Tim Gowers asked a really great question on MathOverflow recently, on examples of mathematical “cognitive biases”: false widely held beliefs in (higher) mathematics.  Which naturally enough reminded me of the embarrassing experience yesterday when I realized that, after assuming the contrary for several months, the kernel of A \oplus B \to C is not the same as the direct sum of the kernels of A \to C and B \to C. Whoops. 

It looks like the winning ones so far are about little facts in linear algebra.  It would, indeed, make the proofs of many of the technical results on Lie algebras easier if \mathrm{Tr}(ABC) = \mathrm{Tr}(CBA).

Also, Arrow’s theorem is a scam.  Support range voting!

Finally, on an entirely unrelated note, this quote is ridiculously awesome:

“I’ve had the chance, in the world of mathematics, to meet quite a number of people, both among my elders and amoung young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle — while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”  -Alexandre Grothendieck

Source.

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