# What's a good resource for some of mathematics greatest controversies and debates?

We often think of mathematics as being clean-cut, with questions being settled with theorems, and not much room for disagreement. This isn't true though of course.

Going back to ancient history, we hear the story of how Pythagoras murdered the man that discovered irrational numbers. Jumping to modern mathematics, Cantor's ideas about infinity invoked such harsh criticism, he literally went insane$^1$.

Is there a good website or other resource recounting some of the great controversies and debates of mathematics?

$^{^{^1\text{Although I'm sure going insane isn't fun, you have to admit that's a pretty awesome reason, if one had the choice.}}}$

• Berkeley's potent attack on "The Infidel Mathematicians" is an interesting episode which lead to a significant response. The Newton/Leibniz episode is another well documented affair. – Nick Jul 1 '16 at 19:54
• There is an interesting book on this one: math.stackexchange.com/a/831221/442 – Gerald Edgar Jul 2 '16 at 0:15

One of the greatest mathematical controversies arguably is the one concerning infinitesimals: are they contradictory? are they useful? did Leibniz really use them? is there a connection between Leibniz's infinitesimals and Robinson's? are Euler's infinitesimal procedures sheer madness? are they chimeras, as alleged by authors ranging from Berkeley to Moigno to Connes? Why did Cantor described them as "abominations", "paper numbers", and self-contradictory?

Published related materials can be found here.

You might be interested in "Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever" by Hal Hellman. Personally, I don't think very highly of such books, however.

This is not a list of controversies, but you might be interested in Imre Lakatos' book Proofs and Refutations which describes the mathematical progress. For example it provides a discussion of Euler's Polyhedron Formula as a discussion between a teacher and some students. The arguments the students use in this discussion are arguments which have been brought forward by different mathematicians.

The book also describes different "solutions" which have been used in the history of mathematics if someone found a counterexample to a "Theorem".

Pythagoreas didn't murder the person that found irrational numbers. Actually, we know very little about the person Pythagoreas. There was a sect of men who followed the teachings of a mystic named Pythagoreas. It was a cult with strange rules like not eating beans. Men who joined were given knowledge of mathematics. To remove jealousy, any discovery made by anyone was attributed to Pythagoreas. It's just mathematical apocrypha that the man who discovered irrational numbers was thrown off the ship because it conflicted with their believe that the natural numbers describe everything.

Howard Eves wrote a series of books called Mathematical Circles - Quadrants I, II, III, IV in which he tells apocryphal stories about Mathematics.

• This does not seem to answer the question, which is for resources about mathematical controversies. This would fit better as a comment. – Rory Daulton Jul 3 '16 at 10:05
• @RoryDaulton I cited four books about mathematical apocrypha. Besides, it's too long for a comment. – user230452 Jul 3 '16 at 11:00