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| Mike Kelly |
 Posted: Mon Sep 18, 2006 10:54 am |
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Han de Bruijn wrote:
[quote:1d618df49d]Mike Kelly wrote:
I want to snip all this but Mike doesn't like it:
Han.deBruijn@DTO.TUDelft.NL wrote:
Mike Kelly wrote:
[ ... snip ... ] It's not clear to me that providing finite examples then
saying "obviously this holds for infinite cases too" without any
justification whatsoever should be at all convincing to anyone.
It may be not clear to any mathematician, but it is clear to any
scientist. The reason is that infinities do not really exist.
They only exist as an attempt to make the "very large" rigorous
in some sense. The moment you forget this, you get into trouble.
But we are discussing whether there exists a uniform distribution over
the naturals. If you don't think this claim means anything at all then
why do you dispute it? If you reject the existence of the set of
natural numbers then you reject the set theory probability is based on.
So why bother to argue against individual theorems? You don't accept
*any* of probability theory.
It seem your argument is based on the idea that infinites do not exist
in physical reality. But mathematics is abstract, so this seems an
absurd objection.
End of virtual snipping.
If you refuse the idea of infinite sets, what does it mean to you to
say a function has domain and range R?
I don't really refuse the idea of infinite sets. (How else could I do
my calculus stuff ?) I only refuse the idea that the infinite can be
something which is essentially different from the _finite_.
[/quote:1d618df49d]
Then you refuse the idea of infinite sets. Infinite *means* "not
finite". To claim it means "finite" is just boneheaded.
[quote:1d618df49d]In short:
infinity is just finity in disguise.
[/quote:1d618df49d]
No.
[quote:1d618df49d]That's why I have no problem with limits. But I _have_ a problem with
aleph_0. No, I have no problem with R, because I can add uncertainity
to its members, and then the reals become just floating point numbers
in the PC on my desk. Reals are very handsome idealizations of floats.
But aleph_0 is not an idealization and it's not even handsome ...
[/quote:1d618df49d]
Bully for you.
--
mike. |
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| Han de Bruijn |
| Posted: Mon Sep 18, 2006 10:57 am |
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Mike Kelly wrote:
[quote:701c49adbf]Han de Bruijn wrote:
imaginatorium@despammed.com wrote:
_How_ would you draw a ball from a vase containing an infinite set of
balls.
Yaaawn! This has been discussed, at length, as well:
http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv
Why provide a link that is completely irrelevant to the question asked?
[/quote:701c49adbf]
Irrelevant? Don't think so. Or did I land on the wrong planet?
Han de Bruijn |
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| Mike Kelly |
| Posted: Mon Sep 18, 2006 10:57 am |
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Han de Bruijn wrote:
[quote:f1bbf74da7]Mike Kelly wrote:
Set theory doesn't claim to subsume all of math. People use it in
(almost) every area of math because it works extremely well.
Huh, huh. I use set theory almost nowhere and THAT works extremely well.
[/quote:f1bbf74da7]
You don't do mathematics. You use calculus in physics.
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mike. |
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| Mike Kelly |
| Posted: Mon Sep 18, 2006 10:59 am |
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Han de Bruijn wrote:
[quote:aafbe71754]Mike Kelly wrote:
Han de Bruijn wrote:
imaginatorium@despammed.com wrote:
_How_ would you draw a ball from a vase containing an infinite set of
balls.
Yaaawn! This has been discussed, at length, as well:
http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv
Why provide a link that is completely irrelevant to the question asked?
Irrelevant? Don't think so. Or did I land on the wrong planet?
Han de Bruijn
[/quote:aafbe71754]
You link to a (misleading) summary of the Vase+Balls thread. Maybe you
could point out what part of that link refers to randomly selecting a
ball from a countably infinite collection? Or did you snip so much
context that even you don't know what you're replying to?
--
mike. |
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| Mike Kelly |
| Posted: Mon Sep 18, 2006 11:03 am |
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Han de Bruijn wrote:
[quote:31dd36a28a]Virgil wrote:
In article <1158489723.269348.27860@e3g2000cwe.googlegroups.com>,
Han.deBruijn@DTO.TUDelft.NL wrote:
What's wrong with mathematics ?!
Nothing!!
"Mathematics should be a science" is the answer.
[/quote:31dd36a28a]
Why?
--
mike. |
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| Mike Kelly |
| Posted: Mon Sep 18, 2006 11:04 am |
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Han de Bruijn wrote:
[quote:de76e1253f]Virgil wrote:
As HdB has not been able to counter any of the mainstream arguments to
the satisfaction of any but himself, they are sufficient.
Never underestimate the strength of your opponent.
And the influence of 'sci.math' as a free forum.
Han de Bruijn
[/quote:de76e1253f]
Channelling James Harris?
--
mike. |
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| Guest |
| Posted: Mon Sep 18, 2006 11:54 am |
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Mike Kelly schrieb:
[quote:7e8ed96681]
The mathematics of
the infinite can only be derived from the mathematics of the finite
(because nobody has an idea what "the infinite" is).
Don't extrapolate from yourself so harshly!
[/quote:7e8ed96681]
I talked to a lot of first-rate mathematicians. This in parentheses is
a qoute from many of them.
[quote:7e8ed96681]
What's interesting to me here is that your statement seems rather
Platonic in that it asserts the existence of some "the infinite" and
"the finite" the mathematics of one of which can be observed directly
by humans and one of which cannot. Yet earlier you were arguing against
a literal interpretation of the "existence" of numbers. What's changed?
[/quote:7e8ed96681]
Nothing has changed. There is no complete set of natural numbers. Any
set that can be established is a finite set. Hence, the probability to
select a number divisible by 3 is 1/3 or very very close to 1/3.
[quote:7e8ed96681]
Otherwise the
limit of the sequence 1/n might be 100. Nobody could prove that false.
Babble.
[/quote:7e8ed96681]
No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at
least as close to 2 as we like), not by definition and not by any
axiom, but by rational thought. And the same kind of extrapolation is
appropriate if we investigate the infinite, be it the sequence 1/n or
the "bijection" N <--> Q.
Regards, WM |
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| Guest |
| Posted: Mon Sep 18, 2006 12:01 pm |
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Han.deBruijn@DTO.TUDelft.NL schrieb:
[quote:805790f08b]Mike Kelly wrote:
You claimed that you have a very much better understanding of
probability than me. Since you know nothing of my knowledge of
probability other than that I disagree that it is meaningful to discuss
the probability of "a natural" being divisible by 3, [ ... snip ... ]
What more evidence do we need, huh?
The good news is that you are doing wrong only _one_ thing: infinitary
reasoning. You think that completed infinities do exist. Once you stop
thinking this way, everything falls in its place and you will see that
it is quite meaningful to discuss the probability of "a natural" being
divisible by 3.
[/quote:805790f08b]
Bravo. That's it.
Regards, WM |
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| Randy Poe |
| Posted: Mon Sep 18, 2006 12:04 pm |
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mueckenh@rz.fh-augsburg.de wrote:
[quote:304795d8d5]Mike Kelly schrieb:
The mathematics of
the infinite can only be derived from the mathematics of the finite
(because nobody has an idea what "the infinite" is).
Don't extrapolate from yourself so harshly!
I talked to a lot of first-rate mathematicians. This in parentheses is
a qoute from many of them.
[/quote:304795d8d5]
Is that your idea of a citation?
- Randy |
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| Guest |
| Posted: Mon Sep 18, 2006 2:33 pm |
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Randy Poe schrieb:
[quote:348ee4b661]mueckenh@rz.fh-augsburg.de wrote:
Mike Kelly schrieb:
The mathematics of
the infinite can only be derived from the mathematics of the finite
(because nobody has an idea what "the infinite" is).
Don't extrapolate from yourself so harshly!
I talked to a lot of first-rate mathematicians. This in parentheses is
a qoute from many of them.
Is that your idea of a citation?
[/quote:348ee4b661]
It is my idea of trust and honour. I do not publish names of my private
correspondents.
Regards, WM |
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| Guest |
| Posted: Mon Sep 18, 2006 2:41 pm |
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Mike Kelly schrieb:
[quote:7e004341b4]Han.deBruijn@DTO.TUDelft.NL wrote:
Mike Kelly wrote:
[ ... snip ... ] It's not clear to me that providing finite examples then
saying "obviously this holds for infinite cases too" without any
justification whatsoever should be at all convincing to anyone.
It may be not clear to any mathematician, but it is clear to any
scientist. The reason is that infinities do not really exist.
They only exist as an attempt to make the "very large" rigorous
in some sense. The moment you forget this, you get into trouble.
But we are discussing whether there exists a uniform distribution over
the naturals.
[/quote:7e004341b4]
Please make just the experiment. Choose at random 30 natural numbers
from the whole set N. What is the result? How many of these 30 numbers
are in fact divisible by 3? (In case you have problems with large
numbers: It is easy to check the divisibility of the number by checking
the divisibility of the sum of its decimal digits.)
Now, it there a distribution lacking, or is the complete set of natural
numbers lacking?
[quote:7e004341b4]If you don't think this claim means anything at all then
why do you dispute it? If you reject the existence of the set of
natural numbers then you reject the set theory probability is based on.
[/quote:7e004341b4]
In order to calculate probability we do not need set theory. Pascal and
Fermat, for instance, did it without set theory very well. Your result
shows only that set theory is not useful in any branch of useful
mathematics.
[quote:7e004341b4]So why bother to argue against individual theorems? You don't accept
*any* of probability theory.
[/quote:7e004341b4]
We have a better probability theory.
[quote:7e004341b4]
It seem your argument is based on the idea that infinites do not exist
in physical reality. But mathematics is abstract, so this seems an
absurd objection.
If you refuse the idea of infinite sets, what does it mean to you to
say a function has domain and range R?
[/quote:7e004341b4]
As an argument you can choose any real which you really can choose.
See the experiment above. You don't really believe that you can choose
a natural from the whole set N, do you? But if so, what then is N god
for in probability theory (and elsewhere)?
Regards, WM |
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| MoeBlee |
| Posted: Mon Sep 18, 2006 3:07 pm |
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Tony Orlow wrote:
[quote:60e3397224]MoeBlee wrote:
Tony Orlow wrote:
David R Tribble wrote:
Tony Orlow wrote:
Yes, I am including infinite values on the number line, since it's
"infinitely long".
David R Tribble wrote:
Yet another thing you have to define or prove. Where do these
"infinite values" appear on your real number line?
Tony Orlow wrote:
Further from 0 than any finite number.
Then how are they "on" the same "line"?
By trichotomy. For all x ad y on the line, either x>y, x=y or x<y.
That's what makes a line in concept.
That's question begging. Indeed, trichotomy is necessary for an
ordering to be linear. But you have not proven the existence of such an
ordering with the values you claim to be in its field. Just saying that
your claim follows from trichotomy is just assuming what you are being
asked to show, viz. that there is such a linear ordering with the
values you claim to be in its field. But more basically, it doesn't
matter, since you have no axiomatization nor rules of inference upon
which to prove anything whatsoever in a mathematical system.
Well, MoeBlee, I have been trying to get that axiom together so that it
properly ties together count with measure, and trichotomy is one of the
axioms that defines the values along the line. If such a rule is
declared for all pairs of members of a set, then that can be considered
the definition of a linear set,
[/quote:60e3397224]
I'm trying to get an axiom that construes identity with position, and
transitivity is one of the axioms that defines iterations in a plane,
and if this rule is posited for all singletons of subsets, then that
can be a definition of a rectangular set. Now, the question is why
think that anyone would care more about your gibberish than about the
gibberish in the previous sentence?
[quote:60e3397224]be it what you would call a sequence, or
a dense linear set like a real interval.
[/quote:60e3397224]
'linear ordering' has a set theoretic definition that does not require
sequence, density, nor interval.
[quote:60e3397224]I guess when I finally publish a set of axioms, then the subject will
suddenly "matter". Funny how that works in transfinitology. My axiom set
must be some sort of limit ordinal or something.
[/quote:60e3397224]
Sure, Tony.
MoeBlee |
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| Randy Poe |
| Posted: Mon Sep 18, 2006 3:12 pm |
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mueckenh@rz.fh-augsburg.de wrote:
[quote:f6537eae62]Mike Kelly schrieb:
Han.deBruijn@DTO.TUDelft.NL wrote:
Mike Kelly wrote:
[ ... snip ... ] It's not clear to me that providing finite examples then
saying "obviously this holds for infinite cases too" without any
justification whatsoever should be at all convincing to anyone.
It may be not clear to any mathematician, but it is clear to any
scientist. The reason is that infinities do not really exist.
They only exist as an attempt to make the "very large" rigorous
in some sense. The moment you forget this, you get into trouble.
But we are discussing whether there exists a uniform distribution over
the naturals.
Please make just the experiment.
[/quote:f6537eae62]
Using what distribution?
[quote:f6537eae62]Choose at random 30 natural numbers
from the whole set N. What is the result?
[/quote:f6537eae62]
OK, I'll use a distribution in which the numbers divisible by 3
occur with probability 0.5.
[quote:f6537eae62]How many of these 30 numbers
are in fact divisible by 3?
[/quote:f6537eae62]
About half of them, i.e., 15.
- Randy |
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| MoeBlee |
| Posted: Mon Sep 18, 2006 3:21 pm |
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Tony Orlow wrote:
[quote:556c4317c9]MoeBlee wrote:
Tony Orlow wrote:
Unbounded but finite may
be considered potentially, but not actually, infinite.
That will be jiffy, once you give axioms and/or our definitions for
'potentially infinite' and 'actual infinite'. Until then, it's pure
handwaving.
MoeBlee
As I said, a potentially infinite set is unbounded, but will all element
indices finite. It's "countable" in standard parlance. An actually
infinite set includes elements with infinite element indices, like 1/3
has in the decimal reals. 
[/quote:556c4317c9]
Usually, 'countable' means bijectable with w or some element of w. On
the other hand, sometimes, 'countable' is used to mean bijectable with
w. When I use 'countable', I mean the former definition, and I use
'denumberable' to mean bijectable with w.
So your 'potentially infinite' just means denumerable, I guess. As to
your 'infinite element indices', I would need a definition from you.
But it is no help for you to give a definition of 'infinite element
indices' if you define it with yet MORE undefined verbiage. None of
your definitions matter, since you never say what your primitives are,
so we just get from you a treatment that is either "POTENTIALLY"
INFINITE or CIRCULAR.
MoeBlee |
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| Randy Poe |
| Posted: Mon Sep 18, 2006 3:33 pm |
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mueckenh@rz.fh-augsburg.de wrote:
[quote:71f8b18282]Randy Poe schrieb:
mueckenh@rz.fh-augsburg.de wrote:
Mike Kelly schrieb:
The mathematics of
the infinite can only be derived from the mathematics of the finite
(because nobody has an idea what "the infinite" is).
Don't extrapolate from yourself so harshly!
I talked to a lot of first-rate mathematicians. This in parentheses is
a qoute from many of them.
Is that your idea of a citation?
It is my idea of trust and honour. I do not publish names of my private
correspondents.
[/quote:71f8b18282]
Nor, apparently, actual quotes.
- Randy |
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