Page 157 - Textos de Matemática Vol. 40

A note on the history of functional self-application 145
Ein solcher Standpunkt ist natu¨rlich unhaltbar. Wenn wir einen ma- thematischen Beweis erst am Resultate auf seine Zula¨ssigkeit pru¨fen ko¨nnen, so brauchen wir u¨berhaupt keinen Beweis.42
Moreover, to consider Hilbert’s paradoxical function f as a reference to the common use of diagonalization would turn the argumentation upside down. Rather than as a formal reason, f can only serve as a form of abstraction from the other paradoxes. But this is not in accordance with the given text.
After all, the alleged foundational role of the paradoxical function f can not be kept to.
Actually, the paragraph is bracketed by pencil and marked with a dele- tion sign in the official copy kept in the library of the Mathematical Institute in G¨ottingen, [Hil05a]. This indicates that Hilbert did not use it any longer in later presentations of the topic. There exist, for example, notes for a lec- ture in the winter term of 1914/15 for a lecture course Probleme und Prinzip- ien der Mathematik (Problems and Principles of Mathematics) [Hil15] in which Hilbert explicitly refers to the Chapter on paradoxes of the 1905 lecture: “Kol- legheft Logische Principien 1905 S. 191–214.”, [Hil15]. Since we know from an- other copy of the lecture of 1905 [Hil05b] that Hilbert indeed presented the paragraph in question in 1905, we can conclude that the marginals in the copy [Hil05a] are only added during a later reuse of the material.
We would like to stress that the mathematical argument presented in the paragraph is completely correct. It shows (only) that the paradoxical function cannot be well-defined—and this is all Hilbert claims mathematically. How- ever, the function does not really serve as a general base for (all) other paradoxes. Therefore, it might be considered a single occurrence of self-application without further impact.
But, in the following, we would like to point out two striking coincidences, one historical and one mathematical: The first one is the role of G¨ottingen as the working place of the main protagonists of self-application; the second one is the similarity of Hilbert’s argumentation with modern undecidability results.
A.5 G¨ottingen
Since the times of Carl Friedrich Gauß (1777–1855) the University of G¨ottingen was a top institution in Mathematics. It maintained its position with mathematicians like Dirichlet, Bernhard Riemann (1826–1866) and Felix Klein (1849–1925), reaching its highest point as an international cen- ter of the mathematical community during Hilbert’s time. After 1917, when
42“Thus, at one time two mathematical operations [addition (union) and self-mapping (power sets)] shall be admissible, because they do not result in a contradiction [in the genera- tion of the second number class]; but at another time two such processes shall be inadmissible, because a [Hilbert’s] paradox is deduced, and only the success shall decide what is forbidden and what not. [. . . ] Such a position is of course indefensible. If we can check a mathematical proof for its admissibility only by the result, then we need no proof at all.”