David Hilbert (1862-1943), a mathematician from the University of Goettingen, launched his program during an address to the International Congress of Mathematicians in the summer of 1900. Hilbert contended that all mathematical principles, or axioms, should be derived from first-order statements, with the logic of deduction and reasoning leading to final conclusions regarding the problem at hand. Using this methodology, the foundations of mathematics could be developed exactly and with certainty; these fundamental axioms would form the basis for all other research into the varied branches of mathematics. In his address, Hilbert famously outlined 23 "unsolved problems" that he characterized as both interesting and important for further study in the coming century. Hilbert hoped that his "program" of formalism would be applied to each.
Hilbert's program, as his initiative as come to be called, was a call to the mathematics community to formalize the basic tenets of the science. Hilbert viewed the new century with great optimism for the study of mathematics, feeling that the 23 unsolved problems demonstrated both the vitality of the field and the opportunity for growth and innovation within it. When a field of study ceases to hold an abundance of interesting, albeit difficult problems, Hilbert felt that the extinction of independent development in the science itself was near. Bernoulli's problem of the "line of quickest descent" (in its simplest form, the idea that an object, when traveling under its own power down a hill or similar surface, will follow the steepest path of travel and thus attain the maximum speed--or rate of change of distance over time--possible) was a difficult problem of its time, yet its usefulness cannot be questioned.
Although his "call to action" involved many areas of mathematics, Hilbert particularly emphasized the relationship of arithmetic axioms to geometric principles and contended that the former could be used to formalize the latter, or at the very least, the formalization of arithmetic could serve as a model for the formalization of geometry. He asserted that development of a mathematical concept would not be complete until it could be easily explained to a layperson--that is, that mathematics should be clear and concise to the average reader. Hilbert further stated that many problems of mathematics were born of the intertwined relationship between pure reason and the reality of observation and intuition. Both pure reason and the observable world pose questions for study; it is through the examination of each that gains in knowledge are made. While Hilbert felt that both reason and phenomena were crucial instruments of learning, he contended that care must be taken that axioms, and thereby proofs, are based on progressive logic. Proof by exhaustion should be avoided at all costs, and problems requiring such proof such be simplified by revisiting from new vantage angles.
Perhaps most importantly, Hilbert declared that the correct solution to any mathematical problem should be solved with a finite number of steps. By solving problems with a finite number of processes, Hilbert asserted that rigor in reasoning would be maintained and thus deduction would apply. To Hilbert, this methodology ensured not only completeness of a concept, but also ensured humanity's philosophical need to understand the concept was attained. New problems, Hilbert felt, must be solved with extensions of axioms developed from old problems, ensuring a continuity of reason and intellectual development.
Hilbert was most ardently concerned with the concept of infinity, especially as it related to the continuum hypothesis. Indeed, the very first of his 23 unsolved problems posed to the mathematics community was based on verifying the concepts of infinity as defined in Cantor's axioms for set theory. At the time of Hilbert's address, the concept of physics and mathematics were reaching significant points of departure; experiments in physics were demonstrating that energy was not infinite; that matter consisted of a finite number of sub-atomic particles and could not be infinitely divided; and that time itself, while extending from an unfathomably distant point in the past and an incomprehensibly distant point in the future, may not be infinite. It was Hilbert's hope that by perfecting the axioms of finite mathematics that the formal system necessary to define infinite mathematics. Thus, Hilbert contended, the axioms of finite mathematics, once robustly stated, could be extended to prove the consistency of infinite mathematics.
Sometimes called a "glorious failure," Hilbert's program, particularly as applied to infinity, did not meet Hilbert's expectations. In 1938, Goedel demonstrated that the continuum hypothesis is consistent with Zermelo-Fraenkel set theory; in 1963, Paul Cohen demonstrated that the negation of the continuum hypothesis is also consistent. Using current axioms of set theory, the question is undecided. Hilbert's program and its emphasis on rigor, however, can hardly be considered immaterial, as it still influences scholars' approach to investigations into contemporary mathematics.
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