(David Hilbert was particularly interested in the foundati...)

David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays. This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Gödel's completeness proof for the predicate calculus has been updated. In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.

Geometry and the Imagination (AMS Chelsea Publishing)

(This remarkable book has endured as a true masterpiece of...)

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half a century! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. "Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before, or had wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader. A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained! The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books.

Theory of Algebraic Invariants (Cambridge Mathematical Library)

(In the summer of 1897, David Hilbert (1862-1943) gave an ...)

In the summer of 1897, David Hilbert (1862-1943) gave an introductory course in Invariant Theory at the University of Gottingen. This book is an English translation of the handwritten notes taken from this course by Hilbert's student Sophus Marxen. At that time his research in the subject had been completed, and his famous finiteness theorem had been proved and published in two papers that changed the course of invariant theory dramatically and that laid the foundation for modern commutative algebra. Thus, these lectures take into account both the old approach of his predecessors and his new ideas. This bridge from nineteenth to twentieth century mathematics makes these lecture notes a special and fascinating account of invariant theory. Hilbert's course was given at a level accessible to graduate students in mathematics, requiring only a familiarity with linear algebra and the basics of ring and group theory. The text will be useful as a self-contained introduction to invariant theory. But it will also be invaluable as a historical source for anyone interested in the foundations of twentieth-century mathematics.

The Theory of Algebraic Number Fields Softcover reprint of hardcover 1st ed. 1998, David Hilbert, F. Lemmermeyer, N. Schappacher, R. Schoof, I.T. Adamson - Amazon.com

(A translation of Hilberts "Theorie der algebraischen Zahl...)

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

David Hilbert was a German mathematician. By the end of his career, David Hilbert was the best-known mathematician in the world, as well as the most influential one. He was also a writer and educator.

Background

David Hilbert was born on January 23, 1862, in Wehlau, Germany, near Königsberg (now Kaliningrad, Russian Federation). Otto Hilbert, his father, was a lawyer of social standing in the society around Königsberg, and his mother’s family name was Erdtmann. The name “David” ran in the family—a fact Hilbert had subsequently to verify to the Nazi regime, which suspected that anyone with the name was of Jewish ancestry.

Education

Hilbert’s early education was in Königsberg, which he would always consider his spiritual home. In 1880 Hilbert entered the University of Königsberg, where he received his Doctor of Philosophy degree in 1885.

Hilbert had became a privat dozent by 1886, and by 1892 had been appointed to the equivalent of an assistant professorship at the University of Königsberg, rising in the ranks to a Professorship the next year. In 1895 he took a chair at Gottingen, where he remained until his retirement. As this rapid progress attests, Hilbert knew enough about academic politics to advance through the complexities of the German system. In this he had the guidance of Felix Klein, mathematician with great political skills who had devoted much of his life to building the University of Gottingen into the world’s Mathematical center.

Hilbert made his mathematical reputation on the strength of his research into invariant theory. The notion of an invariant had been created in the nineteenth century as an expression of something that remains the same under various sorts of transformations. As a simple instance, if all the coefficients in an equation are doubled, the solutions of the equation remain the same. A good deal of work had been done in classifying invariants and in trying to prove what sorts of invariants existed. The results were massive calculations, and books on invariant theory were made up of pages completely filled with symbols. Hilbert rendered most of that work obsolete by taking a path that did not require explicit calculation.

Those who had been practicing invariant theory were taken aback by Hilbert’s effrontery, and one of them described Hilbert’s approach as “not mathematics, but theology.” Invariant theory quickly disappeared from the center of mathematical interest, as Hilbert’s work required some time to be absorbed. Only much later was the field reopened, as invariant theorists at last were ready to proceed from his calculations.

In 1893 the German Mathematical Association appointed Hilbert and Minkowski to summarize the current state of the theory of numbers. Number theory was the oldest branch of mathematics, as it dealt with the properties of whole numbers.

The next direction in which Hilbert pursued his research was somewhat unexpected. After all his work in algebra, he began to look at the foundations of geometry. Euclidean geometry had already laid the foundations more than two thousand years before, but detailed examination of some of Euclid’s proofs revealed gaps in his presentation; he had made assumptions that were neither explicit nor justified by what had been proven earlier. In addition to problems posed by these gaps, another source for a new approach to geometry was the discovery during the nineteenth century of non-Euclidean geometries. These shared some axioms or assumptions with Euclid’s system, but differed in other respects. Hilbert felt that the only way to make progress was to be entirely explicit about each proof and not to trust to unspoken assumptions. The safest way to avoid these assumptions was to regard the terms of the subject as defined only by the axioms in which they were used.

As Hilbert noted, the question of which theorems followed from which axioms had to be unchanged if all the technical terms of the subject (like point, line, or plane) were replaced by words from some other area. It was the form of the axiom that mattered, not what the objects were. This brought Hilbert into conflict with Gottlob Frege, one of the founders of mathematical logic. The controversy between Hilbert and Frege involved issues about the philosophy of mathematics that remained central to the field for much of the twentieth century. In general, it can be claimed that Hilbert’s perspective has been more helpful in enabling mathematicians to pursue the foundations of geometry.

One of the highlights of Hilbert’s career came in 1900, when he was invited to address the International Congress of Mathematicians in Paris. His talk consisted of the statement of twenty-three problems, which he challenged his peers to solve in the twentieth century. Although not all of the problems have proved to be of the same importance, by posing them Hilbert created an agenda that has been followed by many distinguished mathematicians.

In addition to the study of the foundations of geometry, Hilbert turned to mathematical analysis and left a decisive imprint on this field as well. The previous generation of mathematicians had found defects in one of the standard principles from earlier in the century. Hilbert showed that the principle could be preserved, and he proceeded from there to make great progress in the study of integral equations. Hilbert has been credited with the creation of functional analysis, and although there was more foundational work to be done after him, his brief involvement in the area had once again altered it irrevocably.

After a brief dalliance with theoretical physics (an area Hilbert felt too important to leave to the physicists and to which he made few lasting contributions), Hilbert returned to questions of the philosophy of Mathematics that had arisen earlier during his work on geometry. He was eager to pursue a program that could result in the establishment of secure foundations for mathematics. While he was willing to grant some importance to finite mathematics, he felt that the infinite required special treatment. In his account, called formalism in mathematics, he set out to prove the consistency of mathematics.

This enterprise put him in conflict with the other Philosophies of mathematics most frequently advocated. He gave expression to his views most notably in an address “On the Infinite” in 1925; he was challenged by many, including L. E. J. Brouwer and Hermann Weyl, but it was the incompleteness theorem of the young Austrian Mathematician Kurt Godel Friedrich which threatened the entire program Hilbert was pursuing. Certain narrow interpretations of formalism were put to rest by Godel’s work, and some of Hilbert’s views were included among these.

The last years of Hilbert’s life were also darkened by the advent of National Socialism and its dire effects on Germany’s intellectual community. In 1930 Hilbert retired from Gottingen, but nothing could assuage his grief over the sequence of losses that the university suffered from the departure of many of its leading minds. Hilbert turned seventy-one in 1933, the year the Nazis came to power, and it was too late for him to look for a new home. Many of his students had found academic homes abroad, and nothing could rebuild the university in the face of racial laws and hatred of the intellect. It is a measure of the state of German Mathematics and the political atmosphere in Gottingen that at Hilbert’s death on February 14, 1943, no more than a dozen people attended his funeral.

His family was staunchly Protestant, although Hilbert himself was later to leave the church in which he was baptized.

Politics

During the war, Hilbert refused to sign the “Declaration to the Cultural World,” which claimed that Germany was innocent of alleged war crimes. He was also willing to put mathematics before nationality, and he included an obituary for a French mathematician in the journal Mathematische Annalen (the showpiece of German mathematics) during the war. These acts made him unpopular with German nationalists.

Views

Hilbert took pleasure in fighting the academic establishment over the rights of Emmy Noether, who was both a woman and a Jew. Of the sixty-nine students who wrote their theses under Hilbert (an enormous number for any time), there were several women.

His philosophical interests centred upon axiomatization of mathematical theories and related metamathematical questions, such as consistency and completeness.

Quotations:
"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"No one shall expel us from the Paradise that Cantor has created."

"We must know — we will know!"

"The art of doing mathematics consists in finding that special case which contains all the germs of generality."

Membership

Hilbert was a member of the German Mathematical Association. He was also a fellow of the Royal Society of United Kingdom.

Connections

Hilbert married Käthe Jerosch in 1892. While he was willing to be casual with regard to his appearance, his wife helped prevent at least some of his sartorial excesses. She also proved a source of strength to Hilbert in his disappointments, one of which was their only son Franz, who never lived up to his father’s expectations and probably suffered from a mental disorder.