http://www.vervloesem.eu/qed/index.php/2007/08/11/atle-selberg-overleden/ Weblog over wiskunde en computers Sat, 05 Sep 2009 15:46:34 +0000 http://wordpress.org/?v=2.6 http://www.vervloesem.eu/qed/index.php/2007/08/11/atle-selberg-overleden/#comment-81820 Waarom zoeken wiskundigen elementaire bewijzen? at QED Sat, 14 Jun 2008 12:04:26 +0000 http://www.vervloesem.eu/qed/?p=381#comment-81820 [...] Zowel Paul Erdős als Atle Selberg vonden in 1948 een elementair bewijs van de priemgetallenstelling. Voor velen was dit een schok, omdat de priemgetallenstelling als een “diep” resultaat in de getaltheorie werd gezien. Nu duidelijk was dat “elementaire” methodes voldoende waren om de stelling te bewijzen, bleek de stelling volgens velen ineens minder diep. In 1921 nog zei G.H. Hardy tijdens een lezing dat een elementair bewijs van de priemgetallenstelling hem onmogelijk leek door de ‘diepte’ van de stelling (Dorian Goldfeld, “The elementary proof of the Prime Number Theorem: an historical perspective”, p. 3): No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten. [...] [...] Zowel Paul Erdős als Atle Selberg vonden in 1948 een elementair bewijs van de priemgetallenstelling. Voor velen was dit een schok, omdat de priemgetallenstelling als een “diep” resultaat in de getaltheorie werd gezien. Nu duidelijk was dat “elementaire” methodes voldoende waren om de stelling te bewijzen, bleek de stelling volgens velen ineens minder diep. In 1921 nog zei G.H. Hardy tijdens een lezing dat een elementair bewijs van de priemgetallenstelling hem onmogelijk leek door de ‘diepte’ van de stelling (Dorian Goldfeld, “The elementary proof of the Prime Number Theorem: an historical perspective”, p. 3): No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten. [...]

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