Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/19/05 

Date: 19 Mar 2005 17:37:37 GMT

In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:

: <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.

No. It says they have the same cardinality. The term "same number
of elements" does not have an official mathematcial definition.

:>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?

No, it does not count. How do you know if two sets have the same
number of elements? You need to provide a definition that lets
somebody determine if two sets have the same number of elements or
not.

:>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.

Well if you would ever tell me why you think [0,2] has more elements
than [0,1] without mentioning subsets than I will stop bringing it up.

: 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.

The "size" of the sets of octal and decimal strings remain the same
no matter how I define them, and there are many ways to define the same set.
I could "form" the octals as follows: start with the set of all
decimal strings and remove every string that contains an 8 or 9.
Does this mean that the set of octal strings now has fewer elements than
the set of decimal strings? Afterall there are an infinite number of
decimal strings containing an 8 or 9.

I can "form" [0,1] by taking [0,4] and dividing every element by 4.
I can "form" [0,2] by taking [0,4] and dividing every element by 2.
Do [0,1] and [0,2] now have the same number of elements? Or
I could imagine a world with infinitely precise measure cups and
consider all the amounts that fit in a 2 cup measure and all the
amounts that fit in a 1 pint measure. :)

Sets are timeless abstractions. They do not "form", and there are
many equivalent ways to define the same set. It is analogous to
the fact that 5=2+3, but also 5=8-3, and 5=1*5, etc. A general
definition of "size" for sets should be an intrinsic property of
the set and not depend on the particulars of how the set was
defined, ordered or interpreted.

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
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