[We continue with our inventory of Leading Ideas]
(6) Actual Infinity
It is in the first instance odd, that the idea of infinity should ever have occurred to anyone, since everything we have ever actually met with is quite finite. Nor is it certainly the case, that the idea of an actual infinity -- of whatever ordinal type, whether the uncountable number of ideal points on the diameter of this coffee cup, conceived as a segment of the real line, or the orderly spacing of the digits wherewith we may count without limit – lies coiled as it were at the heart of things, and must necessarily in time be dislodged by any sufficiently sophisticated investigations of natural science. For though both sorts of infinity have proved an analytical convenience (perhaps rather as the assumption of the Deity is a moral convenience), the actual particular cosmos in which we find ourselves may well be finite in every respect: finite in time from the Big Bang to the Big Bust; finite in extent, though unbounded (a possibility now more easy to picture, as we understand closed manifolds – the geometry of our cosmos could even be hyperbolic, yet fit comfortably inside one of the baubles at the end of “Men in Black”); and finitely grainy in texture (a notion anticipated by the Greeks, and given more substance by quantum mechanics or by Wolfram’s approach to physics). Yet once one has grasped the idea, it is resplendently independent of how this particular world might happen to be, and might as well have occurred to Og the Troglodyte (staring into the fire, or up at the stars) as to Professor von Milchmustash, staring at the blackboard.
It has often been denied, that we can have any clear idea of an *actual* infinity, as opposed to a process indefinitely prolonged. [***] And in the case of the real line, or even that more modest coffee-cup-contained real interval, I must own that I have none. If, in some state of drunkenness or smugness, I ever did so imagine, thát hubris has been brought to heel by the Cantor Set, replete with paradox. Yet in the case of little-omega, the “dot-dot-dot” in the familiar “1, 2, 3, …”, I believe we do have such a clear idea, or can attain to such. Not, if you wish, a *direct* conception, but as a steadily increasing knowledge of its properties: we know it by its fruits. And here we are in no worse case than in our knowledge of any other ding-an-sich, such as this proverbial, this familiar, coffee cup -- unending in its depth of implications, for which we are perhaps gaining a renewed respect!.
[*** ]Sample animadversions:
Locke Essay II.xvii.13: “Though it be hard, I think, to find anyone so absurd, as to say, he has the positive idea of an actual infinite number … yet there be those who imagine they have positive ideas of infinite duration and space. It would be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no …”