Logicism Today

Program

Tuesday 14 june — Session 1

15.00-16.15
P. Blanchette
Conceptual Analysis and Logicism in Frege

16.30-17.45
S. Valentin Costreie
Frege on contentful arithmetic

Wednesday 15 june — Session 2

9.00-10.15
M. Wilson
Enlarging Ones Stall: How did all of these Sets Get in Here?

10.30-11.45
S. Walsh
TBA

Wednesday 15 juin — Session 3

14.00-15.15
F. Boccuni, M. Carrara and E. Martino
Logicism without Numbers

15.30-16-45
B. Hale
Properties

Thursday 16 june — Session 4

8.45-10.00
O. Linnebo
Grounded abstraction

C. Wright
10.15-11.30
TBA

M. Panza
11.45-13.00
From Natural to real numbers through abstraction

Friday 17 june — Session 5

9.00-10.15
S. Methven
Justification and Logical Truth: Ramsey and the Axiom of Infinity

10.30-11.45
R. Trueman
Propositional Functions in Extension and Abstraction Principles

Friday 17 june — Session 6

14.00-15.15
G. Landini
What is Logic that logicism may be true?

15.30-16.45
S. Gandon/B. Halimi
Structured variable

Abstracts

M. Wilson
Enlarging Ones Stall: How did all of these Sets Get in Here?

In this talk, I will survey two rather different pathways whereby sets came to play a central role within nineteenth century mathematics. The first centers upon the problem, tracing to Euler and Cauchy, of supplying the solutions to differential equations with a proper autonomy and the second traces to attempts to tame the ‘extension element’ problems of geometry and algebra in Dedekind's and Frege’s manners.

O. Linnebo
Grounded abstraction

One of the most serious problems confronting the abstractionist programme is that of distinguishing between acceptable and unacceptable forms of abstraction. After briefly explaining my dissatisfaction with existing approaches to this problem, I explore whether some notion of groundedness may enable us to do better. I canvass three approaches to grounded abstraction, exploring both their philosophical plausibility and their technical merits.

S. Valentin Costreie
Frege on contentful arithmetic

The paper aims to analyze two prima facie disconnected Fregean points: the sense/reference distinction and his view concerning the foundations of logic and mathematics. The main claim is that Frege has introduced his celebrated semantic distinction to endorse his fundamental thesis that mathematics is contentful. Fregean senses are not just the mere result of a mere linguistic analysis, rather they play an important role in securing Frege’s foundational program. Thus, the Fregean thesis that mathematics is contentful, and its defense against formalism and psychologism, provide us the right key to see and understand his works as an organic whole.

M. Panza
From Natural to real numbers through abstraction

An open problem in the neologicist approach is that of providing a definition of real numbers depending on an appropriate system of abstraction principles. It is well known that real numbers can be easily defined using second order arithmetic. Taking Frege Arithmetic (FA) as a basis, it is then possible to define reals quite easily. In my talk, I will show how this can be done by adding a single abstraction principle along with some appropriate explicit definitions. This is not a Fregean definition, since Frege believed that defining reals on the basis of natural numbers (or of rational numbers) would have been contrary to the application constraint, and thus inappropriate. Moreover, it is quite doubtful that such a definition is able to provide a logical foundation of analysis, in Frege's sense. Still, it makes it possible, with a small amount of basic resources, to get the theory of real numbers as an elegant extension of FA.

B. Hale
Properties

According to the abundant or deflationary conception of properties, every meaningful predicate stands for a property or relation, and it is sufficient for the actual existence of a property or relation that there could be a predicate with appropriate satisfaction conditions. In the first part of this talk, I shall argue that purely general properties and relations exist as a matter of (absolute) necessity. In the second, I shall draw out some implications of this conception of properties for the interpretation of higher-order logic. Some of these implications have an obvious bearing on Quine’s charge that higher-order logic is ‘set theory in sheep’s clothing’ and the ingredient charge that it involves ‘staggering existential assumptions’. If we agree that abundant conception’s sufficient condition is also necessary, this approach suggests a clean break between logic and the rest of mathematics. On this view, the so-called ‘standard semantics’ for second-order logic involves a false assimilation of logic to set theory. I shall conclude with some remarks about the implications of my view for the programme of providing a foundation for mathematical theories in higher-order logic plus abstraction principles.

R. Trueman
Propositional Functions in Extension and Abstraction Principles

The Julius Caesar Problem is one of the most important challenges to neo-logicism, but the nature of that problem is still unclear. In my talk I will attempt to bring out one pressing form of this problem by comparing neo-logicism with Ramsey’s logicist project in ‘The Foundations of Mathematics’. By the end we will not only see connections between neo-logicism and Ramsey’s project, but also connections between the Julius Caesar Problem and other well known challenges to neo-logicism.