About the same time as Rudolf Carnap (1929), but apparently independently, Fraenkel (1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complained about its use in Carnap's manuscript so Carnap proposed the word 'Logizismus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502).
Kleene 1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and principles underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894–1908) in expressing mathematical theorems in a logical symbolism" (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).
Indeed he awaits Kronecker's "publishing his reasons for the necessity or merely the expediency of these limitations" (p. 45).
Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903 Principles of Mathematics. Note that he asserts that the belief: "Emily is a rabbit" is non-existent, and yet the truth of this non-existent proposition is independent of any knowing mind; if Emily really is a rabbit, the fact of this truth exists whether or not Russell or any other mind is alive or dead, and the relation of Emily to rabbit-hood is "ultimate" :
"Fictionalism" and Russell's no-class theory: Gödel in his 1944 would disagree with the young Russell of 1903 ("[my premisses] allow mathematics to be true") but would probably agree with Russell's statement quoted above ("something is amiss"); Russell's theory had failed to arrive at a satisfactory foundation of mathematics: the result was "essentially negative; i.e. the classes and concepts introduced this way do not have all the properties required for the use of mathematics" (Gödel 1944:132).
How did Russell arrive in this situation? Gödel observes that Russell is a surprising "realist" with a twist: he cites Russell's 1919:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . [meaning] only that we have no direct perception of them." (Gödel 1944:120)
In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism: extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase; by 1905 he would be in his moderate phase. In a few years he would "dispense with physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book Our knowledge of the External World " (Perry 1997:xxvi).
For Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (assertions of fact about a thing or things). Russell analysed this general notion. He begins with "terms" in sentences, which he analysed as follows:Things are indicated by proper names; concepts are indicated by adjectives or verbs
Concept-adjectives are "predicates"; concept-verbs are "relations": "The former kind will often be called predicates or class-concepts; the latter are always or almost always relations." (1903:44)
Classes (aggregates, complexes): "The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p. 55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "x is a u" or "x is v". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential." (1909 p. 66)
Propositional functions: "The characteristic of a class concept, as distinguished from terms in general, is that "x is a u" is a propositional function when, and only when, u is a class-concept." (1903:56)
Extensional versus intensional definition of a class: "71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal."(1903:69)
And in the second edition of PM (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.
Step 3: Define the null class: Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection.
Step 4: Assign a "numeral" to each bundle: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary.Observe in particular that Russell does not use the unit class of classes "1" to construct the successor
For his definition of successor, Russell will use for his "unit" a single entity or "term" as follows:
Russell's definition requires a new "term" which is "added into" the collections inside the bundles.
Step 8: For every class of equinumerous classes, create its successor.
Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of "asymmetry" i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, he defines the notion of "transitivity" for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of "connected": "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)
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).
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-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).
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.
This sort of definition of I was deemed by Poincaré to be "impredicative". He seems to have considered that only predicative definitions can be allowed in mathematics:
By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle":
To make the function "impredicative", identify the input with the output, yielding α = 1-α
Within the algebra of, say, rational numbers the equation is satisfied when α = 0.5. But within, for instance, a Boolean algebra, where only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied.
But, since the class with numeral 1 is a single object or unit in its own right, it too must be included in the class of unit classes. This inclusion results in an infinite regress of increasing type and increasing content.
Russell avoided this problem by declaring a class to be more or a "fiction". By this he meant that a class could designate only those elements that satisfied its propositional function and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage but is not in Russell's view "worthy of thing-hood":
This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.
But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Frege in his 1903, Russell concludes:
In the following notice the wording "the class as many"—a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a thing-in-itself:
The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom" (PM 1927:xiv).
Gödel 1944 agrees that Russell's logicist project was stymied; he seems to disagree that even the integers survived:
Gödel, in his 1944 work, identifies the place where he considers Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).
He concludes his essay with the following suggestions and observations: