Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 11:07:52 -0500

mmeron@cars3.uchicago.edu said:
> In article <MPG.1c755dd9be3f89e99896ce@newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
> >mmeron@cars3.uchicago.edu said:
> >> In article <MPG.1c740fca635d36559896af@newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
> >> >mmeron@cars3.uchicago.edu said:
> >> >> In article <MPG.1c72f8224885493d9896a1@newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
> >> >>
> >> >> >
> >> >> >If you say one set has the same "cardinality" but half the "measure",
> >> >> >then you are agreeing that Cantor's measure of cardinality is not a true
> >> >> >measure of the "size" of the set.
> >> >>
> >> >> Eh? Whoever told you that there is such a thing as a single "true
> >> >> measure of the size of a se
> >> >>
> >> >> > By basic set theory it is a
> >> >> >contradiction to say one set is a proper subset containing half of all
> >> >> >the elements of another set, and containing the same number of members.
> >> >> >There are half as many even integers as there are integers, even both
> >> >> >sets are infinite, with the same "cardinality". Cantor's cardinality
> >> >> >measure simply indicates a level of infinitude, not an exact size of the
> >> >> >infinite set. When the cardinality is the same, it means there is a
> >> >> >finite ratio between the sizes of the infinite sets.
> >> >> >
> >> >> Not at all. The cardinality of the set of all rationals is same as
> >> >> this of all integers. What is the finite ratio?
> >> >
> >> >Hmmm good question.... The set of rationals would appear to be the
> >> >square of the size of the set of integers. Perhaps, then, the rationals
> >> >contain |I|^2 members.
> >>
> >> Yes, the rationals have a cardinality of (aleph_0)^2. And,
> >> (aleph_0)^2 = aleph_0. In fact, aleph_0 to any finite power is still
> >> aleph_0.
> >>
> >> > That makes me (continue to) think about what
> >> >Cantor's cardinality equivalence really means.
> >>
> >> You see, here is your problem and it is a common problem to many of
> >> the confused souls posting here. It goes as follows:
> >>
> >> 1) Person P hears some term T (be it from science, mathematics or, in
> >> fact any other specialized area of human activity).
> >>
> >> 2) Lacking a complete definition, P forms for himself some mental
> >> image of what T may mean. Said mental image may be based on the sound
> >> of the word, on the context in which it appeared, on a meaning same or
> >> similar word may carry in some other area, etc.
> >>
> >> 3) P has an encounter with a knowledgeable person who, upon hearing
> >> P's interpretation of T, corrects P saying, "no, you got it wrong, The
> >> way T is defined is..."
> >>
> >> at this point we have a bifurcation, and there are two possibilities:
> >>
> >> 4a) P acknowledges the correction, or asks for an explanation, or
> >> asks for references to study the matter further or anything along
> >> these lines.
> >>
> >> or
> >>
> >> 4b) P decides that his interpretation of T *must* be right (since he
> >> formulated it himself), therefore everybody else must be wrong and
> >> that's it.
> >>
> >> These two possible outcomes demonstrate the difference between
> >> "ignorant" and "arrogant-ignorant". There is no problem with
> >> ignorance, we all are ignorant of various things and we can never
> >> reduce our ignorance to zero, only to progressively diminish it
> >> through willingness to learn and be corrected. "Arrogant ignorance",
> >> also referred to (by me:-)) as "second order ignorance", now that's a
> >> different story. That's what we get when somebody is totally ignorant
> >> of being ignorant (hence "second order") and displays total
> >> unwilliness to contemplate the possibility that he may be wrong. And
> >> when somebody consistently and over an extended period of time
> >> displayes arrogant ignorance, all one can (and should) do is to
> >> completely ignore him. So, which of the two types you're?
> >>
> >> Cardinality is a well defined concept. And that the cardinality of
> >> the set of rationals is the same as this of the integers, this is
> >> rigorously proven. There is no "should be" or "perhaps" about it.
> >> Now, it appears that cardinality doesn't quite correspond to some
> >> concept of "sixe" (more about it below) you've formed for yourself.
> >> So? This doesn't mean that there is anything "wrong" with
> >> cardinality, just that it is something different from what you
> >> thought. That's all.
> >>
> >> As for "true notion of size" there ain't no such thing. And it is not
> >> a problem with infinities, "size" is always context dependent. Assume
> >> you're given a box, 1.5 m long, 1 m wide, 0.8 m high. What is its
> >> size? Is the the largest dimension, the smallest one, the footprint,
> >> the surface area, the volume? Which one? Depending on the
> >> application it can be any or none of the above. So?
> >>
> >> > It still doesn't make me
> >> >stop seeing obvious flaws in the current interpretation.
> >> >
> >> "Tony doesn't like it" does not translate to "a flaw", in the language
> >> of mathematics.
> >>
> >> Mati Meron | "When you argue with a fool,
> >> meron@cars.uchicago.edu | chances are he is doing just the same"
> >>
> >So, you don't see any contradiction in the statement that one line
> >segment contains all the infinite points that a subsegment has, plus an
> >additional infinity of points, and yet it has the same number of points
> >as its own subsegment?
>
> Aha. If you think that it ain't so, then you should be able to count
> the points in both cases and show that one number is different than
> the other one. How do you propose to do it?
>
> > This is not only counterintutive, it's self-contradictory.
>
> Ah, then you should've no problem to prove it wrong, within set
> theory. Or to prove that set theory itself is inconsistent. Go
> ahead, and let me know as soon as you have submitted the paper
> containing the proof to a mathematical journal and it had been
> approved for publication. I'll be delighted to see it.
>
> Just, a hint, the fact that it contradicts your notions of "how it
> should be" does not make it a self-contradiction.
> >
> I'll ask again (and I won't be asking much longer), which of the two
> types you're?
>
> Mati Meron | "When you argue with a fool,
> meron@cars.uchicago.edu | chances are he is doing just the same"
>
You want me to answer whether i am an arrogant ignoramus or just an
ignoramus? You're an ignoramus for expecting an answer to that. Ad
hominem attacks are the sign of a weak argument.

I haven't proposed a solution to cardinality. I have simply pointed out
a problem with it. I never said the problem was one with internal
consistency of set theory, but a problem of consistency between set
theory and what it says about geometric reality. If you could provide a
real world situation where the conclusions of Cantorian cardinality are
demonstrated I would be delighted. It has been agreed on that systems
can be internally consistent, and still be wrong or contradict other
internally consistent systems. Overall, we need universal consistency,
and if the conclusions of that untested isolated system aren't borne out
in the real world, then I see a problem. If you don't you're not
looking, or just don't care. Not my problem. 

-- 
Smiles,
Tony

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