Re: Distinct linear orderings on Z

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 03/23/05 

Date: Wed, 23 Mar 2005 07:32:15 +0100

"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?

Alternatively, most of the counter intuitive results are something
like: For all finite sets x, P(x) but *not* for all infinite sets x.
Again, how hard is it to understand that infinite sets can be
different in this respect?

>>
>> That it is not "justified" as a notion of number of elements is
>> false. It is an obvious extension of the equivalence relation
>> generated by counting elements of finite sets. There *is* a clear
>> philosophical argument for cardinality as a notion of set size. If
>> you doubt that argument, then you need to do more than complain about
>> its counter-intuitive consequences.
>
> I've already commented that while counting can be said to be functionally
> equivalent to doing a mapping, that isn't the purpose or importance of
> counting, and also that there are other ways to determine number of elements
> that we consider perfectly valid that do not require that mapping, and also
> that mapping onto the set of integers is violated whenever we count by more
> than one (it would have to be the even integers in the case of counting by
> twos). Basically, it is odd to relate counting directly to mapping onto the
> set of natural numbers when we care nothing about the set we are supposedly
> mapping it onto, and need not even recognize that a set is involved. It
> seems much more likely that all we are going is using the successive numbers
> as placeholders in memory to avoid "losing our place", especially
> considering that when we lose our place we generally end up messing up the
> counting. But that's not an indication that we work directly with "sets"
> instead of individual numbers.
>
> So this philosophical argument seems to fall through as a real justification
> ... and this is BEFORE we ask if cardinality itself can apply in the same
> way to both finite and infinite sets.

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?

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.

Footnotes:
[1] I do not intend to really commit myself on the nature and origin
of intuitions here. 

-- 
"[Criticizing JSH's mathematics will result in] one of the worst debacles in
the history of the world.  It is foretold in most mythologies and religions.
And yes, you are the ones, the cursed ones, who destroy the world."  
    --James S. Harris reads from the Aztec Book of the Damned Mathematicians

Relevant Pages

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