Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

By Amir Alexander

On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world.

Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal.

Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.

• Language: English
• Publisher: Oneworld Publications
• Release date: Jul 3, 2014
• ISBN: 9781780745336

Amir Alexander

Amir Alexander teaches history at the University of California, Los Angeles. He is the author of Infinitesimal, Geometrical Landscapes, and Duel at Dawn. His writing has appeared in The New York Times and the Los Angeles Times, and his work has been featured in Nature and The Guardian, on NPR, and elsewhere. He lives in Los Angeles, California.

Reviews for Infinitesimal

[nbmars · Rating: 4 out of 5 stars]

Infinitesimal is a quirky little book. Its basic thesis is that various responses to an arcane mathematical concept, infinitesimals, or the infinite amount of parts into which a line can be divided, somehow accounts for the political struggles of 16th century Italy and of 17th century Britain. To say the least, this is a bold thesis, and I remained unconvinced at the end, despite enjoying the author’s almost heroic efforts.On a first level, the book is a political history of (a) the Catholic Church’s response to the Protestant Reformation and (b) Britain in the 1600’s, contrasting the two historical periods by their different approaches to mathematics. In the process, the author takes us through the founding and early days of the Society of Jesus (the Jesuit order) in the Counter-reformation of the 16th century, particularly on the Italian peninsula (there being no “Italy” as such in those days). He then moves on to the political and religious struggles in Britain in the next century.Alexander’s retelling of the history is interesting, if very unconventional. In particular, his relation of the debate between Thomas Hobbes (as author of Leviathan) and John Wallis is told from an angle I had never seen before. Hobbes, who fancied himself as a great geometer (he thought he had “squared the circle”—later proved to be impossible), crossed intellectual swords with Wallis, who approached mathematics as an experimental pursuit. Their respective approaches to mathematics probably influenced their political philosophies in the way Alexander describes. Nevertheless, neither actually solved the problem of the infinitesimals—and that story is not included in this book. Alexander has presented a history through a highly unorthodox, but stimulating, prism. He is probably correct that the concept of infinitesimals, which was expanded to become calculus, really did “shape” the modern world. After all, classical physics is expressed in the language of calculus. But the “shaping “ caused by that concept was scientific, mathematical, and intellectual—not political. Alexander’s retelling of the story leaves untold the final rigorous and elegant explication and derivation of calculus by Newton, Leibnitz, and Euler. And that explication and derivation had little or no political consequences. This book is worth reading just for the history even if its attribution of causation is farfetched. (JAB)

[mathmaverick-1 · Rating: 5 out of 5 stars]

A fascinating book that illustrates the impact of politics and religion on mathematics. Very interesting how the rudiments of Calculus were believed by some long before Lenitz and Newton.

[paul_s-289480 · Rating: 4 out of 5 stars]

Amazing history of the interplay of nascent science, entrenched theology and flailing politics of the age culminating in the birth of the modern world. Back when mathematics was truly controversial.

[kukulaj · Rating: 5 out of 5 stars]

Pre-Birth of The Calculus... does metaphysics matter? The outer brackets of this book might be from Luther's pinned-up Theses to the Glorious Revolution in the UK in 1688. The medieval world was stable, even stagnant. The modern world is fertile and chaotic. Alexander understands that metaphysics - the foundations of mathematics - is not the sole determiner of social transformation. But the philosophy of mathematics is right there in the fray. It has a seat at the table. Just like today, Turing's Halting Theorem went from a curiosity of mathematical logic to the seed of our ongoing transformation into some kind of cybernetic society. What kind, it's not easy to predict with any confidence!I especially liked the descriptions of some of Wallis's reasoning here. I'm a math-physics-engineering geek, so this stuff is deep in my bones. But to be able to peek over Newton's shoulder to see where his ideas came from - just a delight!There's not too much math here. Alexander works hard to help the reader stay on track. This is really a history book, with just enough math to be able to follow the action. It'd be fun to do a parallel math book, to explore much more fully why experimental mathematics is not so reliable, and e.g. how did Cauchy convergence develop out of Wallis's experiments with infinite series. But now, in the midst of this coronovirus pandemic, that puts a spear point on the climate change cudgel - people are looking for certainties, when scientists can only offer possibilities and probabilities. We are edging closer to yet another religion versus science war, and the stakes could hardly be higher. This book is a nice refresher course on how ideas matter.