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Topic: Non-Euclidean Arithmetic
Replies: 33   Last Post: Sep 21, 2012 2:48 PM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Non-Euclidean Arithmetic
Posted: Sep 13, 2012 6:17 PM

On Thu, Sep 13, 2012 at 10:36 AM, Joe Niederberger <niederberger@comcast.net> wrote:
> Paul Tanner III says:
>>You better, since the idea that repeated addition is merely a property of natural number multiplication was one of his main themes as to how it is not the case that repeated addition is what natural number multiplication *is*.
>
>>You evidently did not actually read carefully enough what he wrote.
>
> I don't recall him ever saying "repeated addition" was an "algebraic property". Quote him.

I said, "property of natural number multiplication" not algebraic property.

He implies very clearly that repeated is a property of natural number multiplication:

http://www.maa.org/devlin/devlin_0708_08.html

Quote:

"For the record

For the benefit of those readers who want to see the details, the axiom systems for the different number systems are: the axioms for complete ordered fields describe the real number system, the axioms for fields describe the rational number system, and the axioms for integral domains describe the whole numbers. You can find discussions of these systems in any contemporary college-level algebra textbook.

Starting with the reals, which are a complete ordered field, if you restrict to the rational numbers you get a field (which, though ordered, is not complete), and if you restrict further to the whole numbers you get an integral domain (which is not a field). The positive whole numbers do not really constitute a number system, and so mathematicians have had no reason to write down axioms to describe them as such. At the turn of the twentieth century, an Italian mathematician called Peano did formulate what are often called the Peano axioms, but their purpose is to show how the positive whole numbers can be defined from first-order logic; they are not a descriptive axiom system that tells you how to work in the system, as are the other axiom systems I just listed.

The point to bear in mind is that, once you have specified the real number system, everything else follows, whole number arithmetic, rational number arithmetic, and all the relationships between the different subsystems. In particular, there is just one kind of number, real numbers, one addition operation, one multiplication operation, and one exponentiation operator (where the exponent may itself be any real number). You get everything else by restricting to particular subsets of numbers. The axioms do not tell you what the real numbers are or what the addition and multiplication operations are; they simply describe their properties vis a vis arithmetic. The axioms for a complete ordered field describe the properties those operations have when applied to all real numbers, the axioms for a field describe the properties the operations have when restricted to the rational numbers, and the axioms for an integral domain tell you how the operations behave when you restrict them to whole numbers.

As I said earlier, I don't think it would be a sensible thing to teach arithmetic by starting with the real number system; indeed, I find it hard to imagine how that could possibly succeed. But since that is the culmination of the arithmetic learning journey, it would be wise to avoid doing anything that runs counter to that final goal system."

Most specifically, look at these taken together:

Quote:

"Starting with the reals, which are a complete ordered field, if you restrict to the rational numbers you get a field (which, though ordered, is not complete), and if you restrict further to the whole numbers you get an integral domain (which is not a field)."

"The point to bear in mind is that, once you have specified the real number system, everything else follows, whole number arithmetic, rational number arithmetic, and all the relationships between the different subsystems."

"The axioms do not tell you what the real numbers are or what the addition and multiplication operations are; they simply describe their properties vis a vis arithmetic."

"The axioms for a complete ordered field describe the properties those operations have when applied to all real numbers, the axioms for a field describe the properties the operations have when restricted to the rational numbers, and the axioms for an integral domain tell you how the operations behave when you restrict them to whole numbers."

Notice the use of the term "properties" applied to the operations and that these properties of the operations derive from the algebraic properties (closure, associative, identity, inverse, commutative, distributive).

Message was edited by: Paul A. Tanner III