VI.63 David Hilbert
b. Königsberg, Germany, 1862; d. Göttingen, Germany, 1943
Invariant theory; number theory; geometry;
International Congress of Mathematicians; axiomatics
HERMANN WEYL [VI.80] described his teacher Hilbert’s style: “It is as if you were on a swift walk through a sunny open landscape; you look freely around, demarcation lines and connecting roads are pointed out to you, before you must brace yourself to climb the hill; then the path goes straight up. . . .” Several themes balance in Hilbert’s career as a mathematician. He wanted clarity, rigor, simplicity, and depth. Though he loved mathematics for its beauty, a beauty that transcends human failures, Hilbert saw mathematics as a social collaboration. A turning point came when he met MINKOWSKI [VI.64] and Adolf Hurwitz at university in Königsberg.
Hilbert wrote: “On unending walks we engrossed ourselves in the actual problems of the mathematics of the time; exchanged our newly acquired understandings, our thoughts and scientific plans; and formed a friendship for life.” Later Hilbert became professor at Göttingen and, with KLEIN [VI.57], drew mathematicians from all over the world and turned that small city into a crossroads for mathematics—until Hitler destroyed it.
When he was a new Privatdozent, Hilbert decided he would study mathematics as he taught, and he resolved never to repeat lectures. He and Hurwitz decided to embark on a “systematic exploration” of mathematics, and he followed this pattern for the rest of his life. Hilbert’s career divides easily into six periods: (i) algebra and algebraic invariants (1885-93); (ii) algebraic number theory (1893-98); (iii) geometry (1898-1902); (iv) analysis (1902-12); (v) mathematical physics (1910-22); and (vi) foundations (1918-30). Remarkably, there is very little overlap. When Hilbert finished a subject, he was finished with it.
David Hilbert
Hilbert’s first breakthrough came in 1888 when he solved Gordan’s problem, named after Paul Gordan, in a single bold move. Given a polynomial equation with at least two variables, some things about the polynomial change and some do not when you change coordinate systems. For example, consider the real polynomial equation
ax2 + bxy + cy2 + d = 0.
If you rotate the coordinate system, then this equation changes dramatically, but the graph does not, and neither does the discriminant b2 - 4ac. The discriminant is one invariant. In the general case—a more complicated class of polynomials and coordinate changes—there can be many invariants. Mathematicians suspected that a finite number of essentially different invariants existed for any given type of polynomial and class of coordinate changes. Was this so? Many mathematicians calculated individual examples industriously. Instead, Hilbert reasoned indirectly: what if there is no finite basis for a specific class of polynomials and transformations? He found that it was always possible to produce a contradiction. He concluded that there must be such a basis. At first this result was greeted with disbelief because he did not display a basis. Gordan said, “Das ist nicht Mathematik. Das ist Theologie.” However, the result was so powerful that it has been said that it killed algebraic invariant theory.
In 1893 Hilbert and Minkowski were asked by the German Mathematical Society to write a report on number theory. Hilbert chose ALGEBRAIC NUMBER THEORY [IV.1] and transformed the results of the nineteenth century into the study of algebraic NUMBER FIELDS [III.63]. The deep organizing structure Hilbert found eventually led to what has been called “the magnificent edifice of class field theory” (described in [V.28]).
Hilbert’s classic Foundations of Geometry, first published in 1899 and revised many times, starts with real-number arithmetic. He assumes that it is consistent, i.e., that it is free of the possibility of contradictory deductions. Using analytic geometry, he then exhibits a model of EUCLIDEAN GEOMETRY [II.2 §3]. A point is a pair of real numbers; a line is a set of pairs of numbers that satisfy the equation for a line; a circle. . . ; and so on. All of Euclid’s axioms are true statements about these “lines” and “points,” that is, they are true statements about these sets of real numbers. Euclidean geometry is thereby reduced to a fraction of all the true statements about real numbers, and we conclude that if real-number arithmetic is consistent then Euclid’s geometry is consistent. Next Hilbert constructs models of various non-Euclidean geometries in terms of Euclidean geometry, exploring in depth and with great inventiveness which possible axioms follow from which groups of axioms and which are independent yet consistent.
Hilbert was invited to address the Second International Congress of Mathematicians in Paris in 1900. He gave a talk proposing twenty-three problems for the new century. These problems are known today as “Hilbert’s problems”; in a sense they have created a virtual Göttingen where mathematicians have entered into conversation with Hilbert and each other ever since.
Next Hilbert turned to analysis. WEIERSTRASS [VI.44] had found counterexamples to Dirichlet’s principle, which is essentially the assertion that, in variational problems, maxima and minima are always attained. Hilbert proved a modified, but still powerful, version that “salvaged” much of the work that assumed the principle. The larger theme of this period, though, was integral equations and what is now called HILBERT SPACE [III.37]. Newton’s equations for motion are differential equations, and it was natural to phrase equations in physics that way. However, in many cases it was easier to solve problems if the equations were written using integrals rather than derivatives. Between 1902 and 1912 Hilbert attacked a variety of problems from this direction. He viewed the solutions as part of Hilbert space and gave a spectral interpretation analogous to an infinite-dimensional vector space. Thus, an amorphous sea of functions acquired geometric structure.
In 1910 he turned toward mathematical physics and had some successes, but physics was undergoing multiple revolutions and was not ready for mathematical clarification.
When he delivered his problems in 1900, Hilbert was aware that there were contradictions in mathematics as it was then phrased, and specifically in set theory. His second problem asked for a proof that first arithmetic, and then set theory, were consistent. As the debate widened, some mathematicians began to pull back on what they accepted as valid reasoning. Hilbert wanted none of this. By 1918 he was increasingly focused on a program to formally axiomatize mathematics and prove it free of contradictions using proof-theoretic, combinatorial methods. GÖDEL [VI.92] proved his incompleteness theorems in 1930 and thereby showed that Hilbert’s program, at least as initially conceived, could never be successful. Hilbert was wrong here, but even if wrong, his dream of placing mathematics on a formal foundation stimulated some of the most important work of the twentieth century—and mathematics did not pull back.
Further Reading
Reid, C. 1986. Hilbert-Courant. New York: Springer.
Weyl, H. 1944. David Hilbert and his mathematical work. Bulletin of the American Mathematical Society 50:612-54.