Bertrand Russell once wrote that mathematics had a "beauty cold and austere." In this new book, the historian Amir Alexander shows that mathematics can also become entangled in ugliness hot and messy.
The time was the late 16th and 17th centuries, and the mathematics in question was the proper understanding of continua — straight lines, plane figures, solids. Is a line segment, for example, composed of an infinite number of indivisible points?
If so, and if these infinitesimals have zero width, how does the line segment come to have a positive length? And if they have nonzero widths, why isn't the sum of their widths infinite?
For reasons like these, Aristotle had argued that continua could not consist of indivisibles. But new developments were demonstrating that thinking of them this way yielded insights not easily obtained from traditional Euclidean geometry.
Huh? It's natural to wonder how such a seemingly arcane issue could possibly arouse much passion. But this fascinating narrative by Alexander (an occasional book reviewer for Science Times) vivifies the era and the fault lines that the mathematical dispute revealed.
Let's begin with the math. The mathematicians, Cavalieri, Torricelli, Galileo and others, were at the forefront of the new geometric approaches involving infinitesimals. If classical Euclidean geometry is conceived as a top-down approach with all theorems following by pure logic from a few self-evident axioms, the new approaches can be thought of as bottom-up, inspired by experience. For example, just as a piece of cloth can be considered a collection of parallel threads and a book a collection of pages, so too might a geometric plane be considered an infinite number of parallel lines and a solid an infinite number of parallel planes.
Most of the book is devoted to the interrelated stories of many leading mathematical/religious figures during these tumultuous early modern times. They include the priest Christopher Clavius and his mathematical work.
Along the way there are striking geometric insights that buttress the case for infinitesimals. Formulas for the areas and volumes of geometric figures were surprisingly easy to obtain using this principle, which was a precursor of integral calculus.
No one talks of infinitesimals any more: The modern notion of limits accomplishes everything they did, but much more rigorously. One exception is a recent reconstruction of infinitesimals - positive "numbers" smaller than every real number - devised by the logician Abraham Robinson and developed further by H. Jerome Keisler, my adviser at the University of Wisconsin.