Skolem and the löwenheim-skolem theorem: a case study of the philosophical significance of mathematical results
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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History and Philosophy of Logic 6 (1):75-89 (1985)
The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ?a puffed-up proof?. Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21.
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References found in this work BETA
Jean Van Heijenoort (ed.) (1967). From Frege to Gödel. Cambridge, Harvard University Press.
Hilary Putnam (1980). Models and Reality. Journal of Symbolic Logic 45 (3):464-482.
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Citations of this work BETA
Luca Bellotti (2006). Skolem, the Skolem 'Paradox' and Informal Mathematics. Theoria 72 (3):177-212.
Carsten Hansen (1987). Putnam's Indeterminacy Argument: The Skolemization of Absolutely Everything. Philosophical Studies 51 (1):77--99.
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