> _When I was studying in Moscow University, Foundations of Mathematics was really a subject which was extremely unfashionable. We had this different - as we called them "cathedrals" - different groups inside the university, different groups of faculty. So there was the group of algebra, and the group of topology, and then there was also the group of analysis and the group of differential equations. And there was also a group of Foundations of Mathematics. And at the end of the second year of studies, every student had to choose a specialization. Every student had to choose which of the groups to continue to study with. That was of course a big deal, so we discussed things a lot, and different groups had different reputations. Some groups were considered to be very cool and fashionable, and some groups were considered to be very uncool. At my time, the group of algebra was considered to be cool because there were some very famous people there like Manin, and people about whom various legends were going around in the University. The group of topology and geometry was cool, also because there were some very great people there, like Novikov (Sergei Petrovich). Some other groups were kind of less cool. But the group of Foundations of Mathematics was probably the least cool group of all. Worse than that was only the History of Mathematics. Nobody really considered going there. At least nobody from "us" who thought about ourselves as good mathematicians considered going there. And back then I went to the group of algebra and started to study there, and I still consider myself to be mostly an algebraist, and applying the methods of algebra in various other areas of mathematics. So that's in a sense the beginning of the story. Today, I'm working full time on Foundations of Mathematics. So the question is "How did I get here?" and does it mean that I become entirely uncool over all these years, or is there some other reason? So I'm going to try to explain why Foundations of Mathematics the way we're doing them now is really a cool subject again._ > -Vladimir Voevodsky, Fields Medalist 2002, How I became interested in foundations of mathematics. > _What we need are foundations which can be used to construct reliable proofs despite being inconsistent._ > -Vladimir Voevodsky > _We mathematicians will have to learn how to construct reliable proofs using inconsistent formal systems._ > -Vladimir Voevodsky > _It's not easy, but I think it is possible. And such a possibility is emerging for the kind of foundations which are now being developed based on ideas coming from theoretical computer science mostly, at this time. So we need new foundations which will be formulated in a formal system which can be used, despite its inconsistency or possible inconsistency, can be used to construct reliable proofs. So the classical first order logic is not good at it, because of it has an inconsistency then one can prove everything and it stops being informative. However there are other types of formal systems which can be used for the formalization of mathematics. Which react to inconsistency in a much less drastic way in a sense. So inconsistency in such systems doesn't mean the system becomes totally uninformative. And one example of such classes of systems is the so called constructive type theories. And that is a class of formal systems which have been used extensively in the theory of programming languages._ > -Vladimir Voevodsky