When I was very young—I think thirteen or maybe fourteen—I thought I had found a disproof of Cantor’s Diagonal Argument, a famous theorem which demonstrates that the real numbers outnumber the rational numbers. Ah, the dreams of fame and glory that danced in my head!

My idea was that since each whole number can be decomposed into a bag of powers of 2, it was possible to map the whole numbers onto the set of subsets of whole numbers simply by writing out the binary expansion. The number 13, for example, 1101, would map onto {0, 2, 3}. It took a whole week before it occurred to me that perhaps I should apply Cantor’s Diagonal Argument to my clever construction, and of course it found a counterexample—the binary number (. . . 1111), which does not correspond to any finite whole number.

So I found this counterexample, and saw that my attempted disproof was false, along with my dreams of fame and glory.

I was initially a bit disappointed.

The thought went through my mind: “I’ll get that theorem eventually! Someday I’ll disprove Cantor’s Diagonal Argument, even though my first try failed!” I resented the theorem for being obstinately true, for depriving me of my fame and fortune, and I began to look for other disproofs.

And then I realized something. I realized that I had made a mistake, and that, now that I’d spotted my mistake, there was absolutely no reason to suspect the strength of Cantor’s Diagonal Argument any more than other major theorems of mathematics.

I saw then very clearly that I was being offered the opportunity to become a math crank, and to spend the rest of my life writing angry letters in green ink to math professors. (I’d read a book once about math cranks.)

I did not wish this to be my future, so I gave a small laugh, and let it go. I waved Cantor’s Diagonal Argument on with all good wishes, and I did not question it again.

And I don’t remember, now, if I thought this at the time, or if I thought it afterward . . . but what a terribly unfair test to visit upon a child of thirteen. That I had to be that rational, already, at that age, or fail.

The smarter you are, the younger you may be, the first time you have what looks to you like a really revolutionary idea. I was lucky in that I saw the mistake myself; that i...

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