Has philosophy ever clarified mathematics?
&gt; 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&#x27;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 &quot;axiom&quot; suggests, you just have to accept them as true or the whole thing crumbles.  So why is it true that &quot;0 is a natural number&quot;?  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&#x2F;anti-realist debate [2, 3] which seeks to answer the question &quot;are numbers, sets, functions, etc. actual features of the real world, or are they all just in our heads?&quot; (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&#x27;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 &quot;amathists&quot;)?<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Peano_axioms" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Peano_axioms</a><p>[2] <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;platonism-mathematics&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;platonism-mathematics&#x2F;</a><p>[3] <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;scientific-realism&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;scientific-realism&#x2F;</a>