Re: sets of objects, other than numbers ?

From: Julian V. Noble (jvn_at_virginia.edu)
Date: 07/19/04 

Date: Sun, 18 Jul 2004 22:01:27 -0400

ben wrote:
>
> i'm reading the book "Basic Concepts of Mathematics" which is available
> in pdf for free from here <http://www.trillia.com/products.html>.
> pretty good (i'm actually understanding it! :) (so far)).
>
> having stated 9 axioms in these two sections: "Axioms of addition and
> multiplication" (axioms I--VI) and "Axioms of order" (axioms VII--IX)
> the book says:
>
> ...This makes our theory more general. Indeed, our theory also applies
> to any other set of objects (numbers or not numbers), provided only
> that they satisfy our axioms with respect to a certain relation of
> order (<) and certain operations (+) and (*), which may, but need not,
> coincide with ordinary number addition and multiplication. Whatever
> follows logically from the axioms must be true not only for real
> numbers but also for any other set that conforms with these axioms. In
> this connection, we introduce the following definitions.
>
> Definition 1.
> A field F is any set of objects with two operations (+) and (*) (usually
> called "addition" and "multiplication") defined in it, provided that
> these objects and operations satisfy the first six axioms (I--VI)
> listed above. If this set is also equipped with an order relation (<)
> satisfying the additional three axioms VII--IX, it is called an ordered
> field.
>
> i'm just trying to imagine what on earth, or how it could work, a set
> of objects could follow the addition and multiplication axioms but not
> the order axioms? if you can add and multiply them then how could they
> not also follow order? in fact what else apart from numbers follow
> multiplication laws/rules? with numbers, if you say 3 * 2 then you're
> saying three twos. how can that work with anything but numbers? with a
> set of colours: blue multiplied by red :/ the only way to make anything
> like that to work is to assign the colours numerical values in some
> way. so all you've done is mapped a set of colours and a set of numbers
> to each other which makes the colour part irilevent -- you're just
> talking about numbers again.
>
> so how could something be multiplyable, and addable, but not ordered?
> any examples?
>
> i really like the idea of a group of things which aren't numbers being
> manipulated with the axioms which mainly are used with real numbers.
> but surely it does always come down to numbers and nothing else?
>
> thanks, ben.

Matrices, quaternions, linear operators, to name just a few. 

-- 
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
    ^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
   "For there was never yet philosopher that could endure the 
    toothache patiently."  
        -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.