Essays
Long reads on where the mathematics came from. Each one follows the same shape: a technique that obviously worked, used far beyond anyone’s ability to justify it, and the long argument about whether it could be trusted.
They were written for a reader coming back to this material after many years away, so nothing is assumed. Every essay is available as a web page and as a PDF for reading away from the screen.
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The Ghosts of Departed Quantities
How calculus was invented, why the best mathematicians in Europe couldn’t say what it meant, and the two centuries it took to make it safe.
Zeno and Archimedes; Oresme drawing the first graph; Fermat committing the original sin in the 1630s; Newton and Leibniz arriving by different doors; Bishop Berkeley’s devastating and entirely correct objection; Euler summing things that should not be summed; Cauchy and Weierstrass finally closing it. Read this one first if the phrase “dx is zero and also isn’t” ever bothered you.
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Sequence · one of three
The Two Thousand Year Silence
Humanity gambled for twenty centuries without working out the odds — and then took another 279 years to say what a probability is.
Why nobody computed the chance of a seven; Cardano’s suppressed manuscript; the interrupted card game that started everything in 1654; the bet with infinite value that nobody would take; Bertrand’s chord and its three correct answers; probability entering physics; Kolmogorov settling the mathematics in 1933 and deliberately leaving the meaning open.
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Sequence · two of three
Reasoning Backwards
Probability runs from the model to the data. Statistics has to go the other way, and that direction is so much harder that the field is still arguing in public.
A haberdasher counting London’s dead; astronomers inventing the average; the error curve borrowed for human beings and the damage that did; Galton, regression, and the eugenics the founders were not incidental to; Fisher and the lady tasting tea; what a p-value actually says and the four things it does not; the feud that produced the hybrid method nobody designed; the replication crisis.
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Sequence · three of three
Two Questions That Sound the Same
One reversal explains almost every famous statistical disaster: reading “the chance of this evidence if you are innocent” as “the chance you are innocent given this evidence”.
Bayes’ posthumous paper; Laplace calculating the odds on sunrise; the medical test almost every doctor gets wrong, and the counting trick that fixes it; population screening; the prosecutor’s fallacy and the case of Sally Clark; Monty Hall; why a significant result can still be probably false; Turing using the method in secret while it was being dismissed in public.
The last three are a sequence and are best read as one — probability, then statistics, then the essay about the confusion between them. The Ghosts of Departed Quantities stands on its own and can be read at any point.
If you only read one: the calculus essay if you want the oldest and most completely resolved story, or Two Questions That Sound the Same if you want the one most likely to change how you read a newspaper.
Written by Claude as companions to the lecture notes. Suggested further reading — including Thompson’s Calculus Made Easy and the Feynman Lectures — is at the end of the Background page.