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Seeing this question, a related one is to ask when a simple proof or derivation was ever replaced by a complex one, and what the reasons were. This probably makes the most sense in the context of expository material, like textbooks.

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    $\begingroup$ I hope I edited this correctly. Are you asking for examples where a simpler proof was replaced by a more difficult one? This happens all the time in order to extend theorems to broader settings, e.g., to prove theorems in several variables or higher dimensions after they are proved in the single variable or one-dimensional case. $\endgroup$ – KCd Mar 2 '16 at 12:07
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Such cases exist indeed. Sometimes a simple proof is considered unsatisfactory on philosophical grounds. And people search for another proof, which can be more complicated than the original one.

One example is Picard's theorem. The original proof was essentially two lines. However many people thought that this proof does not really reveal the reasons why it is true. Very sophisticated theories were constructed to "explain" (and generalize) Picard's theorem. See for example "Nevanlinna theory". This research had the general title: search of an "elementary" proof of Picard's theorem. It is somewhat difficult to explain what the word "elementary" exactly means in this context.

A different example is known under the title "Elementary methods in number theory". Since Euler, Dirichlet and Riemann, the most powerful methods in the study of distribution of primes were based on the theory of functions of a complex variable. However it was considered desirable to give "elementary" proofs of theorems like the asymptotic law of prime numbers. Such proof was obtained by Erdos and Selberg in 1940-s, and this brought to Selberg a Fields medal. "Elementary" here means "not using complex function theory".

In both cases, the reasons for search of new proof were philosophical. But at least in the first case, the search of a new proof really led to much progress and interesting and important new results. I am not so sure about the second example.

Reference: Gelʹfond, A. O.; Linnik, Yu. V. Elementary methods in the analytic theory of numbers, Pergamon Press, Oxford-New York-Toronto, Ont. 1966.

EDIT. Another classical example is the Quadratic Reciprocity Law of Legendre and Gauss in number theory. Gauss himself published 6 proofs, and according to Wikipedia 200 various proofs have been published since.

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  • $\begingroup$ Thanks! But the idea of "elementary proofs" in number theory is to find proofs that don't use "higher mathematics" (calculus, generating functions, analytic number theory). Such proofs often are much harder than the "non-elementary" ones. A similar phenomenon is seen in combinatorics, where often "biyection" proofs are being asked for, instead of much simpler proofs using e.g. generating functions. Yes, it is interesting to explore how fast you can run with your feet bound, but it isn't the best we can do. $\endgroup$ – vonbrand Feb 28 '16 at 16:42
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    $\begingroup$ Some mathematicians do not approve this activities. But I do. (PIcard's theorem is my prime example, of how useful it can be). $\endgroup$ – Alexandre Eremenko Feb 28 '16 at 16:51
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    $\begingroup$ Euler's proof of the infinitude of the primes could be considered an example of this, "replacing" Euclid's proof. Without Euler's more sophisticated argument via analysis Dirichlet would not have proved his theorem on primes in arithmetic progressions. $\endgroup$ – KCd Mar 2 '16 at 12:10
  • $\begingroup$ @KCd, right. But Euler's gem did not replace Euclid's proof as the traditional way to prove it. $\endgroup$ – vonbrand Mar 2 '16 at 12:15

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