By foundations of mathematics I am referring to the mathematical, logical, and philosophical foundations of the subject. I'm interested in seeing which of these have active research going on within them, and what those active areas of research are, i.e. what questions they deal with. Another way to put it; what questions in the foundations of mathematics remain unanswered, and which of these unanswered questions remain active fields of research? Any information you can share on the subject would be greatly appreciated!
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1[Philosophy of mathematics](http://plato.stanford.edu/entries/philosophy-mathematics/) is still "alive and well": you can see there ref to many detailed entries with biblio (many titles from 2000-on). – Mauro ALLEGRANZA May 11 '16 at 14:08
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Foundations in mathematics are understood far more pragmatically today than at the time of Hilbert, Russell and Brouwer. The latest trend is univalent foundations, see http://philosophy.stackexchange.com/questions/28353/are-univalent-foundations-of-mathematics-a-revamped-version-of-logicism/28380#28380 As a result, "foundations" are more technical and quite separate from philosophy of mathematics in the traditional sense, but check out recent philosophical/mathematical workshop on ontology of large cardinals http://logic.harvard.edu/efi.php#multimedia – Conifold May 12 '16 at 04:38
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You may find the introduction to _Set Theory, Arithmetic, and the Foundations of Mathematics: Theorems, Philosophies_, Kennedy and Kossak (editors), CUP, 2011 illuminating. – dwolfeu Oct 07 '22 at 11:49