[GW-C] Re:numbers [was Re: understanding probability]

[D. Hugh Redelmeier]

| From: Lennart Sorensen <lsorense-1wCw9BSqJbv44Nm34jS7GywD8/FfD2ys at public.gmane.org>

| If the matrices have contents you can do arithmetic on, then you can do it
| on the matrices too.

As explicitly as I can say it:

- what does "arithmetic" mean to you?

- what does "number" mean to you?

Surely your answer should be with respect to an algebraic ofject
having elements and operators on those elements.  Just talking about
operators and not elements or elements and not operators cannot be
rigorous enough to make your meaning clear and specific.

|  I am not sure you can multiply two matrices that are
| full of letters without first defining what it means to multiply letters.

Letters?  Do you mean formal variables, as in polynomials?  Or as
elements in an algebra?

| > IEEE 754 floating point includes infinities.  I was surprised to be
| > reminded recently that its infinities are not NaNs (non-a-number).
| 
| Their infinities are still invalid to do calculations with.  You can
| only really use them to compare against (not that that makes that much
| sense either).

The brains behind 754, William Kahan, certainly had clear (and
complicated) ideas behind these infinities.  He had plenty of uses for
them.  I've always thought that he wanted too many properties from
numerical hardware and elementary functions, overconstraining the
implementation.  But his talks are pretty entertaining and convincing.

Amazing (to me) story about Kahan.  He showed that the original
IBM/360 floating point hardware specifications yielded significantly
lower precision than could be achieved.  IBM took his advice and
changed the specs and retrofitted a "guard digit" into every /360 in
the field!  This was not a trivial reflash-the-firmware kind of thing.
It must have cost a fortune.

| Not very useful,  Seems simpler to just not allow doing operations on it.
| It's a great limit, but not a great number.

If I remember correctly, Kahan wanted infinities so that he could
safely write strraight-line code for (for example) elementary
operations.  Eliminating conditionals makes the code simpler and
faster.

| > "They often consist of numbers" is a pretty strong statement.  Do you
| > mean "their elements can be denoted by numerals"?  That's more
| > conservative.  Essentially, your original statement seems to imply
| > that those groups are numbers.
| 
| I am sure someone has defined what it should mean.  Most groups I have
| seen have elements that are numbers.

Are the elments of Z mod (2) numbers?  Or are the elements the same as
(isomorphic to) Heads and Tails, or Naughts and Crosses, or...

It's not reasonable to say that a number is anything written down
using only digits.
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