Sunday, February 03, 2008

Rough Jottings on Syncategorematic and Categorematic Infinites

And I do mean rough. It has been known since the Middle Ages that talk of infinites admits of two basic types of interpretation, the categorematic and the syncategorematic. Roughly, if I say "The numbers for counting are infinite," I can mean that for any number or magnitude to which one counts (which is always finite), there is a greater to which he counts (which is always finite). That's syncategorematic. Or I can mean that the counting numbers together constitute something greater than any number or magnitude to which one can count. They are not the same because they have different logical properties. If I take "An infinity of numbers will be counted" categorematically, I mean that all the numbers actually counted constitute a magnitude greater than any finite number; if I take it syncategorematically, I mean that any given finite number of numbers counted will be surpassed by some larger finite number of numbers counted. The former gives me an infinite magnitude; the latter gives me an unending series of finite magnitudes.

Most people assume that syncategorematic infinites are necessarily potential. This is not so clear: Leibniz, for instance, argues for an actual syncategorematic infinite. If a syncategorematic infinite is potential, it indicates a count or increase that is always finite but does not stop. The count is actually infinite (in the sense that it has no finite stopping point) but is not an actual infinite (because it is always finite). But not every syncategorematic infinite is obviously potential.

Here is a possible example. If every traversal requires a beginning and an end, and an infinite past has no beginning, this is a problem only if we already assume that traversal of an infinite past would require traversal of infinite days. But on the infinite past view, every day in the past is finitely distant from the present; it's just that for every finitely distant day there's a day that is more distant. Thus this is true: For every day in the past, traversal of the days from that day to today is traversal of a finite number of days. The fact that there are infinite such days doesn't change this. This is true just as much as it is true that the fact that every integer is a finite distant from zero is not made a problem by the fact that there are infinite numbers. This is, arguably, Aquinas's point in response to arguments that purport to demonstrate the impossibility of an infinite past, although he does not use the terminology. Such arguments cannot show that an actual syncategorematically infinite past is impossible; at most they show that an actual categorematically infinite past is impossible. We can think of it as a count (backwards in time): every day is counted, but the count never ceases to be finite, because every day is a finite distance from the day at which we begin. Just as every natural number is a finite distance from zero (and thus counting never gets you anything other than a finite number), so every day in the past is a finite period of time before today. What makes the infinite is not that there is ever any infinite distance actually covered, but that there is no finite limit to the finite distances covered.

On the other hand, the only thing that makes this an actual infinite is that we have asserted by fiat that the syncategorematically infinite count is completely actual --the days have already gone by. If someone insisted that every syncategorematic infinite is potential, there's very little grounds on which one could argue that they are wrong. Leibniz, for instance, argues that extension is an instance of actual syncategorematic infinity: every part of an extended body has an actual division into other parts of that body. If, indeed, the latter is true, the parts of the body are an actual syncategorematic infinite: no part would be infinitely small, since very part would be finitely small, but there is no limit to how finitely small you could get. But if someone simply denied this, holding instead that the division is always potential, there's no clear argument you could give otherwise; it's not as if you could say, "Here, look, see for yourself, these are all the infinite divisions of the extended body." Thus there doesn't seem to be any available (non-question-begging) argument that a syncategorematic infinite is actual; one either postulates it or not, and it would seem there's an end on it.