John Gabriel <firstname.lastname@example.org> wrote in news:email@example.com:
> Unless you know what is a fraction, you never really understand what > is a natural number. Natural numbers are ratios of magnitudes to > units, where the magnitudes are exact multiples of units. Let's see > the construction of rational numbers from scratch, in 5 easy steps. > > Construction of rational numbers: > > 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).
No definition of an exent, or the size of extent, nor the idea of the size of extent.
No procedure given for comparing magnitudes. Say the height of a tree in my yard, vs on in a distant forest. How do I compare those extents?
No definition of what it means for one extent to be bigger than another either.
> 2. We can form ratios of magnitudes. AB : CD where AB and CD are line > segments.
All of a sudden we have line segments. Before we have ideas of sizes of extents.
> The expression AB : CD means the comparison of magnitudes AB > and CD.
It doesn't say what "comparison" means. The only things so far is that we can say they are equal, or one is smaller and the other bigger.
So it appear AB : CD is "equal", "smaller" or "bigger"
> 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 anything that is the same magnitude as itself is "the" unit.
So the unit is a magnitude
> The unit > is a ratio of equal magnitudes.
So the unit is NOT a magnitude.
What the hell is a unit then?
> 4. The unit enables us now to compare AB and CD if both are exact > multiples of the unit that measures both.
No definition of HOW we use a unit to compare AB and CD. Nor what it means for a magnitude is measured by a unit. Not what multiples of a unit means
> 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.
No definition of what "how much greater" means. If AB is 3 and CD is 6, then is CD greater than AB by 2 or by 3?
> 5. Finally, if a magnitude is only part of a unit,
No defintion of "part of a unit" given
> then we arrive at a > ratio of numbers, say AB : CD where AB and CD are multiples of the > unit.
Where did these come from?
> AB : CD now means the comparison of numbers AB and CD.
But it was only defined for magnitudes. And where did these numbers come from? All of a sudden numbers exist when we are trying to define them
> When we > write AB/CD, it is called a fraction.
We would we write that?
> So, in five steps I have derived the concept of number for you. There > is one thing left - what happens when you can't measure a magnitude > that is not a multiple of a unit and can't be expressed exactly using > any part of a unit? This is called an incommensurable magnitude and > the best you can do is provide an approximation such as 3.14159... or > 1.414..., etc. > > Euclid's Elements: > Definition of magnitude: Bk V. > My definition of magnitude is better than Euclid's because it is not > circular. Definition of number: Bk. VII > My definition of unit is better than Euclid's also because Euclid's > definition is vague and ethereal.