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Blanchette, Patricia, , . Frege and Hilbert on Consistency
1996, Journal of Philosophy 93 (7):317
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Added by: Clotilde Torregrossa, Contributed by: Alex Yates

Abstract: Gottlob Frege’s work in logic and the foundations of mathemat- ics centers on claims of logical entailment; most important among these is the claim that arithmetical truths are entailed by purely logical principles. Occupying a less central but nonetheless important role in Frege’s work are claims about failures of entailment. Here, the clearest examples are his theses that the truths of geometry are not entailed by the truths of logic or of arithmetic, and that some of them are not entailed by each other. As he, and we, would put it: the truths of Eluclidean geometry are independent of the truths of logic, and some of them are independent of one another.’ Frege’s talk of independence and related notions sounds familiar to a modern ear: a proposition is independent of a collection of propositions just in case it is not a consequence of that collection, and a proposition or collection of propositions is consistent just in case no contradiction is a consequence of it. But some of Frege’s views and procedures are decidedly tinmodern. Despite developing an extremely sophisticated apparattus for demonstrating that one claim is a consequience of others, Frege offers not a single demon- stration that one claim is not a conseqtuence of others. Thus, in par- tictular, he gives no proofs of independence or of consistency. This is no accident. Despite his firm commitment to the independence and consistency claims just mentioned, Frege holds that independence and consistency cannot systematically be demonstrated.2 Frege’s view here is particularly striking in light of the fact that his contemporaries had a fruitful and systematic method for proving consistency and independence, a method which was well known to him. One of the clearest applications of this method in Frege’s day came in David Hilbert’s 1899 Foundations of Geometry,3 in which he es- tablishes via essentially our own modern method the consistency and independence of various axioms and axiom systems for Euclidean geometry. Frege’s reaction to Hilbert’s work was that it was simply a failure: that its central methods were incapable of demonstrating consistency and independence, and that its usefulness in the founda- tions of mathematics was highly questionable.4 Regarding the general usefulness of the method, it is clear that Frege was wrong; the last one hundred years of work in logic and mathemat- ics gives ample evidence of the fruitfulness of those techniques which grow directly from the Hilbert-style approach. The standard view today is that Frege was also wrong in his claim that Hilbert’s methods fail to demonstrate consistency and independence. The view would seem to be that Frege largely missed Hilbert’s point, and that a better under- standing of Hilbert’s techniques would have revealed to Frege their success. Despite Frege’s historic role as the founder of the methods we now use to demonstrate positive consequence-results, he simply failed, on this account, to understand the ways in which Hilbert’s methods could be used to demonstrate negative consequence-results. The purpose of this paper is to question this account of the Frege- Hilbert disagreement. By 1899, Frege had a well-developed view of log- ical consequence, consistency, and independence, a view which was central to his foundational work in arithmetic and to the epistemologi- cal significance of that work. Given this understanding of the logical relations, I shall argue, Hilbert’s demonstrations do fail. Successful as they were in demonstrating significant metatheoretic results, Hilbert’s proofs do not establish the consistency and independence, in Frege’s sense, of geometrical axioms. This point is important, I think, both for an understanding of the basis of Frege’s epistemological claims about mathematics, and for an understanding of just how different Frege’s conception of logic is from the modern model-theoretic conception that has grown out of the Hilbert-style approach to consistency.

Comment: Good for a historically-based course on philosophy of logic or mathematics.
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Blanchette, Patricia, , . Frege’s Conception of Logic
2012, New York: Oxford University Press.
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Added by: Clotilde Torregrossa, Contributed by: Alex Yates

Publisher’s Note: In Frege’s Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege’s understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation.

The first part of the book locates the role of conceptual analysis in Frege’s logicist project. Blanchette argues that despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals.

In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

Comment: This book would be a suitable resource for independent study, or for a historically oriented course on philosophy of logic, of math, or on early analytic philosophy, especially one which looks at philosophical approaches to axiomatic systems.

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Jeshion, Robin, , . Frege’s Notions of Self-Evidence
2001, Mind 110 (440):937-976
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Added by: Simon Fokt, Contributed by: Alexander Yates

Abstract: Controversy remains over exactly why Frege aimed to estabish logicism. In this essay, I argue that the most influential interpretations of Frege’s motivations fall short because they misunderstand or neglect Frege’s claims that axioms must be self-evident. I offer an interpretation of his appeals to self-evidence and attempt to show that they reveal a previously overlooked motivation for establishing logicism, one which has roots in the Euclidean rationalist tradition. More specifically, my view is that Frege had two notions of self-evidence. One notion is that of a truth being foundationally secure, yet not grounded on any other truth. The second notion is that of a truth that requires only clearly grasping its content for rational, a priori justified recognition of its truth. The overarching thesis I develop is that Frege required that axioms be self-evident in both senses, and he relied on judging propositions to be self-evident as part of his fallibilist method for identifying a foundation of arithmetic. Consequently, we must recognize both notions in order to understand how Frege construes ultimate foundational proofs, his methodology for discovering and identifying such proofs, and why he thought the propositions of arithmetic required proof.

Comment: A nice discussion of what sort of epistemic status Frege thought axioms needed to have. A nice historical example of foundationalist epistemology – good for a course on Frege or analytic philosophy more generally, or as further reading in a course on epistemology, to give students a historical example of certain epistemological subtleties.

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