Monday, October 01, 2007

Lines, making a point

One objection to the existence of points (that goes way back) is, How could they could make up an extended line? Two points, if they do not coincide, must be separated from each other, since they themselves have zero extension; so it seems impossible to pile them up to make a line of points. The problem is that there is no point next to another point in a line of points; whereas our intuitions, about piling things up, tell us that there must be. The solution is to note that such intuitions are always of (pseudophysical) things with nonzero extensions. It is their spatial arrangement that makes a lot of points into a line, and since the former is just the line of points, so the line (even if it is full of points) must be (in some sense) prior to those points, as Poincaré said.
......So that was not really an intuition against the existence of points, nor to lines being full of points, but only to lines being made of points (in too pseudophysical a way). And another objection to lines being full of points fails for a similar reason; the objection I have in mind is that if a finite line segment, say AB, is full of points, then we can surely consider all those points minus B, and we are then considering all of AB up to but excluding B, which seems impossible, for the following reason: If our new line goes all the way up to B, up to zero distance away from B, then how could B be excluded? (Still, maybe it is just that our intuitions are all of closed line intervals, not such half-open ones.) Or, when none of the points of our new line get so close to B, how can our line (which is supposedly nothing but those points) get so close? Still, how could we see or imagine that? Presumably only by being given a picture in which the points were represented by nonzero dots, could we possibly see what was going on... and yet that must give us the wrong picture.
......So, since such a thing is naturally inconceivable, such inconceivability hardly counts against it, against lines being full of points. Nonetheless, a similar argument may have some force against the actuality of the infinitude of the natural numbers, against the completability of an endless reiteration: Our line AB can be bisected, and then the section next to B bisected, and that last step repeated endlessly (as in Zeno's Dichotomy, e.g. see the recent discussions at Maverick Philosopher). The resulting intervals are (this time) naturally all next to each other, but again, while they all go, collectively, up to B (if they actually exist), none of them are next to B. In other words, although B is not next to any interval, the line that it is next to is nothing but those intervals (cf. how each natural number is in the first 0%, so to speak, of the naturally ordered sequence of all and only those numbers:)

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