(Two volumes in one. Minkowski's Collected Works are issue...)

Two volumes in one. Minkowski's Collected Works are issued under the editorship of David Hilbert, with the assistance of Andreas Speiser and Hermann Weyl. Minkowski was one of those great mathematicians who did important work both in pure mathematics and in mathematical physics. The papers in this volume are in four fields: the theory of Quadratic Forms, the Geometry of Numbers, Geometry, and Physics. The text is in German.

Hermann Minkowski was a German mathematician. He created and developed the geometry of numbers. Minkowski is best known for his work in relativity.

Background

Hermann Minkowski was born on June 22, 1864, in Alexotas, Russian Empire (now Kaunas, Lithuania) to the family of Lewin Boruch Minkowski, a businessman, and Rachel Taubmann. They both were of Jewish descent. He was his parents' third son. His oldest brother Max took over the family business, but he was also an art collector and the French consul in Königsberg. The second brother Oskar was a physician, best known for his work on diabetes, and the father of astrophysicist Rudolph Minkowski. Apart from Max and Oskar, Minkowski also had an older sister, Fanny and a younger brother, Toby. Lewin and Rachel Minkowski were Germans 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.

Education

Minkowski first showed his talent for mathematics while studying at the Altstadt 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. 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 1882-1883 there. He 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.

In 1881 the French Academy of Sciences 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 proof in 1867. However, the Academy of Sciences was 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 to 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 which was not published at the time but in 1991 the lecture was finally published.

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's 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 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 learned the latest results and theories in electrodynamics.

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.

Minkowski grew in religious atmosphere of Judaism. His father was a sponsor of the synagogue construction on Kovno.

Politics

Hermann Minkowski was never involved in politics and there are no political remarks of him are left.

Views

Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity. By 1907 Minkowski realized 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). 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.

Quotations:
"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

"The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines."

"The word postulate of relativity... appears to me very stale... I should rather like to give this statement the name Postulate of the absolute world (or briefly, world-postulate)."

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity."

"Oh, that Einstein, always cutting lectures. I really would not believe him capable of it."

"It came as a tremendous surprise, for in his student days Einstein had been a lazy dog. He never bothered about mathematics at all."

"The assumption of the contraction of the electron in Lorentz's theory must be introduced at an earlier stage than Lorentz has actually done."

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone."

"It would be very unsatisfactory if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics."

"Many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basic of the new Electro-dynamics."

Membership

The first International Congress of Mathematicians was held in Zürich in 1897. Minkowski joined the organizing committee in December 1896 - he might not yet have been in Zürich for the preliminary meeting in July. He joined the amusement committee and was appointed to the sub-committee that was responsible for choosing the speakers. He suggested inviting Hilbert to give a talk in case Klein could not attend, as it was, Klein did attend the congress but Hilbert did not. Minkowski also offered to give a talk himself in one of the section meetings, but for reasons that are not explained in the minutes, he did not after all. At the congress, he chaired section I: Arithmetic and Algebra.

Minkowski acted as one of the secretaries at the 1900 ICM in Paris and gave a talk in section I at the 1904 ICM in Heidelberg, entitled Zur Geometrie der Zahlen (On the Geometry of Numbers). At this point, he represented the University of Göttingen, likewise at the 1908 ICM in Rome.

Minkowski was a member of the Göttingen Academy of Sciences.

Göttingen Academy of Sciences
,
Germany

Personality

Minkowski was a loyal and sincere friend.

Physical Characteristics:
Minkowski suffered from appendicitis which became a cause of his death in 1909 at the age of 44 when his appendix ruptured.

Quotes from others about the person

"Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us." - David Hilbert, German mathematician

Interests

Philosophers & Thinkers

Hendrik Lorentz, Henri Poincaré

Connections

Minkowski married Auguste Adler in Strasbourg 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.