Re: Axioms for addition

[Jeremy Weissmann]

Dear Eric (and the google group lurking),

   First, let me quote your message in full:

In Gries & Schneider, we encounter the following axiom for addition:

(0)     (E x :: x + a = 0)

I wanted to replace it with the following axiom:

 (1)    a + b = z == a = z + (-b)

One benefit is that we can then prove that a+(-a) = 0 which Gries & Schneider have to postulate.

What do you think of this idea?

(End of quote.)

   So, I'm not exactly sure how to answer your question, because I don't think  (0)  and  (1)  are equivalent, and I don't know what other axioms G&S propose.

   The whole business of what to postulate vs what to prove is a tricky one, because a lot of it can be arbitrary.  The traditional logical view, aligned with philosophy, would want the postulates to be intuitively plausible.  A more sober calculational view would just want the postulates to be useful for developing the theory.

   And that is the heart of it, really:  the distinction of postulate vs theorem is only really relevant in an exposition of a theory.  (Or for a logician studying alternate models, etc.)  Maybe you could present a little fragment of that theory for us here, and show us how  (1)  might make for a smoother exposition than  (0) .   Since  (1)  is a Galois connection and  (0)  is an existential quantification which mentions a specific constant, I have no doubt that you can!

   One final thing:  As a mathematician, and an educator of young people, I find formulations like  (1)  absolutely fundamental .   Unintuitive, but fundamental.  Teachers like to break  (1)  down into something like  "add -b to both sides, apply associativity, use the definition of inverses, use the definition of 0" .   This seems like empty syntactic virtuosity to me.  A competent mathematician knows in their gut that a 7 added on one side of an equation is as good as a 7 subtracted from the other.  Or that a 7 multiplied on one side can be traded for a "divided by 7".   This is effortless symbol dynamics, which we know are crucial to competent calculation.

   Incidentally, do you know what a  "7 minus ..."  can be traded for?  Or a  "7 divided by..." ?

   Looking forward to seeing a simple theory of algebra based on  (1) !

   Best,

+j

[Jeremy Weissmann]

---------- Forwarded message ----------
From: Eric <ee...@yahoo.com>
Date: Wed, Mar 21, 2012 at 9:11 PM
Subject: Re: Axioms for addition
To: Jeremy Weissmann <jer...@mathmeth.com>

Attached are the first steps into a theory of algebra using (1). By the way, I think the trading rules you asked for are

a - b = x  ==  b = a - x

a / b = x == b = a / x

Grace be to you,

eric

---

From: Jeremy Weissmann <jer...@mathmeth.com>
To: Eric <ee...@yahoo.com>; calculationa...@googlegroups.com
Sent: Wednesday, March 21, 2012 12:48 AM
Subject: Re: Axioms for addition

[Jeremy Weissmann]

By the way, I think the trading rules you asked for are

a - b = x  ==  b = a - x

a / b = x == b = a / x

Indeed... that means that  (a-)  is traded for  (a-)  and  (a/)  is traded for  (a/) .   They are self-inverting operations!  I don't think many people are consciously aware of these rules or of their symbol dynamics.

+j