Re: [Phys-l] Types of scalars

[John Denker <jsd@av8n.com>]

On 01/04/2011 07:11 AM, Moses Fayngold wrote:
>    I think that in discussing this topic, we have to agree on some conventional
> definition of the notion of scalar. In one such definition a "scalar" is synonym
> of an "invariant". 

But that's not the definition.  Never has been.  Never will be.

Not even the examples given below conform to this "definition".

> If this is accepted, 

But what if it isn't accepted?

> then it is relatively straightforward to
> indicate types of scalars by the kind of transformation under which they remain
> invariant. But this taxonomy is rather fuzzy since a characteristics which is a
> scalar under one kind of transformation may be not a scalar under another
> transformation. Here are some familiar examples:
>
> 1)  a vector component - a scalar under translations, not a scalar under
> rotations or inversions (reflections)

Hmmmm.  A velocity vector is invariant with respect to translation.
Does that mean velocity is a vector?

On the other side of the same coin, the X-component of position is
not invariant with respect to translation.  Does that mean it is not
a scalar?

Oh, and what about scalars in thermodynamic state-space.  The scalars
exist, even though we don't have any notion of 3-rotation, let alone
4-rotation.

The whole idea of scalar=invariant is Dead on Arrival.

=======================

More generally:  As previously mentioned, you shouldn't expect the number
system to do all the work for you.  The whole point of mathematics is
that it is abstract.  This allows us to use the *same* type of numbers
to mean different things in different situations.  Numbers are re-usable.
We can count apples with the *same* type of numbers we use for counting
oranges.

Inventing new "types of scalars" -- in situations where the old types work
fine -- is profoundly misguided.