Re: Epistemology 201: The Science of Science

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/19/05 

Date: Sat, 19 Mar 2005 07:19:28 -0500

<stephen@nomail.com> wrote in message
news:d1gdr8$2luh$1@msunews.cl.msu.edu...
> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
>
> : Hmmm. So you're telling me that someone stating that a certain theory
is to
> : be preferred in mathematics or science simply because it is more
interesting
> : is something that you are okay with?
>
> Why would I have a problem with that, especially with regards to
> mathematics? Science is constrained by having to make predictions
> about the physical world, but mathematics has no such constraint.
> Consistency is the only constraint, and some people apparently even
> tinker with that one.

Apparently, since this is partly what the debate was about [grin].

I don't know, but generally the mindset that differentiated fields like
science, engineering, computer science, and mathematics from fields like
sociology, philosophy, and social sciences was the idea that simply because
an idea sounded cool wasn't the ultimate arbiter in whether the idea had any
value or not (I think that SHOULD be wrong about philosophy, but concede
that a lot of the time it is). It seems to me that this idea is what people
like Wolf, at least, rail against in philosophy. If you don't share that
opinion, then I apologize.

>
> : People are
> :> free to define whatever terms they find interesting, and
> :> if others too find them interesting, they will use them. However
> :> that will not change anything about existing definitions.
> :> Other definitions of "size" exist for infinite sets but that
> :> does not change the definition of cardinality or the interesting
> :> implications of cardinality.
>
> : Then I think you have missed the objections that at least I was making,
> : since my objection was NEVER to the definition of "cardinality", but to
the
> : conflation of that to simply "number of elements", and as such those who
> : claimed that number of elements seemed different by the more common
notion
> : of that were at least somewhat misguided.
>
> As I have repeatedly said, you need to define what you mean by
> "number of elements". If you do not have a definition for
> "number of elements", how do you know that cardinality is not it?

Because it is clear that what I mean by "number of elements" does not simply
mean "bijection", as per the fact that I have pointed out that claiming that
the set [0,1] and the set [0,2] have the same number of elements.

What I mean by "number of elements" is the classic and traditional
definition of "number of things in the set". And it seems clear to me that
even though we can't count them in an infinite set, we can still reasonably
claim that if one set contains exactly half of the elements in another set,
even if infinite the relative number of elements has to be half of the other
set. Claim that this is relying on a proper subset argument if you like,
but that would merely be a shallow view of what I'm doing.

> You and others keep implying that there is some "real" definition
> of "number of elements" that differs from cardinality, but you
> seem unable articulate what that definition is or why it is "real".

The only argument I've made on that point is that we are using the
definition of number of elements that gave rise to the bijection approach
that you posit, and so it seems odd to conflate the bijection approach in a
way that contradicts that initial definition. How did we know that it was
the case for finite sets that you could only map one set onto another if
they had the same number of elements? We counted the finite set in
accordance with the definition of "number of things in the set" and
discovered this. If that notion conflicts with the bijection or mapping
approach, why should the mapping approach be considered reasonable? Doesn't
it violate its own basis?

> If you can come up with a consistent and interesting definition
> of "number of elements" that applies to infinite sets then people
> might find it interesting and useful. If it is interesting and
> useful enough, it might even replace cardinality as the default
> meaning of "number of elements".

But cardinality is NOT the default meaning of "number of elements" EXCEPT
among mathematicians who hold that it just IS the number of elements. I
guarantee you that most people who do not hold that axiom who are asked if
there are more integers than even integers will agree that there are more
integers -- as I have attempted to show by appealing to our intuitions about
counting.

>
> : If you want to play with cardinality in that way, be my guest. Just
don't
> : then imply that I can't reject the idea that cardinality just means
"number
> : of elements". And remember that I repeatedly gave you the opportunity
to
> : claim the bijection theory as an axiom that others could reject, and
that
> : you failed to take it.
>
> I do not know what you mean by "the bijection theory". It is not
> a matter of "theory" whether or not a bijection exists between
> two sets or not.

The bijection theory that says that even for infinite sets if there is a
bijection between them they have the same number of elements. Let's not get
off track.

>If you are going to argue about what
> the "size" of a set really is, then you need to define mathematically
> what you mean by the "size" of a set.

And thus you conflate the method for calculating the number of elements with
the definition of number of elements itself. Or else why insist that it be
"mathematically"? Does "number of things in the set" not count?

>So far you have claimed
> that sometimes two sets can have the same "size" even if
> one is a proper subset of the others, and at the same time have
> claimed that proper subsets cannot be the same size as their
> proper superset (although you never specified the reason).

I NEVER claimed the latter, and have consistently denied that that is what I
am saying. So please stop harping on that.

I appeal to the definition of sets. In the octal and decimal string case,
you defined them as representing all integers. Therefore, if there is a
proper subset relation there, then by definition they must have the same
number of elements. In the other case, I argued that if you would form the
set by taking precisely half the elements that are in the other set, then by
definition again it must have half the elements of the superset, since for
every element in the subset there are two in the superset. I fail to see
why you have such trouble understanding the notion of using the definition
of sets to figure this out.

Relevant Pages

  • Re: Attempts to Refute Cantors Uncountability Proof?
    ... produce an explicit bijection. ... there is a bijective mapping between the set of rational ... onto infinite sets. ... However, in a field like mathematics, ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... be preferred in mathematics or science simply because it is more ... Science is constrained by having to make predictions ... Claim that this is relying on a proper subset argument if you like, ... definition of number of elements that gave rise to the bijection approach ...
    (sci.physics)