19/03/2025

Folha: What is the most famous mathematical argument?

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Some time ago I wrote the preface to a book on plane curves authored by three dear colleagues. I began by stating that "mathematicians are happy people because they do what they love." That's true, at least in my experience, but it doesn't mean everything is peace and love in the world of numbers: sometimes things get tough.

There's even a book on the subject, "Great Math Fights," in which the American writer Hal Hellman describes in detail a dozen more or less heated disagreements between mathematicians. Hellman had no prior training in the field, and his selection criteria aren't always perfect, but the book is nonetheless an entertaining and instructive read.

From Niccolò Tartaglia's virulent attacks on Girolamo Cardano regarding the solution of the cubic equation in the 16th century, to the tragic discord between Leopold Kronecker and Georg Cantor over the concept of infinity in the 19th century, these are ten excellent opportunities to see that, even though it is universal and absolute in its content, mathematics remains a human activity, subject to the moods (and egos) of those who practice it.

The most famous mathematical rivalry is probably the infamous dispute between the German Gottfried Leibniz (1646–1716) and the Englishman Isaac Newton (1643–1727) over the credit for the discovery of calculus. The correct answer to the question “Who discovered it first?” is “Both.” Furthermore, some of their ideas had been anticipated by others, even by Archimedes in the 3rd century BC. Still, the unpleasant disagreement between the two grew to the proportions of an international conflict, with England and its sphere of influence siding with Newton, and most of continental Europe defending Leibniz's primacy. Some see among its harmful effects the sharp decline of English mathematics in the following century.

The theory is that, although Newton's and Leibniz's ideas were equally correct and equivalent, the language used by the German was far superior, much more intuitive, which is certainly true. By refraining from using it, to show their loyalty to Newton, the English mathematicians relinquished a powerful tool, which their French, Italian, German, etc. colleagues used masterfully.

I believe that the decline of English mathematics during this period had more than one cause, but I accept that this was one of them. Especially since England only regained its place among the European mathematical powers after Leibniz's notation became popular in the country in the first decades of the 19th century.

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