Infinite Powers: How Calculus Reveals the Secrets of the Universe – Steven Strogatz

Thoughts: Steven Strogatz is such a good communicator! While there were passages that I struggled through, on the whole Strogatz succeeds admirably in communicating the import and impact of calculus, as well as conveying his excitement about the topic.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering. I read this as an e-book, so page numbers are as they appeared in the app I used, Libby.)

Strogatz, Steven. 2019. Infinite Powers: How Calculus Reveals the Secrets of the Universe. Houghton Mifflin Harcourt.

Introduction

• xiii: essay mentioned: Eugene Wigener’s “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences”, in which he wrote “The miracle of the appropriateness of the language of mathematics for the formation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

1. Infinity

2. The Man Who Harnessed Infinity

3. Discovering the Laws of Motion

• 70: In his treatise Two New Sciences, Galileo tells readers who might note small discrepancies between measured values and values predicted by theory (due to things like air resistance, friction, etc. in his experiments rolling balls down slanted boards) not to “divert the discussion from its main intent and fasten upon some statement of mind which lacks a hair’s-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship’s cable.” Zing!

4. The Dawn of Differential Calculus

5. The Crossroads

6. The Vocabulary of Change

• 144: Strogatz characterizes calculus as dealing with three “central problems”:
  • 1 - the “forward problem”: “Given a curve, find its slope everywhere.”
  • 2 - the “backward problem”: “Given a curve’s slope everywhere, find the curve.”
  • 3 - the “area problem”: “Given a curve, find the area under it.”

7. The Secret Fountain

• 167: calculus comes from Latin calx: small stone. It’s the same root as calcium, chalk and caulk. It’s a linguistic remnant of when people used pebbles to count/do calculations
• 193: “More than two centuries [before Newton calculated power series for trig functions], mathematicians in Kerala, India had discovered power series for the sine, cosine and arctangent functions.”

8. Fictions of the Mind

9. The Logical Universe

10. Making Waves

11. The Future of Calculus

• 278: Short story mentioned: Alice Munro’s Too Much Happiness, about Sofia Kovalevskaya.
• 278: Kovalevskaya proved that there exist only certain spinning tops (specific instances of rigid bodies) whose behavior can be “completely analyzed” and thus predicted indefinitely
• 278-279: “If even a spinning top could defy Laplace’s demon, there was no hope—even in principle—of finding a formula for the fate of the universe”
• 279: Linear vs non-linear systems:
  • in linear systems, component elements “simply add up.” “The whole is equal to the sum of its parts…. Cause and effect are proportional.”
  • “Whenever parts of a system interfere or cooperate or compete with each other, there are nonlinear interactions taking place.”
• 280: when a system is linear, it can be modelled using reductionist approaches, i.e. by looking at tiny bits of it and then summing all the bits. I.e., Strogatz points out, linear systems are amenable to analysis by calculus.
• 280: “Sofia Kovalevskaya helped us understand how different the world appears when we finally face up to nonlinearity. She realized that nonlinearity places limits on human hubris. When a system is nonlinear, its behavior can be impossible to forecast with formulas, even though that behavior is completely determined. In other words, determinism does not imply predictability.”
• 281: “Chaotic systems can be predicted perfectly well up to a time known as the predictability horizon. Before that, the determinism of the system makes it predictable. [Once we pass the predictability horizon, however, interactions within the system] accumulate until we can no longer forecast the system accurately.”
• 282: in non-linear systems, errors grow exponentially in time - this is what leads to the predictability horizon.

Conclusion