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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
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