Hardy and Wright, Lang, Stolarsky, Dieudonné

From Hardy & Wright 's classic The Theory of Numbers [my 1962 edition, which I bought, and read, while at school] I quote: It is not immediately obvious that there are any transcendental numbers ...

Hardy & Wright do not give a casual reader any idea as to why it is 'not obvious'. I hope to do so.


Serge Lang (1966, Introduction to Transcendental Numbers) ... it is remarkable that a mathematical theory as old as the theory of transcendental numbers (dating back to Hermite's first result of 1873, the transcendence of e) is still in what can only be called an under-developed state [ Baker was just about to change all of that; it's interesting - and unfair, I believe - that Lang doesn't date back to Liouville]


Kenneth Stollarsky (1978, Bulletin American Mathematical Society review of books on transcendental numbers by Baker, Mahler, and Waldschmidt) The last dozen years have been a golden age for transcendental number theory . It has scored successes on its own ground, while its methods have triumphed over problems in classical number theory involving exponential sums, class numbers, and Diophantine equations... The result [of Gelfond and Schneider] has now been brilliantly extended by Baker in several useful ways... At present the high point of the theory [of work that began with Liouville, and then was revolutionised by Thue, Siegel, ...] is the following consequence of a deep [1970] n-dimensional theorem of W. Schmidt ... (When the manuscript of Schmidt's proof first became available, it provided a Diophantine approximation seminar at the University of Illinois with material for an entire semester...)


From Jean Dieudonné 's A PANORAMA OF PURE MATHEMATICS (1982): This book [ 's ] ... aim is to provide an extremely sketchy survey of a rather large area of modern mathematics, and a guide to the literature for those who wish to embark on a more serious study of any of the subjects surveyed .

At the end of each surveyed topic, Dieudonné has an 'originators' section. I quote from the end of his Theory of Numbers chapter [I include only names of mathematicians who figure in my talk, however peripherally]: The principal ideas in the theory of numbers are due to the following mathematicians:

Algebraic number theory. L. Euler (1707-1783), J. L. Lagrange (1736-1813), C. Hermite (1822-1901), D. Hilbert (1862-1943), C. Siegel (1896-1081) (and nineteen others)

Diophantine approximations and transcendental numbers. C. Hermite (1822-1901), A. Thue (1863-1922), C. Siegel (1896-1981), A. Gelfond (1906-1968), T. Schneider, K. Roth, A. Baker, W. Schmidt. [complete list of all those named by Dieudonné. Klaus Roth and Alan Baker are Fields Medallists (1958 and 1970). Wolfgang Schmidt must surely be one of the greatest mathematicians (Andrew Wiles being another) not to have been awarded a Fields medal on the foolish age-restriction.]

Diophantine geometry. L. Mordell (1888-1972), C. Siegel (1896-1981) (and five others)

Arithmetic groups. C. Hermite (1822-1901), C. Siegel (1896-1981) (and seven others)

Analytic number theory. G. Hardy (1877-1947) (and seven others)

The following have also contributed substantially to these theories: H. Davenport (1907-1969), N. Feldman, E. Landau (1877-1938), S. Lang, C. Lindemann (1852-1939), Ju. Linnik (1915-1972), J. Liouville (1809-1882), K. Mahler (1903-1988), C. Rogers, S. Schanuel, P. Turán (1910-1976) (and ninety-eight others)

Incidentally, Dieudonné placed A. Wiles in the later group (remember it was 1982). Where would his name be in a new edition? I think we all know the answer to that one.

Contact details 

After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com

This page was last updated 18 February 2005 15:10:02 -0000