InWittgenstein(1976 and 1987) I located what I took to be the key move in Wittgenstein’s reflections on the possibility of a private language inPI198. There Wittgen-stein presents the following problem concerning rule-following:

PI198. “But how can a rule shew me what I have to do at this point? Whatever I do is, on some interpretation, in accord with the rule.”—That is not what we ought to say, but rather: any interpretation still hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning....

Chapter 1 lays out the central theme of this work: Witt-genstein’s rejection, in his later philosophy, of aninterpretationalaccount of rule-following, replacing it with what I have labeled adefactoistaccount. This theme shapes Wittgenstein’s later thought, going back at least to thePhilosophical Grammar,appearing over and over again in a wide variety of philosophical contexts. This aspect of Wittgenstein’s later philosophy has not been wholly ignored—indeed, as noted in the introduction, it has often been decried. Chapter 1 gives it the prominence I think it deserves.

Chapter 2 offers a brisk application of this discussion of...

Readers ofPhilosophical Investigationsare familiar with the story of the child being taught to produce the series of even numbers starting with 2. She starts out well enough, writing down 2, 4, 6, 8. However, when asked to pick up the series at 1000, she writes down 1000, 1004, 1008, 1012 (PI183).¹ Told that she is no longer following the instructions we gave her—no longer doing the same thing—she replies that she is, perhaps saying, “Look, see for yourself!” The rub is this: Whatever she writes down, there will be some interpretation of the instructions we...

As indicated in the introduction, I no longer think that Wittgenstein was attempting to establish the strong negative claim that a private language (as specified inPI243) is not possible. I previously based my interpretation of the so-called private language argument on what might be called astraightreading ofPI202.

PI202. ‘Obeying a rule’ is a practice. And tothinkone is obeying a rule is not to obey a rule. Hence it is not possible to obey a rule ‘privately’: otherwise thinking one was obeying a rule would be the same thing as obeying it....

In this, the closing passage ofPhilosophical Investigations, Wittgenstein draws a parallel—seemingly an exact parallel—between the way one should deal with conceptual confusions in psychology and the way one should deal with them in mathematics. Broadly speaking, the connection is this: In both areas confusions arise from employing a referential model for understanding a class of expressions where that model is inherently misleading. With respect to mathematical propositions, his primary target is, as it is commonly called, platonism in mathematics.¹ This view, or family of views, admits of wide variation, but, broadly sketched, it embodies the claim that...

One of Wittgenstein’s central ideas is that philosophers are often misled by taking similarities in syntactical structure as indicators of a similarity ingrammar—that is, as Wittgenstein uses this notion, as similarities inuse. Often this assumption is correct, but not always. In the previous chapter we saw that Wittgenstein held that this assumption produces conceptual muddles when applied to fi rstperson ascriptions of mental properties. In the passage cited above, he makes a parallel claim with respect to mathematical expressions. Wittgenstein’s suggestion is that we have a tendency, indeed a strong tendency, to treat “2 + 2 =...

The previous chapter concerned what might be calledglobalproblems that arise in the philosophy of mathematics, problems, for example, concerning the status of numbers and the status of arithmetic propositions. This chapter will deal with a philosophical problem that arises with respect to a specific result within mathematics: Cantor’s discovery or introduction of transfinite cardinals.¹ Here, as before, I will take Wittgenstein’s methodological remarks at face value. I will assume that he means it when he says such things as: “It will be most important not to interfere with the mathematicians.” (LFM, p. 13)

I will try to show...

Wittgenstein has a reputation for having a laissez-faire attitude toward inconsistencies, paradoxes, contradictions, and the like. He doesn’t. As the passage given above indicates, his refl ections on these notions have a specific target: the idea that the occurrence of a contradiction in a calculus (for example, a logical system) destroys it. Wittgenstein’s central claim is that this, as a matter of fact, is false. Wittgenstein’s fullest defense of this claim and related matters occurs in the lectures on the foundations of mathematics that he presented at Cambridge in 1939.

In these lectures, Wittgenstein presents examples that illustrate the following...

This work offers a reading of Wittgenstein’s later philosophy that emerges from taking his methodological statements at face value. Taking him at his word has, to a large extent, involved letting him speak for himself. This approach involves stressing certain passages at the expense of others, and that, of course, itself imposes a strong interpretation on the text. I have tried to mitigate this problem by concentrating on persistent themes in Wittgenstein’s later philosophy—themes that were already falling into place in the early 1930s, for example, in the material found inPhilosophical Grammar. To borrow one of Wittgenstein’s expressions,...