
Re: Euclid's Elements Book 1 Proposition 1  something Euclid missed?
Posted:
Feb 17, 2014 9:02 PM


On Monday, February 17, 2014 8:44:54 PM UTC5, John Gabriel wrote: > KQ: What is a reifiable point? > > > > http://www.spacetimeandtheuniverse.com/math/45070999equalone353.html#post22093 > > > > A reifiable point is a point that corresponds to the measure of a given line segment. For example, a straightedge calibrated with natural numbers. The natural numbers marked on the ruler reify those points. For example, > > > > 0_________1 > > > > 0 and 1 are reifiable points. 1 is the measure of the magnitude of the line segment between 0 and 1. > > > > 0_____________sqrt(2) > > > > 0 is reifiable but sqrt(2) is not, because sqrt(2) is not a number. It is incommensurable. > > > > I can take two line segments and physically place an invisible marker between them to describe 0, 1 or sqrt(2). In the case of 0 and 1, both my line segments are measurable. In the case of sqrt(2), the line segment is not measurable. The point between the line segment sqrt(2) and the line segment following it does not correspond to any number. > >
OK, that partially clarifies it for me.
However it still leaves some fuzziness in my mind. You admit that a line segment can have a length of sqrt(2). Yet you say it does not correspond to any number. If something has a length of sqrt(2), doesn't that mean that the number sqrt(2) exists? What is a length if not a number? If you mean you can't specfy all the digits beforehand, like you can for any rational (since the digits repeat, even if the repeating group is very long), so what? If you want digit #56908124897987367512 of sqrt(2) it can be produced right at your table if you give the waiter enuf time to run the appropriate program on his computer. Any digit you want, we'll give you. Doesn't that give sqrt(2) the right to exist?
I'm going to talk to the American Numerical Rights Union about this.
> > Points are reified not by only marking off distances with an invisible marker(point), but by associating a number with each invisible marker. > > > > The above link explains this.

