Article ID: CBB858610340

What did Frege take Russell to have proved? (2021)


In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege.

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Authors & Contributors
Grattan-Guinness, Ivor
Beaney, Michael
Gandon, Sébastien
Künne, Wolfgang
Schirn, Matthias
Guay, Alexandre
History and Philosophy of Logic
Logica Universalis
Metascience: An International Review Journal for the History, Philosophy and Social Studies of Science
Revue d'Histoire des Sciences
The Review of Modern Logic
Princeton University Press
Cambridge University Press
Palgrave Macmillan
Vittorio Klostermann
Éd. universitaires de Dijon
Philosophy of mathematics
Philosophy of science
Set theory
Russell, Bertrand Arthur William
Frege, Gottlob
Cantor, Georg Ferdinand Ludwig
Gödel, Kurt
Dedekind, Richard
Hilbert, David
Time Periods
20th century, early
19th century
20th century, late
20th century
21st century
Great Britain
North America

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