希尔伯特论无限.pdf

BERTRAND RUSSELL alabour which is, in fact, much less than might be thought. As the above hasty Survey must have made evident, there are innumerable unsolvad problems in the subject, and much work nerds to be done. If any student is led into a serious study of mathematical logic by this little book, it vill have served the chief purpose for which it has been written. I On the infinite DAVID HILBERT As a result of his penetrating critique, Weierstrass has provided a solid foundation for mathematical analysis. By elucidating many notions, in particular those of minimum, function, and differential quotient, he removed the defects which were still found in the infinitesimal calculus, rid it of all confused notions about the infinitesimal, and thereby com- pletely resolved the difficulties which stem from that concept. If in analy- sis today there is complete agreement and certitude in employing the deductive methods which are based on the concepts of irrational number / and limit, and if in even the most complex questions of the theory of dif- 1 ferential and integral equations, notwithstanding the use of the most I ingenious and varied combinations of the different kinds of limits, there nevertheless is unanimity with respect to the results obtained, then this happy state of affairs is due primarily to Weierstrass's scientific work. And yet in spite of the foundation Weierstrass has provided for the infinitesimal calculus, disputes about the foundations of analysis still 1 goon. These disputes have not terminated because the meaning of the in- I finite, as that concept is used in mathematics, has never been completely clarified. Wejerstrass's analysis did indeed eliminate the infinitely large ! and the infinitely small by reducing statements about them to [statements about] relations between finite magnitudes. Nevertheless the infinite still appears in the infinite numerical series which defines the real numbers and in the concept of the real number system which is thought of as a completed totality existing all at once. In his foundation for analysis, Weierstrass accepted unreservedly and used repeatedly those forms of logical deduction in which the concept of the infinite comes into play, as when one treats of all real numbers with a certain property or when one argues that there exist real numbers with a certain property. Delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, in honor of Karl Weierstrass. Translated by Erna Putnam and Gerald J. Masse~ from Mathernatische Annolen (Berlin) vol. 95 (1926), pp. 161-90. Permlsslon for the.trans- lation and inciusion of the article in this volume was kindly granted by the publishers, Springer Verlag. DAVID HILBERT On the infinite Hence the infinite can reappear in another guise in Weierstrass's theory and thus escape the precision imposed by his critique. It is, therefore, the problem of the infinite in the sense just indicated which we need to resolve once and for all. Just as in the limit processes of the infinitesimal calculus, the infinite in the sense of the infinitely large and the infinitely small proved to be merely a figure of speech, so too we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is an illusion. Just as operations with the infinitely small were replaced by operations with the finite which vielded exactly the same results and led to exactly the same elegant formal relationships, so in general must deductive methods based on the infinite be replaced by finite procedures which yield exactly the same results; i.e., which make possible the same chains of proofs and the same methods of getting formulas and theorems. The goal of my theory is to establish once and for all the certitude of mathematical methods. This is a task which was not accomplished even during the critical period of the infinitesimal calculus. This theory should thus complete what Weierstrass hoped to achieve by his foundation for analysis and toward the accomplishment of which he has taken a neces- sary and important step. But a still more general perspective is relevant for clarifying the con- cept of the infinite. A careful reader will find that the literature of mathe- matics is glutted with inanities and absurdities which have had their source in the infinite. For example, we find writers insisting, as though it were a restrictive condition, that in rigorous mathematics only a /inite number of deductions are admissible in a proof - as if someone had ceded in making an infinite number of them. Also old objections which we supposcd long abandoned still reapwar in different forms. For example, the following recently appeared: ~lthough it may be possible to introduce a concept without risk, i.e., without get- ting contradictions, and even though one can prove that its intmd~ction no contradictions to arise, 。