(There’s first a sermon version of this, then at the end a seminar version.)

Borel said you don’t know when you’re wrong.

He said “the inner certainty of someone who is making a mistake and answering wrongly is exactly the same as the inner certainty of someone who knows and is not making a mistake.”

But you do know when you’re wrong.

When your mistake is exposed, and your judgment is revealed to be wrong, another thing is also revealed: that you already knew the right answer.

Some recent mistakes I made:

I under-estimated the share of users who performed a certain action over a 28-day period.

I dumped the milk straight into the flour and so made the waffles lumpy.

I bought a pair of ugly shoes

I thought the Uber would be quick.

I said something tart to someone I love.

I kicked the ball too hard.

I didn’t mention a counterexample to a generalization I made in a paper.

I drew a pair of eyes too almond-like, they didn’t fit the nose.

I picked up my phone to check if I had new messages.

I bought an expensive croissant to take home to my family.

I thought “cuando” meant while.

When I discover that I made a mistake I also recognize the mistake.

I recognize that this is not new information, I already knew it. I also recognize that I have made that same mistake before, sometimes 100 times, sometimes my whole life.

Some people say mistakes are residuals.

Some people will say the mistake is the difference between the information you have and the truth, and that they average to zero. But this is just the formal model of a mistake and I think this type of mistake is rare. Almost all mistakes of consequence are not from lack of information but from lack of clarity.

Imagine Borel in his study in Paris in 1941.

Quite pleased with himself, having formalized probability he is now calmly explaining others’ errors. We can see that his intensity of concentration on one thing had blinded him to the other aspects. He had an inner certainty that is qualitatively different from the inner certainty which people have who are right.

When Aristotle is reincarnated he will not say his problem was lack of information.

Aristotle will see that the dolphin is next to the dog, not the fish. The far East is next to the far West. The evening star is next to the morning star, and the sun fits in among one set of stars, and the moon a different set of stars. The races and sexes of humans are closer to each other than he had thought, and they also are all surprisingly close to the apes and the monkeys.

He will see that we have collected more jigsaw pieces than he had but he will also observe that our arrangement of the pieces was already latent in his own smaller collection, and he will kick himself for his mistake.

There are three types of mistake.

Lack of information. These are Borel’s mistakes: just residuals from too little information. They are rare.

Mistakes of confusion. You have enough information but it is not well organized in your brain. You know it implicitly.

Mistakes of passion. Your instincts drag you towards some judgment and then run away when it is revealed to be wrong.

Cutting scones.

I was cutting a roll of dough into squares to make scones. Once the knife fell I immediately felt that I’d made a mistake but I couldn’t put my finger on it. I had been hesitating on whether there was enough dough left for 3 strips or for 2, should I aim to cut down at 1/2 or 1/3? Instead I cut down between 1/2 and 1/3, gratuitous spoiling indecision. Like a signwriter whose letters become progressively bunched up towards the end of the sign, a slow motion traffic accident.

Seminar Version

Appendix

Borel (1950) Probability and Certainty

The problem of error has preoccupied philosophers since earliest antiquity. According to the subtle remark made by a famous Greek philosopher, the man who makes a mistake is twice ignorant, for he does not know the correct answer, and he does not know that he does not know it. This second ignorance is the really serious one, for the inner certainty of someone who is making a mistake and answering wrongly is exactly the same as the inner certainty of someone who knows and is not making a mistake. We all know that we are subject to error and that we sometimes make mistakes, entirely in good faith.