
Re: Euclid's Elements Book 1 Proposition 1  something Euclid missed?
Posted:
Feb 18, 2014 10:49 AM


KQ:'Fraid not.
You use the term 'reified' simply to say that one point is rational and another is not. There's no need for this extra word, altho people do seem to like creating new words to express the same thing considered from another point of view. I'm all for that if the new point of view brings something new to the table, but if it's just putting old wine in new bottles it seems to confuse things rather than clarify them.
JG: Well, I only used the term because it makes no sense to call a point rational or irrational. The attributes apply to commensurable or incommensurable magnitudes. But a point is not a magnitude. Nevertheless, I can see that you do understand the concept.
KQ: You do grant that a point which is the marker for a rational distance can be removed and leave a hole in the line, but a point which is the marker for an irrational distance does not leave a hole if it is removed because it does not represent a rational distance (leaving aside for the moment the more philosophical question of how can you remove something and NOT leave a hole, if you have already granted its existence).
JG: Yes.
KQ: But consider a line segment AC which contains the intermediate point B, so that AB is length sqrt(2) and BC is length 1. Then B is both from your point of view irrational as a distance measured from A, and rational, as a distance measured from C.
JG: Yes, but the line segments are different, and not to sound picky, but you can't place the origin in more than one location. :)
KQ: So the point is both reifiable and nonreifiable. This seems to imply that the notion of reifiable as meaning anything more than rational vs irrational, at least as you have so far characterized it, is unusable since it is internally contradictory  in ALL cases where it can be applied.
JG: Sorry, it does not imply that at all. You can't place the origin at more than one point. The concept of reification depends on the origin being fixed. :) > > > > Thanks & Regards, > > > > Ken

