Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/21/05 

Date: 21 Feb 2005 20:52:41 GMT

In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
: stephen@nomail.com said:
:> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
:> : stephen@nomail.com said:
:> :> Whoever said anyone other than mathematicians should care about
:> :> cardinality? Tony and others were claiming that the mathematicians
:> :> are wrong and their definitions are contradictory.
:> : My claim was that Catorian cardinality is only a very rough measure of
:> : infinity, and that it doesn't equate to any fine definition of the size
:> : of a set. It only distinguishes between sets that are on the order of a
:> : power set above the other. It's like counting 1,infinity, and skipping
:> : all the other integers and saying they're the same as 1. The problem
:> : here is that the mathematicians among us fall back on Cantor and
:> : cardinality and claim his is the final word, and that there is no
:> : improvement to be done in the area of classifying infinities. If they
:> : haven't heard the math from a book or professor, it's automatically
:> : wrong.
:>
:> You are being dishonest. Several people, myself included, have
:> told you that there are other mathematical definitions that
:> roughly correspond to the notion of "size". Cardinality is
:> just one of them. Measure theory is another. You are free to invent
:> other definitions if you wish, and if others find them
:> useful, they will use them. Of course that will not change the
:> definition of cardinality, or change the fact that cardinality
:> has its uses. You ignored this and instead were just ranting
:> about cardinality contradicting geometric reality. I suppose
:> you just find it more enjoyable and easier to bash mathematicians
:> than to construct a logical alternative.
: Someone mentioned measure theory in passing without elaborating, and it
: didn't sound like it applied to infinite sets anyway. It sounded like
: yet another distraction. If this kind of finer picture of infinite sets
: has been developed, then talk about it. If measure theory has nothing to
: do with infinite sets then don't talk about it. Does it or not?

Google on "measure theory". Measure theory does apply to infinite sets.
One of the nice thing about all the various math terms people
use in these discussions is that their definitions and much more
can be found with a simple on-line search. I wish the same
could be said for the non-math terminology people have been throwing
about.

: I also never said cardinality as a measure contradicted geometric
: reality, but that the equating of cardinality to number of members or
: absolute size of an infinite set is inaccurate, and that mathematicians
: need to distinguish between this measure and the size of an infinite
: set. Just because two sets have the same cardinality does not mean that
: one isn't larger than the other, in the sense that it contains more
: members. You have ignored this repeatedly, regurgitated what you learned
: in school at me, and then run back to Cantor's skirts.

I have never ignored this. I have repeatedly pointed out that
you need to define what you mean when you say "more", or what
you mean when you say "size of a set". "More" is not a precise
mathematical term. "Proper subset" and "same cardinality" are
precise terms. Likewise "size of a set" does not have a precise
meaning. If you mean something other than "cardinality" when
you say "size of a set" you need to precisely define what you mean.

: If you say I said otherwise, quote accurately. From my perspective, you
: are the dishonest, and dense, one.

http://groups-beta.google.com/group/comp.ai.philosophy/msg/e2acc05655d356f2
        "I never said that just because a result is counterintuitive that
         makes it wrong. There are accepted implications of Cantor's
     cardinalities that strike me as not just surprising and
     counterintutive, but contradictory to reality."

As you do nothing but rant below this, I will snip here.

Stephen

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Tony Orlow wrote: ... :>:> cardinality? ... Tony and others were claiming that the mathematicians ... didn't sound like it applied to infinite sets anyway. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :> In sci.math Tony Orlow wrote: ... :>:> cardinality? ... Tony and others were claiming that the mathematicians ... didn't sound like it applied to infinite sets anyway. ...
    (sci.cognitive)
  • Re: Distinct linear orderings on Z
    ... I am not insisting that mathematicians cannot use cardinality to ... basic misunderstanding of infinite sets that it is hard to explain ... > capture what we feel is IMPORTANT about our counting. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... The purpose of a set is to act as a collection. ... >> It applies equally well to any measure of a set, such as cardinality. ... Is it something different for infinite sets? ... >> It signifies an ordering of sets based on whether or not ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... of "number" to the case of infinite sets. ... >mathematicians decided to apply cardinality as just being that as well ... ... who study ordered sets would write these as X = N + (0,1] or something. ...
    (sci.math)