Date: Sun, 02 Aug 1998 12:03:29 -0700 From: Steve Callihan <callihan-AT-callihan.seanet.com> Subject: Re: god and gramm At 08:33 AM 8/2/98 -0400, Malgosia Askansas wrote: >Steve wrote: > >> Peirce asks the question whether a line is composed of the points along >> that line, and answers that that cannot be the case, for even if you have >> an infinite number of points, there would still be gaps between the points. >> Therefore the line must be something other than the points (even infinite) >> that are arrayed along it. > >This somehow keeps haunting me, so let me yield and go back to it. Could >you say more about it? I think that the haunting has to do mainly with >the fact that both my history of math and my history-of-math books are >in storage. So for example, is the idea of "gaps" consistent with Euclidean >geometry as it was in Peirce's time? Can the Euclidean plane (as it was >axiomatized then) admit of two kind of entities -- points and gaps -- without >contradictions such as, for example, that one would then be able to deduce >the existence of two non-parallel lines that do not intersect at any point? >Is the notion of "continuity" ensured only by putting forth the mutual >mappability between the Euclidean line and the real numbers? Is this >putting forth what constitutes the Peircean "something other"? _Why_ did >Pierce think that there would be gaps between the points? It seems much >easier to _not_ think that. I came across the idea in an online article on Peirce's idea of continuity, "Continuity: An Integrated Introduction" by Cathy Legg. (I believe that Cathy was at one time a lurker on this list, and may still be.) If you are interested in the full article, you can find it at http://coombs.anu.edu.au/Depts/RSSS/Philosophy/People/Cathy/Continuity.html. Here's a short excerpt from that article, which I think says it better than I can manage, which can be taken to be a general commentary on Peirce's definition, "A true continuum is something whole possibilities of determination no multitude of individuals can exhaust." "Peirce thought there had to be more to a line than a multitude of points. For given tht points have no extension, how can ne get a line from assembling points, not matter how many points one puts together? And given this, how can one think of the line as composed of points, (even as many points as there are real numbers)? Points are more properly thought of as breaks in a line productive of dicontinuity than as any sort of ultimate constituent of linear continuity: 'Breaking grains of sand more and more will only make the sand more broken. It will not weld the grains into unbroken continuity.' (Peirce, Collected Papers, 6.186) -- Peirce thought that if a true continuum may properly be spoken of as having parts, they must be capable of being what he called 'welded together.'. Thus, referring to the continuum as a 'supermultitudinous collection', he wrote that, 'A supermultidinous collection is so great that its individuals are no longer distinct from one another.' (Peirce, New Elements of Mathematics) Another way of making this point is to say that continuity is a notion that has come down to us through geometry, and which is actually not reducible to or expressible in the language of arithmetric, which quantifies over numbers, for numbers are necessarily discrete." If this reminds you of Zeno's paradox (one cannot get from A to B by halves), it is no accident. In other words, the points along a line, even an infinite array of points, can at best only be a representation of the line, but not the line itself (the distance or relation between A and B). Mixed in with this, also, is the notion of transfinite numbers, "the recognition that some infinities are bigger than others," as Legg describes it. She, however, contrasts Peirce's thought here as distinct from Georg Cantor's: "His approach to transfinite numbers and to continuity differed from Cantor's, however, in that Cantor assumed that with the Reals he had reached continuity as examplified by the geometric line. He thus identified contnuity with the Reals, which as been mathematical orthodoxy ever since. Peirce on the other hand was not satisfied with this, which can be seen in his definition of continuity as something whose determination no possible multitude of individuals can exhaust." This is also related to Peirce's notion of "Thirdness." To put it quite roughly: A in relation to A would be Firstness (every point in relation to itself would be a firstness). B in relation to A would be a Secondness (A --> B). The relation between A and B, however, would itself be a Thirdness (A <--> B). The notion here is that the relation between A and B is implicative of something other than A or B. I suppose one might say that the gap between any two points cannot itself be a point. Therefore, the something other that is implicated by the relation between A and B is, itself, gapless. The direct link with Nietzsche's thought (and there are lots of subterranean links between Peirce and Nietzsche) here would be with the notion of reason and logic as the "equating of the inequatable." Thirdness would thus be inequatability, the always remaining remainder, if you will. Anyway, wish I had more time to delve into this line of thought. Perhaps, Malgosia, if you can find the time to read the whole article, you'll have some very interesting thoughts to provide in return. (Just as an aside, the question of music comes into play here. A piece of music, in other words, cannot be reduced simply to the notes that compose it. Rather, it is the relations between the notes that form the primality of music. The valency of the notes, in other words, is determined by their relation to each other and to the whole. How do we tell the difference between a wrong and a right note, for instance? When listening to Thelonious Monk, for instance, every note may seem wrong when we hear it, but right thereafter. The following note makes the previous note correct, in other words, but itself sounds wrong again (is a dissonance). Or a sequence of notes may all sound wrong (dissonant), but is rendered melodic in relation to a follwing sequence. This, of course, all plays into Nietzsche's idea of the world as music.) Best, Steve C. --- from list nietzsche-AT-lists.village.virginia.edu ---
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