Page 149 - Textos de Matemática Vol. 40

Appendix
A note on the history of functional self-application
The introduction of functional self-application is usually attributed to Moses Scho¨nfinkel (1889–1942), [Sch24]15. Here we will discuss a paragraph of un- published lecture notes of David Hilbert (1862–1943) from 1905 which con- tains an example of functional self-application used to explain the set-theoretic paradoxes. Thus, this first use of self-application appears 15 years before Scho¨nfinkel presented his work.
Although Hilbert’s argument did not contribute to the discussion of the paradoxes, we can argue that it might have inspired later work on self-application. Besides this, it is of interest in the present context that the argument al- ready contains the core of modern undecidability results. Thereby, it shows the fruitfulness of self-application and gives a justification why theories with self-application have to be considered in logic.
A.1 Self-application
Nowadays a mathematical function is usually identified with its graph. For- mally, a function can be considered as a set of tuples of objects, where each tuple contains a list of arguments and the corresponding value which the func- tion returns for these arguments. From a foundational point of view, functions are therefore set-theoretical objects. Looking at the history of the notion of function in mathematics, it can be traced back from its introduction by Got- tfried Wilhelm Leibniz (1646–1716) via Leonhard Euler (1707–1783) to Peter Gustav Lejeune Dirichlet (1805–1859) (see, e.g., [Thi95a]). How- ever, an abstract definition of the notion of function was established only in terms of set theory.16 It is of interest that Artur Schoenflies (1853–1928)
This appendix is a preliminary version of the paper [Kah0x].
15The publication of 1924 goes back to a talk given on Dec 7th, 1920 in the Mathematische
Gesellschaft in G¨ottingen, cf. [Sch24, footnote 1, p. 305].
16As for other elementary notions in mathematics, it seems to have been Gottlob Frege
(1848–1925) who first addressed the question of the nature of functions, see Function und Begriff [Fre91] (english translation in [Fre52]), even if at first his analysis was not considered
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