Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

Bernard Linsky, The Evolution of Principia Mathematica; Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press. 2011. 407 pp. + two plates. $150.00/£...

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot (...) recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered. (shrink)

Leon Chwistek's 1924 paper ?The Theory of Constructive Types? is cited in the list of recent ?contributions to mathematical logic? in the second edition of Principia Mathematica, yet its prefatory criticisms of the no-classes theory have been seldom noticed. This paper presents a transcription of the relevant section of Chwistek's paper, comments on the significance of his arguments, and traces the reception of the paper. It is suggested that while Russell was aware of Chwistek's points, they were not important in (...) leading him to the adoption of extensionality that marks the second edition of PM. Rudolf Carnap seems to have independently rediscovered Chwistek's issue about the scope of class expressions in identity contexts in his Meaning and Necessity in 1947. (shrink)

This volume portrays the Polish or Lvov-Warsaw School, one of the most influential schools in analytic philosophy, which, as discussed in the thorough introduction, presented an alternative working picture of the unity of science.

It is generally acknowledged that the 1910 Principia does not deny the existence of classes, but claims only that the theory it advances can be developed so that any apparent commitment to them is eliminable by the method of contextual analysis. The application of contextual analysis to ontological questions is widely viewed as the central philosophical innovation of Russell’s theory of descriptions. Principia’s “no-classes theory of classes” is a striking example of such an application. The present paper develops a reconstruction (...) of Principia’s theory of functions and classes that is based on Russell’s epistemological applications of the method of contextual analysis. Such a reconstruction is not eliminativist—indeed, it explicitly assumes the existence of classes—and possesses certain advantages over the no–classes theory advocated by Whitehead and Russell. (shrink)