*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

*non*zero 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 PoincarĂ© 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

*non*zero 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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