Russell’s Paradox: a historical study about the paradox in Frege’s theories

For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work wasdone, Bertrand Russell sent him a letter pointing out a paradox, known as Russell‟s paradox. It is often considered that Russell identified the paradox in Frege‟s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributedsignificantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor‟s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege‟s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell‟s paradox.

Russell's Paradox: A Historical Study about the Paradox in Frege's Theories

For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work was done, Bertrand Russell sent him a letter pointing out a paradox, known as Russell’s paradox. It is often considered that Russell identified the paradox in Frege’s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributed significantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor’s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege’s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell’s paradox.

Generality and Objectivity in Frege’s Foundations of Arithmetic

This chapter argues for two principal contentions, both of which mark points of divergence from the neo-Fregean position first developed in Crispin Wright’s monograph Frege’s Conception of Numbers as Objects, and developed further in an extended series of works by Wright and Bob Hale. First, that Frege can be regarded as addressing the apriority of arithmetic in a manner that is independent of the ideas that numbers are logical objects or that arithmetic is analytic or a part of logic. Second, that Frege can secure the objectivity of arithmetic in a way that is independent of the idea that numbers are logical objects.

Pasch’s Empiricism as Methodological Structuralism

This chapter presents Pasch’s structuralist methodology within his empiricist philosophy. Two criteria for a minimal version of methodological structuralism are proposed, and it is argued that they are met in Pasch’s work on projective geometry and the foundations of arithmetic, despite the fact that Pasch firmly held an empiricist standpoint according to which only empirical objects can ultimately serve as a foundation for mathematics. What drove Pasch toward his version of structuralism were his focus on rigorous deductions, the duality of projective geometry, and the independence of arithmetical results from the referents of numerals.

Is incompatibilism compatible with Fregeanism?

This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can both be upheld simultaneously. Both views are attractive on their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the problem of epistemic access to numbers. For an incompatibilist, proofs of numerical non-identities must appeal to primitive incompatibilities. I argue that no analytic primitive incompatibilities are forthcoming. Hence incompatibilists cannot be Fregeans.

Semantic Typology and Composition

How many types of expression meaning are there, and are some types more basic than others? According to a familiar tripartite proposal, languages like English generate (i) denoters of a basic type <e>; (ii) truth-evaluable sentences of a basic type <t>; and (iii) expressions of nonbasic types that are characterized recursively: if <A> and <B> are types, so is <A,B>; where expressions of type <A,B> signify functions, from things of the sort signified with expressions of type <A> to things of the sort signified with expressions of type <B>. On this view, human languages are importantly like the language that Frege invented to study the foundations of arithmetic. In this chapter it is argued that each third of the tripartite proposal is wrong. An alternative is then sketched according to which there are exactly two semantic types, corresponding to monadic and dyadic concepts.