Epistemology Behind Logicism
TBD:
Dedekind and Frege: The epistemology of Dedekind and Frege is not as well-defined as that of the philosopher 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" rebels against 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 thougt concerning it" (p. 44). In a subsequent paragraph Dedekind is discusses what a "system S is: it is an aggregate, a manifold, a totality of associated elements (things) a, b, c; 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" had his foes, among them the formidable 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" and equate 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". Hilbert then baldly states that "mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . ."(p. 479).
Russell the realist: Russell's Realism served him as an antidote to British Idealism, with portions borrowed from European Rationalism and British empiricism. 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, while empiricism would contribute the role of experiential knowledge (induction from experience). Russell would credit Kant with the idea of "a priori" knowledge, but he offers an objection to Kant he deems "fatal": "The facts 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)
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 and the paradox: In 1902 Russell discovered of a "vicious circle" (the so-called Russell's paradox) in Frege's Begriffsschrift and he was determined not to repeat it in his 1903 Principles of Mathematics. In two Appendices that he tacked on at the last minute he devotes 28 pages to a detailed analysis of, first Frege's theory contrasted against his own, and secondly a fix for the paradox. Unfortunately 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 (" 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" . . . only that we have no direct perception of them." (Gödel 1944:120)
In an observation pertainent 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).
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.
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