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Set Theory: Should you believe?
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Thu Jul 13, 2006 1:00 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

guenther vonKnakspot wrote:

Quote:
the ever more present wrong notion that mathematics is dependent on
computability and the expanding belief that mathematics is somehow
subjected to the constrains of physical reality.

I know this was addressed to someone else, but I would also like to offer my
thoughts on this matter. I contend that mathematics _is_ constrained by
physical reality. The underlying logic which determines mathematics is a
manifestation of physical reality. I believe what you are asserting is
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic. It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

I note that you object to the idea that "mathematics is dependent on
computability". I don't know if you mean that exclusively in terms of
solving equations and/or finding approximate numerical solutions, or if you
also object to the idea the mathematical proofs should be machine
verifiable. I curious to know what you think of this: http://metamath.org/

--
Nil conscire sibi
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Richard Herring
science forum Guru Wannabe


Joined: 19 May 2005
Posts: 194

PostPosted: Thu Jul 13, 2006 1:11 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In message <1152759284.458199.84440@h48g2000cwc.googlegroups.com>, david
petry <david_lawrence_petry@yahoo.com> writes
Quote:
Norman Wildberger wrote:
[...]

Analysts and set theorists are welcome to send me reasoned responses.

"Send" them to you? Why would anyone do that?

Well, after the initial hit-and-run posting he doesn't appear to be

replying to anything else in this thread.

--
Richard Herring
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Thu Jul 13, 2006 1:33 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Virgil wrote:

Quote:
If the technicalities of set theory were as easy to learn as Russell
seems to think it ought to be then everyone would learn it in grade
school. In fact, it, like many specialities, usually takes years of
study for one to become really good at it.

A good number of people can master concepts of mathematics sufficiently to
solve difficult problems in, say, fluid mechanics without much grasp of set
theory. That makes me wonder if set theory really is fundamental to
mathematics. I ask what it is that underlies the pragmatic application of
sophisticated mathematics. What are the intuitive assumptions these people
have made, and how are they manipulating ideas? I believe what I'm asking
is, what are the anthropological foundations of mathematics?
--
Nil conscire sibi
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Robert Kolker
science forum Guru


Joined: 23 Apr 2005
Posts: 1756

PostPosted: Thu Jul 13, 2006 1:56 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Hatto von Aquitanien wrote:

Quote:

I know this was addressed to someone else, but I would also like to offer my
thoughts on this matter. I contend that mathematics _is_ constrained by
physical reality. The underlying logic which determines mathematics is a
manifestation of physical reality. I believe what you are asserting is
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic. It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

But formal proofs generally cannot be discovered by finitary algorithmic
means. We still need Inspiration. If you regard all, so-called "mental"
processes, as really physical then your assertion may have some basis.

Bob Kolker
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Thu Jul 13, 2006 2:29 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Hatto von Aquitanien wrote:
Quote:
guenther vonKnakspot wrote:

the ever more present wrong notion that mathematics is dependent on
computability and the expanding belief that mathematics is somehow
subjected to the constrains of physical reality.

I know this was addressed to someone else, but I would also like to offer my
thoughts on this matter. I contend that mathematics _is_ constrained by
physical reality. The underlying logic which determines mathematics is a
manifestation of physical reality. I believe what you are asserting is
That is an issue for Philosophers which I don't believe can be resolved

definitely. I do not subscribe to it, but can not refute it either, so
let us please agree to exclude it from this particular discussion.This
is not howewer the flawed reasoning that I am refering to. I am talking
much baser contentions made by the ill educated like denying the
existence of certain mathematical objects on the ground that they can
not be physically constructed. An example would be the set of Natural
Numbers whichs existence is denied because there are not sufficient
atoms in the universe to build a tangible physical representation of
it, or specific irrational numbers on on the same grounds pertaining to
their decimal base representation.

Quote:
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic.

If you make such a requirement, then you will not get very far. What is
a physically measurable result that gives validity to the number 2 ? Or
to the law of distributivity? or to the differentiability of a given
function? Mathematical concepts have no consequences in physical
reality and the laws of the physical universe have no consequence for
mathematical concepts. (as long as we keep to the agreement I requested
above).

Quote:
It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

I note that you object to the idea that "mathematics is dependent on
computability". I don't know if you mean that exclusively in terms of
solving equations and/or finding approximate numerical solutions, or if you
also object to the idea the mathematical proofs should be machine
verifiable. I curious to know what you think of this: http://metamath.org/

I agree with the idea that mathematical proofs should be machine
verifiable; as long as this is a statement about computer science and
not about mathematics.
Regards.
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Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

PostPosted: Thu Jul 13, 2006 2:29 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Hatto von Aquitanien wrote:

Quote:
I know this was addressed to someone else, but I would also
like to offer my thoughts on this matter. I contend that
mathematics _is_ constrained by physical reality. The
underlying logic which determines mathematics is a
manifestation of physical reality. I believe what
you are asserting is that mathematics should not be
required to produce physically measurable results as
a test of its validity. I really have to wonder if
such a requirement is unrealistic. It's interesting
to observe that some people are wont to point to the
fact that formal proofs can be verified by computer programs.

I've often wondered about similar issues myself.

For one thing, we could argue that there is a difference
between our interpretation of certain mathematical notions,
such as completed infinite sets, and what we're actually
doing, which is writing finitely many symbols down on
paper in certain ways. There's no reason for which I can
see that, because we can write certain symbols down in
certain ways, that certain interpretations of what we're
doing, above and beyond this, must follow. Add to this
the fact that the act of writing down these symbols is
only possible by the nature of our reality. Or at least,
I don't see how we could prove its independence from our
reality in a way that we would understand, because it seems
to me that any such metaproof must also be within, and
hence a feature of, our reality.

Dave L. Renfro
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Thu Jul 13, 2006 2:31 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Robert J. Kolker wrote:

Quote:
Hatto von Aquitanien wrote:


I know this was addressed to someone else, but I would also like to offer
my
thoughts on this matter. I contend that mathematics _is_ constrained by
physical reality. The underlying logic which determines mathematics is a
manifestation of physical reality. I believe what you are asserting is
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic. It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

But formal proofs generally cannot be discovered by finitary algorithmic
means.

Isn't that Chruch's theorem?

Quote:
We still need Inspiration. If you regard all, so-called "mental"
processes, as really physical then your assertion may have some basis.

The only thing I am asserting with absolute conviction that will never be
shaken is that the thought processes which we call mathematics are governed
by the Laws of Nature. That is to say Physical Laws. Whether that amounts
to "finitary algorithmic means" is less certain. Everything else was
intended contingently.

Note that my comment regarding proofs involved verification, not production.
--
Nil conscire sibi
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stephen@nomail.com
science forum Guru


Joined: 11 Sep 2005
Posts: 681

PostPosted: Thu Jul 13, 2006 3:04 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

In sci.math Gene Ward Smith <genewardsmith@gmail.com> wrote:

Quote:
Russell Easterly wrote:

Most branches of mathematics will accept any reasonable proof.
Set theorists demand proofs in set theory.

What the hell does this mean, if anything?

It means Russell has repeatedly failed to prove anything. :)

Stephen
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kunzmilan@atlas.cz
science forum beginner


Joined: 21 Feb 2006
Posts: 42

PostPosted: Thu Jul 13, 2006 3:06 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

1) most of `elementary mathematics' is not sufficiently
well understood by the mathematical establishment.
Well, a simple example:
1x3(aaa,bbb,ccc)
3x6(aab,abb,aac,acc,bbc,bcc)
6x1(abc)
Which column is the Newton's one? What meaning has
the second column?
There are two polynomial coefficients. try it wor any similar products.
Did you studied combinatorics? Why you do not know it?
If you know this fact, why you did not used it?
kunzmilan
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Thu Jul 13, 2006 3:18 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

guenther vonKnakspot wrote:

Quote:
Hatto von Aquitanien wrote:
This
is not howewer the flawed reasoning that I am refering to. I am talking
much baser contentions made by the ill educated like denying the
existence of certain mathematical objects on the ground that they can
not be physically constructed. An example would be the set of Natural
Numbers whichs existence is denied because there are not sufficient
atoms in the universe to build a tangible physical representation of
it, or specific irrational numbers on on the same grounds pertaining to
their decimal base representation.

I don't know that such an opinion is ill-educated or merely unpersuasive.
OTOH, I don't believe that is the essence of most objections to the concept
of infinite sets. For myself, I am willing to consider the existence of
infinite sets without much reservation. At the same time, I believe that
arguments and reasoning applicable to finite sets should be applied with
caution to infinite sets, and carefully scrutinized to determine if it
makes sense in any given instance.

Quote:
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic.

If you make such a requirement, then you will not get very far. What is
a physically measurable result that gives validity to the number 2 ? Or
to the law of distributivity? or to the differentiability of a given
function? Mathematical concepts have no consequences in physical
reality and the laws of the physical universe have no consequence for
mathematical concepts. (as long as we keep to the agreement I requested
above).

I am not stating this conclusively, but consider Cantor's transfinite
induction, in comparison to the calculus of Newton and Leibnitz. We have
ample evidence that the results of differentiation and integration
correspond in many cases to physical experiments in convincing ways. Is
there any result from Cantor's theory which leads to a verifiable
prediction in terms of physical experiment?

That is merely one aspect of the idea that mathematics might be physically
testable. Another is whether the actual reasoning can be reproduced or
verified using computers.

Quote:
It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

I note that you object to the idea that "mathematics is dependent on
computability". I don't know if you mean that exclusively in terms of
solving equations and/or finding approximate numerical solutions, or if
you also object to the idea the mathematical proofs should be machine
verifiable. I curious to know what you think of this:
http://metamath.org/

I agree with the idea that mathematical proofs should be machine
verifiable; as long as this is a statement about computer science and
not about mathematics.

One might argue that this is merely using the computer as a tool.
--
Nil conscire sibi
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Thu Jul 13, 2006 3:39 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Hatto von Aquitanien wrote:
Quote:
guenther vonKnakspot wrote:

Hatto von Aquitanien wrote:
This
is not howewer the flawed reasoning that I am refering to. I am talking
much baser contentions made by the ill educated like denying the
existence of certain mathematical objects on the ground that they can
not be physically constructed. An example would be the set of Natural
Numbers whichs existence is denied because there are not sufficient
atoms in the universe to build a tangible physical representation of
it, or specific irrational numbers on on the same grounds pertaining to
their decimal base representation.

I don't know that such an opinion is ill-educated or merely unpersuasive.
It is not a matter of what the opinion itself is. The problem is with

the person holding such an opinion. That person must have received a
flawed or no mathematical education at all.

Quote:
OTOH, I don't believe that is the essence of most objections to the concept
of infinite sets. For myself, I am willing to consider the existence of
Of course it is not the essence of most objections to to the concept of

infinite sets. It is an example of the flawed assumption that
mathematics are constrained by physical reality in such a manner.


Quote:
infinite sets without much reservation. At the same time, I believe that
arguments and reasoning applicable to finite sets should be applied with
caution to infinite sets, and carefully scrutinized to determine if it
makes sense in any given instance.
That is not related to this discussion.



Quote:
that mathematics should not be required to produce physically measurable
results as a test of its validity. I really have to wonder if such a
requirement is unrealistic.

If you make such a requirement, then you will not get very far. What is
a physically measurable result that gives validity to the number 2 ? Or
to the law of distributivity? or to the differentiability of a given
function? Mathematical concepts have no consequences in physical
reality and the laws of the physical universe have no consequence for
mathematical concepts. (as long as we keep to the agreement I requested
above).

I am not stating this conclusively, but consider Cantor's transfinite
induction, in comparison to the calculus of Newton and Leibnitz. We have
ample evidence that the results of differentiation and integration
correspond in many cases to physical experiments in convincing ways. Is
there any result from Cantor's theory which leads to a verifiable
prediction in terms of physical experiment?
You were the one arguing in favour of the requirement that mathematical

concepts be subjected to verification by physical measurability.
Obviously you have found a further difficulty in supporting that case.


Quote:
That is merely one aspect of the idea that mathematics might be physically
testable. Another is whether the actual reasoning can be reproduced or
verified using computers.

It's interesting to observe that some people
are wont to point to the fact that formal proofs can be verified by
computer programs.

I note that you object to the idea that "mathematics is dependent on
computability". I don't know if you mean that exclusively in terms of
solving equations and/or finding approximate numerical solutions, or if
you also object to the idea the mathematical proofs should be machine
verifiable. I curious to know what you think of this:
http://metamath.org/

I agree with the idea that mathematical proofs should be machine
verifiable; as long as this is a statement about computer science and
not about mathematics.

One might argue that this is merely using the computer as a tool.
Your use of the word 'merely' suggests that you might be thinking of

manners of using a computer trascending those of a tool. What would
those be ?
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Thu Jul 13, 2006 3:44 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

"Gene Ward Smith" <genewardsmith@gmail.com> writes:


Quote:
Gerry Myerson wrote:

I personally don't put set theory in the same category as astrology
or creation science. Maybe Norm does. I don't know.

Norm apparently puts number theory in that category.

Quite right, too. We have, on at least as good authority as that which
either the astrolgers or "creation scientists" have for their belief
systems, that God created both the firmament (stars included) and the
integers.

Then there's that bit where Jesus is asserted to be both a generic cardinal
*and* the first infinite ordinal.

Lee Rudolph
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Thu Jul 13, 2006 3:52 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

Lee Rudolph wrote:
Quote:
Then there's that bit where Jesus is asserted to be both a generic cardinal
*and* the first infinite ordinal.

But most Christians don't find him quite inaccessible, right?
Remarkable, subtle and strong, perhaps, somewhat ineffable and
indescribable at times - though probably not totally indescribable. I
think I can see a set theoretical proof of the non-existence of Jesus.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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lesterzick@cableone.net
science forum beginner


Joined: 12 Jul 2006
Posts: 3

PostPosted: Thu Jul 13, 2006 4:42 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On Wed, 12 Jul 2006 21:37:30 -0600, Virgil <virgil@comcast.net> wrote:

Quote:
In article <ggebb2dj42j589uhhsojj0gbr0ddgec2u1@4ax.com>,
lesterzick@cableone.net wrote:

What about the set of all points on a perfect circle (ie all solutions to
x^2 + y^2 = 1).

Well technically of course all solutions you point out define a
perfect sphere not a perfect circle.

With only two variables, x and y, there is no need to presume three
dimensions, but if one does, one gets a right circular cylinder, not a
sphere.

And as long as one gets to assume whatever one wants one gets whatever
one wants to assume. There is no unambiguous right angle dimension to
any other when fewer than three dimensions are involved. If you're
talking about the set of all points equidistant from any other the
figure is a sphere.

Quote:
If one doesn't assume three (or more) dimensions, one does get a circle.

Well the basic problem here is that it's a little difficult to discern
what the author is complaining about in set theory. Sets of properties
are perfectly useful. However typical set theory definitions which run
along the lines of a "set of all points which . . ." do turn out to be
a joke because they invariably rely on various geometric assumptions
regarding figures such as planes, lines, etc.
~v~~
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lDontBother@nowhere.net
science forum beginner


Joined: 13 Jul 2006
Posts: 2

PostPosted: Thu Jul 13, 2006 5:15 pm    Post subject: Re: Set Theory: Should you believe? Reply with quote

On 12 Jul 2006 23:34:03 -0700, "Gene Ward Smith"
<genewardsmith@gmail.com> wrote:

Quote:

Gerry Myerson wrote:

I think Norm would say, 1) you can't form the set (and Norm's reasons
are given in the article), and 2) you don't need to - there's no good
mathematics you can do with the completed infinite set that you can't
do without it.

Norm is also opposed to axioms.

Isn't everyone? The only reason for axioms is that people are too lazy
or stupid to demonstrate the truth of their assumptions.

Quote:
Without axioms, how do we know when we
are "forming" an infinte set? If I state Euclid's theorem on the
infinitude of primes, am I "forming" a set? Am I forming a set just by
referencing the integers at all? If Norm won't give a set of axioms he
finds acceptable, we can't very well say that measureable cardinals
contradict his foundations for mathematics, because he hasn't really
given a foundation. He has, in fact, claimed that infinite sets are
metaphysics; but if they are metaphysics, he's not talking mathematics
at all, but metaphysics. In which case, so what? What do his
metaphysical beliefs have to do with mathematics?

More ad hominem. Norm accepts the axioms of group theory as the
definition of what a group is, and has no problem with them because
he can construct (finite) models of them.

In which case, he can hardly say he is rejecting axioms, and ought to
step forward and say what his proposed axioms are. Would ditching the
axiom of infinity do it? If not, what would?
~v~~
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