12
$\begingroup$

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of foundations of mathematics. Reading about Kurt Gödel and the story of Hilbert's programme really inspired me. My plans now are to find a mathematical logic program to go into foundational research, but I saw a post on a forum that really discouraged and shocked me.

It can be found here.

Similar posts in the thread also offer a bleak outlook.

Is it true? Is it mostly a dead field filled with quacks and not much going on?

  • 5
    [related](http://math.stackexchange.com/q/24854/5363)2012-04-13
  • 2
    Do you plan on studying *logic* or *set theory* or *model theory*? All three are different and related to foundations. I don't know about logic, but the other two have quite active research nowadays.2012-04-13
  • 0
    @Asaf: Not entirely sure. I just know I love them and don't have to come down on any side until I apply for grad school!2012-04-13
  • 0
    I have added relevant tags to those fields. I would contribute my answer as a set theory grad student, but I cannot do it right now. Most of what I have to say is in the link t.b. posted anyway.2012-04-13
  • 0
    I'm not really qualified to answer, but my understanding is that foundational research is still alive and well, but what constitutes foundations has broadened beyond set theory to include category theory, etc. It seems similar to the fact that number theory is alive and well, but that elementary number theory is the focus of a small subset of number theorists, while many others use modern tools from algebraic geometry, representation theory, etc. (Ctd.)2012-04-13
  • 4
    Imho, if you are young, it's generally a bad career move to decide you want to enter a field but only use classical methods. Rightfully so, few agencies are going to support such a research program. There's nothing wrong with being captivated by "classical" mathematics (if that's the right word for Godel's work), but that should motivate you to be interested in discovering in what ways mathematicians have reinterpreted the probelms being studied using more recently discovered knowledge. So as general advice, if you become passionate about something, study it ... but with an open mind.2012-04-13
  • 0
    @Michael, while categories are certainly foundational there is a gap between set theory and category theory. I had a conversation with one of the post-docs in my department and he asked me why won't set theorists switch to topos theory and the language of categories, because it seemed to him as that would greatly improve the ability of the common mathematician to understand set theory research. I do agree with your addition, that categories are a contemporary foundational research field which may also fit the OP.2012-04-13
  • 0
    @Asaf: I certainly don't know enough about the specifics of the question, but some of the best advice I have received (though not always taken!) is to be willing to broaden one's conception of a problem that one is working on. Ultimately, it is practicing set theorists such as yourself who determine how that broadening take place in matters relevant to the OP, but it's important for the OP to resist the somewhat natural desire to fall in love with a "classical" problem to the point that you don't develop the motivation to learn more modern (often abstract) perspectives.2012-04-13
  • 2
    @Asaf: the interesting thing about that request is that, outside of a few fields, I doubt that many mathematicians would be any more comfortable with toposes than with axioms for set theory. For example consider someone studying numerical methods for PDEs. Everyone finds their familiar theories to be the most natural.2012-04-13
  • 0
    @Carl: True, but what is even more interesting is that the person is a model-theorist. :-)2012-04-13
  • 0
    It will be a dead field if you (and others like you) decide to do something else instead of what most interests you, on the basis of opinions offered on some website (this or any other).2012-04-14
  • 1
    In my experience it is certainly not a dead field, and I find foundational questions fascinating. However, the question of how easy it is to get a job in this area is a notable question. At all the departments I've been associated with, the logic group has been very small compared with other research groups in the department. I'm not saying this is justified, but it is something to keep in mind.2012-04-14
  • 0
    Foundational research is just getting started; just have a look at [categorial logic](http://en.wikipedia.org/wiki/Categorical_logic) and you'll see what I mean. The coming century will surely hold great things for mathematicians with an interest in foundations.2013-10-29

3 Answers 3