Mathematicians (of whom I am one) drive me crazy. The other day at a conference the speaker said

“A rigorous mathematical proof of [such-and-such] is not known.”

The thing is, the speaker knew that such-and-such was false! So of course, no amount of rigor was ever going to make it true. Mathematicians are so obsessed with theorems and proofs as the gold standard of knowledge that “we can’t prove it” becomes their way of saying “it isn’t so”.

[26 July 2023]

On a rainy day last March I marveled at how many Oxford bicycles have no fenders. So the water sprays up and leaves muddy stripes on shirts and trousers.

I decided to examine the next 100 bikes I passed. It turned out that 39 had no fenders. (About half of these had skinny tires, cyclists who fancy themselves racers I guess, and half had fat ones, the off-roaders.)

The next week I was in Leuven, and it was rainy again. I decided to examine 100 bikes. This time, just 4 were without fenders.

[14 July 2023]

Suppose you are serving at tennis, and your probability of winning each serve is p. What’s the probability P that you’ll win this game?

Well, let W(n) be the probability of winning the game at serve n. The possible values are n = 4, 5, 6, 8, 10, 12,…. Let T(n) be the probability that the score is tied after serve n, and define s = 2p(1-p).

We calculate W(4) = p⁴, W(5) = 4(1-p)p⁴ = 2p³s, W(6) = 10p⁴(1-p)² = (5/2)p²s², and T(6) = (5/2)s³. After this, the pattern is completely regular. For any k>0, we have T(6+2k) = sT(4+2k) = s^kT(6) = (5/2)s^k+3 and W(6+2k) = p²T(4+2k) = (5/2)p²s^k+2. So W(8)+W(10)+W(12)+… = (5/2)p²s³/(1-s).

Grand total: P = p⁴+2p³s+(5/2)p²s²+(5/2)p²s³/(1-s). With a bit of work this simplifies to P = 1/2 + (10r-5r³+4r⁵–r⁷)/(8+8r²) with r = 2p-1.

For example, if you win 2/3 of your serves, your probability of winning the game is 208/243.

[13 July 2023]

Something funny happened in a talk yesterday that reminded me of my 2007 note “CAUTION: RACE HORSES”. Sven Leyffer, a senior figure and indeed currently the President of SIAM, showed a photo of his research group and said

“This is joint work with a lot of young people that make me look really really –“

at which point I assumed the next word would be a self-deprecating “old”.

Instead, the next word was “good”. So self-deprecation was achieved by the reverse of the logic I expected.

In Cultural Amnesia Clive James says Edward Gibbon is exasperating because he perpetually constructs sentences whose meanings you can’t figure out until the last few words. In this case, charmingly, I knew from the start what the deeper meaning had to be, just not the details of how Sven was going to get us there.

[30 June 2023]