David Hilbert Sets an Agenda for Twentieth-Century Mathematics

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David Hilbert Sets an Agenda for Twentieth-Century Mathematics

Overview

In 1900 David Hilbert (1862-1943), one of the acknowledged leaders of pure mathematics at the turn of the century, identified what he considered to be the most important problems facing contemporary mathematicians at an address to the Second International Congress of Mathematicians. While some of the questions concerned purely technical issues, a number addressed the foundations of mathematics. These led to major changes in the philosophy of mathematics and ultimately to the development of digital computers and artificial intelligence.

Background

The nineteenth century was a period of restructuring in mathematics. Once considered a study of self-evident truths about quantity and space, the paradoxes of set theory and the discovery of non-Euclidean geometries and generalized kinds of numbers motivated mathematicians to reexamine the most fundamental ideas in each area of mathematics. Many mathematicians thought that the best way to avoid any possible problems was to adopt the axiomatic method, allowing each field to have very few undefined concepts and unproved statements about it, called axioms, which were taken to be true. All other statements could only be considered true if they could be proved in a strictly logical way from the axioms.

One of the masters of this new, more formal, mathematics was David Hilbert, Professor of Mathematics at the University of Göttingen. Hilbert had established his reputation with a treatise on algebraic number theory in 1897. In 1899 he published a book, The Foundations of Geometry, which placed the subject on a far more rigorous footing than the classic Elements of Euclid (c. 330-260 b.c.), which had dominated geometric thinking since antiquity. In doing so he emphasized the importance of considering the simplest geometric concepts—point, line and plane—to be undefined terms.

In 1900 Hilbert was invited to present the inaugural address at the Second International Congress of Mathematics, to be held in Paris, France. Hilbert used the occasion to list the most important issues he considered to be facing mathematicians at the beginning of the twentieth century. Hilbert's list of 23 problems, published in a German mathematical journal and translated into English and republished by the American Mathematical Society, would serve as a stimulus to mathematical research for decades to come. A number of these problems are described below.

Hilbert's first problem came from the theory of sets with an infinite number of elements. Georg Cantor (1845-1918) had shown that the set of all real numbers could not be put into a one-to-one correspondence with the counting numbers, 1, 2, 3.... In contrast, the set of positive rational numbers (fractions) could be placed in countable order by expressing each fraction in lowest terms, and then listing them in the order of the total of numerator and denominator, 0/1, 1/1, 1/2, 2/1, 1/3, 3/1, and so on. Cantor guessed that there were only two types of infinite sets of numbers—those that could be put into correspondence with the counting numbers and those that could be put into correspondence with the real numbers, which includes rational numbers and irrational numbers as well. Hilbert called for a proof or disproof of Cantor's conjecture.

The second problem posed was a challenge to mathematicians to prove that the axioms of arithmetic could never lead to a contradiction, or, more precisely, that there existed a set of axioms for arithmetic that would yield the familiar properties of the whole numbers without leading to a contradiction. As mathematicians had worked on axiomatic formulations of calculus and geometry in the nineteenth century, it became clear that these fields would be free of contradiction provided that arithmetic could be put on such an axiomatic basis. Hilbert had been particularly interested in the attempt of Italian mathematician Giuseppe Peano (1858-1932) to provide an axiomatic basis for arithmetic, and he was fairly confident that this approach would eventually eliminate any possible contradiction.

Hilbert's sixth question concerned the possibility of providing a fundamental set of rules, or axioms, for the science of physics. As established by Sir Isaac Newton (1642-1727), the laws of motion provided a powerful means of analyzing what was happening in any system of bodies, but as in the case of arithmetic it was not clear whether one might come up with contradictory predictions about the same physical situation. The question had become more important as new laws of physics had been discovered to describe electric, magnetic, and thermal phenomena.

The eighth question posed by Hilbert involved the distribution of prime numbers. A prime number is one that can only be divided by the number one and itself. It had been proved in ancient times that there was no largest prime number, but no one had a definite way of calculating the number of prime numbers between any two numbers, short of testing each number by dividing it by all smaller prime numbers.

Hilbert's tenth question asked if it was possible to determine whether a given set of Diophantine equations could be solved in whole numbers. Diophantine equations, named after the Greek mathematician Diophantus (fl. c. 250 a.d.), are sets of equations involving whole number multiples of several unknown quantities and their squares or cubes or higher powers. Mathematicians thought that it should be possible by manipulating the numbers appearing in the set of equations to determine if a solution in integers could be found, but no one had discovered such a method.

The twenty-third and last challenge posed by Hilbert was to further develop the calculus of variations, that branch of mathematics concerned with finding the one function of space and time for which a specified overall property takes on a minimum or maximum value. Problems of this type include that of the catenary, the shape taken by a chain suspended at the ends, and the geodesic, the shortest curve between two points on a given surface. Nineteenth-century mathematicians had demonstrated that Newton's laws of mechanics could be cast in the form of a minimum principle, further stimulating the development of this mathematical field.

Impact

The status of a scientific discipline can often be gauged by the social organization of the field: the number of professional organizations, the character of professional meetings, the number of journals, and the number of universities offering advanced degrees in the field. Hilbert's list of 23 questions appeared at a time when the field of mathematics had moved from a matter of correspondence by letter to periodic meetings of leading mathematicians. The First International Congress of Mathematicians had been held in Zurich, Switzerland, in 1897. Subsequent meetings have been held every four years with the exception of the periods during the two world wars (1914-1918 and 1939-1945).

Possibly the question of the greatest fundamental importance posed by Hilbert was the second, about the possibility of contradictions arising from the axioms of arithmetic. Austrian (later American) mathematician Kurt Gödel (1906-1978) in 1931 ruled out the possibility that a set of axioms for arithmetic could be found that would never lead to a meaningful statement being both true and not true. He did this by proving that any set of axioms that led to the accepted properties of integers would necessarily allow for statements that could neither be proved nor disproved. Gödel's work caused many mathematicians and philosophers to reexamine their own concepts of mathematical truth.

Gödel also provided half of the answer to Hilbert's first question in 1938 by proving that the continuum hypothesis could not be disproved. The other half was provided in 1963 by American mathematician Paul J. Cohen (1934- ), a former student of Gödel who showed that the hypothesis could not be proved, either.

The question that may have had the greatest impact on twentieth-century technology may have been Hilbert's tenth, about being able to decide whether a set of Diophantine equations had an integer solution. Hilbert generalized this at the Eighth International Congress, held in Bologna, Italy, to a general question of decidability—that is, whether there might exist a generally effective procedure that would allow a mathematician to determine whether any mathematical statement was true by manipulating the symbols appearing in it. To answer this question English mathematician Alan Matheson Turing (1912-1954) developed the idea of a machine that could implement any effective procedure for manipulating symbols. This machine, which could read the symbols on a tape one at a time and, following an internal set of instructions, compute a new symbol to be written on a tape and move the tape forward or backward, would become the model for the digital computer. With this type of machine carefully described, Turing converted Hilbert's question about a problem being decidable to a question about whether such a machine would halt or continue in an endless loop, given any problem translated into a suitable code. Turing was then able to prove that the desired general effective procedure could not possibly exist.

Hilbert's sixth question would turn out to be of great significance for twentieth-century physics. As physicists were forced to abandon familiar notions of location and trajectory in describing the behavior of subatomic particles, many came to accept the necessity of an axiomatic approach. Hilbert himself would play an important role in developing a sound mathematical formulation for the quantum theory. In this theory, each state of a physical system corresponds to a direction in an abstract geometrical space, called Hilbert space in his honor, and all observable quantities are given geometrical interpretation in this space.

DONALD R. FRANCESCHETTI