Papers by Patricia Blanchette
Formalism and Beyond, 2014
Dame. Her research centers on the history and philosophy of logic and mathematics, and the histor... more Dame. Her research centers on the history and philosophy of logic and mathematics, and the history of analytic philosophy. Recent work includes Frege's Conception of Logic, Oxford University Press 2012.
Similar remarks appear in Russell’s Principles of Mathematics, in Principia Mathematica, and in t... more Similar remarks appear in Russell’s Principles of Mathematics, in Principia Mathematica, and in the 1906 “The Theory of Implication.” The sentiment expressed in these passages is a strange one, especially given its setting in the first decade of the twentieth century. The modern method of demonstrating independence had become, by this point, standard fare, having been applied already in geometry, arithmetic, analysis, and class theory. And despite Russell’s concerns, the method was soon to be applied (by Bernays) to demonstrate the independence of Russell’s own fundamental principles of logic as expressed in Principia Mathematica.
Origins and Varieties of Logicism, 2021
Philosophia Mathematica, 2018
Canadian Journal of Philosophy, 1999
Peter Geach famously holds that there is no such thing as absolute identity. There are rather, as... more Peter Geach famously holds that there is no such thing as absolute identity. There are rather, as Geach sees it, a variety of relative identity relations, each essentially connected with a particular monadic predicate. Though we can strictly and meaningfully say that an individual a is the same man as the individual b, or that a is the same statue as b, we cannot, on this view, strictly and meaningfully say that the individual a simply is b.It is difficult to find anything like a persuasive argument for this doctrine in Geach’s work. But one claim made by Geach is that his account of identity is the account most naturally aligned with Frege's widely admired treatment of cardinality. And though this claim of an affinity between Frege's and Geach's doctrines has been challenged, the challenge has been resisted. William Alston and Jonathan Bennett, indeed, go further than Geach to argue that Frege's doctrine implies Geach's.
Philosophia Mathematica, 2015
This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of ex... more This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in (HP). It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: (i) what's required for proofs of existence; (ii) the connection between models, consistency, and existence; and (iii) the prospects for a logical grounding of arithmetic in the wake of the paradox.
Journal of Philosophy, 2012
Journal for the History of Analytical Philosophy, 2015
All contributions included in the present issue were originally presented at an 'Author Meets Cri... more All contributions included in the present issue were originally presented at an 'Author Meets Critics' session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.
The Oxford Handbook of German Philosophy in the Nineteenth Century, 2015
Notre Dame Journal of Formal Logic, 2000
This essay addresses the question of the effect of Russell's paradox on Frege's d... more This essay addresses the question of the effect of Russell's paradox on Frege's distinctive brand of arithmetical realism. It is argued that the effect is not just to undermine Frege's specific account of numbers as extensions (courses of value) but more importantly to undermine his general means of explaining the object-directedness of arithmetical discourse. It is argued that contemporary neo-Fregean
The Palgrave Centenary Companion to Principia Mathematica
From 1914, when Behmann first lectured on Principia in Göttingen, to 1930, when Gödel proved the ... more From 1914, when Behmann first lectured on Principia in Göttingen, to 1930, when Gödel proved the incompleteness of its system, Principia Mathematica played a large role in the development of modern metatheory. ii The Principia system, with its explicit axiomatic approach to the fundamental principles of logic, was just what was needed in the early years of the 20 th century to make possible the precise formulation and treatment of meta-logical questions. One might have thought, then, that at least by the time of finishing his work on Principia, Russell would have been in just the right position to appreciate such straightforward metatheoretical issues as those of the completeness and soundness of a logical system, of the independence of its axioms, and so on. But, notoriously, he seems curiously far removed from anything like modern metatheory. Russell never formulates a completeness theorem, or even raises anything like a modern completeness question about his system. He even seems strangely confused about what we now take to be an entirely straightforward method of proving the independence of logical axioms. In Principles of Mathematics, Russell remarks that [W]e require certain indemonstrable propositions, which hitherto I have not succeeded in reducing to less than ten. Some indemonstrables there must be; and some propositions, such as the syllogism, must be of the number, since no demonstration is possible without them. But concerning others, it may be doubted whether they are indemonstrable or merely undemonstrated; and it should be observed that the method of supposing an axiom false, and deducing the consequences of this assumption, which has been found admirable in such cases as the axiom of parallels, is here not universally available. For all our axioms are principles of deduction; and if they are true, the consequences which appear to follow from the employment of an opposite principle will not really follow, so that arguments from the supposition of the falsity of an axiom are here subject to special fallacies. Thus the number of indemonstrable propositions may be capable of further reduction, and in regard to some of them I know of no grounds for regarding them as indemonstrable except that they have hitherto remained undemonstrated. iii This view, that we can't use standard methods to demonstrate the independence of logical axioms, is one that Russell maintains up to and including the period of writing Principia. iv Why doesn't Russell, apparently well placed to appreciate modern metatheoretical questions and techniques, ever raise, employ, or even appear to understand them? One answer to this question has been proposed by a group of scholars including Burt Dreben and Jean van Heijenoort, Warren Goldfarb, and Tom Ricketts. To quote the first pair: [N]either in the tradition in logic that stemmed from Frege through Russell and Whitehead, that is, logicism, nor in the tradition that stemmed from Boole through Peirce and Schröder, that is, algebra of logic, could the question of the completeness of a formal system arise. For Frege, and then for Russell and Whitehead, logic was universal: within each explicit formulation of logic all deductive reasoning, including all of classical analysis and much of Cantorian set theory, was to be formalized. Hence not only was pure quantification theory never at the center of their attention, but metasystematic questions as such, for example the question of completeness,
Synthese, 2000
This paper examines the connection between model-theoretic truth and necessary truth. It is argue... more This paper examines the connection between model-theoretic truth and necessary truth. It is argued that though the model-theoretic truths of some standard languages are demonstrably ''necessary'' (in a precise sense), the widespread view of model-theoretic truth as providing a general guarantee of necessity is mistaken. Several arguments to the contrary are criticized.
Philosophia Mathematica, 2007
Journal of Philosophy, 1996
G ottlob Frege's work in logic and the foundations of mathemat- ics centers on claims of log... more G ottlob 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 ...
History and Philosophy of Logic, 1994
Frege's logicism, the thesis that "the laws of arithmetic are analytic'' 1 is standardly taken to... more Frege's logicism, the thesis that "the laws of arithmetic are analytic'' 1 is standardly taken to be an important epistemological thesis. The traditional view of Frege's work is that his "reduction'' of arithmetic to logic was intended to provide the cornerstone of an argument that the truths of arithmetic are knowable a priori and independently of anything which Kant would have labelled "intuition". The truth of Fregean logicism would, on this view, have repudiated the explanations of arithmetical knowledge offered by Kant and by Mill. It would have provided an explanation of arithmetical knowledge which was acceptable from a generally empiricist perspective, and which preserved the intuition that arithmetical truths are necessary and knowable a priori. As against this received view, Paul Benacerraf has recently argued in "Frege: The Last Logicist" 2 that Frege's project was not an epistemological one, and was in particular not an attempt to counter Kant's view of the nature of arithmetical knowledge. Though Frege explicitly claims to be engaged in demonstrating the analytic, a priori nature of arithmetical truth, Benacerraf claims that Frege has so re-construed the notions of analyticity and a priori truth that the entire project is, from the very beginning, a nonepistemological one. As Benacerraf puts it, Frege's "attempt to establish the analyticity of arithmetic [is] not to be construed as an attempt to enter an ongoing philosophical debate between Kant and the empiricists, and indeed ... his very construal of the question took it out of that arena." Benacerraf's claim is an alarming one. First of all, Frege's project clearly looks like an epistemological one, and one intended to provide an alternative to Kant's view of the nature of arithmetical knowledge. In the conclusion of the Grundlagen, Frege sums up that work as follows: "I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori,'' 3 and that in showing this result, "we achieved an improvement on the view of Kant.'' 4 The early Grundlagen discussion of the "distinctions between a priori and a posteriori, analytic and synthetic'' are accompanied by the note: "I do not, of course, mean to assign a new sense to these terms, but only to state accurately what earlier writers, Kant in particular, have meant by them. '' 5 Secondly, the incompatibility of Frege's project with epistemological goals is due largely, on Benacerraf's reading, to the fact that Frege allows multiple reductions of the numbers to logical objects. But if the inference here is warranted, then it is difficult to see how any reductionist project, whether
Early Analytic Philosophy - New Perspectives on the Tradition, 2016
Frege claims that mathematical theories are collections of thoughts, and that scientific continui... more Frege claims that mathematical theories are collections of thoughts, and that scientific continuity turns on thought-identity. This essay explores the difficulties posed for this conception of mathematics by the conceptual development canonically involved in mathematical progress. The central difficulties are (i) that mathematical development often involves sufficient conceptual progress that mature versions of theories do not involve easily-recognizable synonyms of their earlier versions, and (ii) that the introduction of new elements in the domains of mathematical theories would seem to conflict with Frege’s view that the original theories involved determinate reference. It is argued here that the difficulties apparently posed to Frege’s central views stem from an overly-simple view of Frege’s understanding of mathematical objects and of reference. The positive view recommended is one on which Frege’s view of mathematical theories is largely consistent with, and helps make sense of, the phenomenon of theoretical unity across conceptual development.
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Papers by Patricia Blanchette