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

[D. Hugh Redelmeier]

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

| If I can do arithmetic with them, they are numbers.  That certainly
| includes i but not infinity.

For example, you can do arithmetic with matrices: +, -, *, / are
defined on them.  Are these numbers?  (The arithmethic isn't exactly
the same as in natural numbers, so it is up to the observer to decide
if it counts as real arithmetic to him or her.)

IEEE 754 floating point includes infinities.  I was surprised to be
reminded recently that its infinities are not NaNs (non-a-number).

| > | And infitity to the power of infinity makes no sense.
| > 
| > In most systems.
| > 
| > The way we understand infinities today is dominated by Cantor's
| > approach.  It isn't the only consistent version.
| 
| What other option is consistent?  I guess being the commonly used one,
| I am used to Cantor's approach.

Any system with infinities loses some properties that we assumed were
universal.  That usually happens whenever you generalize a numeric
system.  Classic example: complex numbers cannot be ordered in a
useful way.

One simple version of infinity: just one infinity; any operation on
infinity yields infinity.  What do we lose: lots of things, for
example:
	x + 1 != x
Infinity to the power infinity would then be infinity.

| > I'm rather inclusive with the term number, at least sometimes.  I
| > accept lots of groups/rings/fields as sorts of numbers.  Like naturals
| > mod 2 or 3 or 18446744073709551616.

| I am not quite sure how to think of groups/rings/fields as numbers.
| They often consist of numbers though (although they don't have to).

"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.

"The price of metaphor is eternal vigilance."
(Arturo Rosenblueth and Norbert Wiener)
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