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in the introduction to the Mengenlehrebericht [Sch13] explicitly states that the development of set theory is linked to the notion of function: “Die Entwicke- lung der Mengenlehre hat ihre Quelle in dem Bestreben, fu¨r zwei grundlegende mathematische Begriffe eine kla¨rende Analyse zu schaffen, na¨mlich fu¨r die Be- griffe des Arguments und der Funktion.”17 And, as a matter of fact, set theory mastered this challenge.
However, there is an alternative view. Functions can be considered as primary objects mapping an argument (or several) to its value.18 This seems to come closer to the general view of a function one hundred years ago when set theory was still premature and far from being accepted as the foundational framework of mathematics.
Let us consider in such an approach functions like the following: f1(x) = 0 and f2(x) = x. In this context we can pose the question of self-application: f1(f1) seems to work fine, since f1 always returns 0, independently from the ar- gument. Also, f2(f2) seems to be a meaningful case of self-application, returning f2 as result.
Self-application, however, was discredited, first of all, by Russell who tried to block his paradox19 by imposing the Vicious Circle principle which pro- hibits (functional) self-application, cf. the discussion in the Principia [WR25–27, Vol. I, p. 62f].20
In spite of this negative attitude towards self-application, it turned out that it is possible to define meaningful and fruitful theories allowing it. This was first carried out in 1920 by Scho¨nfinkel in his seminal paper Bausteine der Mathematischen Logik [Sch24] (On the Building Blocks of Mathematical Logic [Sch67]). In this paper the identity function (f2 above) is introduced as I x = x and the possibility of self-application is explicitely stated: “[...] d.h. daß der Funktionswert I x stets derselbe ist wie der Argumentwert x, was man auch fu¨r x einsetzen mag. (So w¨are z. B. I I = I.)” [Sch24, p. 309]21. And, later on, Sch¨onfinkel makes substantial use of self-applications. By the work of Haskell Curry (1900–1982) this approach became popular under the name
in the mathematical development of this notion.
17“The development of set theory has its source in the endeavor to provide a clear analysis
for two fundamental mathematical notions, namely the notions of argument and of function.” 18Nowadays, this view is best represented in the λ-notation for functions, using, for instance, λx.x + 2 as an expression for the function f(x) = x + 2. A brief overview of notations for function abstraction of Giuseppe Peano (1858–1932), Frege, as well as Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947) is given in [CH0x, §2]. For the
origin of the λ-notation in Whitehead and Russell’s “caret” notation see [Ros84, p. 338]. 19His very first presentation of the paradox in the Principles of Mathematics, [Rus03, §101], does not involve functions; in §102 the paradox is discussed in relation to propositional func-
tions, using a form of diagonalization rather than self-application.
20A strict ban of functional self-application was also imposed by Ludwig Wittgenstein
(1889–1951) in the Tractatus, cf. [Wit21, 3.333].
21“[...] that is, that the function value I x is always the same as the argument value x,
whatever we may substitute for x. (Thus, for instance, I I would be equal to I.)” [Sch67, p. 360].