Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World – Amir Alexander

Thoughts: For a book about the early history of calculus, Infinitesimal was a surprisingly engaging read – Amir Alexander shows that infinitely small quantities were a surprisingly contentious idea, describing the people, conflicts and concepts involved in vivid detail. I was surprised at how, in the later part of the book, science was held up as an opposite approach to mathematics/logic – I'm used to thinking of them as being in very close alignment. If you're interested in the history of mathematics or the history of science, this is likely a good book for you!

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Alexander, Amir. 2014. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Scientific American/Farrar, Straus and Giroux.

Introduction

• 13: Main argument: throughout the Middle Ages / early Renaissance, hierarchies had been justified with an appeal to strict Aristotelian and the unquestionable conclusions of geometry. “By demonstrating that reality can never be reduced to strict mathematical reasoning, the infinitely small liberated the social and political order from the need for inflexible hierarchies.”

Part I: The War Against Disorder: The Jesuits Against the Infinitely Small

1. The Children of Ignatius

• 61-62: The Gregorian Calendar was adopted in different places at different times. Reticence to adopt the new system was due in part to countries not wanting to switch to a system that had been developed by the Catholics (even as the Julian calendar had drifted by several weeks by that time).

2. Mathematical Order

• 90: Aristotle’s wheel: imagine two hexagons with the same center, one inside the other, the inner hexagon having half the “circumference” of the outer. Roll the outer hexagon, while keeping the inner hexagon in the center of the outer hexagon. By the time the outer hexagon has turned by one full rotation, the outer hexagon will have travelled the distance of its perimeter. The inner hexagon will have also travelled the same distance, but it only has half of the perimeter of the outer hexagon: if we note where its sides were whenever they were horizontal, we will see a series of line segments along the same line, but with gaps between them.
  • Galileo pointed out that this will be true of circumscribed polygons with any number of sides, up to infinity (i.e. an infinitely-close approximation of a circle) - the number of sides will be equal between the two polygons, but there will also be that number of small gaps filling out the distance the inner polygon has travelled.
• 96: Nice metaphor drawn from real life: Cavalieri argued that in the same way that a flat sheet of fabric is composed of many linear threads, so planes are composed of infinitely many lines, and in the same way as books are composed of many flat pages, geometric solids are composed of infinitely many planes.

3. Mathematical Disorder

4. “Destroy or Be Destroyed”: The War on the Infinitely Small

5. The Battle of the Mathematicians

Part II: Leviathan and the Infinitesimal

6. The Coming of Leviathan

• 204: Hobbes’s Leviathan: “much more than an absolute ruler, or even an absolute state.” “In their desperation to escape the state of nature, men (sic.) conclude that the only way out is for each of them to give up his own free will and invest it in the sovereign.” Hobbes: “[It] is more than consent, or concord: it is a real unity of them all, in one and the same person.” “The multitude so united is called a COMMONWEALTH.”

7. Thomas Hobbes, Geometer

8. Who Was John Wallis?

• 248: curious practice of the early Royal Society: rather than conduct an experiment and report on its results, fellows would perform experiments in front of other fellows. “All those present would then discuss what they had seen, examining its meaning and significance.”
• 252: There was a stark contrast between the “dogmatic rationalism of Descartes and Hobbes” and the experimental philosophy of the Royal Society - those of the Royal Society argued that the best way to learn about the world is by observing it through experiments, not by trying to reason from unquestionable axioms.
• 253: book mentioned: Francis Bacon’s New Atlantis: “To the Royal Society, Bacon was the prophet of the experimental method, and the spiritual father of the Society itself, though he died many years before its founding. In fact, the Society considered itself the true incarnation of Bacok’s ‘Salomon’s House,’ a state institution for the study of nature that he proposed in his utopian work New Atlantis.”

9. Mathematics for a New World

Epilogue: Two Modernities