Article ID: CBB837308592

Hobson’s Conception of Definable Numbers (2020)

unapi

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.

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Authors & Contributors
Grattan-Guinness, Ivor
Gray, Jeremy
Parshall, Karen V. Hunger
Yap, Audrey
Ricketts, Tom
Potter, Michael D.
Journals
History and Philosophy of Logic
Historia Mathematica
Studies in History and Philosophy of Science
Revue d'Histoire des Sciences
Archive for History of Exact Sciences
British Journal for the History of Philosophy
Publishers
Princeton University Press
American Mathematical Society
Cambridge University Press
Southern Illinois University at Carbondale
Concepts
Philosophy of mathematics
Number theory; number concept
Mathematics
Logic
Philosophy of science
Philosophy of science, as a discipline
People
Poincaré, Jules Henri
Frege, Gottlob
Cantor, Georg Ferdinand Ludwig
Peano, Giuseppe
Russell, Bertrand Arthur William
Gödel, Kurt
Places
France
Germany
Times
19th century
20th century, early
18th century
20th century
21st century
Institutions
University of Chicago
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