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Poincaré, Richard's Paradox and Indefinite Extensibility

Published online by Cambridge University Press:  28 February 2022

Peter Clark
Peter Clark*
Affiliation:
University of St. Andrews
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Extract

Sometimes themes exist only in the eye of the beholder and emerge from the history of mathematics only with the benefit (or otherwise) of hindsight. There is however at least one issue (no doubt there are others) which was clearly present throughout the foundational debates of the early Twentieth century. It formed the core component of one of the major traditions that grew up as responses to the paradoxes, though the intuition behind it has never been fully articulated nor has it been possible to give a fully fledged mathematical articulation to that intuition. Essentially the central idea is that of ‘indefinite extensibility’ generated by an appropriate form of diagonalisation and it has its fundamental source in Richard's paradox and in particular in Poincaré's response to that paradox.

Of the four great epistemological concerns around which much of the work in the foundations of mathematics this century has concentrated viz the notions of demonstrability, definability, set or class and computability only the latter clearly falls outside the scope of the Richard type reasoning.

Type
Part VII. Foundational Projects in Mathematics at the Beginning of the 20th Century in Their Systematic and Historical Contexts
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1994 , Issue 2: Volume Two: Symposia and Invited Papers 1994 , 1994 , pp. 227 - 235
Copyright
Copyright © 1995 by the Philosophy of Science Association

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