Theory, due to Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970), that the concepts and theories of mathematics (in particular of arithmetic) can be derived from those of logic.

This, if feasible, would support logical positivism and reductionism in general.

Arithmetic was in fact reduced to set theory – developed by Georg Cantor (1845-1918) – as a first step, but set theory itself has never successfully been derived from pure logic, and the enterprise was frustrated by K Gddel’s (1906-1978) proof in 1931 that for any system rich enough to formalize arithmetic there will always be truths that can be stated in the system (and so form part of it) but cannot be proved within it.

S Korner, Philosophy of Mathematics (1960), chs 2, 3


Dedekind’s path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a “logic” of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.

The philosophical impetus behind Frege’s logicist programme from the Grundlagen der Arithmetik onwards was in part his dissatisfaction with the epistemological and ontological commitments of then-extant accounts of the natural numbers, and his conviction that Kant’s use of truths about the natural numbers as examples of synthetic a priori truth was incorrect.

This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an inconsistency in Frege’s system set out in the Grundgesetze der Arithmetik. Note that naive set theory also suffers from this difficulty.

On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano’s school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica.[2]

Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of Zermelo–Fraenkel set theory (or its extension ZFC), from which no inconsistencies have as yet been derived. 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.

Kurt Gödel’s incompleteness theorems show that no formal system from which the Peano axioms for the natural numbers may be derived — such as Russell’s systems in PM — can decide all the well-formed sentences of that system.[3] This result damaged Hilbert’s programme for foundations of mathematics whereby ‘infinitary’ theories — such as that of PM — were to be proved consistent from finitary theories, with the aim that those uneasy about ‘infinitary methods’ could be reassurred that their use should provably not result in the derivation of a contradiction. 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. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning ‘infinitary methods’. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel’s result.

One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are ‘proved with logic just like any other theorems’. However, that argument appears not to acknowledge the distinction between theorems of first-order logic and theorems of higher-order logic. The former can be proven using finistic methods, while the latter — in general — cannot. Tarski’s undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.[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.

Origin of the name ‘logicism’

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.” (G-G 2000:501).

Apparently the first (and only) usage by Russell appeared in his 1919: “Russell referred several time [sic] to Frege, introducing him as one ‘who first succeeded in “logicising” mathematics’ (p. 7). Apart from the misrepresentation (which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that ‘logicism’ did not emerge until the later 1920s” (G-G 2002:434).[6]

About the same time as 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).

Intent, or goal, of logicism

Symbolic logic: The overt intent of Logicism is to derive all of mathematics from symbolic logic (Frege, Dedekind, Peano, Russell.) As contrasted with algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a few “logical” axioms that embody the “laws of thought”, and rules of inference that dictate how the marks are to be assembled and manipulated — for instance substitution and modus ponens (ie from [1] A materially implies B and [2] A, one may derive B). 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”.

In a logicist derivation of the natural numbers and their properties, no “intuition” of number should “sneak in” either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the real numbers, from some chosen “laws of thought” alone, without any tacit assumptions of “before” and “after” or “less” and “more” or to the point: “successor” and “predecessor”. Gödel 1944 summarized Russell’s logicistic “constructions”, when compared to “constructions” in the foundational systems of Intuitionism and Formalism (“the Hilbert School”) as follows: “Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell’s constructivism” (Gödel 1944 in Collected Works 1990:119).

History: Gödel 1944 summarized the historical background from Leibniz’s in Characteristica universalis, through Frege and Peano to Russell: “Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic”, whereas Peano “was more interested in its applications within mathematics”. 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. In addition, the young science was enriched by a new instrument, the abstract theory of relations” (p. 120-121).

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).

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”?

“I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thought that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bend every effort to keep the chain of inferences free of gaps . . . I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed” (Frege 1879 in van Heijenoort 1967:5).

Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning of Numbers. 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”:

In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediate result from the laws of thought . . . numbers are free creations of the human mind . . . [and] only through the purely logical process of building up the science of numbers . . . are we prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind” (Dedekind 1887 Dover republication 1963 :31).

Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That is why it is of the utmost importance to examine attentively the very words we use. My goal has been to undertake this examination” (Peano 1889 in van Heijenoort 1967:85).

Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:

“THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles” (Preface 1903:vi).
“A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. . . . [From two questions — acceleration and absolute motion in a “relational theory of space”] I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic” (Preface 1903:vi-vii).

Epistemology, ontology and logicism

Dedekind and Frege: 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 “free formations of the human mind” in contrast to the “strictures” of Kronecker: Dedekind’s argument begins with “1. In what follows I understand by thing every object of our thought”; we humans use symbols to discuss these “things” of our minds; “A thing is completely determined by all that can be affirmed or thought concerning it” (p. 44). In a subsequent paragraph Dedekind discusses what a “system S is: it is an aggregate, a manifold, a totality of associated elements (things) abc“; he asserts that “such a system S . . . as an object of our thought is likewise a thing (1); it is completely determined when with respect to every thing it is determined whether it is an element of S or not.*” (p. 45, italics added). The * indicates a footnote where he states that:

“Kronecker not long ago (Crelle’s Journal, Vol. 99, pp. 334-336) has endeavored to impose certain limitations upon the free formation of concepts in mathematics which I do not believe to be justified” (p. 45).

Indeed he awaits Kronecker’s “publishing his reasons for the necessity or merely the expediency of these limitations” (p. 45).

Leopold Kronecker, famous for his assertion that “God made the integers, all else is the work of man”[7] had his foes, among them Hilbert. 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”.[9] Hilbert then states that “mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . .” (p. 479).

Russell as realist: Russell’s Realism served him as an antidote to British Idealism,[10] with portions borrowed from European Rationalism and British empiricism.[11] To begin with, “Russell was a realist about two key issues: universals and material objects” (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of a priori knowledge,[12] while empiricism would contribute the role of experiential knowledge (induction from experience).[13] Russell would credit Kant with the idea of “a priori” knowledge, but he offers an objection to Kant he deems “fatal”: “The facts [of the world] must always conform to logic and arithmetic. 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). And in this Russell’s epistemology seems different from that of Dedekind’s belief that “numbers are free creations of the human mind” (Dedekind 1887:31)[14]

But his epistemology about the innate (he prefers the word a priori when applied to logical principles, cf. 1912:74) is intricate. He would strongly, unambiguously express support for the Platonic “universals” (cf. 1912:91-118) and he would conclude that truth and falsity are “out there”; minds create beliefs and what makes a belief true is a fact, “and this fact does not (except in exceptional cases) involve the mind of the person who has the belief” (1912:130).

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” :

“On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. . . . Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour.” (Preface 1903:viii)

Russell’s paradox: In 1902 Russell discovered a “vicious circle” (Russell’s paradox) in Frege’s Grundgesetze der Arithmetik, derived from Frege’s Basic Law V and he was determined not to repeat it in his 1903 Principles of Mathematics. 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 was not optimistic about the outcome:

“In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)”

“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 [1914]” (Perry 1997:xxvi).

These constructions in what Gödel 1944 would call “nominalistic constructivism . . . which might better be called fictionalism” derived from Russell’s “more radical idea, the no-class theory” (p. 125):

“according to which classes or concepts never exist as real objects, and sentences containing these terms are meaningful only as they can be interpreted as . . . a manner of speaking about other things” (p. 125).

See more in the Criticism sections, below.

An example of a logicist construction of the natural numbers: Russell’s construction in the Principia

The logicism of Frege and Dedekind is similar to that of Russell, but with differences in the particulars (see Criticisms, below). Overall, the logicist derivations of the natural numbers are different from derivations from, for example, Zermelo’s axioms for set theory (‘Z’). Whereas, in derivations from Z, one definition of “number” uses an axiom of that system — the axiom of pairing — that leads to the definition of “ordered pair” — no overt number axiom exists in the various logicist axiom systems allowing the derivation of the natural numbers. Note that the axioms needed to derive the definition of a number may differ between axiom systems for set theory in any case. For instance, in ZF and ZFC, the axiom of pairing, and hence ultimately the notion of an ordered pair is derivable from the Axiom of Infinity and the Axiom of Replacement and is required in the definition of the Von Neumann numerals (but not the Zermelo numerals), whereas in NFU the Frege numerals may be derived in an analogous way to their derivation in the Grundgesetze.

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)”.[15] The logicistic derivation equates the cardinal numbers constructed this way to the natural numbers, and these numbers end up all of the same “type” — as classes of classes — whereas in some set theoretical constructions — for instance the von Neumman and the Zermelo numerals — each number has its predecessor as a subset. Kleene observes the following. (Kleene’s assumptions (1) and (2) state that 0 has property P and n+1 has property P whenever n has property P.)

“The viewpoint here is very different from that of [Kronecker]’s maxim that ‘God made the integers’ plus Peano’s axioms of number and mathematical induction], where we presupposed an intuitive conception of the natural number sequence, and elicited from it the principle that, whenever a particular property P of natural numbers is given such that (1) and (2), then any given natural number must have the property P.” (Kleene 1952:44).

The importance to the logicist programme of the construction of the natural numbers derives from Russell’s contention that “That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected” (1919:4). One derivation of the real numbers derives from the theory of Dedekind cuts on the rational numbers, rational numbers in turn being derived from the naturals. While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophical difficulties appear in a logicist derivation of the natural numbers, these problems should be sufficient to stop the program until these are resolved (see Criticisms, below).

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