Re: [Phys-l] Types of scalars

[John Denker <jsd@av8n.com>]

On 01/01/2011 10:37 PM, Scott Hill wrote:
> I've been reflecting on fundamental mathematics that I gloss over in
> my introductory physics classes, and it occurred to me that there are
> actually three types of scalars that we use in introductory physics:

Only three?  There are dozens of "types" of scalars in common use,
and literally an infinite number of less-common ones.
  Integers
  Non-negative integers
  Positive integers
  Odd integers
  Even integers
  Integers modulo 13
  Infinitely many other finite rings and fields
  Square numbers
  Prime numbers
  Perfect numbers
  IEEE floating-point numbers
  Rationals
  Non-negative rationals
  Positive rationals
  Reals
  Non-negative reals
  Positive reals
  Gaussian integers
  Complex numbers
  Infinitely many other extension fields
  Gödel numbers of things that are true but cannot be proved.
  ... et cetera ..............................

> 1) Never-negative scalars, like mass or length or distance.
> 2) Scalars which can be negative, and the zero is significant: like
> charge or flux or current, or changes in temperature, or vector
> components.
> 3) Scalars which can be negative, and the zero is not significant:
> like temperature (in Celsius) and potential energy.

I'm not so sure about item (3).  I multiply Celsius temperatures
all the time.  Sometimes I multiply them by 1.8 and then add 32.
Similarly I multiply energies all the time.

There is an interesting physical distinction between things that
are gauge invariant (e.g. classical potential energy) and things
that are not (e.g. number of apples in a basket) ... but this has 
got nothing to do with "scalars" as you can see from the fact that
there are plenty of gauge-invariant vectors (e.g. position and 
velocity).

Temperature is not gauge invariant, so it does not belong in the
same category as classical potential energy.

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

I have my doubts about the whole approach.  Sometimes it pays to
use the exact-right kind of number ... but sometimes it doesn't.
For example, in my book, gauge invariance is a property of the
thing being measured, not a property of the number itself.  

To say the same thing another way, don't ask the number system to
do all the work for you.  Sometimes you need to use adjectives or
even complete sentences to describe what you want the number to
mean.

The whole project becomes even more dubious when you realize that
there's not even a clear-cut distinction between scalars and non-
scalars;  for instance a complex scalar can equally well be 
considered a two-dimensional vector over the real number field.


> More importantly, are there NAMES for these three categories?

1) There are positive numbers (and non-negative numbers) of 
various kinds.

3) Certainly gauge invariance has a name, although it is not 
conventional or helpful to define a whole new number system to
express gauge invariance.

> They can be broken up even further if one has a mind:
> 1a) Quantities like mass or length or duration, which can be added and
> subtracted directly.

Supplying units is formally equivalent to supplying basis vectors.  
Technically you don't have scalars at all once you start doing this.
Example  5 apples plus 3 oranges is a vector in a two dimensional
space.

> Too many categories may be more confusing than helpful, however.

Yup.