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October 25, 2024 - May 4, 2025
Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn’t. “You had better learn it,” said Feynman. “It’s the language God talks.”
differential equations. Such equations describe the difference between something right now and the same thing an instant later or between something right here and the same thing infinitesimally close by.
an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses it.
By inadvertently discovering this strange language, first in a corner of geometry and later in the code of the universe, then by learning to speak it fluently and decipher its idioms and nuances, and finally by harnessing its forecasting powers, humans have used calculus to remake the world.
That’s the central argument of this book.
I’m going to try to explain everything we’ll need with the help of pictures, metaphors, and anecdotes. I’ll also walk us through some of the finest equations and proofs ever created, because how could we visit a gallery without seeing its masterpieces?
To be an applied mathematician is to be outward-looking and intellectually promiscuous.
pas de deux.
It’s often said that mathematics is the language of science.
is much more than a language; it’s also an incredibly powerful system of reasoning.
palpable.
This is what Einstein marveled at when he wrote, “The eternal mystery of the world is its comprehensibility.”
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”
unexpected
link between music (the harmony of this world) and numbers (the harmony of an imagined world) led the Pythagoreans to the mystical belief that all is number.
Let’s call it the Infinity Principle. It
inch our way up to it by asking what calculus wants . . . and how it gets what it wants.
calculus wants to make hard problems simpler.
Calculus succeeds by breaking complicated problems down into simpler parts.
The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme—all the way out to infinity.
Instead of cutting a big problem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them. Once that’s done, it solves the original problem for all the tiny parts, which is usually a much easier task than solving the initial giant problem. The remaining challenge at that point is to put all the tiny answers back together again.
Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus.
The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.
This strategy can be used on anything that we can imagine slicing endlessly. Such infinitely divisible things are called cont...
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More generally, the kinds of entities modeled as continua by calculus include almost anything one can think of.
The Infinity Principle To shed light on any continuous shape, object, motion, process, or phenomenon—no matter how wild and complicated it may appear—reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole.
Calculus gave the right answers, even when its creators couldn’t explain why. The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five-hundred-year
that mathematics is usually portrayed as exact and impeccably rational. It is rational, but not always initially.
Curves, Motion, and Change
Three mysteries above all have spurred its development: the mystery of curves, the mystery of motion, and the mystery of change.
So this is how calculus began. It grew out of geometers’ curiosity and frustration with roundness.
The breakthrough came from insisting that curves were actually made of straight pieces. It wasn’t true, but one could pretend that it was.
integral calculus was born.
the second great obsession: a fascination with the mysteries of motion on Earth and in the solar system.
Motion was proving to be as much of a conceptual Mount Everest as curves were.
This time the act of wishful fantasy was to pretend that motion at a changing speed was made up of infinitely many, infinitesimally brief motions at a constant speed.
he found a way to represent any curve as a sum of infinitely many simpler curves described by powers of a variable x, like x2, x3, x4, and so on.
His astonishing “system of the world” unified heaven and earth, launched the Enlightenment, and changed Western culture.
1917 Albert Einstein applied calculus to a simple model of atomic transitions to predict a remarkable effect called stimulated emission
Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center.
Limits are often simpler than the approximations leading up to them.
Doing so would be to commit the sin of completed infinity.
So infinity times zero would have to be both the circumference
That’s the real reason why we can’t divide by zero. The faint of heart say the answer is undefined, but the truth is it’s infinite.
The question naturally arises: Do such infinitesimally small things exist in the real world? Quantum mechanics has something to say about that.
If the walker were an electron, there’s a chance it might walk right through the wall. This effect is known as quantum tunneling.
At such small scales, space and time might seethe and roil at random. They might fluctuate like bubbling foam.
One of them is the gravitational constant, G. It measures the strength of gravity in the universe.
ħ (pronounced “h bar”), reflects the strength of quantum effects.
the speed of light,
