Letters to the Editor: János Bolyai and Mathematical Bombshells

January 1, 2005

To the Editor:

I read with pleasure Phil Davis's book review in the December 2004 issue of SIAM News ("Running Counter to Inert Crystallized Opinion"), but I wish to make some corrections. Davis writes that Transylvania, the birthplace of the Bolyais, has been oscillating between Hungary and Romania for ages. Not so: Romania as a state was created by the Treaty of Versailles in 1919; Transylvania had been part of Hungary for a thousand years, except for a period in the 16th and 17th centuries, when Hungary was under Turkish occupation and Transylvania became an independent principality. Under enlightened rulers it had its renaissance flowering; it was in Transylvania that the Unitarian faith grew into a sect.

Davis is not quite right that Bolyai's Nachlass contains little of mathematical interest. Bolyai had studied the arithmetic of Gaussian integers, and used them to give an elegant proof that primes of form 4n + 1 can be written as a sum of two squares. This material can be found in Elemér Kiss's Mathematical Gems from the Bolyai Chests (Akadémiai Kiadó, 1999).

Davis classes the discovery of non-Euclidean geometry among four mathematical bombshells that have seriously altered the philosophy of mathematics, the other three being Newton's Principia, Cantor's set theory, and Gödel's incompleteness theorem. I would add to this list John Milnor's discovery of differential topology, which has altered our notion of how space hangs together.---Peter D. Lax, Courant Institute of Mathematical Sciences.

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