Logicism

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Logicism

 

a trend in the foundations of mathematics and the philosophy of mathematics whose fundamental thesis is the assertion of the “reducibility of mathematics to logic,” that is, the possibility (and necessity) of defining all primitive mathematical concepts (not definable within the framework of mathematics itself) in terms of “pure” logic and of proving all mathematical propositions, including the axioms, by logical methods.

The concepts of logicism were first advanced by G. W. Leibniz, but in its expanded form the doctrine was first formulated in the works of G. Frege, who proposed reducing the fundamental mathematical concept—that of the natural number—to the extensions of concepts. Frege also worked out in detail a logical system by means of which he was able to prove all the theorems of arithmetic. Since by that time in mathematics the work on reducing, in the same sense as above, the fundamental concepts of mathematical analysis, geometry, and algebra to arithmetic had been practically completed, by carrying out partial reductions of them to one another and by expressing their concepts in terms of set theory, Frege believed that the program of logicism had thereby been basically carried out.

However, even before the publication of Frege’s two-volume work Grundgesetze der Arithmetik (The Fundamental Laws of Arithmetic; 1893–1903), B. Russell had discovered a contradiction (now usually called Russell’s paradox) in Frege’s system. Russell himself, however, shared the fundamental theses of the program of logicism. He made an attempt to “remedy” Frege’s system and to “rescue” it from contradictions. The solution of this problem required much work on the consistent and detailed formalization not only of mathematics but of the logic that lay at its foundation (according to the program of logicism). The result of this work was the three-volume Principia Mathematica (1910–13) written by Russell together with A. N. Whitehead.

The chief novelty in the Russell-Whitehead system (called the PM system below) was the construction of logic in the form of a “stage-by-stage” calculus,” or “theory of types.” The formal objects of this theory were divided into types (stages), and this “hierarchy of types” (in other modifications of the PM system, an additional “hierarchy of levels”) made possible the elimination of all known paradoxes. However, in order to construct classical mathematics by means of the PM system, it was necessary to add to it certain axioms that intuitively characterize the important properties of the given concrete “world of mathematics” (and, of course, the world of real things corresponding to it) and that are not in any way “analytical truths,” or, in the sense of Leibniz, true “in all possible worlds.” Thus, not all of Russellian mathematics is derivable from logic. Furthermore, this mathematics does not constitute all of mathematics: as was shown by K. Godel in 1931, PM-type systems, as well as all systems as powerful as it is, are essentially incomplete, that is, it is always possible to formulate by their means intuitively true but undecidable (neither provable nor disprovable) mathematical assertions.

Thus, the program of logicism for a “purely logical” foundation of mathematics proved to be impracticable. Nevertheless, Russell’s results as well as the work of other scientists who later proposed various improvements of the PM system (such as the American mathematician W. V. O. Quine) have exerted an enormous positive influence on the development of mathematical logic and of science as a whole because they facilitate the formation and refinement of the most important logicomathematical and general methodological concepts and the construction of a corresponding exact mathematical apparatus.

REFERENCES

Kleene, S. C. Vvedenie ν metamatematiku. Moscow, 1957. Chapter 3. (Translated from English.)
Fraenkel, A., and Y. Bar-Hillel. Osnovaniia teorii mnozhestv. Moscow, 1966. Chapter 3. (Translated from English.)

IU. A. GASTEV

References in periodicals archive ?
or logicists, expect.., to wring from these axioms ...
Thus, the main logicist contention is that either all the truths of arithmetic or a core set of arithmetic truths are true in virtue of the meaning of certain logical terms alone.
For present purposes, we assume that full, standard second-order logic and even full type theory (as in Martin-Lof 1984, for example) is logic and so is available to a prospective logicist. In particular, we presuppose a language with a "ground-type" of variables ranging over ordinary objects like cars, cats, and continents, as well as variables ranging over properties (or sets) and relations on ordinary objects, and possibly also variables ranging over properties of properties, etc.
If someone accepts the logicist thesis (Basic Truths) and takes the language of arithmetic at face value, then she will hold that the existence of numbers is a logical truth, and is thus analytic.
It seems clear that the sentence 0 [not equal to] 0 should be a basic truth of arithmetic, and so a theorem in any logicist system.
Since (Two) is a logical consequence of 0 [not equal to] 0, then according to our logicist (Two) is itself analytic and logically true, provided that analyticity and logical truth are closed under logical consequence, or at least the introduction rule for the first-order existential quantifier.
A logicist might retort that a structure whose domain has only one element is not a logically possible model, since it does not contain the necessarily existing natural numbers.
The latter, of course, is held by any Fregean logicist, Tennant in particular.
So if our logicist restricts groundtype identity to avoid the violation of separability, then he must restrict the second-order quantifiers and variables as well.
Thus, a logicist who takes this rescue cannot adopt unrestricted second-order logic, or at least he cannot have variables ranging over all ground-type sorts.
Our logicist can maintain a single ground-type with only one first-order sort, but locate numbers elsewhere in the typehierarchy.
However, none of the variations on this rescue-theme succeed in resolving the dogma of existence in favor of the logicist. Frege, Wright, Hale, and Tennant all develop the arithmetic of natural numbers in terms of identity at the ground-type and first-order quantifiers.