There was a time when, following Aristotle, most educated people thought that the cosmos (the totality of space and everything in it) was finite and bounded. This changed in the sixteenth and seventeenth centuries under the attack of various radical thinkers. Even in Antiquity, however, the view had its critics: notably Archytas and the later atomists (see Chapter 8 of Sorabji 1989 and Bailey 1970). A persuasive argument used against Aristotle went as follows. The cosmos cannot be bounded. For a boundary has things on both sides of it. Hence, if there were a boundary to the cosmos there would have to be something on the far side of it. But this, by definition, is impossible.2 The ultimate fate and validity of this argument I do not intend to discuss. I give it to introduce a similar argument concerning not the universe of space but the universe of thought. Suppose that there were some boundaries to the conceivable, some things that were too complex or recondite to be conceived of (maybe due to the finitude of our brains). Then there must be things on the far side of the boundary, and these must be inconceivable. But to conceive of the boundary is to conceive of things on either side of it; and for the far side this is, by definition, impossible. We have here an argument for idealism-at least in one sense of that word. A realist-in the corresponding sense-holds that there are things which exist, and which are what they are, quite independently of thought. They would exist even if they were not thought about. It must be possible, therefore, for there to be things that are not conceived of, indeed, inconceivable. But then there must be a boundary between that which is conceivable and that which is not. And this is precisely what this argument rules out.