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Why is probability an under-emphasized subject in most math programs? Why does it seem that the hot topics these days are category theory and algebra? What do you think about the following: A student learns probability (both on the applied side and the pure side) and picks up the real analysis on the way. That way there is a purpose for learning the real analysis.

Can we do this with many mathematical subjects (e.g. first teach from an applied perspective, and learn the pure stuff because it is needed)?

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This is just my personal opinion, but I think learning pure mathematics just because it is necessary to do something applied detracts an awful lot from its beauty and elegance. I certainly would have avoided real analysis if I was told it was just some sort of advanced probability class. –  Zev Chonoles Feb 6 '10 at 19:15
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In what parallel universe is category theory emphasized more than probability? –  Felipe Voloch Feb 6 '10 at 19:31
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The question seems ambiguous to me: Undergraduate program or graduate program? I agree that every math major should have some exposure to probability and statistics, but that cannot be based on measure theory. At the graduate level breadth is usually enforced by requiring qualifying exams. Under the pick-and-choose model, there could be a qualifying exam in probability, which only a minority of graduate students would take. Under the older model, there would be qualifying exams in algebra, analysis and geometry/topology. It think it is unrealistic to require probability at the grad level. –  engelbrekt Feb 6 '10 at 20:06
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@Everybody: I agree that probability is an excellent motivation for measure theory (and certainly that measure theory is an essential tool for serious probability). In the last few years I have come to feel a little cheated that my early education in mathematics included a large dose of abstract measure theory (for what, really?) without making the link to probability. Thus I think there is something interesting in this question... –  Pete L. Clark Feb 6 '10 at 20:40
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...BUT I also agree that it is stated in a vague and contentious way: it's almost designed to instigate a pure v. applied argument. I'm voting to close and at the same time stating my interest in seeing a retooled version of the question. –  Pete L. Clark Feb 6 '10 at 20:41

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