Ivor Grattan-Guinness
Logicism in the context of the development of set theory
Set theory began to be developed on a large scale from the late 1890s onwards; for example, it was part of the mathematical logic that grounded logicism, and for convenience much of Principia mathematica was elaborated in its terms. Several different parts and features of set theory became prominent, and logicism was supposed to embrace all of them. In practice, though, how many did it tackle...?
- Point set topology
- Applications to real-variable mathematical analysis (measure theory, functional analysis and integral equations)
- Applications to complex-variable mathematical analysis
- The axioms of choice and their consequences
- Transfinite arithmetic
- General set theory and the absolute \
- Order-types
- Topology, especially dimension theory
- Axiomatisation of set theory
- Relationship of set theory to logic
- Relationship of set theory to model theory.
Michael Potter
The Role of Classes in Principia
The axiom of reducibility in Principia can be thought of as a class existence axiom. Yet class terms are incomplete symbols which disappear on analysis. I shall discuss the interaction between these two features of the system and relate them to later criticisms of Principia by Wittgenstein and Ramsey.
James Levine
Russell on the Paradoxes: Uniform Structure? Uniform Solution?
Using J.L. Mackie’s distinction between “absolute” and “operational” self-refutation, I wll argue for recognizing a distinction in kind between the set-theoretic paradoxes Russell discusses in Principia Mathematica and the so-called “semantic” paradoxes he discusses there. In doing so, I will argue against Graham Priest’s suggestion that his “Inclosure Schema” reveals all these paradoxes as both exhibiting the same structure and requiring the same sort of solution.
Graham Stevens
Types, Kinds, and the Unity of Judgements
This paper defends an interpretation of Wittgenstein’s criticisms of Russell’s 1913 multiple relation theory of judgement. On the interpretation I defend, Wittgenstein showed Russell that the only way to secure the unity of judgement on the 1913 theory was to impose the theory of logical types onto his ontology. Russell’s refusal to take this route, preferring instead to abandon the theory of judgement, is taken as evidence for a linguistic interpretation of Principia Mathematica’s type-theory.
Peter Simons
Whitehead and Russell on Quantity: The Final Part of Principia Mathematica
Russell repeatedly complained that almost no-one read all of PM and very few people took note of the later parts. We take note of the last part published: Part VI: “Quantity”. This encompasses the W–R account of the real numbers and their subsystems, as well as the quantity families which the reals coordinate. Their investigation was but one of a host of studies of quantity undertaken around this time, by Frege, Burali-Forti, Hölder, and Huntington. Nevertheless the W–R approach is distinctive in two ways. Firstly, the theory is embedded in the already rich formal system of PM Parts I–V. And secondly, perhaps as a result, the specifically quantitative aspect is added in gentle doses to their existing logic of relations, rather than postulated in one go as by others. The result is subtle and embodies several advantages as a theory of quantity and the reals.
