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Hermann Minkowski

Hermann Minkowski
Born: 22 June 1864 in Alexotas, Russian Empire (now Kaunas, Lithuania)
Died: 12 Jan 1909 in Göttingen, Germany

Hermann Minkowski's parents were Lewin Minkowski, a businessman, and
Rachel Taubmann. Hermann was his parents' second son, the eldest being
Oskar Minkowski who went on to become a famous pathologist. Lewin and
Rachel Minkowski were Germans ( Jews) although their son Hermann was
born while they were living in Russia. When Hermann was eight years
old the family returned to Germany and settled in Königsberg where
Lewin Minkowski conducted his business.

Minkowski first showed his talent for mathematics while studying at
the Gymnasium in Königsberg. Already at this stage in his education he
was reading the work of Dedekind, Dirichlet and Gauss. The outstanding
abilities he showed at this time were noted in a letter that Heinrich
Weber, then at Königsberg University, wrote to Dedekind in 1881 (see
[13]). He studied at the University of Königsberg, entering the
university in April 1880. He spent three semesters at the University
of Berlin, for example spending the winter semester of the academic
year 1882-83 there. His became close friends with Hilbert while at
Königsberg, for Hilbert was an undergraduate at the same time as
Minkowski. In 1884, while he was a student at Königsberg, Hurwitz was
appointed to the staff. The student Minkowski soon became close
friends with the newly appointed academic Hurwitz. He received his
doctorate in 1885 from Königsberg for a thesis entitled Untersuchungen
über quadratische Formen, Bestimmung der Anzahl verschiedener Formen,
welche ein gegebenes Genus enthält. Minkowski became interested in
quadratic forms early in his university studies. In 1881 the Academy
of Sciences (Paris) announced that the Grand Prix for mathematical
science to be awarded in 1883 would be for a solution to the problem
of the number of representations of an integer as the sum of five
squares. Eisenstein had given a formula for the number of such
representations in 1847, but he had not given a proof of the result.
In fact the Academy of Sciences had set a problem for the Grand Prix
which had already been solved, for Henry Smith had published an
outline of a proof in 1867. However the Academy of Sciences were
unaware of Smith's contributions when the prize topic was set.

Eisenstein had been studying quadratic forms in n variables with
integer coefficients at the time he published his unproved formula in
1847 but as he was already ill by this time details were never
published. Minkowski, although only eighteen years old at the time,
reconstructed Eisenstein's theory of quadratic forms and produced a
beautiful solution to the Grand Prix problem. Smith reworked his
earlier proof, adding detail and submitted that to the Academy. The
decision was that the prize be shared between Minkowski and Smith but
this was a stunning beginning to Minkowski's mathematical career. On 2
April 1883 the Academy granted the Grand Prize in Mathematics jointly
to the young Minkowski at the start of his career and the elderly
Smith at the end of his. Minkowski's doctoral thesis, submitted in
1885, was a continuation of this prize winning work involving his
natural definition of the genus of a form. After the award of his
doctorate, he continued undertaking research at Königsberg.

In 1887, a professorship became vacant at the University of Bonn, and
Minkowski applied for that position; according to the regulations of
German universities, he had to submit orally to the faculty an
original paper, as an Habilitationsschrift. Minkowski presented
Räumliche Anschauung und Minima positiv definiter quadratischer Formen
(Spatial visualization and minima of positive definite quadratic
forms) which was not published at the time but in 1991 the lecture was
published in [11]. Dieudonné writes:-

This lecture is particularly interesting, for it contains the first
example of the method which Minkowski would develop some years later
in his famous "geometry of numbers".

Minkowski taught at Bonn from 1887, being promoted to assistant
professor in 1892. Two years later he moved back to Königsberg where
he taught for two years before being appointed to the Eidgenössische
Polytechnikum Zürich. There he became a colleague of his friend
Hurwitz who had been appointed to fill Frobenius' chair after he left
Zürich for Berlin in 1892. Einstein was a student in several of the
courses he gave and the two would later become interested in similar
problems in relativity theory. Minkowski married Auguste Adler in
Strasburg in 1897; they had two daughters, Lily born in 1898 and Ruth
born in 1902.

The family left Zürich in the year that their second daughter was born
for Minkowski accepted a chair at the University of Göttingen in 1902.
It was Hilbert who arranged for the chair to be created specially for
Minkowski and he held it for the rest of his life. At Göttingen he
became interested in mathematical physics gaining enthusiasm from
Hilbert and his associates. He participated in a seminar on electron
theory in 1905 and he learnt the latest results and theories in

Minkowski developed a new view of space and time and laid the
mathematical foundation of the theory of relativity. By 1907 Minkowski
realised that the work of Lorentz and Einstein could be best
understood in a non-euclidean space. He considered space and time,
which were formerly thought to be independent, to be coupled together
in a four-dimensional 'space-time continuum'. Minkowski worked out a
four-dimensional treatment of electrodynamics. His major works in this
area are Raum und Zeit (1907) and Zwei Abhand lungen über die
Grundgleichungen der Elektrodynamik (1909). Kline, reviewing [9]

A key point of the paper is the difference in approach to physical
problems taken by mathematical physicists as opposed to theoretical
physicists. In a paper published in 1908 Minkowski reformulated
Einstein's 1905 paper by introducing the four-dimensional (space-time)
non-Euclidean geometry, a step which Einstein did not think much of at
the time. But more important is the attitude or philosophy that
Minkowski, Hilbert - with whom Minkowski worked for a few years -
Felix Klein and Hermann Weyl pursued, namely, that purely mathematical
considerations, including harmony and elegance of ideas, should
dominate in embracing new physical facts. Mathematics so to speak was
to be master and physical theory could be made to bow to the master.
Put otherwise, theoretical physics was a subdomain of mathematical
physics, which in turn was a subdiscipline of pure mathematics. In
this view Minkowski followed Poincaré whose philosophy was that
mathematical physics, as opposed to theoretical physics, can furnish
new physical principles. This philosophy would seem to be a carry-over
(modified of course) from the Eighteenth Century view that the world
is designed mathematically and hence that the world must obey
principles and laws which mathematicians uncover, such as the
principle of least action of Maupertuis, Lagrange and Hamilton.
Einstein was a theoretical physicist and for him mathematics must be
suited to the physics.

This space-time continuum provided a framework for all later
mathematical work in relativity. These ideas were used by Einstein in
developing the general theory of relativity. In fact Minkowski had a
major influence on Einstein as Corry points out in [7]:-

In the early years of his scientific career, Albert Einstein
considered mathematics to be a mere tool in the service of physical
intuition. In later years, he came to consider mathematics as the very
source of scientific creativity. A main motive behind this change was
the influence of two prominent German mathematicians: David Hilbert
and Hermann Minkowski.

We have mentioned several times in this biography that Minkowski and
Hilbert were close friends. Less well known is the fact that Minkowski
actually suggested to Hilbert what he should take as the theme for his
famous 1900 lecture in Paris. Minkowski, in a letter to Hilbert
written on 5 January 1900, writes:-

What would have the greatest impact would be an attempt to give a
preview of the future, i.e. a sketch of the problems with which future
mathematicians should occupy themselves. In this way you could perhaps
make sure that people would talk about your lecture for decades in the

Time has certainly proved Minkowski correct!

Minkowski's original mathematical interests were in pure mathematics
and he spent much of his time investigating quadratic forms and
continued fractions. His most original achievement, however, was his
'geometry of numbers' which he initiated in 1890. Geometrie der Zahlen
was first published in 1910 but the first 240 pages (of the 256)
appeared as the first section in 1896. Geometrie der Zahlen was
reprinted in 1953 by Chelsea, New York, and reprinted again in 1968.
Minkowski published Diophantische Approximationen: Eine Einführung in
die Zahlentheorie in 1907. It gave an elementary account of his work
on the geometry of numbers and of its applications to the theories of
Diophantine approximation and of algebraic numbers. Work on the
geometry of numbers led on to work on convex bodies and to questions
about packing problems, the ways in which figures of a given shape can
be placed within another given figure.

At the young age of 44, Minkowski died suddenly from a ruptured

Article by: J J O'Connor and E F Robertson