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It is a tragedy that all too often mathematicians, when asked about foundations, may be of the opinion that this was all sorted out in the earlier part of the 20th century. This is certainly not the case. Active research in this area is happening in many places, by a small but dedicated (and spreading) following.
An example of research on foundations in the classical arena is a group led by Hugh Woodin in the USA; his team is investigating possibilities for a "golden axiom of set theory", with hopes that there will be interesting outcomes concerning big problems like the continuum hypothesis.
Constructive mathematics is the "typical" response by those who take issue with the classical conception of mathematical proof, and there are groups currently looking at intuitionistic type theory, set theory, and the mathematics that is built on them - there are active groups in Germany, Scandinavia, Japan, and New Zealand (to name a few). A Google/Wikipedia search for constructive mathematics will bring up many references. I am currently working with Douglas Bridges (and others) on various constructive ideas.
Paraconsistent mathematics, being fairly new on the block, has not developed as large a following as yet but there is active research going on in South America, Australia and New Zealand - it seems to be a largely southern-hemisphere phenomenon so far. I'm currently working with Zach Weber from Melbourne on characterising the real line using paraconsistent reasoning.
You typically don't have to look far!