I was asked to write some advice for young mathematicians. The first observation is that each mathematician is a special case, and in general mathematicians tend to behave like "fermions" i.e. avoid working in areas which are too trendy whereas physicists behave a lot more like "bosons" which coalesce in large packs and are often "overselling" their doings, an attitude which mathematicians despise.and also:
My main point is that mathematicians are so "singular", (and behave like fermions as opposed to the physicists who behave like bosons) that making general statements about them often produces something obviously wrong or devoid of any content.You need a little quantum field theory to understand this. There are two kinds of quantum particles, bosons and fermions. Bosonic fields result from commuting operators, while fermionic fields anticommute. It is the difference between light and matter. Physical forces are intermediated by photons and other bosons, while matter is made of quarks and leptons (both fermions). Bosons in the same state are statistically distinguishable, while the Pauli exclusion principle bars more than one fermion being in the same state.
Mathematical discoveries complement other discoveries, and slowly build a substantial and enduring body of wisdom. Physicists jump from fad to fad like String Theory, and they all say more or less the same thing. We need both, of course. Connes has made a useful analogy.
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