Title (eng)

Nonstandard Logicism

Abstract (eng)

Logicism was a dominant position in the foundations of mathematics of the late nineteenth and
early twentieth century. Roughly put, it is the view that pure mathematics is reducible to higherorder logic. More specifically, the logicist thesis is usually taken to consist of two claims. First, all primitive terms of an axiomatized mathematical theory can be explicitly defined by using only logical vocabulary. Second, all axioms of the theory can be deduced from purely logical principles.
It follows from these two claims that all theorems of a mathematical theory are also derivable from purely logical principles. Let us call this the classical or standard logicist thesis.
It is well known that the pioneering logicists Frege, Russell, Whitehead as well as subsequent philosophers such as Ramsey or Carnap defended variants of this view. However, the contributions of second generation logicists often differed from classical logicism in important respects, in particular concerning (i) the mathematical theories considered, (ii) the logical principles adopted, and (iii) the very concept of a logicist reduction. Thus, based on different accounts of what is meant by “logic,” “mathematics,” and “reducible,” one can identify a number of nonstandard theories of logicism developed in the 1920s and later on.
Logicism should thus not be viewed as a monolithic research program, but rather as a family of different approaches on how the general project of reducing mathematics to logic can be made precise. The focus of this entry will be on different theories of logicism developed in the heyday of logical empiricism, that is, roughly between 1920 and 1940. The central aim here is to survey how Frege’s and Russell’s logicist programs were modified in the period in question. The changes concern not only formal details of the underlying logic such as the adoption of a simple theory of logical types, but also the kind of mathematical theories considered for the reduction to logic.
Whereas classical logicism focused mainly on arithmetic, logical empiricists such as Carnap and Hahn were interested in a generalized logicist thesis which is applicable to any axiomatic theory of pure mathematics.

Keywords (eng)

Mathematical Philosophy, Logicism, Philosophy

Type (eng)

Language

[eng]

Persistent identifier

https://phaidra.univie.ac.at/o:1427949

Is contained in

Title

The Routledge Handbook of Logical Empiricism

ISBN

978-1138122000

Editor

Thomas Uebel

Christoph Limbeck-Lilienau

Publication

Routledge , 2021-12-28

From page

1

To page

424

Version type (eng)

Date available

2022-12-28

Date accepted

2021-03-17

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