Birthday Special: Who Was David Hilbert & Why Is He Called A Top Rate Mathematician?

"The Infinite!! No other question has ever moved so profoundly the spirit of man."

On 23 January 1862 in Königsberg, Prussia (now Kaliningrad, Russia) one of the most influential mathematician David Hilbert was born. He was the one behind the discovery and development of broad range of fundamental ideas in many areas, including invariant theory, calculus of variations, commutative algebra, algebraic number theory, the foundation of Geometry, spectral theory of operators and its application to sntegral equations. Also known as one of the founders of proof theory and mathematical logic, as well as for being amongst the first to distinguish between Mathematics and Metamathematics.

David Hilbert (1912)

Life and Education

Back in those days the usual age for someone to begin schooling was six but David did not enter his first school, the Royal Friedrichskolleg, until he was eight years old. It is almost certain that his mother taught him at home until he was eight. The Friedrichskolleg, also known as the Collegium Fridericianum, had a junior section which David attended for two years before entering the gymnasium of the Friedrichskolleg in 1872. Although this was reputed to be the best school in Königsberg, the emphasis was on Latin and Greek with mathematics considered as less important. Science was not taught at all in the Friedrichskolleg. The main approach to learning was having pupils memorise large amounts of material, something David was not particularly good at. Perhaps surprisingly for someone who was to make a gigantic impact on mathematics, he did not shine at school. In later life he described himself as a "dull and silly" boy at the Friedrichskolleg.

In September 1879 he transferred from the Friedrichskolleg to the Wilhelm Gymnasium where he spent his final year of schooling. Here there was more emphasis on mathematics and the teachers encouraged original thinking in a way that had not happened at the Friedrichskolleg. Hilbert was much happier and his performance in all his subjects improved. He received the top grade for mathematics and his final report stated:-F

"For Mathematics he always showed a very lively interest and a penetrating understanding: he mastered all the material taught in the school in a very pleasing manner and was able to apply it with sureness and ingenuity."

Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions").

Hilbert remained at the University  as a Privatdozent (senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of
Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for the rest of his life.

On his birth-anniversary here are some great works of David Hilbert that influenced and shaped the world of Mathematics.

Gordon's Problem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem, showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. 

The 23 Problems

Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathemticians in Paris in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.

The mathematical community as a whole could enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key. The problem set was launched as a talk "The Problems of Mathematics" presented during the course of the Second International Congress of Mathematicians held in Paris.

Hilbert's Program

In 1920 he proposed explicitly a research project (in Metamathematics, as it was then termed) that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. This program is still recognizable in the most popular Philosophy of Mathematics, where it is usually called formalism.

Physics

Until 1912, Hilbert was almost exclusively a "pure" mathematician. When planning a visit from Bonn, where he was immersed in studying physics.
Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself. He started studying Kinetic Gas Theory and moved on to elementary radiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of Albert Einstein and others were followed closely. He invited Einstein to Göttingen to deliver a week of lectures on the subject. When published several papers culminating in "The Field Equations of Gravitation" simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations. Hilbert fully credited Einstein as the originator of the theory, and no public priority dispute concerning the field equations ever arose between the two men during their lives.

The Number Theory

Hilbert unified the field of Algebraic Number Theory with his 1897 treatise Zahlbericht(literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with The Finiteness Theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory.

Hilbert received many honors and awards in his lifetime, including the Bolyai Prize in 1910. He was also elected a fellow of the Royal Society of London in 1928. He died in 1943 at age 81. He was indeed a true legend.

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