On Sunday, July 6, 2014 7:55:52 AM UTC-4, John Gabriel wrote: > A unit is the result of a ratio where the magnitudes are equal, that is, x:x, y:y, u:u, etc. > > > > This was a quantum leap in the development of number. From the abstraction of the unit, one now has a standard of measurement.
This is a false statement. The unit of the real line is assigned arbitrarily. One group chose the inch, and another the meter. When I draw a real line on paper I arbitrarily tick off positive unity, and then other positions can be graphed.
The ratio does nothing for this. By your logic the people working in meters will forever be pointing their finger at a subgroup who works in inches, and vice versa, and both will be attempting to resolve why the other's parts always scale by 0.024, while both groups (under your theory) will believe that they have a standard of measurement that is universal.
It is a fair criticism of the real numbers (in my opinion) to point out this fact that unity is arbitrarily assigned, so it seems from my position that you've detailed a valid topic, but have not actually resolved the issue. Perhaps from your own context you have, so I am open to being corrected. I think though that your language will need revision.