Wednesday, April 23, 2014

History of infinitesimals

A new book on the history of math, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, has an Amazon review starting:
The opening chapters of "Infinitesimal" are about a board of Jesuits in the 17th century ruling on legitimacy of a mathematical topic. That which would be binding on every university of the Jesuit order-the most prestigious of the time.
Another reviewer says:
This is a gripping story of human passion, that further debunks the myth that mathematics is "objective". While the "true" mathematics may be objective, the mathematics created by humans is almost as subjective as poetry or religion, and the engaging story told so masterfully by Amir Alexander, will not only teach you about the historical roots and the not-so-hidden agendas of the pioneers that lead to the invention of calculus, but would also challenge the conventional wisdom that mathematics is objective and ideologically neutral.
The NY Times review says:
To the Jesuits, tradition, resoluteness and authority seemed bound up with Euclid and Catholicism; chaos, confusion and paradoxes were associated with infinitesimals and the motley array of proliferating Protestant sects.

Indeed, Galileo, later to be found guilty of heresy, supported some of his ideas with infinitesimal-flavored arguments. Dr. Alexander writes that he was as courageous as possible under the circumstances, not only a great mathematician and physicist but a witty and compelling writer willing to take on the Jesuits. ...

Since the Jesuits succeeded in banning infinitesimals in Italy, the last part of Dr. Alexander’s finely detailed, dramatic story traces their subsequent history north to England. There one of the key figures is Thomas Hobbes, the 17th-century philosopher of authoritarianism, a strong advocate of law, order — and, like the Jesuits, of the top-down hierarchical nature of Euclidean geometry.

Hobbes’s hated antagonist, the mathematician John Wallis, used infinitesimals freely, along with any other ideas he thought might further mathematical insight. And further it they did, leading over time to calculus, differential equations, and science and technology that have truly shaped the modern world.
That makes it sound as if the infinitesimal was heresy. But Galileo was not convicted of heresy, and infinitesimals had nothing to do with it.

This book appears to have a lot of good math history, but you might get the wrong impression about infinitesimals. They are essential to analysis. Math was not axiomatized and made properly rigorous until the 20th century, so early infinitesimal arguments were often non-rigorous. But textbooks today explain how infinitesimal arguments can be a shorthand for completely rigorous arguments using limits or other constructions. The Jesuits were right to be suspicious about infinitesimals, and they are sometimes abused today, but they are now known as good math, if done properly.

I did not know about this alternative development:
A major development in the refounding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis (SIA). Based on the ideas of the American mathematician F. W. Lawvere, ...

Since in SIA all functions are continuous, it embodies in a striking way Leibniz's principle of continuity Natura non facit saltus.
SIA uses some goofy logic, but is apparently a consistent treatment of infinitesimals (in addition to the preferred treatments).


  1. This comment has been removed by the author.

  2. Rog, how is it possible for 'logic' to have any aspect whatsoever callable as goofy?

  3. In regular logic, statements are either true or false, and true is the opposite of false. In goofy logic, there are other possibilities.