numbers [was Re: understanding probability]

[Lennart Sorensen]

On Fri, Aug 09, 2013 at 02:40:04PM -0400, D. Hugh Redelmeier wrote:
> 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)?

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

> | 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.

> 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)

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).

-- 
Len Sorensen
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