Re: Epistemology 201: The Science of Science

From: Allan C Cybulskie (allan.c.cybulskie_at_yahoo.ca)
Date: 03/13/05 

Date: Sun, 13 Mar 2005 08:33:09 -0500

<stephen@nomail.com> wrote in message
news:d0j3vd$fdl$3@msunews.cl.msu.edu...
> In sci.math Allan C Cybulskie <allan.c.cybulskie@yahoo.ca> wrote:
> :> No. You have to define by what you mean by "less elements".
> :> I already gave you the definition for "same number of elements" that
> :> I am using, which is that there is a bijection between two sets.
>
> : And what is your justification for saying that this has any relation to
any
> : meaning of "less elements" that anyone who does not accept that
definition
> : carte blanche would accept?
>
> : I think that saying "I form this set by taking half the elements from
the
> : superset, so it has less elements" fits everyone's intuitional and
learned
> : definitions of "less elements". I doubt your bijection theory is so
easily
> : justified.
>
> Yes it is so easily justified. In many cases our intutions
> agree with the bijection theory more readily than the subset
> theory. Most people would agree that the set of strings that
> represent octal integers is a subset of the set of strings that
> represent decimal integers, but that the two sets have the same
> size because any integer can be represented in any base.
> Cardinality captures this intuition quite well.

Well, I never used subset theory, as I have said many, many times, so this
is kind of irrelevant. Note that in my analysis I also pointed out that the
definition of the sets also intuitively leads us to believe that the two
sets are of the same size. How did we ever figure out that cardinality
COULD work without appeal to cases where the definition makes it obvious?

>
> Consider a 2 cup measure. The possible values you could
> measure with it are in the range [0,2]. Consider a 1 pint
> measure. The possible values you could measure with it
> are in the range [0,1]. Of course, 2 cups equals 1 pint in the
> English system, and it is clear that any amount that fits
> in the 2 cup measure also fits in the 1 pint measure. There
> are not twice as many measurements possible with the 2 cup
> measure. Similarly, using meters instead of feet does
> not result in fewer possible results. The fact that
> the number of possible measurements is independent of the
> units used is quite intuitive.

The example is also totally wrong. I can get far more accuracy of
measurement from the 2 cup example than the pint example. Why? Because
there are more numbers that I can retrieve from the 2 cup measure than the 1
cup measure, allowing me to subdivide the measure further than I can with
the pint measure. Yes, indeed there are different measures that I can get.
Yes, in theory every measure is represented in both sets, but the fact of
the matter is that I have twice as many numbers to use in measuring with
cups than with pints. Only when you play with infinity does your example
even come close to working ... but that's what we are arguing over, so it
doesn't count [grin].

> : I argued it on the basis of the definition of the set, as I said. This
is a
> : concept that you are having much difficulty with for some reason.
>
> : First, since I did not necessarily accept that a proper subset has less
> : elements than its superset, you cannot claim that I was justifying my
claim
> : on the basis of the generalization about proper subsets, which is what
you
> : seem to want to do -- and tried to do. Now, it turns out that, in this
> : case, appealing to the definition of the sets ends up in an argument
that
> : sounds a lot like the definition of proper subset ie that all of the
> : elements that are in the set [0,1] are also in the set [0,2], but there
are
> : extra elements -- precisely twice as many, in fact -- in the set [0,2]
(the
> : ones from [1,2]). And this meets the intuitional and definitional
> : requirements for "more number of elements" in most definitions. But
that
> : does not mean that I ever claimed that that just WAS the case. It just
so
> : happens that in this case the proper subset we are talking about really
DOES
> : have less elements than the superset. But I never claimed that that was
> : generally true, nor based my reasoning on that fact.
>
> Your justification for why [0,2] has more elements than [0,1]
> still sounds like "[0,1] is a proper subset of [0,2]".

Do you read posts? Didn't I just say above that it sounds a lot like the
proper subset argument?

> It is like saying "The reason Bob cannot drive is because
> he cannot see, not because he is blind." Honestly I do not
> see what distinction you see.

The distinction that is critical is that I am saying that while Bob, being
blind, may not be able to see to drive a car I am not claiming that there is
no blind person who could possibly drive a car. You wanted to conflate my
argument to the proper subset one to trot out the octal and decimal strings
example and catch me in a contradiction. Yet I do not fall into that
contradiction because it is as obvious by definition that two sets that
reflect all possible integers have the same number of elements that if one
set is formed by taking precisely half the elements of the other that it has
less elements than the other set.

>
> Does [0,1] have more elements than (0,1)? The first includes
> 0 and 1 and everything in between, the second includes everything
> between 0 and 1 but does not include 0 and 1. [0,1] contains
> everything in (0,1), but there are extra elements. In this
> case, not twice as many, but just two more.

And yes, it does have more elements.

> How about [0,1)
> and (0,1)? [0,1) contains just one extra element. Does
> it have more elements?

Yes.

> If so, can you give me an example
> of a set of numbers and a proper subset that do contain
> the same number of elements?

I'm not sure, but since I never claimed that it had to I don't have to, now
do I?

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