As I mentioned before, I'm reading A.W. Moore's The Infinite. (I found out about it through the webpage of a member of the Syndey School of Mathematics.) The book is much clearer than the Zellini book I commented on previously. Whereas Zellini makes a single distinction between the actual infinite and the potential infinite (which are synonymous with "true" and "false" infinites), Moore adds an additional distinction between true and false versions of the actual and potential infinites. The true versions of these he calls metaphysical and mathematical infinites.
Part of Moore's clarity comes from starting off (in the introduction) discussing the paradoxes of the infinite and defining terms.
[O]ne of the central issues concerning the infinite is whether it can be defined. Many have felt that it cannot; for if we try to define the infinite as that which is thus ans so, we fall foul of the fact that being thus and so is already a way of being limited or conditioned. (It is as if the infinite cannot, by definition, be defined....)1
Two clusters of concepts nevertheless dominate, and much of the dialectic in the history of the topic has taken the form of oscillation between them. Within the first cluster we find: boundlessness; endlessness; unlimitedness; immeasurability; eternity; that which is such that, given any determinate part of it, there is always more to come; that which is greater than any assignable quantity. Within the second cluster we find: completeness; wholeness; unity; universality; absoluteness; perfection; self-sufficiency; autonomy. The concepts in the first cluster are more negative and convey a sense of potentiality. They are the concepts that might be expected to inform a more mathematical or logical discussion of the infinite. The concepts in the second cluster are more positive and convey a sense of actuality. They are concepts that might be expected to inform a more metaphysical or theological discussion of the infinite. Let us label the concepts 'mathematical' and 'metaphysical' respectively. (1-2)
The book is divided into two parts. The first part is an overview of the historical of thought on the infinite, and the second part is an assessment of the various strains of thought. Aristotle has a foundational role in both. Moore points out that Aristotle wasn't saying that the (mathematical) infinite was false, but that it only exists potentially, not actually:
I said at the beginning of §2 that Aristotle appeared to abhor the mathematical infinite. We can now see how profoundly false such an appearance was. What he abhorred was the metaphysically infinite, and (relatedly) the actual infinite—a kind of incoherent compromise between the metaphysical and the mathematical, whereby endlessness was supposed to be wholly and completely present all at once. It was the mathematically infinite that he was urging us to take seriously. Properly understood, the mathematically infinite and the potentially infinite were, for Aristotle, one and the same. Far from abhorring the mathematically infinite, he was the first philosopher who seriously championed it. In so doing he recoiled from earlier thinking in such a way that he set the scene for nearly all subsequent discussion of this topic. (44)
According to Moore, the pre-Socratics had given voice to what was essentially the metaphysical infinite, but Plotinus first articulated it clearly and distinctly:
He called it self-sufficient, perfect, and omnipotent, a complete and pure unity, utterly beyond our finite experience. He also said that it was 'supremely adequate, autonomous, all-transcending, most utterly without need.' Sometimes he spoke of it in a Parmenidean way, implying that it had internal limits. 'Its manner of being is settled for it,' he said, 'by itself alone.' But elsewhere he emphasized its lack of limits, either exeternal or internal. Indeed, in line with this, he insisted that all our attempts to talk about it or derfine it were strictly speaking, and inevitably, inadeqaute. This, in truth, it even transcended such descriptions of it as 'The Good' or 'God'. Its ineffability meant that we had to be content with mystical insight into it. He nevertheless tried to convey as much as possible with words. And in so doing he supplied one of the first explicit identifications of the infinite with God. (46)
Moore very clearly explains the categorematic/syncategorematic distinction originated by Peter of Spain and taken up by Jean Buridan and Gregory of Rimini:
Roughly: to use 'infinite' categorematically is to say that there is something [an actual whole] which has a property that surpasses any finite measure; to use it syncategorematically is to say that, give any finite measure, there is something [another individual] which has a property that surpasses it. (51)
This distinction carlifies an important point about uses of infinite, and furthermore subsumes Aristotle's actual/potential distinction.
By way of illustration, consider the following application of the new distinction in a temporal context, noted by Gregory [of Rimini]. If I say, 'An infinity of men will be dead,' and use 'infinity' categorematically, then I mean that there will come a time when infinitely many men are dead; there will then be an actual infinity of dead men. If I say the same thing, and use 'infinity' sycategorematically, then I mean that there is no end to the number of men who will, each in his own time, be dead; there is a potential infinity of dead men. This explains, I think, why so many philosophers have thought that there was something deeper and more abstract underlying straight-forward temporal accounts of the actual/potential distinction. It seems they were right. There is something—something grammatical [cf. later invocation of Wittgenstein]. (Working with this new distinction also has the advantage that one can avoid the false implication in Aristotle's terminology, noted by Aristotle himself, that what is potentially infinite must be capable of being actually infinite.) (52)
Moore includes some provocative reactions of contemporaries to Cantor's transfinite mathematics:
[French mathematician Henri Poincaré] challenged Cantor's claim to have proved that R [the set of real numbers] was bigger than N [the set of natural numbers]. Cantor's proof could just as well be taken to establish merely that we could not devise a way of pairing off the natural numbers with the real numbers, or indeed that R was not a genuine set at all—presumably because the real numbers were somehow too unwieldy to be grouped together into one determinate totality.
This, incidentally, was something urged by the American philosopher and mathematician C.S. Peirce (1938-1914). He had independently discovered that there was no way of pairing off the natural numbers with the real numbers, but he concluded that R did not exist as a completed whole. At most it existed as something potentially infinite. However many reals had been actualized, there were always more waiting to be. A continuum, he felt, was precisely not just a set of points. It was something absolute, consisting of unactualized possibilities, cenmented together in a way that defied description but of which we were aware in experience.
I'm not sure how much credence to give these ideas, but they do seem to lead us back to Aristotle's notion of the continuum not being composed of points.
I'm still re-reading and digesting the book. Moore's advocacy of Kant and Wittgenstein's positions on the subject sound reasonable to me (from what he says), but I'm not entirely certain how kosher they are (especially Kant) and I need to examine them more closely because they have a big part to play in the second half of the book. Being a good Englishman, Moore ends (as I read him) with the empiricist (or more-or-less Aristotelian) position of denying the reality of the metaphysical infinite and affirming that while we can see things about the mathematical infinite, we cannot really say anything about it as actual.
1. Not sure this is the right way to note the ellipsis.
A.W. Moore, The Infinite (New York: Routledge, 1991). All emphases in original.
Paolo Zellini, A Brief History of Infinity, trans. David Marsh (New York: Penguin Books, 2004).