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What is Neologicism?

Published online by Cambridge University Press:  15 January 2014

Bernard Linsky  and
Edward N. Zalta
Bernard Linsky
Affiliation:
Department of Philosophy, University of Alberta, Edmonton, Alberta T6G 2E5, Canada. E-mail: [email protected]
Edward N. Zalta
Affiliation:
CSLI, Ventura Hall, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected]
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Extract

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 1 , March 2006 , pp. 60 - 99
Copyright
Copyright © Association for Symbolic Logic 2006

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