Editing Logicism (section)

===Criticism===
'''The presumption of an 'extralogical' notion of iteration''': Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)

Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, "a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be an elementary ''structural concept''  . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of "logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).

Hilbert 1931:266-7, like Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive ''a priori'' mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The ''a priori'' is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain ''extra-logical concrete objects'' that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).

In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an ''a priori'' notion that lies outside symbolic logic.

Hilbert dismissed logicism as a "false path": "Some tried to define the  numbers purely logically; others simply took the usual [[number theory|number-theoretic]] modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267).  The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.

Mancosu states that Brouwer concluded that: "the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation]; they are derived from the post factum record of mathematical constructions . . . Theoretical logic  . . . [is] an empirical science and an application of mathematics" (Brouwer quoted by Mancosu 1998:9).

With respect to the ''technical'' aspects of Russellian logicism as it appears in ''Principia Mathematica'' (either edition), Gödel in 1944 was disappointed:
:"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foundations (contained in *1–*21 of ''Principia'') that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944 ''Collected Works'' 1990:120).

In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their ''definiens''" (Russell 1944:120)

With respect to the philosophy that might underlie these foundations, Gödel considered Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) – to be faulty. See more in "Gödel's criticism and suggestions" below.

A complicated theory of relations continued to strangle Russell's explanatory 1919 ''Introduction to Mathematical Philosophy'' and his 1927 second edition of ''Principia''. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets. [[Grattan-Guinness]] observes that in the second edition of ''Principia'' Russell ignored this reduction that had been achieved by his own student [[Norbert Wiener]] (1914). Perhaps because of "residual annoyance, Russell did not react at all".<ref>Russell deemed Wiener "the infant phenomenon . . . more infant than phenomenon"; see ''Russell's confrontation with Wiener'' in Grattan-Guinness 2000:419ff.</ref> By 1914 [[Felix Hausdorff|Hausdorff]] would provide another, equivalent definition, and [[Kuratowski]] in 1921 would provide [[Kuratowski ordered pair|the one in use today]].<ref>See van Heijenoort's commentary and Norbert Wiener's 1914 ''A simplification of the logic of relations'' in van Heijenoort 1967:224ff.</ref>