CALCULUS Understanding Its Concepts and Methods

David Hilbert (1862--1943) --- Historical Sketch

David Hilbert was born in 1862 in Königsberg, a city that is well known among mathematicians, then a part of Germany but now in Russia. To see one reason mathematicians know about Königsberg, go to http://math.youngzones.org/Konigsberg.html. Another reason that mathematicians are familiar with this city is that so many mathematicians were born there, the most illustrious of whom are Carl Neumann, Alfred Clebsch, Christian Goldbach, Kurt Hensel, Ludwig Otto Hesse, David Hilbert, Gustav Kirchhoff, Rudolf Lipschitz, Johann Rosenhain, Heinrich Schroeter, and Arnold Sommerfeld.

One measure of the importance of Hilbert's place in mathematics is that Carl Boyer, in his beautiful book, A History of Mathematics, devotes ten pages to describe his work.

Hilbert attended the University of Königsberg, obtaining his Ph.D. under the direction of Ferdinand von Lindemann in 1885. He then taught there until 1895. He was subsequently appointed to a chair of mathematics at the University of Göttingen, where he spent the rest of his career (and life, dying in early 1943). When Hitler came to power in Germany, Hilbert reportedly used his prestige and influence to help younger mathematicians emigrate.

Hilbert formulated axioms for Euclidean geometry. On the basis of his success in geometry, he set about in the 1920s to produce a formal foundation for all of mathematics. This came to be known as Hilbert's Program. Hilbert wanted to construct a setting in which any true theorem of mathematics could be established in a mechanical way in a finite number of steps. However, Kurt Gödel proved in 1931 that such a goal is unattainable: there will always be true theorems that cannot be proved.

One of the many things that Hilbert did was to propose, at the 1900 International Congress of Mathematicians held in Paris, a list of 23 problems for mathematicians in the 20th century. Their current status can be found in http://en.wikipedia.org/wiki/Hilbert's_problems.

One of the best known of Hilbert's many constructs is Hilbert space. This is an infinite-dimensional space, where the points are infinite sequences that are square summable, that is, for which the series converges to some finite number. Show that the harmonic sequence is a point in Hilbert space.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.