CHAPTER 13

Saying and Showing

The one which is is both one and many, whole and parts, limited and yet unlimited in number. (Plato)

A question with no answer is a barrier that cannot be breached…[It] is questions with no answers that set the limits of human possibilities, describe the boundaries of human existence. (Milan Kundera)

All finite things reveal infinitude. (Theodore Roethke)

At various points in the last three chapters there was a sense of the ineffable in the background. Twice in particular it came to the fore. First, in Chapter 11, §3, I suggested that the sceptical or relativist reaction to the Löwenheim-Skolem theorem might be seen as (arising from) an ill-fated attempt to express an insight that somehow cannot be expressed. And later, in §5, I suggested that the same might be true of our continual allusions to the hierarchy of Sets—or indeed to the Set of Sets, if we treat the concept of the Set of Sets as a Kantian Idea of reason, and try to squeeze a regulative use out of it. Kant himself, when he allowed Ideas a regulative use, certainly seemed to be skirting the ineffable. It was very much as if there was something that he wanted to say (that the physical world exists as an infinite, unconditioned whole) though he had debarred himself from saying it, and this was his way of easing the tension. 1 We too want to say that there is a Set of all Sets. Is it satisfactory—is it, for that matter, legitimate—to speak as if there were such a Set, and then somehow to reconstrue what we have been saying so that it is not to be taken at face value? Maybe the right thing to do is simply to admit that we have an insight to which (we find) we cannot properly give voice.

My aim in this chapter is to lend some respectability to this suggestion. I shall try to make sense of the idea that there are certain insights that cannot be articulated; there are certain things that can be known though they cannot be put into words. Although this idea has so far been inchoate, there has already been enough to suggest an important link with self-consciousness (see above, Chapter 11, §5 and Chapter 12, §5). We have

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