(Sektion 208: Philosophische Untersuchungen von Ludwig Wittgenstein).
"That we cannot write all the digits of pi is not a human shortcoming as mathematicians sometimes believe."
In Philosophische Untersuchungen section 208, Wittgenstein continues elaboration of our processes of recognizing and defining language and common linguistic processes. He concentrates (as in Philosophische Untersuchungen 207) on the notion of regularity and repetition in language -- one of the questions driving this passage is, 'Is regularity a necessary attribute of language?'. In section 208, Wittgenstein considers how we might attempt to define 'regularity' -- the section is reminiscent of the early sections of the Untersuchungen in which Wittgenstein elaborates on our processes of defining the meanings of common names -- we can read 208 as an extension of the lines of thoughts and rhetorical tropes developed in the first fifty or so sections of the book insofar as Wittgenstein tackles the question 'How do we define the meaning of 'regularity'' in the same way that he tackled the question 'How do we define the meaning of a common name, such as 'five red applies'. Namely, he attempts to write out our actual processes of explaining these things (as, for example, in the teaching of children) rather than attempting to elaborate on them by way of theorizing on their foundations and necessary conditions (which would be a rather epistemological approach to the issue). In this, Wittgenstein is conflating philosophical activity with sociological and anthropological activity; on this most commentators (particularly those that are not professors of philosophy) agree.
In this section, Wittgenstein also makes the interesting claim that it is not a human shortcoming that we cannot write out all the digits of pi -- it is not an imperfection inherent in being human, nor does it highlight a lack in us or our social practices. Within the context of this section, we understand that Wittgenstein is drawing our attention to the value and practical importance, at least as concerns rule-following and the extension of rules, of expressions such as and so on and the gesture of, for example, rolling our hand before our body as if to indicate "continue, continue". That we cannot write out pi to the end (or, at least, further out than we normally do) is not an indication that we are inferior to the mathematical entity we have discovered (Wittgenstein would question the value of calling it a discovery anyways), rather it is an indication that we are satisfied to end the writing with an ellipsis (...) or an 'ad infinitum'. And so, to call it a shortcoming is to privelege the mathematical entity in favor of our own usage of pi.
The mathematicians are mistaken in considering it a human deficiency insofar as, practically, it is not a deficiency--... because we do not have any need or want of a further elaboration of pi, and if we did we would probably develop a program for researching such an elaboration (and the problem would quickly become practical). This sort of move is typical in Wittgenstein's later philosophy of language; we often find him dissolving tough philosophical problems (such as the extension of patterns, the sorites paradox (vagueness), metaphor, and meaning/reference) by referring them to the practical application of these problems. The end result is often that: these are not problems for anyone except philosophers. And, to the extent that there are problems, these problems are always practical (for example, we cannot adequately dissolve the vague border between two frequencies of color that are used in some light-sensitive mechanism or we cannot compute pi to enough digits sufficient for the micro-fine fabrication of a part for an advanced nuclear reactor). In cases like this, we are not faced with a theoretical limit, but rather a practical one -- one that we have certain explanations for ('well, these color frequencies are so close, that it's really difficult to tell them apart, even with a machine').(For more on this: see Wittgenstein on Vagueness (forthcoming)).
208. Am I therefore defining (erkläre) what "order" and what "rule" mean through "regularity"?--How do I define the meaning of "regular", "uniform", "same" to anybody? To someone who only speaks French, we say, we would define these words by means of the corresponding French words. But if someone does not possess these concepts, then I would teach them the use of the words through examples and practice.--And in doing this I share with them no less than I myself know.
In this instruction I will show him the same colors, the same lengths, and the same shapes, he will find them and produce them, and so on. I will give him an order to continue an ornamental pattern 'uniformly' (gleichmäßig).--And also to continue progressions. Thus something like . . . . . . and he will go on: . . . . . . . . . . . . . . . .
I do it before him, he does it after me; and I influence him through expressions of agreement, rejection, expectation and encouragement. I let him go his way or hold him back; and so on.
Think about witnessing such an instruction. It defines no words by means of themselves, it makes no logical circles.
The expressions "and so on" and "and so on ad infinitum" are also defined in this instruction. A gesture (Gebärde) as well as other things might serve here. The gesture that refers to (bedeutet) "go on like this!" or "and so on" has a function much like that which pointing to objects or to places has.
There is a difference: between the "and so on" which is an abbreviated notion and the "and so on" which is not. The "and so on ad inf." is not an abbreviated notation (Abkürzung der Schreibweise). That we cannot write all the digits of pi is not a human shortcoming as mathematicians sometimes believe.
An instruction that is meant to apply to the examples that are given is different from the one that 'points beyond' them.