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.)