Roger, Hmmm! Interesting. Well, I feel I am just being picky, now. But pickiness is what formalisms are about, isnt it???? In my language, I would say that a point is just a position. It's a point of reference. Just as a point of view is a place from where something is seen. I dont know what it would mean to say that the point 0,0 moved? So, to some extent, I am liking this particle, thing, because a particle could be something that could occupy a point. But wait a minute!!! A particle has extension, so lets go with "pointicle". Now we can talk about the motion of this pointicle as we look at smaller and smaller samples of that motion... in fact, make those samples as small as anybody would care to imagine without making them FREEZE the motion of the pointicle. At that point, we could talk about the direction and speed of the particle over that miniscule distance. This seems wiser than to start redefining "particle" which is inherently something that has extension.
But I should be hasty to say, in talking about a category error, and insisting that there was one there, I was not mounting some sort of challange to Newton from my lofty position as a psychologist. On the contrary, I was noticing that psychologists were not the first people to encounter a category error. Newton, did, and on the whole he did rather well with it. I mean, he had a reasonably good career, don't ya think? So, perhaps, category errors play a different role than the devilish one that Ryle and others assigned to them. Perhaps there are good and bad category errors or good or bad USES of category errors. I need to THINK about this. You have to know that I am inclined to worship mathematicians. My brother was (is, actually) one, and as I was growing up, my parents would beam encouragingly at me and tell me every day that they hoped I would be as smart as my brother. And behaviorists psychologists has always been accused of wanting to reduce psychology to mathematical physics. So, if anything, this project is about casting off these youthful illusions and coming to understand what mathematics REALLY is. In that connection, did you have a chance to look at the passage cited at www.sfcomplex.org/wiki/MathematicsAndMusic? Here Rothstein quotes Reuben Hersh in support of the idea that mathematics has styles (Kuhnian paradigms?) just like music (or history, or art or psychology, or any of the sloppy disreputable ways that non-physicist intellectuals make their living. Because of my long standing argument with Owen Densmore about formalisms, I think Rothstein is probably taking this point of view too far, but up till now, no mathematician has read the passage that I posted and commented on it, so I don't know for sure. Reading tonight about non-Euclidian geometries tonight, I was struck by the fact that the Peter Wolff wanted me to know that these geometries were not designed for different surfaces as I had always supposed... rather, they were explorations of what happens when one relaxes certain crucial postulates and propositions of the euclidian system. That they might have relevance to other sorts of surfaces is apparently a SECONDARY consequence. Although the switch between geometries might be a big jump for mathematicians, it is NOT the sort of thing I would identify with a Kuhnian paradigm shift. In fact, everybody seems to have been meticulous in their attempts to maintain continuity as much as possible from one geometry to another. Left to make my own judgement, I think Rothstein is just wrong. All of this would mean that Owen has been right all along. Damn! By the way, I got through all the mathematical parts of Rothstein's book, but when he started to talk about music I had to bail. As a formalism, sheet music defeated me. Sorry to go on. Wednesday, I have to make a run to eastern canada to be examined by my older sister who is concerned for my health, so I shall go silent soon, I promise. I hear there have been deluges in Santa Fe. Acequias that have been dry for a decade brim full with water. Flash floods in the streets! And I am in Massachusetts. I always miss the good stuff. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([EMAIL PROTECTED]) ----- Original Message ----- From: Roger Frye To: [EMAIL PROTECTED];The Friday Morning Applied Complexity Coffee Group Sent: 7/14/2008 10:01:58 PM Subject: Re: [FRIAM] Mentalism and Calculus Nick, I think I am beginning to get a glimmer of what you are complaining about. The wording of your definition is ambiguous. How about this one from Google: a geometric element that has position but no extension; "a point is defined by its coordinates" I think you are arguing that since a point has a fixed position, it can't move. The rest of us are talking about a particle (again with no extension) that is moving from one point to another. -Roger On Jul 14, 2008, at 9:28 PM, Nicholas Thompson wrote: Robert, Some how this message got caught in my outbox and you went unchastised for a whole 48 hours. No! You have gone a bridge to far, unless you are willing to rewrite the role of definitions in axiom systems. In a system in which a definition is, "a point is a position in space lacking dimension"
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