best philosophy of math would need to be written by a mathematician. An issue with philosophy is that it's quite difficult to write it well. There is a lot of philosophy out there that's poorly-written. It might suffer from a lack of clarity or simply be inaccurate. To have a brilliant philosophy of math, you would need a person who is both a skilled mathematician and a skilled writer. That is a rare thing.
Most of the famous mathematicians were driven by philosophy. For example Kepler wanted to keep perfection in the new model universe. He thought he could do it by using Plato's solids for the relative distances between the planets. He was wrong, but the mathematical attempts he tired help to form a more correct answer. The motivation was philosophical.
Also, philosophy has been dogged since Descartes with the self-imposed goal of providing a "foundation" for all knowledge enterprises (e.g. science, math) — by attempting to, put crudely, solve the mind-body problem and legitimatize that our thoughts and connection to the world are valid.
> Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians?<p>I would hope so! The short answer is the philosophy of Math will help you determine whether what you're researching is <i>true</i>! Surely it would be very bizarre to research something with complete apathy regarding its truth value. A few examples:<p>The famous Peano axioms [1] are widely used to prove such things as the commutative property of multiplication (ab=ba). But as the name "axiom" suggests, you just have to accept them as true or the whole thing crumbles. So why is it true that "0 is a natural number"? If this is false, much (all?) of math research is in big trouble! Does this suggest a sort of mathematical epistemic foundationalism? If so, what are its limits? When is mathematical research warranted, and when can we simply regard mathematical beliefs as properly basic?<p>Also, consider the realist/anti-realist debate [2, 3] which seeks to answer the question "are numbers, sets, functions, etc. actual features of the real world, or are they all just in our heads?" (or some refined variation thereof). If they are real entities, how is it that these non-causal things (like 5) lie at the very heart of the laws governing the physical, causal universe? But if they aren't real, then what possible explanation can one give for the perfect harmony of the physical world and these functions, that are ultimately all in my head? Moreover, why is belief in these unreal entities so widespread (I know of no "amathists")?<p>[1] <a href="https://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow">https://en.wikipedia.org/wiki/Peano_axioms</a><p>[2] <a href="https://plato.stanford.edu/entries/platonism-mathematics/" rel="nofollow">https://plato.stanford.edu/entries/platonism-mathematics/</a><p>[3] <a href="https://plato.stanford.edu/entries/scientific-realism/" rel="nofollow">https://plato.stanford.edu/entries/scientific-realism/</a>
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if you asked practicing mathematicians whether or not the philosophy of foundational mathematics, the hilbert programme, and wittgensteins consternation on the notion of equality were at all relevant...you would get a pretty dispassionate 'of course'. for whatever reasons most of these topics are taught in the philosophy departments and not the mathematics.
I think it's worth making it clear that Plato's contributions to political science, maths, physics, and religious thought were all framed at the time as philosophy.
> Secondly, remember that broadly the point of philosophy is to make things not philosophy. In extremely simplistic historical terms, once natural philosophy becomes rigorous it becomes science, once philosophy of language became rigorous it became linguistics, and today we're seeing philosophy of mind turn to neuroscience.<p>This is absolutely not correct, and elucidates little but the prejudices of the answerer. Philosophy of language has /not/ become linguistics, philosophy of mind has /not/ become neuroscience, and only a subset of natural philosophy has become natural science.<p>The philosophical questions discussed by Socrates have elided the grasp of both dogmatic rigor and empiricism for twenty-four hundred years, and there seems to be absolutely no reason to expect this to change.<p>The answerer has either no actual grasp of the history or content of philosophy, or has simply decided, apparently by fiat, to discard all but the most narrow positivism-flavored slice as nonsense.
Has philosophy ever clarified mathematics?<p>Let's frame this question properly for analysis.<p><pre><code> All work of Aristotle is philosophy.
Aristotle's work includes syllogistic logic.
Syllogistic logic clarifies mathematical inference.
Mathematical inference is part of mathematics.
Therefore, some philosophy clarifies mathematics.
</code></pre>
QED
It strikes me that this is similar to the 'no true ai' paradox, where as soon as a program can do something, such as beat a human at chess, then it stops being AI and starts being just computation. As soon as something philosophical becomes rigerous or well understood, we stop calling it philosophy and start calling it mathematics.