numbers [was Re: understanding probability]

[D. Hugh Redelmeier]

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

| Well infinity is NOT a number as such, at least not the way everything
| else you listed is (well except the aleph ones, which are infinity
| related).

It depends.  The word "number" means slightly different things to
different people, usually with a large and useful overlap.

Certainly to you (with a math degree) the word has a richer meaning
than to the average person.

Transfinite numbers include infinities.  That's kind of the standard
model in math.  Not the only model.

| pi is a specific value, as is e and i.  Sure they are irrational (or
| complex in the case of i) but they are still specific numbers.

The normal use of the English word "number" is unlikely to include
imaginary numbers.  If you include them, where do you draw the line?
Are surreal numbers numbers?  Transfinite numbers?  Hypercomplex
numbers (eg. quaternions)?

| 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.

|  Power of isn't
| defined for things that aren't numbers as far as I know.

In math (not English) one is free to define or redefine whatever you
want, at the risk of
(a) having difficulty communicating with others, and
(b) having a not very useful definition.

Unless you are with Plato in thinking that there is one true class
"number".  Probably not including irrationals.

"God made the integers; all else is the work of man."
or "God made natural numbers; all else is the work of man"
(Kronecker)

My view: There are lots of kinds of things that I would classify as
numbers.  They are a useful abstraction of a bunch of properties of
things in the physical (or imagined) world.  Manipulating numbers lets
us predict or understand properties of things in the physical (or
imagined) world.  Number isn't in the real world, but it is often
about it.

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.

This is an interesting and classic paper on this topic from a
physicist.  The title itself has been influential "The Unreasonable
Effectiveness of Mathematics in the Natural Sciences".
<https://dtrinkle.matse.illinois.edu/_media/unreasonable-effectiveness-cpam1960.pdf>

I like this quotation: ... philosophy is the misuse of a terminology which
was invented just for this purpose.  (Dubislav)
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