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| Aatu Koskensilta |
 Posted: Sat Sep 16, 2006 4:27 pm |
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David R Tribble wrote:
[quote:528351aa1a]The problem is that using too simple a language can lead to
further confusion, or at least maintain the misunderstandings.
I've been chastised for talking about "set size" instead of using the
more specific term "cardinality", for instance.
[/quote:528351aa1a]
When discussing technical subjects there is no hope of avoiding
technicalities and technical concepts and terminology. But if someone
wonders about a feature of, say, a proof of Cantor's theorem it is
seldom useful, and often counterproductive, to bring in the heavy
machinery of first order logic and this or that formal set theory.
[quote:528351aa1a]So there may be
times when a more technically precise meaning is called for in
order to cut through the confusion of multiple meanings.
[/quote:528351aa1a]
Of course, clarity is to be always sought, and in making clear exactly
what is meant by such ambiguous words as "size" in some context is often
necessary.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Tony Orlow |
| Posted: Sat Sep 16, 2006 5:45 pm |
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Virgil wrote:
[quote:26cc9b44d8]In article <450bf9ae@news2.lightlink.com>,
Tony Orlow <tony@lightlink.com> wrote:
Virgil wrote:
In article <450b4cb2$1@news2.lightlink.com>,
Tony Orlow <tony@lightlink.com> wrote:
Virgil wrote:
In article <49edf$450aacfc$82a1e228$14539@news2.tudelft.nl>,
Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> wrote:
Mike Kelly wrote:
Han de Bruijn wrote:
Plagiarism? I don't get it. Who is plagiarising what?
"Your" would-be arguments against mine are not really yours. They are
just a _plagiary_ of well-known "arguments" employed by the mainstream
mathematics community.
Han de Bruijn
What is common knowledge can be used by anyone without plagiarizing.
Otherwise only its original author could use "2 + 2 = 4".
Yes, and Virgil would be collecting the royalties from every first-grade
class.
Does TO credit me with discovering/inventing/creating "2 + 2 = 4" ?
That fact has been around for millennia.
How old does TO take me for?
I couldn't guess, but I heard you fart dust devils and used to date
Methuselah's sister.
You must have heard such things from someone else whose pretentions I
have also been puncturing.
[/quote:26cc9b44d8]
Yes, I think it was the dirt by the road, who claims you refer to it as
a "young'un" or "greenhorn" or something. |
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| Tony Orlow |
| Posted: Sat Sep 16, 2006 5:48 pm |
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Virgil wrote:
[quote:51432573c0]In article <450c3c65@news2.lightlink.com>,
Tony Orlow <tony@lightlink.com> wrote:
What is the average value of the reals in [0,1]?
By what definition of average?
[/quote:51432573c0]
Expected value of an element chose at random. Is it impossible to choose
a real in [0,1] at random?
[quote:51432573c0]
There are a whole bunch of such definitions.
[/quote:51432573c0]
And, what are the "whole bunch" of answers to that question?
[quote:51432573c0]
Note that many such definitions only apply to finite sets of numbers,
and thus won't work.
[/quote:51432573c0]
Says you.
[quote:51432573c0]
Also, since the set [0,1] is invariant under x --> x^n, its average
should be also.
[/quote:51432573c0]
What does average have to do with exponentiation. (You could use a good
opportunity to strut your stuff, so remind me what "invariant" means  |
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| Aatu Koskensilta |
| Posted: Sat Sep 16, 2006 10:47 pm |
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Virgil wrote:
[quote:13a5a5d73c]In article <450c71a1@news2.lightlink.com>,
Tony Orlow <tony@lightlink.com> wrote:
Aatu Koskensilta wrote:
Given the axioms and rules of inference, the conclusions are provably
true or false.
Soundness is another issue, regarding the fundamental justification for
the logical axioms themselves, and whether they are "correct", meaning
"objectively verifiable".
[/quote:13a5a5d73c]
I didn't write the above, nor did Tony in his post partially quoted by
Virgil claim I did. Do be careful with the attributions and quotations.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Aatu Koskensilta |
| Posted: Sat Sep 16, 2006 10:47 pm |
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Tony Orlow wrote:
[quote:b7fcff3857]Hi Aatu -
I appreciate your desire to accommodate the simply naive and confused by
addressing issues in terms they can understand regarding mathematical
questions. I think that's very human and generous of you, and good
advice to anyone trying to teach. I do think that once we get into
foundational arguments of the sort going on here, we really can't avoid
such technicalities, since the validity, if not the soundness, of the
arguments hinges on such fundamental issues.
[/quote:b7fcff3857]
First order logic, rules of inference, and all that form a mathematical
tool that correctly captures, to an extent, the informal notion of
something logically following from a set of premises. This is shown by
the completeness theorem for first order logic with a few conceptual
considerations - famously due to Kreisel in his informal rigour paper.
Now, one might consider the basic idea of logical consequence in first
order context suspect or inadequate in some way, in which case an
alternative notion of logical validity etc. must be provided - most
likely not in the form of any formal system of logic with rules of
inferences and formal axioms, but in informal terms similarly as one
presents the classical or the intuitionistic picture. It is then up to
the individual mathematicians to evaluate the fruitfulness,
plausibility, coherence, applicability and so forth of the presented
idea of logic - in this process formalization might or might not be of
some help, e.g. by enabling a proof that the alternative picture is in
some sense incompatible but still intertranslatable with the classical
picture, or enabling one to establish conclusively that arguments of
this or that kind are not valid under the alternative conception of logic.
Now, in addition to the question of logic one might simply choose to
reject this or that mathematical principle either as outright false or
simply as unjustified. Of course, without tweaking one's logic it is not
possible to accept a mathematical principle and reject some of its
consequences. Here too formal theories might be of some limited use,
e.g. allowing one to establish that some principle is indeed independent
of some other principle (over some suitable background assumptions).
[quote:b7fcff3857]Yes, this is my point. When we speak of the "size" of a set, for finite
sets it's the count. For infinite sets, some generalization becomes
necessary. The issue for me is which tenets of finite sets do we
consider most important to preserve as we generalize.
[/quote:b7fcff3857]
The concept of "size" when applied to non-finite sets bifurcates into
many different concepts that just happen to agree on finite sets. The
most general such notion of size is provided by cardinality in the
Cantorian sense, since it doesn't presuppose any numerical ordering or a
notion of density or some such be provided along with the sets compared.
It also leads to a highly fruitful and beautiful mathematical theory
with applications in almost every area of modern mathematics. This is
not to say that other notions of "size" as applied to sets are ignored;
the idea that there are twice as many naturals as there are odd naturals
can be captured mathematically, although this notion is less general and
applies only in case the sets in question are equipped with additional
structure.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Virgil |
| Posted: Sun Sep 17, 2006 1:54 am |
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In article <Yz2Pg.13567$VX1.6175@reader1.news.jippii.net>,
Aatu Koskensilta <aatu.koskensilta@xortec.fi> wrote:
[quote:42e452ca11]Tony Orlow wrote:
Hi Aatu -
I appreciate your desire to accommodate the simply naive and confused by
addressing issues in terms they can understand regarding mathematical
questions. I think that's very human and generous of you, and good
advice to anyone trying to teach. I do think that once we get into
foundational arguments of the sort going on here, we really can't avoid
such technicalities, since the validity, if not the soundness, of the
arguments hinges on such fundamental issues.
First order logic, rules of inference, and all that form a mathematical
tool that correctly captures, to an extent, the informal notion of
something logically following from a set of premises. This is shown by
the completeness theorem for first order logic with a few conceptual
considerations - famously due to Kreisel in his informal rigour paper.
Now, one might consider the basic idea of logical consequence in first
order context suspect or inadequate in some way, in which case an
alternative notion of logical validity etc. must be provided - most
likely not in the form of any formal system of logic with rules of
inferences and formal axioms, but in informal terms similarly as one
presents the classical or the intuitionistic picture. It is then up to
the individual mathematicians to evaluate the fruitfulness,
plausibility, coherence, applicability and so forth of the presented
idea of logic - in this process formalization might or might not be of
some help, e.g. by enabling a proof that the alternative picture is in
some sense incompatible but still intertranslatable with the classical
picture, or enabling one to establish conclusively that arguments of
this or that kind are not valid under the alternative conception of logic.
Now, in addition to the question of logic one might simply choose to
reject this or that mathematical principle either as outright false or
simply as unjustified. Of course, without tweaking one's logic it is not
possible to accept a mathematical principle and reject some of its
consequences. Here too formal theories might be of some limited use,
e.g. allowing one to establish that some principle is indeed independent
of some other principle (over some suitable background assumptions).
Yes, this is my point. When we speak of the "size" of a set, for finite
sets it's the count. For infinite sets, some generalization becomes
necessary. The issue for me is which tenets of finite sets do we
consider most important to preserve as we generalize.
The concept of "size" when applied to non-finite sets bifurcates into
many different concepts that just happen to agree on finite sets. The
most general such notion of size is provided by cardinality in the
Cantorian sense, since it doesn't presuppose any numerical ordering or a
notion of density or some such be provided along with the sets compared.
It also leads to a highly fruitful and beautiful mathematical theory
with applications in almost every area of modern mathematics. This is
not to say that other notions of "size" as applied to sets are ignored;
the idea that there are twice as many naturals as there are odd naturals
can be captured mathematically, although this notion is less general and
applies only in case the sets in question are equipped with additional
structure.
[/quote:42e452ca11]
Exactly the point that Tony Orlow rejects. |
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| Guest |
| Posted: Sun Sep 17, 2006 5:06 am |
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Mike Kelly <mk4284@bris.ac.uk> wrote:
[quote:8db8dd01ca]Han.deBruijn@DTO.TUDelft.NL wrote:
Mike Kelly wrote:
The good news is that you are doing wrong only _one_ thing: infinitary
reasoning. You think that completed infinities do exist.
If you don't accept the existence of a set of natural numbers then you
don't accept the set theory that probability theory is based upon and
you haven't suggested an alternative. Indeed, it seems somewhat odd to
complain about the conclusion of a theorem discussing an object you
don't accept even exists.
[/quote:8db8dd01ca]
You gotta love Han's claim:
The probability of picking a number divisble by 3 from the
set of all natural numbers, which does not exist, is 1/3.
<snip>
[quote:8db8dd01ca]It is meaningful to say that a natural drawn uniformly at random from a
set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being
divisible by 3. Nobody disputes this. But talking about the probability
of "a natural" being divisible by 3 implies a uniform distribution over
the naturals. Such a thing does not exist.
[/quote:8db8dd01ca]
But that is just fine according to Han's logic. No uniform
distribution exists over the set of all naturals because
the set of all naturals does not exist. Therefore the probability
of choosing a number divisible by 3 from the set of all naturals
is 1/3, according to the non-existent uniform distribution
on the non-existent set.
Stephen |
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| Guest |
| Posted: Sun Sep 17, 2006 7:42 am |
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stephen@nomail.com wrote:
[quote:5f9c0b0b8d]Han de Bruijn <Han.deBruijn@dto.tudelft.nl> wrote:
Mike Kelly wrote:
Han de Bruijn wrote:
Plagiarism? I don't get it. Who is plagiarising what?
"Your" would-be arguments against mine are not really yours. They are
just a _plagiary_ of well-known "arguments" employed by the mainstream
mathematics community.
That is pretty pathetic Han. So any mainstream argument is
plagiarism? That is a convenient way to dismiss anyone who
disagrees with you.
[/quote:5f9c0b0b8d]
No. It's a convenient way to avoid hearing again the arguments
that I have already heard.
[quote:5f9c0b0b8d]What is wrong with you? What is the source of
your hostility towards mathematics?
[/quote:5f9c0b0b8d]
What's wrong with mathematics ?!
Han de Bruijn |
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| Guest |
| Posted: Sun Sep 17, 2006 7:56 am |
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Mike Kelly wrote:
[quote:25d31de4af]astounded that you are claiming that employing a mathematical argument
that is not your own invention is plagiarism. Perhaps you are simply
unaware of the meaning and connotations of the word. Plagiarism is
dishonest and in many cases criminal. A fairly hefty accusation.
[/quote:25d31de4af]
I've been looking for a good English equivalent of the Dutch word
"meeloperij" and found "plagiarism" as my best match. I think that you
are right and that it's actuallty a mismatch. I apologize for this
fact but I don't know what shorthand expression to substitute instead.
What I meant to express is that you are about to be parrotting
mainstream arguments, without adding to it much thoughts of
yourself. And that is quite senseless because we have gone
through all this already.
Han de Bruijn |
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| Guest |
| Posted: Sun Sep 17, 2006 8:07 am |
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Mike Kelly wrote:
[quote:7e4f669028]Given that any second-year student of probability theory knows that
there are no uniform distributions over countable sample spaces, [ ... ]
[/quote:7e4f669028]
This "given" is most disturbing. Mainstream mathematics is so certain
about its own right that no sensible debate is possible.
[quote:7e4f669028]Finally, please stop with the scare quotes. They make you look like a
"tool".
[/quote:7e4f669028]
Sorry. I don't know what "scare quotes" are
and I don't know what I'm doing wrong here.
Han de Bruijn |
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| Aatu Koskensilta |
| Posted: Sun Sep 17, 2006 8:21 am |
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Virgil wrote:
[quote:5b26c1ddcd]In article <Yz2Pg.13567$VX1.6175@reader1.news.jippii.net>,
Aatu Koskensilta <aatu.koskensilta@xortec.fi> wrote:
This is not to say that other notions of "size" as applied to sets are ignored;
the idea that there are twice as many naturals as there are odd naturals
can be captured mathematically, although this notion is less general and
applies only in case the sets in question are equipped with additional
structure.
Exactly the point that Tony Orlow rejects.
[/quote:5b26c1ddcd]
Quite possibly. Since he's an obvious crank there's really very little
point in caring about what he thinks or rejects, and even less point in
engaging him in endless "debates". Of course, this is USENET and there's
very little point to anything in any case; hence my few observations on
the rhetorical tactics in these debates.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Guest |
| Posted: Sun Sep 17, 2006 8:44 am |
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Mike Kelly wrote:
[quote:b8ab86831d]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 ... ]
[/quote:b8ab86831d]
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.
Han de Bruijn |
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| Mike Kelly |
| Posted: Sun Sep 17, 2006 8:54 am |
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Han.deBruijn@DTO.TUDelft.NL wrote:
[quote:800d3d63ce]stephen@nomail.com wrote:
Han de Bruijn <Han.deBruijn@dto.tudelft.nl> wrote:
Mike Kelly wrote:
Han de Bruijn wrote:
Plagiarism? I don't get it. Who is plagiarising what?
"Your" would-be arguments against mine are not really yours. They are
just a _plagiary_ of well-known "arguments" employed by the mainstream
mathematics community.
That is pretty pathetic Han. So any mainstream argument is
plagiarism? That is a convenient way to dismiss anyone who
disagrees with you.
No. It's a convenient way to avoid hearing again the arguments
that I have already heard.
[/quote:800d3d63ce]
If you are not willing to defend your absurdly smug claims then do not
make them.
--
mike. |
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| Guest |
| Posted: Sun Sep 17, 2006 8:54 am |
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Mike Kelly wrote:
[quote:d6c1b3e133][ ... 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.
[/quote:d6c1b3e133]
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.
Han de Bruijn |
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| Mike Kelly |
| Posted: Sun Sep 17, 2006 9:02 am |
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Han.deBruijn@DTO.TUDelft.NL wrote:
[quote:c145d8db3a]Mike Kelly wrote:
astounded that you are claiming that employing a mathematical argument
that is not your own invention is plagiarism. Perhaps you are simply
unaware of the meaning and connotations of the word. Plagiarism is
dishonest and in many cases criminal. A fairly hefty accusation.
I've been looking for a good English equivalent of the Dutch word
"meeloperij" and found "plagiarism" as my best match. I think that you
are right and that it's actuallty a mismatch. I apologize for this
fact but I don't know what shorthand expression to substitute instead.
What I meant to express is that you are about to be parrotting
mainstream arguments, without adding to it much thoughts of
yourself. And that is quite senseless because we have gone
through all this already.
[/quote:c145d8db3a]
Yes, and your position was utterly ripped apart in the very first
response to you by David C. Ullrich, and then by several others. You
were unable to defend your claim. So, why repeat it as though it were
in any way valid?
--
mike. |
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