Strict finitism

Conclusion Dummett's objections to the coherence of the strict finitist philosophy of mathematics are thus, at the present time at least, ill-taken. We have so far no definitive treatment of Sorites paradoxes; so no conclusive ground for dismissing Dummett's response — the response of simply writing off a large class of familiar, confidently handled expressions as semantically incoherent. I believe that cannot be the right response, if only because it threatens to open an unacceptable gulf between the insight into his own understanding available to a philosophically reflective speaker and the conclusions available to one confined to observing the former's linguistic practice; for an observer of our linguistic practice could never justifiably arrive at the conclusion that red, child, etc., are governed by inconsistent rules. But the Sorites is not the subject of this paper. The points I hope to have made plausible are: that a generalized intuitionist position cannot be so much as formulated and that even a most local intuitionism, argued for the special case of arithmetic, is hard pressed effectively to stabilize and defend itself; that strict finitism remains the natural outcome of the anti-realism which Dummett has propounded by way of support for the intuitionist philosophy of mathematics; that it is powerfully buttressed by the ideas of the latter Wittgenstein on rule-following; and that there is no extant compelling reason to suppose that its involvement with predicates of surveyability calls its coherence into question. The correct philosophical assessment of strict finitism, and its proper mathematical exegesis, remain absolutely open, almost virgin issues. This is not a situation which philosophers of mathematics should tolerate very much longer.The term was introduced by Kreisel in [6] to denote what he took to be an aspect of Wittgenstein's later philosophy of mathematics; and taken over by Kielkopf (Strict Finitism, Mouton 1970) — misunderstanding, as it seems to me, both Kreisel and Wittgenstein — as a label for Wittgenstein's later philosophy of maths. in its entirety. It is not a happy label for the ideas I am concerned with, since it is only from non-strict finitist points of view that the strict finitist can be straightforwardly seen as stressing the finitude of human capacities, countenancing only finite sets, etc. (See subsections 5 and 6 below). But we need a labeel; and Dummett in [3] has already followed Kreisel's lead. Anyway, a rose by any other name,...     

Dummett's intuitionism is not strict finitism

Michael Dummett's anti-realism is founded on the semantics of natural language which, he argues, can only be satisfactorily given in mathematics by intuitionism. It has been objected that an analog of Dummett's argument will collapse intuitionism into strict finitism. My purpose in this paper is to refute this objection, which I argue Dummett does not successfully do. I link the coherence of strict finitism to a view of confirmation — that our actual practical abilities cannot confirm we know what would happen if we could compute impracticably vast problems. But to state his case, the strict finitists have to suppose that we grasp the truth conditions of sentences we can't actually decide. This comprehension must be practically demonstrable, or the analogy with Dummett's argument fails. So, our actual abilities must be capable of confirming that we know what would be the case if actually undecidable sentences were true, contradicting the view of confirmation. I end by considering objections.I especially want to thank Alex George and Philip Kitcher for their help on this paper. I'd also like to thank the members of the Propositional Attitudes Task Force, Jane Braaten, Jay Garfield, Lee Bowie, Murray Kitely and Tom Tymoczko. My thanks also to Peter Godfrey-Smith and the anonymous reviewers of Synthese, one of whom was particularly helpful.     

维特根斯坦和传统

维特根斯坦在奥地利-维也纳传统中长大,并坚定地植根于这一传统之中。任何理解他的人格的尝试都必须设法把他置于这一背景中加以考察。但维特根斯坦作为哲学家的工作,几乎完全是在他并不分享其文化传统、并且对他本人是非常不相宜的环境中完成的。     

Wittgenstein and criteria

An essay to develop some of Wittgenstein's remarks about the notion of ‘criteria’ and to give the concept clarity even at the expense of some features Wittgenstein claimed for it. This effort was made because of the important role ‘criteria’ plays in Wittgenstein's discussions of feelings and mental states, and it is hoped that a defense of ‘criteria’ will make those discussions more coherent. An attempt is made to relate this notion of ‘criteria’ to the definition and expression of mental states, following some of Wittgenstein's suggestions, and to rebut skepticism about other minds.     

Wittgenstein and Ebersole

Frank B. Ebersole died recently. Here I remind philosophers of the thinking of this reclusive philosopher who brought out the value of Wittgenstein's dictum that philosophers should “bring words back from their metaphysical to their everyday use.” I illustrate Ebersole's singular thinking by focusing on his philosophical investigation of Wittgenstein's family resemblance metaphor.     

维特根斯坦与宗教

维特根斯坦是一个伟大的分析哲学家,但是一生与宗教有非常密切的联系。本文分为两部分。第一部分从维特根斯坦的家庭背景、成长经历、性格特征、所受影响等方面分析了维特根斯坦与宗教(基督教)的联系;第二部分从“关于宗教的本质是否可以言说”、“为什么会有宗教?”“宗教与激情”、“上帝是谁?”“信仰上帝是为了给人生寻找意义”等五个方面分析和评论了维特根斯坦的宗教观点。     

Wittgenstein and Friendship

In his article “It's a Wonderful Life,” Ronald Hall connects Wittgenstein's last words with Frank Capra's 1946 film. His analysis focuses on the concept of wonder, but he misses one of the most important aspects of both the film and Wittgenstein's last words: the significance of friendship. This is philosophically (and biographically) important because it raises questions about aspect‐seeing, friendship and everyday life. Wittgenstein's final words provide a striking example of the philosophical complexity of his life and work.     

Wittgenstein and logic

In his Tractatus Logico-Philosophicus Ludwig Wittgenstein (1889–1951) presents the concept of order in terms of a notational iteration that is completely logical but not part of logic. Logic for him is not the foundation of mathematical concepts but rather a purely formal way of reflecting the world that at the minimum adds absolutely no content. Order for him is not based on the concepts of logic but is instead revealed through an ideal notational series. He states that logic is “transcendental”. As such it requires an ideal that his philosophical method eventually forces him to reject. I argue that Wittgenstein’s philosophy is more dialectical than transcendental.     

Wittgenstein and Brouwer

In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic.     

Wittgenstein and logical necessity

An attempt is made to show that Wittgenstein's later philosophy of logic is not the kind of conventionalism which is often ascribed to him. On the contrary, Wittgenstein gives expression to a “mixed” theory which is not only interesting but tends to resolve the perplexities usually associated with the question of the a priori character of logical truth. I try to show that Wittgenstein is better understood not as denying that there are such things as “logical rules” nor as denying that the results of applying such rules are “logically necessary,” but as trying to understand what it is to appeal to a logical rule and what it means to say that the results of applying such a rule are “necessary.” He is not so much overthrowing standard accounts of logical necessity as discovering the limits of the concept.     

Wittgenstein and not-just-in-the-head cognition

Wittgenstein's later writings criticize the idea that the mind is an intrinsically representational device. More or less clearly apprehending certain limitations of identifying all aspects of cognition with ‘internal representation’, certain theorists in a variety of disciplines have mostly independently reached similar conclusions, which are reflected in notions like ‘distributed,’ ‘extended,’ and ‘situated’ cognition. We explore the variety of purposes for which these terms have been introduced into psychological theorizing and relate these to Wittgenstein's philosophy of psychology.     

Reading Wittgenstein

Books reviewed in this essay:
Robert Arrington and Hans-Johann Glock (eds), Wittgenstein & Quine
John Koethe, The Continuity of Wittgenstein's Thought
P.M.S. Hacker, Wittgenstein's Place in Twentieth-Century Analytic Philosophy
Hans Sluga and David G. Stern (eds), The Cambridge Companion to Wittgenstein
Marie McGinn, Wittgenstein and the Philosophical Investigations