David Hilbert: the architect of modern mathematics

In July 1915, Albert Einstein paid a visit to the University of Göttingen (Germany) on the invitation of the mathematician David Hilbert (1862-1943). It was a fruitful encounter for both men that continued over the following months with an intense scientific correspondence. Einstein described that period as the most exhausting and stimulating of his entire life, the result of which was a series of studies and articles, authored by one or the other scientist, in which they formulated the equations of the gravitational field of the General Theory of Relativity (GTR).


Hilbert believed in the ability of mathematics to find answers to all questions. Credit: Universität Göttingen

In December 1915, both geniuses presented and published, almost simultaneously, articles containing these equations. As a result of this, the question has been raised as to whether it was Hilbert who preceded Einstein in the discovery of them. However, Hilbert himself was responsible for resolving the debate by recognizing in his article that the fundamental ideas of the theory were the work of his colleague: “The resulting differential equations of gravitation are, it seems to me, in agreement with the magnificent general theory of relativity established by Einstein in his later papers.”

Two positions about GTR

Hilbert’s position was, incidentally, consistent with the true purpose that led him to tackle the GTR. While finding such equations was the priority for Einstein, Hilbert on the other hand intended to establish a minimum set of fundamental principles that would allow him to deduce not only the mathematical equations to validate the GTR, but any other theory of physics as well. He was looking for the minimum number of axioms on which to base all mathematical physics.

It was one more link in his colossal project of constructing a theoretical framework, by means of the axiomatic method, to develop the tools (methods and techniques) necessary to solve any mathematical problem. It was a goal consistent with the vision and unshakable faith that Hilbert had in the ability of his discipline to find answers to all questions. This was the motor and common thread of a successful career that began in 1886 when he obtained the position of Privatdozent (Assistant Lecturer) at the University of Königsberg, and that would eventually lead him to become the architect of modern mathematics.

This search for axioms led him to tackle successively—and establish the foundations of—the Theory of Invariants (1886-1893), the Theory of Numbers (1893-1898), Geometry (1898-1902), integral analysis and equations (1902-1912) —thereby laying the cornerstone of Functional Analysis—and finally physical mathematics (1910-1922).

Mathematical challenges for the XXI century

Beyond all that, if there is one reason why Hilbert continues to be so popular today it is because in 1900 he enunciated a list of 23 mathematical problems that were to constitute the field of study of his colleagues during the twentieth century. And these problems are still keeping mathematicians busy today.

Apart from his enormous academic work, Hilbert was also distinguished for his activism. Thus, in 1914, he refused (as did Einstein) to add his name to the “Manifesto for a Civilized World,” signed by 93 German intellectuals and scientists, in which the motives of Germany to declare war were justified and argued, a decision that for a time condemned him to isolation by his colleagues and students. Shortly thereafter, he engaged in a struggle so that the University of Göttingen, where he spent most of his career, would hire mathematician Emmy Noether, claiming that “the sex of a candidate should not be an argument against her admission.”  After failing in his attempt, he persisted in keeping her at the university by advertising her courses and lectures under his own name.

In 1928, already at the end of his career, when the International Congress of Mathematicians met, Hilbert confronted many of his German colleagues who refused to attend and he headed the German delegation in the name of the universality of knowledge. He also positioned himself against the measures taken by the Nazi party to expel the teachers of Jewish descent.

By then, his career had already come to an end. And not only because of the inevitable loss of vitality caused by age, but also because of the blow that, in 1931, the young Austrian mathematician Kurt Godel had dealt to Hilbert’s ideas. Godel postulated the existence of undecidable statements, i.e. that cannot be denied or affirmed within a formal system, or in a more intuitive way, he showed that mathematics could not answer all the questions.

Words in Hilbert's tombstone have turned into a famous inspiring quote.

Words in Hilbert’s tombstone have turned into a famous inspiring quote.

Shortly before, in 1930, Hilbert delivered a speech in which he reiterated his unshakable faith in mathematics, and his last words were “We must know, we will know.” After his death in 1943, these same words were etched as an epitaph on his gravestone in the cemetery of Gotinga.

Miguel Barral