Re: Distinct linear orderings on Z

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/23/05 

Date: Wed, 23 Mar 2005 17:25:17 -0500

"Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
news:87vf7ihnyo.fsf@phiwumbda.org...
> "Allan C Cybulskie" <allan.c.cybulskie@yahoo.ca> writes:
>
> > "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
> > news:87br9cciq5.fsf@phiwumbda.org...
> >>
> >> That it provides counter-intuitive results on infinite sets is not a
> >> particularly negative feature. Our intuitions of the infinite are
> >> much less trustworthy than derived results from a well-motivated
> >> definition.
> >
> > But that doesn't mean they can be just swept under the rug. They
> > need to be explained.
>
> Perhaps, but how hard is the explanation? Our intuitions primarily
> arise from our experiences[1], I imagine, and we have no particular
> experience with infinite collections of objects. Why trust our
> intuitions there?

Well, the issue would be that cardinality is derived from the same
experiences that justify our intuitions, so to attack them should be to
attack cardinality as well. However, I have no problem with saying that
number of elements is derived from experience and that experience doesn't
apply to infinite sets, and so "number of elements" isn't a concept that can
have meaning for infinite sets. And then we can say things this way: For
finite sets, "size" defaults to "number of elements" (which functionally
works the same as cardinality). For infinite sets "size" defaults to
"cardinality" and if you want to refer to "number of elements" you end up
talking about a term not formally defined or formalized.

> I disagree here. As far as I'm concerned, cardinality is the
> obviously *right* abstraction from comparing finite sets. It's not
> just an abstraction from counting. Anytime we are asked to determine
> that two finite sets are the same size, we do so by showing that there
> is a correspondence between the two sets. What else would it mean?

I disagree that this is what we do (but this would be an epistemological
claim, not a philosophy of mathematics claim). For example, one way to
determine if two bags of coins have the same number of elements is to weigh
them, which is not about correspondence or bijections between sets.

>
> But, tell you what. You show me that your analysis of counting gives
> a generalization to number of elements and then it will have borne
> fruit.
>
> Heck, I'd be impressed if you could just show me a real live
> philosopher of mathematics that argues against cardinality as a
> measure of set size. Honest.

Set size isn't the issue here, since for the most part we've agreed that
size can have different meanings, and so cardinality can be one good notion
for set size. It just doesn't seem to be "number of elements".

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