On Sunday, March 2, 2014 9:11:22 AM UTC+2, Dan Christensen wrote:
> > On Sunday, 2 March 2014 07:06:44 UTC+2, Dan Christensen wrote:
> 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).
> Just answer the question, John Gabriel. You listed several "axioms of arithmetic" here. Which ones in particular did you use to derive the above statement? Did you use your 1st axiom 1? Your 2nd axiom? Which one(s) did you use? This is not a "trick question".
How can I use axioms if that is the first one? The words are the definition of axiom 1. It first tells us what is a "magnitude" (the concept of size). Then it tells us we can either determine equality or not through qualitative measurement. The first axiom is *qualitative measurement*. There are no numbers yet. We can only say smaller, bigger or equal, but can't say how much smaller or how much bigger. We can tell there is a *DIFFERENCE*. This is the most primitive operator. Not addition and not multiplication. Look, I can derive every arithmetical operator directly from subtraction. But I cannot do this with any of the other operators. No, you cannot start with addition. NEVER.
Tell me Dannie boy, which axiom did Peano use when he wrote "0 is a natural number" ???
Sum: m - (-n) = m+n
Quotient: m/n m measured in terms of n or n measured in terms of m.
Product: m*n = n / (1/m) or m / (1/n)
As you can see, every operator is defined directly from the primitive operator that is, subtraction.