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M. R. SCHROEDER

In mathematics proper, Hermann Minkowski, in the preface to his

introductory book, on number theory, Diophantische Approximationen;

published in 1907 (the year he gave special relativity its proper four-

dimensional clothing in preparation for its journey into general covariance and

cosmology) expressed his conviction that the "deepest interrelationships in

analysis are of an arithmetical nature.''

Yet much of our schooling concentrates on analysis and other branches of

continuum mathematics to the virtual exclusion of number theory, group

theory, combinatorics and graph theory. As an illustration, at a recent

symposium on information theory, the author met several young

mathematicians working in the field of primality testing, who - in all their

studies up to the Ph.D. - had not heard a single lecture on number theory!

Or, to give an earlier example, when Werner Heisenberg discovered

"matrix" mechanics in 1925, he did not know what a matrix was (Max Bom

had to tell him), and neither Heisenberg nor Born knew what to make of the

appearance of matrices in the context of the atom. (David Hilbert is reported

to have told them to go look for a differential equation with the same

eigenvalues, if that would make them happier. They did not follow Hilbert's

well-meant advice and thereby may have missed discovering the Schrodinger

wave equation.)

Integers have repeatedly played a crucial role in the evolution of the

natural sciences. Thus, in the 18th century, Lavoisier discovered that

chemical compounds are composed of fixed proportions of their constituents

which, when expressed in proper weights, correspond to the ratios of small

integers. This was one of the strongest hints to the existence of atoms; but

chemists, for a long time, ignored the evidence and continued to treat atoms as

a conceptual convenience devoid of physical meaning. (Ironically, it was

from the statistical laws of large numbers, in Einstein's and Smoluchowski's

analysis of Brownian motion at the beginning of our own century, that the

irrefutable reality of atoms and molecules finally emerged.)

In the analysis of optical spectra, certain integer relationships between the

wavelengths of spectral lines emitted by excited atoms gave early clues to the

structure of atoms, culminating in the creation of matrix mechanics in 1925,

an important year in the growth of integer physics.

In 1882, Rayleigh discovered that the ratio of atomic weights of oxygen

and hydrogen is not 16:1 but 15.882:1. These near-integer ratios of atomic

weights suggested to physicists that the atomic nucleus must be made up of

integer numbers of similar nucleons. The deviations from integer ratios later