Excerpt from Math Odyssey 2000

David Hilbert was born in Koenigsberg, East Prussia in 1862 and received his doctorate from his home town university in 1885. His knowledge of mathematics was broad and he excelled in most areas. His early work was in a field called the theory of algebraic invariants. In this subject his contributions equaled that of Eduard Study, a mathematician who, according to Hilbert, "knows only one field of mathematics." Next after looking over the work done by French mathematicians, Hilbert concentrated on theories involving algebraic and transfinite numbers.

In 1899 he published his little book * The Foundations of Geometry *, in which he stated a set of axioms that finally removed the flaws from Euclidean geometry. At the same time and independently, the American mathematician Robert L. Moore (who was then 19 years old) also published an equivalent set of axioms for Euclidean geometry. Some of the axioms in both systems were the same, but there was an interesting feature about those axioms that were different. Hilbert's axioms could be proved as theorems from Moore's and conversely, Moore's axioms could be proved as
theorems from Hilbert's.

After these successes with the axiomatization of geometry, Hilbert was inspired to try to develop a program to axiomatize all of mathematics. With his attempt to achieve this goal, he began what is known as the "formalist school" of mathematics. In the meantime, he was expanding his contributions to mathematics in several directions partial differential equations, calculus of variations and mathematical physics. It was clear to him that he could not do all this alone; so in 1900, when he was 38 years old, Hilbert gave a massive homework assignment to all the mathematicians of the world.

This was done when he presented a lecture, entitled "MATHEMATICAL PROBLEMS" before the International Congress of Mathematicians in Paris. Here is the introduction to his lecture.

** "Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?**

David Hilbert, Mathematical Problems , Paris, 1900, Translation by Mary Winston published in the Bulletin of the American Mathematical Society volume 8 (1902), pp 437-479.

After a brief discussion about the nature of mathematical problems (that they should be difficult but not completely inaccessible), and what mathematicians should do about them (look them over and select the ones which seem to show promise of being solvable or important), he stated 23 problems.

We list the titles of these twenty-three problems.

- Cantor's Problem of the Cardinal Number of the Continuum
- The Compatibility of the Arithmetical Axioms
- The Equality of the Volumes of Two Tetrahedra of Equal Bases and Equal Altitudes
- Problem of the Straight Line as the Shortest Distance Between Two Points
- Lie's Concept of a Continuous Group of Transformations Without the Assumption of the Differentiability of the Function Defining the Group
- Mathematical Treatment of the Axioms of Physics
- Irrationality and Transcendence of Certain Numbers
- Problems of Prime Numbers
- Proof of the Most General Law of Reciprocity in any Number Field
- Determination of the Solvability of a Diophantine Equation
- Quadratic Forms With Any Algebraic Numerical Coefficients
- Extension of Kronecker's Theorem on Abelian Fields to Any Algebraic Realm of Rationality
- Impossibility of Solution of the General Equation of the 7th Degree by Means of Functions of Only Two Arguments
- Proof of the Finiteness of Certain Complete Systems of Functions
- Rigorous Foundations of Schubert's Enumerative Calculus
- Problem of the Topology of Algebraic Curves and Surfaces
- Expressions of Definite Forms by Squares
- Building up of Space From Congruent Polyhedra
- Are The Solutions of Regular Problems in the Calculus of Variations Always Necessarily Analytic?
- The General Problem of Boundary Values
- Proof of the Existence of Linear Differential Equations Having a Prescribed Monodromic Group
- Uniformization of Analytic Relations by Means of Automorphic Functions
- Further Development of the Methods of the Calculus of Variations

Some of these problems were already long standing and Hilbert himself had made some progress in them, so he knew something about their difficulties. It takes a certain amount of skill and knowledge to be able to ask the the type of questions that people will want to answer. Each problem had to have some kernel that made it important (and difficult) enough to be interesting, and it had to have the potential to lead to some wider results.

Have the mathematicians of the twentieth century been working on these problems for the past 90 years?

Oh, yes! Every so often a rumor (sometimes true, sometimes "partially true") will run wild among the mathematical community. "Did you hear that So & So has just solved a part of Hilbert's 4th problem?" Or, "Somebody just got a negative result to a question in Hilbert's 10th problem.", Then everyone would anxiously await the next issue of the journal in which the paper was to be published. Sometimes a mimeographed copy of the the paper would circulate before the actual publication and you would try to get your hands on a copy of it as quickly as possible, especially if you, yourself, had been working on the problem and had been leaning toward a result that was contrary to the current rumor.

At times the American Mathematical Society, or the Soviet Academy of Sciences or the French Mathematical Society, etc would publish an article updating progress on one of the Hilbert Problems. Attached to each such article would be a bibliography ranging from 5 to 100 papers written in this century on the subject of this particular problem. In 1975 The American Mathematical Society Published a two volume collection of articles updating the progress on every problem except #3 and #16.

Have the mathematicians of the twentieth century been successful in solving Hilbert's problems? To some degree, yes. Most of the problems have been partially solved; some have been restated and the new interpretations have been solved; Problem #1 is thought to be solved by some and not by others. Problem #10 is solved, negatively. This means that the collective work of the mathematician working on this problem has proved that it is impossible to derive the process that Hilbert wanted for solving Diophantine equations. Several other problems have partial solutions (some almost completely solved and some just barely started upon).

In any case, such a large body of new mathematics has arisen from Hilbert's problems that we can say that the 23 problems he posed in 1900 became the backbone for mathematical research in the 20th century and now we are awaiting the year 2000 and another young mathematician who will soon give us our homework assignment for the twenty-first century.

EXERCISES

- For what field of mathematics did David Hilbert and Robert L. Moore both write an equivalent set of axioms ?
- What major lecture did Hilbert present at the International Congress of Mathematicians in 1900 and how did it affect mathematics in the twentieth century?
- Find out a little bit about one of Hilbert's Problems and discuss it.