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Matt, I understand that you are writing primarily to your graduate students, but I also have to object to your first two points in general. I am a counterexample, as I took 12+ unnecessary math classes (getting A's in them) on my way to a successful PhD in theoretical computer science/crypto. I've found that my background in math has given me some tools and perspective that improve my research (I recognize when we are constructing a finite field in a round-about way, or when the abstraction we really want is a group action, or when Borel-Cantelli is really needed to make an argument rigorous, etc etc). I wouldn't have obtained these tools without a huge investment in time at some point.

Actually, I still find myself reading and learning unnecessary math papers most evenings as a hobby. I think it makes a huge difference.

Perhaps this is unique to mathematics, where theorem proving skills are portable. I certainly wouldn't use my situation to argue that PL grad students need to study French. But maybe the PL students should take an extra distributed systems course - and TA while they do it!

In theory, nothing is unnecessary, because everything in theory is unnecessary.

(j/k, of course; I love theory.)

More seriously, folks like you are the reason I phrased it "almost always." If you take the right "unnecessary" courses, it can help a lot. (So, are they really unnecessary then?)

My quantum mechanics class got me to think about the PL problem I was working on in terms of group theory. That turned out the be the key serendipitous insight that solved the problem.

But, I'm watching two students at two universities go down in flames right now because they're too distracted by taking lots of "fun" classes. They're like kids that locked themselves inside the candy store.

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