Date: Jan 1, 2013 12:09 PM Author: Paul A. Tanner III Subject: Re: A Point on Understanding On Sun, Dec 30, 2012 at 8:27 PM, Joe Niederberger

<niederberger@comcast.net> wrote:

> Paul Tanner III says:

>>This asymmetry in the non-dual distributive property that "connects" the two binary operations of this type of ringoid fully explains the asymmetry between the binary operations we see in such examples of such a ringoid as the real numbers and its subsets like the integers.

>

> I've mentioned before Tarski's axioms for real numbers: they say nothing outright about either multiplication (existence and properties thereof are proved as theorems later), or (obviously) distributivity.

>

> So, under Tarskian real numbers, Paul, you are wrong, again.

>

> Cheers,

> Joe N

No. I am not the one wrong here. Those who hold to the idea that

"addition" is inherently fundamental to "multiplication" because of

some asymmetry in their behavior in some ringoid are the ones who are

wrong. This notion that there exists a ringoid in which one of the

operations of the ringoid is fundamental to the other is a mirage

caused by the defining properties of the ringoid. Just because Tarski

derived one operation from assuming the other does not mean that one

of the operations is fundamental to the other. Read on to see why.

Fact: A ringoid is simply a set under two binary operations such that

one of them distributes over the other. This distributivity can go in

one direction only, in which case we can call it non-dual distributive

property, or it can go in both directions, in which case we can call

it a dual distributive property. In the non-dual case, if one wants to

define the operation being distributed over as being fundamental to

the other one simply because it is the one being distributed over,

then that's one thing, but it's another thing to say that the one

being distributed over is inherently fundamental to the other one.

(Notice that I did not name these operations. This is because the

operations are named as they are only by convention. There is nothing

magical about the word "addition" so as to make the operation of a

ringoid named in such a way fundamental to the other one named

"multiplication".)

Fact: Tarski's axioms assume addition among things and yield a ringoid

under a non-dual distributive property, this particular yielded

ringoid under the non-dual distributive property being the set of real

numbers. (See

http://en.wikipedia.org/wiki/Tarski's_axiomatization_of_the_reals

for more.) That is, they yield a Dedekind-complete ordered field,

which is isomorphic to the set of areal numbers. (See

http://en.wikipedia.org/wiki/Ordered_field

for more.) What Tarski did was made possible by the defining

properties of the target object to be derived, a ringoid under a

non-dual distributive property, and the non-dual nature of this

property is the key as to why there we have the mirage in question.

(Can Tarski's axioms yield a ringoid under the dual distributive

property?)

Fact: For every ringoid that is under the dual distributive property,

because of the symmetry of the dual distributive property, the

asymmetries in question do not exist unless there some set of axioms

for some such ringoid that defines an asymmetry or from which we can

derive an asymmetry. (The reason they do not exist is because we can

perform the derivations in question for the non-dual distributive

property context in both directions, not in just one direction. I give

an example below.)

Fact: For every ringoid that is under the non-dual distributive

property, because of the asymmetry of the non-dual distributive

property, the asymmetries in question exist and we can derive the

asymmetries in question using only algebraic properties.

To see these derivations, see my post

"Re: A Point on Understanding"

http://mathforum.org/kb/message.jspa?messageID=7945385

in which not only to I quote what Devlin actually said, I link to my post

"Re: Bringing the Discussion to Order"

http://mathforum.org/kb/message.jspa?messageID=7043327

in which I derive the behavior of multiplication as repeated addition

for every ringoid under the barest minimum of algebraic properties,

and adding more and more properties until we reach a field, that

barest minimum starting with the ringoid having only a multiplicative

identity, where, for all b in a specific subset of the set of all sums

in the ringoid and for all a, I derive in that minimal context the

equality

ab = a + a,

multiplication behaving as repeated addition.

In fact, even just the distributive property itself, which is the only

algebraic property that is part of the definition of a ringoid, is a

form of one operation as another operation.

For every ringoid that is under the dual distributive property and

that contains (ever merely) an additive identity, we can derive

a + b = aa,

addition behaving as repeated multiplication, using the same method I

used in that post above but with things appropriately reversed.

To sum up: For any ringoid (including the set of real numbers), there

is an asymmetry in the behavior of the two operations if and only if

the defining properties of the ringoid are such that they define some

asymmetry in question or are such that we can derive some asymmetry in

question. (The defining property most important in this regard is the

defining property of the distributive property: If it is an

asymmetrical non-dual distributive property, then we have the

asymmetries in question, and if it the symmetrical dual distributive

property, then we do not have the asymmetries in question unless there

is some additional defining properties of the given ringoid that

defines some asymmetry in question or from which can derive some

asymmetry in question.)