[1] Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.
However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor's 1896, Zermelo and Russell's 1900–1901).
Since he treated the subject of primitive notions in geometry and set theory as well as the calculus of relations, this text is a watershed in the development of logicism.
Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with Henkin semantics have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought.
Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of infinity as part of logic.
[4] Logicism – especially through the influence of Frege on Russell and Wittgenstein[5] and later Dummett – was a significant contributor to the development of analytic philosophy during the twentieth century.
Ivor Grattan-Guinness states that the French word 'Logistique' was "introduced by Couturat and others at the 1904 International Congress of Philosophy, and was used by Russell and others from then on, in versions appropriate for various languages."
[6] 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).
Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"—the notions "all", "some", "class" (collection, aggregate) and "relation".
But "It was only [Russell's] Principia Mathematica that full use was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms.
Frege 1879 describes his intent in the Preface to his 1879 Begriffsschrift: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?
He believed that in the "foundations of the simplest science; viz., that part of logic which deals with the theory of numbers" had not been properly argued – "nothing capable of proof ought to be accepted without proof": Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic: Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics: The epistemologies of Dedekind and of Frege seem less well-defined than that of Russell, but both seem accepting as a priori the customary "laws of thought" concerning simple propositional statements (usually of belief); these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g. x R y) between individuals x and y linked by the generalization R. Dedekind's argument begins with "1.
Hilbert called Kronecker a "dogmatist, to the extent that he accepts the integer with its essential properties as a dogma and does not look back"[8] and equated his extreme constructivist stance with that of Brouwer's intuitionism, accusing both of "subjectivism": "It is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism".
To say that logic and arithmetic are contributed by us does not account for this" (1912:87); Russell concludes that the a priori knowledge that we possess is "about things, and not merely about thoughts" (1912:89).
In two Appendices added at the last minute he devoted 28 pages to both a detailed analysis of Frege's theory contrasted against his own, and a fix for the paradox.
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" .
The Principia, like its forerunner the Grundgesetze, begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("equinumerosity": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)".
This is a proposition, an assertion of the speaker's belief, which is to be tested against the "facts" of the outer world: "Minds do not create truth or falsehood.
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.
(1919:35) 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.
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.
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 .
With respect to the technical aspects of Russellian logicism as it appears in Principia Mathematica (either edition), Gödel in 1944 was disappointed: 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 .
A complicated theory of relations continued to strangle Russell's explanatory 1919 Introduction to Mathematical Philosophy and his 1927 second edition of Principia.
In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle": To illustrate what a pernicious instance of impredicativity might be, consider the consequence of inputting argument α into the function f with output ω = 1−α.
[28] A careful reading of the first edition indicates that an nth order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions.
Gödel concluded that it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (all quotes and content derived from page 135).
Russell's problem derives from his "constructivistic (or nominalistic"[29]) standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions .
[32] Proponents of neo-Fregeanism include Crispin Wright and Bob Hale, sometimes also called the Scottish School or abstractionist Platonism,[33] who espouse a form of epistemic foundationalism.
In this kind of amendment to the Grundgesetze, BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's NF and related systems.