Date: Jan 1, 2013 5:26 PM
Author: Paul A. Tanner III
Subject: Re: A Point on Understanding
In reply to my post
"Re: A Point on Understanding"
in which I outlined (but presented in detail via links to proofs) how Keith Devlin was right, that it is wrong to think that, because of asymmetry in behavior (such behavior as multiplication as repeated addition) in some ringoids (a set under two operations such that the operations are related by a distributive property), thee is a ringoid in which one operation is inherently more fundamental than the other operation,
On Tue, Jan 1, 2013 at 3:28 PM, Joe Niederberger <email@example.com> wrote:
> Your wrongness has to do with your view that one perspective exhausts the subject.
I do not have the view that one perspective exhausts the subject. That's a straw man.
> Addition and multiplication on N thru C are not what they are because those things are classified with many disparate algebraic sructures as ringoids, either.
They are in part. And that is something you admit here:
> Ringoid theory will tell you many things about those structures, but not everything about some particular structure.
It does not have to tell you everything about them.
Your mistake it to think that what can be known to be a fact for all ringoids via the algebra is negated somehow by learning more detail by looking at the structure of some ringoid. Once a fact is established, that's it. One cannot legitimately deny the facts.
> is more to be learned about the particular relationships of those operations in the unique individual structures they are studied in. For example, ringoid theory says nothing about logical derivability of one operation from the other in some particular structure, or about computational perspectives.
And this is irrelevant, since as I showed in my post above including the links to the proofs in question, such facts as there being a set of axioms including some definition of some operation from which we can derive some particular ringoid does not negate the fact that in all ringoids, one operation is not more fundamental than the other one - the fact that in all ringoids, the existence or nonexistence of the asymmetries in question between the operations (including when one operation can or cannot be derived from the other operation) can be shown to be entirely determined by the existence or nonexistence of asymmetries in the algebra.