John Gabriel <email@example.com> wrote in news:firstname.lastname@example.org:
> Listed below are the Axioms of magnitude: > > > Book V, Def. 1: > 1. A magnitude is the idea of size of extent. We can either tell that > two magnitudes are equal or not. If we can tell they are not equal, > then we know which is smaller or bigger, but we can't tell how much > bigger or smaller. This is called qualitative measurement (without > numbers).
How can we tell that two different magnitudes are equal? Or if one is greater. Define what 'greater' means. How can I tell if a tree in my back yard is the same magnitude (height) as a tree in your backyard?
> Book V, Def. 3: > 2. We can form ratios of magnitudes. AB : CD where AB and CD are line > segments.
So you can only use ratios for line segments .. not any other magnitudes?
> The expression AB : CD means the comparison of magnitudes AB > and CD.
What does "comparison of magnitudes" means .. what is its value? Is it a boolean which is true if they are the same or false otherwise?
> Book VII, Def. 1 > 3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to > use either as the standard of measurement, that is, the unit.
So either AB or CD is a unit .. in fact any magnitude can be the unit. Including (as you've not excluded it) AA.
> The unit > is a ratio of equal magnitudes.
So now a unit is NOT a magnitude .. it is a ratio.
> Book VII, Def. 2 > 4. The unit enables us now to compare AB and CD if both are exact > multiples of the unit that measures both.
But a unit is a ratio, not a magnitude. Or is it a magnitude again now
Define a multiple without the concept of number .. you are trying to construct numbers, so you cannot use them.
How can you measure with a unit without there being numbers?
> We can now perform > quantitative measurement, because we can tell how much greater AB is > than CD or how much less AB is than CD.
Only because numbers already exist
> Book VII, Def. 3 - 5 > 5. Finally, if a magnitude is only part of a unit, then we arrive at a > ratio of numbers, say AB : CD where AB and CD are multiples of the > unit.
How do we 'arrive' at that? What does it mean and how is it related to this magnitude that is part of a unit
> AB : CD now means the comparison of numbers AB and CD.
Hang on .. when did numbers suddenly pop into existence .. you're supposed to be constructing them. Now you just assume they exist and are taking ratios of them
> When we > write AB/CD, it is called a fraction.
OK .. so if we already have numbers, we can create a fraction by writing one number a "/" and another number. What does that mean? You need to define it.
> So in fact, anyone who disagreed with me, was actually disagreeing > with Euclid himself! Tsk, tsk. You should now learn from the terrible > folly of your ways. :-)
I'm not disagreeing with Euaclid, but with your poorly set out axioms and supposed construction of number.
And are you saying now that your wonderful new way to construct numbers is just a copy of Euclid? That's cheating and makes you a theif.