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Set Theory: Should you believe?
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Russell Easterly
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Joined: 27 Jun 2005
Posts: 199

PostPosted: Thu Jul 13, 2006 7:14 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

"Gene Ward Smith" <genewardsmith@gmail.com> wrote in message
news:1152770679.597975.88170@p79g2000cwp.googlegroups.com...
Quote:

Gerry Myerson wrote:

I'm dismayed by the level of vituperation in some of the posts in
this thread.

Norm starts out his paper, which I didn't read because the beginning
was so extremely unpromising, in what seems to me to be a very
insulting way. If he has ideas he wants to be taken seriously I suggest
he remove the sneers directed at set theorists, who apparently are
beneath contempt, and wild remarks about physics and the like. Present
a reasoned argument in a reasonable way and people are likely to react
more positively, and less likely to conclude that you are an idiot and
simply quit reading.

He's adopting a finitistic, or
constructivist, or computational view of mathematics.

He's also spitting on people who don't. I think it is terribly arrogant
to dismiss people like
Shelah or Woodin with such utter contempt like this, and I didn't see
any signs, as far as I had gotten, that he even knows anything about
modern set theory. Does he?

I think he is all too familiar with modern set theorists.
Set theorists have written the book on how to treat
others with contempt.

Most branches of mathematics will accept any reasonable proof.
Set theorists demand proofs in set theory.
This is like the Catholic Church requiring Mass be given in Latin.
It is a method of guaranteeing only the priests (the true
believers) know the Church's doctrines.
It is designed to prevent skeptics (non-believers) from being
able to question Church doctrine, since you need to know
a dead language to have any idea what that doctrine is.

Imagine if Einstein had been told he had to prove relativity
in Newtonian physics before anyone would consider his ideas.


Russell
- 2 many 2 count
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Gene Ward Smith
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Joined: 08 Jul 2005
Posts: 409

PostPosted: Thu Jul 13, 2006 7:18 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gerry Myerson wrote:

Quote:
I don't know, and if you really want to know, you could try
asking him. But perhaps he is only saying, "the axioms
of group theory define an interesting set of structures, which
I can construct, and which help me answer questions about physics,
while the axioms of set theory only help me study set theory,
which is not where the real value of mathematics is."

The Heine-Borel theorem is awfully useful for developing real analysis.
Is it a good thing or a bad one in Norm's view to have a system strong
enough to prove it? How the hell can anyone know, unless he bites the
bullet and says what he thinks you can assume?
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Gene Ward Smith
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Joined: 08 Jul 2005
Posts: 409

PostPosted: Thu Jul 13, 2006 7:19 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Russell Easterly wrote:

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

What the hell does this mean, if anything?
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Stephen Montgomery-Smith1
science forum Guru


Joined: 01 May 2005
Posts: 487

PostPosted: Thu Jul 13, 2006 7:48 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Norman Wildberger wrote:
Quote:
I have posted an article at http://web.maths.unsw.edu.au/~norman/views.htm
that has caused a bit of discussion in some logic circles.

My claims in short: 1) most of `elementary mathematics' is not sufficiently
well understood by the mathematical establishment, leading to weaknesses in
K12 and college curriculum, 2) the current theory of `real numbers' is a
joke, and sidesteps the crucial issue of understanding the computational
specification of the continuum, and 3) `infinite sets' are a metaphysical
concept, and unnecessary for correct mathematics.

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

Assoc Prof N J Wildberger
School of Maths
UNSW

After casually reading his notes, I think that he is saying this. The
axioms of modern set theory are too burdensome for actual
mathematicians, who in practice take a somewhat Platonic view of their
subject. But he is unsatisfied with the pragmatic Platonist approach
that we take, particular in the manner in which mathematics is taught at
pre-college levels. He thinks we need to roll up our sleeves and
rethink foundations so that we get something that really is usuable, so
that we can finally truly rid ourselves of Platonism, superstition,
instinct, gut reaction, religion, etc, etc.

Personally I really like the Platonic approach, using set theory as a
highly convenient crutch. I'm not going to preclude the possibility
that one day a set of foundations for mathematics will be found, that
will greatly simplify and advance the extent that we will be able to
think about mathematics, but I think it will take a great genius, and
also some crisis of cicumstance (perhaps the discovery of some horribly
unresolvable contradiction).

But on the whole, even if his tone was not exactly politic, I liked a
lot of what he said. But I don't think it is going to change the world.

Stephen
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Kevin Karn
science forum beginner


Joined: 13 Jul 2006
Posts: 6

PostPosted: Thu Jul 13, 2006 7:58 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gene Ward Smith wrote:
Quote:
Gerry Myerson wrote:

I'm dismayed by the level of vituperation in some of the posts in
this thread.

Norm starts out his paper, which I didn't read because the beginning
was so extremely unpromising, in what seems to me to be a very
insulting way. If he has ideas he wants to be taken seriously I suggest
he remove the sneers directed at set theorists, who apparently are
beneath contempt, and wild remarks about physics and the like. Present
a reasoned argument in a reasonable way and people are likely to react
more positively, and less likely to conclude that you are an idiot and
simply quit reading.

He's adopting a finitistic, or
constructivist, or computational view of mathematics.

He's also spitting on people who don't. I think it is terribly arrogant
to dismiss people like
Shelah or Woodin with such utter contempt like this,

Hugh Woodin is a con-artist/leech who needs to get a real job.
Ideally, what we need to do with people like Woodin, is haul them into
the dock for public hearings.

"Where did that $1 million in federal grants you sucked down in the
last 15 years go, Mr. Woodin? What were the practical spin-offs? Why
should we fund you, as opposed to someone working on real-world
problems, like bird flu etc.? What practical benefit does your research
have? We're going to need an explanation, Mr. Woodin, otherwise we
can't sign the check. We simply can't fund research which has NO
practical applications."

It's worse than the toilet seat scandals. When the government buys a
toilet seat for $1 million, at least you get the toilet seat. When you
give Hugh Woodin $1 million, you get nothing of practical value, not
even a toilet seat. That public money should be rerouted to people
doing work which actually benefits society. If you wanna do theology,
do it on your own dime.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Thu Jul 13, 2006 8:01 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

In article <UbWdnZXmzLV8byjZnZ2dnUVZ_ridnZ2d@comcast.com>,
"Russell Easterly" <logiclab@comcast.net> wrote:

Quote:
"Gene Ward Smith" <genewardsmith@gmail.com> wrote in message
news:1152770679.597975.88170@p79g2000cwp.googlegroups.com...

Gerry Myerson wrote:

I'm dismayed by the level of vituperation in some of the posts in
this thread.

Norm starts out his paper, which I didn't read because the beginning
was so extremely unpromising, in what seems to me to be a very
insulting way. If he has ideas he wants to be taken seriously I suggest
he remove the sneers directed at set theorists, who apparently are
beneath contempt, and wild remarks about physics and the like. Present
a reasoned argument in a reasonable way and people are likely to react
more positively, and less likely to conclude that you are an idiot and
simply quit reading.

He's adopting a finitistic, or
constructivist, or computational view of mathematics.

He's also spitting on people who don't. I think it is terribly arrogant
to dismiss people like
Shelah or Woodin with such utter contempt like this, and I didn't see
any signs, as far as I had gotten, that he even knows anything about
modern set theory. Does he?

I think he is all too familiar with modern set theorists.
Set theorists have written the book on how to treat
others with contempt.

Most of the contempt is reserved for those whose criticisms exhibit
profound ignorance of what they are criticizing and those whose
criticisms are so contemptuous as to inspire countercontempt.


Quote:

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

They will accept reasonable proofs of reasonable claims but require
extraordinary, or at least rigorous, proofs of extraordinary claims. And
their standard of judging other's proof is no stricter than that they
apply to their own proofs.

Quote:
This is like the Catholic Church requiring Mass be given in Latin.
It is a method of guaranteeing only the priests (the true
believers) know the Church's doctrines.

It may seem like that to those unfamiliar with set theory.
Any speciality tends to look arcane to those outside it.




Quote:
It is designed to prevent skeptics (non-believers) from being
able to question Church doctrine, since you need to know
a dead language to have any idea what that doctrine is.

This is precisely the attitude of those like Russell who dump on set
theorists which inspires the set theorists to dump back.

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.

Russell seems to expect a "Set Theory for Dummies" short course which
will bring him up to PhD levels in an afternoons reading.
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Thu Jul 13, 2006 8:40 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Stephen Montgomery-Smith wrote:

Quote:
After casually reading his notes, I think that he is saying this. The
axioms of modern set theory are too burdensome for actual
mathematicians, who in practice take a somewhat Platonic view of their
subject. But he is unsatisfied with the pragmatic Platonist approach
that we take, particular in the manner in which mathematics is taught at
pre-college levels.

How do you get from there to his claim that you can't prove the
fundamental theorem of arithmetic?

He thinks we need to roll up our sleeves and
Quote:
rethink foundations so that we get something that really is usuable, so
that we can finally truly rid ourselves of Platonism, superstition,
instinct, gut reaction, religion, etc, etc.

It seems to me what he's saying is simply incoherent. He likes Lie
groups, so it's OK to talk about them, so long as you cross your
fingers and say you are really talking about constructable numbers. But
integers, which are about as constructable as it gets, he is willing to
blow off, apparently merely out of disinterest in number theory as
opposed to Lie groups.

Quote:
But on the whole, even if his tone was not exactly politic, I liked a
lot of what he said. But I don't think it is going to change the world.

"Let's make things way harder to prove for no reason, and do it in an
incoherent way" is a tough sell.
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Thu Jul 13, 2006 8:42 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gerry Myerson wrote:

Quote:
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.
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Peter Webb
science forum Guru Wannabe


Joined: 05 May 2005
Posts: 192

PostPosted: Thu Jul 13, 2006 8:42 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

"Gerry Myerson" <gerry@maths.mq.edi.ai.i2u4email> wrote in message
news:gerry-2088DC.14022113072006@sunb.ocs.mq.edu.au...
Quote:
In article <44b5abd7$0$1207$afc38c87@news.optusnet.com.au>,
"Peter Webb" <webbfamily-diespamdie@optusnet.com.au> wrote:

malbrain@yahoo.com> wrote in message
news:1152744973.140793.242750@h48g2000cwc.googlegroups.com...
Norman Wildberger wrote:
I have posted an article at
http://web.maths.unsw.edu.au/~norman/views.htm
that has caused a bit of discussion in some logic circles.


Well, the discussion that I have seen - on this newsgroup (sci.math or
sci.logic, I can't remember) - is that it is bullshit.

Ad hominem.

Not ad hominem. I talk about his post, not him as a person.

Quote:

My claims in short: 1) most of `elementary mathematics' is not
sufficiently well understood by the mathematical establishment,
leading to weaknesses in K12 and college curriculum,

I don't know about the "mathematical establishment" (as a whole) not
understanding "elementary mathematics", but your own writings on set
theory
and the axiomatic method don't fill me with confidence.

Ad hominem.



His argument is ad hominem. Mine is just a cheap shot.


Quote:
2) the current theory of `real numbers' is a
joke, and sidesteps the crucial issue of understanding the
computational
specification of the continuum, and

This is pure crank stuff. Describing a huge and extremely rigorously
defined
area such as the construction of the Reals as a "joke" without any
mathematical justification is flakey at best; the phrase "computational
specification of the continuum" (a phrase that gets exactly zero matches
on
Google) is crank babble.

The mathematical justification for describing the current theory of
real numbers as a joke is given in the paper. You may not find it
convincing - I may not find it convincing - but it's there.

"Crank babble" is ad hominem.


Not ad hominem. I talk about his post, not him as a person.

Quote:
3) `infinite sets' are a metaphysical
concept, and unnecessary for correct mathematics.

No, infinite sets are a mathematical concept, not unlike perfect circles
and
the exact value of the sqrt(2).

Tell me, is the set of all natural numbers finite or infinite? Or if you
can't form the set, why not?

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.


Ohh, so there's no "good" mathematics that can be done with the axiom of
infinity that can't be done without. So infinite set theory is not "good"
mathematics? How about computability theory and Turing Machines? Not "good"
mathematics?

Quote:
What about the set of all points on a perfect circle (ie all solutions to
x^2 + y^2 = 1). Finite or infinite? Or don't you believe that I can
define a
set as being all points on the unit circle. If not, why not? What is
wrong
with {(x,y) | x^2 + y^2 = 1} as a set? Infinite or finite?

Again, I think Norm is arguing against the completed infinite. For
reasons discussed in some detail in the article, you can't (he contends)
sensibly write about the set of all natural numbers, or any of these
other sets that most mathematicians are quite happy with - moreover,
you don't lose anything valuable if you discard them.

Your paper has no mathematical content, and is pure crank stuff. The
stuff
about Axioms somehow being irrelevant to mathematics is just your own
philosophical ramblings. Surprising, since you seem to accept the axioms
of
group theory, but not set theory (because they are too complicated and
too
abstract for your liking)? Do groups with an infinite number of elements
exist, by the way?

More ad hominem.


Not ad hominem. I talk about his post, not him as a person.

Quote:
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. He argues that the axioms
of set theory (in particular, ZFC) don't define what a set is and
don't lead to sensible constructions of infinite sets.

I would have ignored this post - and previous posts on your "mathematical
insights" - in much the same way that I ignore posts about "Einstein was
wrong" or "Cantor's diagonal proof is flawed" - as pure crank material.
The
only reason I haven't, is that you are a mathematics teacher, and it
worries
me that somebody (eg your students) may be being taught this stuff about
set
theory.

I'm not sure how you think your method of not ignoring Norm's post
will prevent his students from being taught his ideas.


Well, maybe if he attempted to justify his ideas, and decided he couldn't,
maybe he would drop them.

Quote:
Tell me, do you accept that there are models for the ZF Axioms if we drop
Axiom 6 ? Do you accept that there is a model for the ZF axioms if we
include an additional axiom:

Exists S such that { } is an element of S, and x elements of S implies x
union {x) is an element of S ???
(informally known as the axiom of infinity)

If you don't, I can certainly show you a model (the von-Neumann
construction
of N).

How is this axiom fundamentally different from the other axioms?

It seems to me Norm is making several points, two of which are
that ZFC sucks and that mathematics doesn't need axioms in the first
place. Your suggestions may or may not have any bearing on the first
point, but they don't address the second.


His "paper" is on ZFC. His "philosophy" that axioms are not needed is
secondary. I also address this philosophy, by asking whether axioms are
needed in group theory.

Quote:
I'm dismayed by the level of vituperation in some of the posts in
this thread. Norm is not presenting a high-school algebra proof of
Fermat's Last Theorem, nor is he insisting that the reals are countable
because you can always take that real number that you left off your
list and stick it on at the end. He's adopting a finitistic, or
constructivist, or computational view of mathematics. It's an unpopular
view, it doesn't particularly appeal to me, but I don't see the need
to go ballistic in response.


Because his "paper"s that I have seen either display a profound lack of
knowledge of set theory, or are deliberately false/misleading. The guy is a
professor of mathematics. I want to know which it is. I know you must know
him professionally, which is presumably why you are concerned. I would hope
that he can defend his ideas and papers for himself.

So lets go. These are neccessarily out of context, but I don't think this
misrepresents his position:

For each of these I would expect the statement is true, it is wrong out of
Norman's ingorance (eg crank material), or it is wrong and he knows it is
wrong (deliberately misleading):

--------------------------
To get you used to the modern magic of Cauchy sequences, here is one I just
made up:

[2/3, 2/3, 2/3 ...]
Anyone want to guess what the limit is? Oh, you want some more information
first? The initial billion terms are all 2/3. Now would you like to guess?
No, you want more information. All right, the billion and first term is 2/3
Now would you like to guess? No, you want more information. Fine, the next
trillion terms are all 2/3 You are getting tired of asking for more
information, so you want me to tell you the pattern once and for all? Ha Ha!
Modern mathematics doesn't require it! There doesn't need to be a pattern,
and in this case, there isn't, because I say so. You are getting tired of
this game, so you guess Good effort, but sadly you are wrong. The actual
answer is -17. That's right, after the first trillion and billion and one
terms, the entries start doing really wild and crazy things, which I don't
need to describe to you, and then `eventually' they start heading towards
but how they do so and at what rate is not known by anyone. Isn't modern
religion fun?
---------------------------

Compare and contrast:

To get you used to the modern magic of decimal notation, here is one I just
made up:
0.666...

Anyone want to guess what the limit is? Oh, you want some more information
first? The initial billion digits are all 6. Now would you like to guess?
No, you want more information. All right, the billion and first digit is 6.
Now would you like to guess? No, you want more information. Fine, the next
trillion terms are all 6. You are getting tired of asking for more
information, so you want me to tell you the pattern once and for all? Ha Ha!
Modern mathematics doesn't require it! There doesn't need to be a pattern,
and in this case, there isn't, because I say so. You are getting tired of
this game, so you guess Good effort, but sadly you are wrong. The actual
answer is 2/3 - 10^billion. That's right, after the first trillion and
billion and one terms, the entries start doing really wild and crazy things,
which I don't need to describe to you. Isn't modern religion fun?

These seem exactly the same argument to me. True, crank or deliberately
misleading?

How about:

---------------------------
Now that you are comfortable with the definition of real numbers, perhaps
you would like to know how to do arithmetic with them? How to add them, and
multiply them? And perhaps you might want to check that once you have
defined these operations, they obey the properties you would like, such as
associativity etc. Well, all I can say is---good luck. If you write this all
down coherently, you will certainly be the first to have done so.
--------------------------

True, crank or deliberately misleading?



On Cauchy sequences:

--------------------
On top of the manifold ugliness and complexity of the situation, you will be
continually dogged by the difficulty that in all these sequences there does
not have to be a pattern---they are allowed to be completely `arbitrary'.
That means you are unable to say when two given real numbers are the same,
or when a particular arithmetical statement involving real numbers is
correct.
----------------------

True, crank or deliberately misleading?

-----------------------
A set of rational numbers is essentially a sequence of zeros and ones, and
such a sequence is specified properly when you have a finite function or
computer program which generates it. Otherwise `it' is not accessible in a
finite universe.
-------------------------

(My mind boggles at what he thinks he means here).

True, crank or deliberately misleading?

--------------------------
Even the `computable real numbers' are quite misunderstood. Most
mathematicians reading this paper suffer from the impression that the
`computable real numbers' are countable, and that they are not complete. As
I mention in my recent book, this is quite wrong. Think clearly about the
subject for a few days, and you will see that the computable real numbers
are not countable , and are complete.
-------------------------

Oh, my God. He has written a book on it. I hope its not part of the UNSW
pure mathemtics syllabus.

True, crank or deliberately misleading?

He then goes on:
-------------------------
Think for a few more days, and you will be able to see how to make these
statements without any reference to `infinite sets', and that this suffices
for Cantor's proof that not all irrational numbers are algebraic.
-------------------------

Sorry, I am having a lot of trouble understanding how I can think about
"all" irrational numbers without thinking about an infinite set. I have even
more trouble comparing the cardinality of "all computable numbers" and "all
algebriac numbers" without using set theoretic constructions.

True, crank or deliberately misleading?

------------------------------------
In my studies of Lie theory, hypergroups and geometry, there has never been
a point at which I have pondered---should I assume this postulate about the
mathematical world, or that postulate?
---------------------------------

Hmm. Never seen a Group theorem that assumes a Group is commutative?

-----------------------------
but the nature of the mathematical world that I investigate appears to me to
be absolutely fixed. Either G2 has an eleven dimensional irreducible
representation or it doesn't (in fact it doesn't).
-----------------------------

The guy is a group theory expert. Presumably he has heard of the Whitehead
problem. http://en.wikipedia.org/wiki/Whitehead_problem

True, crank or deliberately misleading?



Quote:
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

I could go on, almost every sentence is a jewel of mininformation or
ignorance.

He walks like a crank, talks like a crank, and sqwaks like a crank. No
problem, unless you also run around telling people you are a Professor of
Mathematics.
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sradhakr
science forum addict


Joined: 19 Jul 2005
Posts: 57

PostPosted: Thu Jul 13, 2006 9:26 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gerry Myerson wrote:
Quote:
In article <1152772442.981716.222160@m73g2000cwd.googlegroups.com>,
"Gene Ward Smith" <genewardsmith@gmail.com> wrote:

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. 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?

All these questions are better directed to Norm than to me,
but I'll make believe I know what he is on about, and answer
thus:

If I remember right, Euclid never said "there's an infinitude of
primes." He just said, "given any prime, there's a bigger one."
You and I are accustomed to interpreting the second as meaning
the same thing as the first, but I think that until quite recent
times, mathematicians didn't. They didn't accept "the completed
infinity," and they were still able to develop the theory of
numbers, prove the quadratic reciprocity theorem, the four squares
theorem, etc. If you state the theorem the way Euclid did,
you are not forming an infinite set, and you can get on perfectly
well that way.
[...]


Well, the Greeks were not able to resolve Zeno's paradoxes, were they?
The point is that the Greek way of thinking leads to a paradox that
Achilles can never catch up with a tortoise (moving ahead of him in a
straight line at a slower velocity). This is the problem with the Greek
concept of "potential infinity" -- which will not permit Achilles to
"complete" the infinitely many conscious acts needed to catch up with
the tortoise (first Achilles has to reach a point where the tortoise
was previuously located, but now has moved ahead -- this step repeats
itself ad infinitum). I don't think classical real analysis resolves
Zeno's paradoxes with the concept of limits either -- this leads to the
issue of how infinitely many finite, non-zero and non-infinitesimal
intervals of reals can sum to a finite interval.

Norman Wildberger, like the Greeks, has the problem of explaining what
he means by an "arbitrary" natural number or prime number, etc. For
example, when NW says that some proposition holds for an arbitrary
natural number, is that not an unverifiable assertion about infinitely
many naturals that cannot be proven unless one makes postulates (or
axioms) to which he is so opposed? And such axioms *must* take for
granted the existence of an infinite class of natural numbers for which
the said proposition holds -- otherwise we don't have a proof of this
proposition. And NW does oppose the concept of an "arbitrary" real
number-- here is a quote from his paper "Set theory: should you
believe?" at <http://web.maths.unsw.edu.au/~norman/views2.htm>:

"But here is a very important point: we are not obliged, in modern
mathematics, to actually have a rule or algorithm that specifies the
sequence r1, r2, r3, . In other words, 'arbitrary'
sequences are allowed, as long as they have the Cauchy convergence
property. This removes the obligation to specify concretely the objects
which you are talking about. Sequences generated by algorithms can be
specified by those algorithms, but what possibly could it mean to
discuss a 'sequence' which is not generated by such a finite rule?
Such an object would contain an 'infinite amount' of information,
and there are no concrete examples of such things in the known
universe. This is metaphysics masquerading as mathematics."

I agree with NW's conclusion that an arbitrary real number, in the
classical sense, is problematic. But definitely not with his stated
reason for this conclusion -- that such an arbitrary real number (not
specified by a finite rule) contains an "infinite amount of
information". The fact is that even, say, a Cauchy sequence converging
to Pi, specified by a finite rule, does contain an infinite amount of
information -- no human mind can complete the task of running through
all the terms of this sequence generated by this finite rule. And NW
will have to explain how he can accept statements like "For any real
number x>0, (1/x)>0", if he does not accept the concept of an arbitrary
Cauchy sequence of rationals. For x is an arbitrary real number, is it
not?

My point of view, explained in my arxiv paper
<http://arxiv.org/abs/math.LO/0506475>, is that the problem is not with
the existence of an inifnite class of naturals -- one can accept that
an arbitrary prime can only be defined as that belonging to an infinite
class of primes, for example. What consititutes infinitary reasoning
(in my view) is quantifying over these infinite classes, i.e., formally
referring to infinitely many such infinite classes in a single formula.
Thus an arbitrary real number x, in the classical sense, is not allowed
in my proposed logic NAFL-- because x can only be defined as belonging
to an infinite class of reals, which requires quantification over reals
-- not permitted in NAFL because each real is an infinite object.
Infinite sets do not exist in NAFL, but infinite classes can exist (in
fact MUST exist whenever the infinitely many finite objects belonging
to that class exist). Infinite classes, like real numbers, are proper
classes-- so the reals do not consititute a class and you can't
quantify over reals in NAFL.

I believe that the correct resolution of Zeno's paradoxes is in the
version of real analysis proposed in the above-cited arxiv paper
<math.LO/0506475>. In particular, see Sec. 4, which is more or less
self-contained.

Regards, RS
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Peter Webb
science forum Guru Wannabe


Joined: 05 May 2005
Posts: 192

PostPosted: Thu Jul 13, 2006 9:46 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

"Gerry Myerson" <gerry@maths.mq.edi.ai.i2u4email> wrote in message
news:gerry-D682F9.17131513072006@sunb.ocs.mq.edu.au...
Quote:
In article <1152770679.597975.88170@p79g2000cwp.googlegroups.com>,
"Gene Ward Smith" <genewardsmith@gmail.com> wrote:

I didn't see any signs, as far as I had gotten, that he even knows
anything about modern set theory. Does he?

I don't know.

I reject astrology, even though I don't know anything about modern
astrology (I don't even know if there is such a thing). I reject
"creation science" and "intelligent design," even though I haven't
read any recent writings of their advocates. I don't have to; I
know where they're going, and I know they're never going to get
anywhere useful, going in that direction.

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

Doesn't this undermine your whole analogy? Why didn't you pick an orthodox
theory like Evolution, Special Realtivity or Plate Techtonics as being the
theory he is attacking? (Set theory is every bit as well accepted as any of
these other topics). Because he looks less of a crank if you compare him to
attacking astrology than him attacking (say) the Theory of Evolution, even
though this is a much closer analogy?

Quote:
Maybe Norm does. I don't know.

--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

I don't see you publishing any papers on astrology. Nor do you sign your
posts as a "Professor of Astronomy". And finally, I don't see you saying
"Astronomers and cosmologists are welcome to send me reasoned responses.",
as if your level of knowledge of astrology was so advanced you didn't think
non-specialists should be able to respond.

Doesn't it worry you that a professional mathematician can write a paper on
set theory (that has "caused a bit of discussion in some logic circles") and
you can't tell from the paper if he actually "knows anything about modern
set theory" ?
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Daryl McCullough
science forum Guru


Joined: 24 Mar 2005
Posts: 1167

PostPosted: Thu Jul 13, 2006 10:10 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gerry Myerson says...

Quote:
I'm dismayed by the level of vituperation in some of the posts in
this thread. Norm is not presenting a high-school algebra proof of
Fermat's Last Theorem, nor is he insisting that the reals are countable
because you can always take that real number that you left off your
list and stick it on at the end. He's adopting a finitistic, or
constructivist, or computational view of mathematics. It's an unpopular
view, it doesn't particularly appeal to me, but I don't see the need
to go ballistic in response.

Norm's paper is not a discussion of finitistic methods. It's
not a discussion of the computational view of mathematics. It
is a mean-spirited, sneering attacking on a huge swath of
modern mathematics and modern mathematicians. It is *full*
of vituperation.

When it comes to reasonableness, I don't see why Norm's equating
set theory with a religious is *more* reasonable than Peter Webb's
equating Norm's paper with crank babble. Norm's paper is not a
mathematical paper, it is a polemic. There is really no way to
give a reasoned, mathematical response to it.

In contrast, a paper that starts off saying that the author is
rejecting the axiom of infinity because he wishes to see how
much of mathematics can be done with minimal ontological commitment
could be the start of a reasonable mathematical paper. An exploration
of finitistic mathematics could be interesting mathematics. Nobody
would accuse Norm of being a crank for writing such a paper, or
even for dedicating his life to the development of finitistic
methods. People accuse him of being a crank when he says things
such as comparing set theory with a religious cult. *That's*
what's crank material, not finitistic methods, and not the
computational view of mathematics.

However, I understand why Norm might prefer to write the
inflamatory type of paper: because at least it generates
a response, while the more reasonable exploration of finitistic
methods would generate polite indifference and yawns.

--
Daryl McCullough
Ithaca, NY
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Thu Jul 13, 2006 10:20 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Norman Wildberger wrote:

Quote:
I have posted an article at http://web.maths.unsw.edu.au/~norman/views.htm
that has caused a bit of discussion in some logic circles.

My claims in short: 1) most of `elementary mathematics' is not
sufficiently well understood by the mathematical establishment, leading to
weaknesses in K12 and college curriculum, 2) the current theory of `real
numbers' is a joke, and sidesteps the crucial issue of understanding the
computational specification of the continuum, and 3) `infinite sets' are a
metaphysical concept, and unnecessary for correct mathematics.

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

Assoc Prof N J Wildberger
School of Maths
UNSW

I need to mull over what your paper says before I attempt to provide much in
the way of analysis. I tend to agree with the spirit of the presentation,
though I am not quite as dismissive of the established dogma.

Somewhere in the dust piles of my neglected papers is a one or two page
derivation of Kepler's laws beginning with no other assumption than basic
arithmetic and the belief that I could reasonably well communicate such
concepts as the Pythagorean theorem through simple drawings. The
presentation provides an illustration of what taking a derivative means;
IIRC using epsilon delta arguments. It gives a definition of an ellipse,
of Newton's laws of motion, and of the inverse square law of gravitation.
When I consider that Russell and Whitehead took some 360 pages to get
around to 1+1=2, I have to wonder if they really chose the best set of
axioms. I acknowledge that their objective was "perfect" rigor, where mine
was intuitive clarity, but I am inclined to believe the two objectives are
far closer to one another than is suggested by the current formalisms of
so-called foundational mathematics.

For now I will make only one comment about your paper. Regarding the proof
that multiplication is associative; that fact is "proved" in the volume I
am currently struggling through. This set of volumes attempts to outline
the state of the art of foundational mathematics in its day. Perhaps it is
nothing more than the formalization of that to which you object, but it was
intended to address many of the concerns you appear to be expressing.

<quote
url='http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=9431'>

Fundamentals of Mathematics, Volume I
Foundations of Mathematics: The Real Number System and Algebra
Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suss and H. Kunle
Translated by S. H. Gould

Fundamentals of Mathematics represents a new kind of mathematical
publication. While excellent technical treatises have been written about
specialized fields, they provide little help for the nonspecialist; and
other books, some of them semipopular in nature, give an overview of
mathematics while omitting some necessary details. Fundamentals of
Mathematics strikes a unique balance, presenting an irreproachable
treatment of specialized fields and at the same time providing a very clear
view of their interrelations, a feature of great value to students,
instructors, and those who use mathematics in applied and scientific
endeavors. Moreover, as noted in a review of the German edition in
Mathematical Reviews, the work is ?designed to acquaint [the student] with
modern viewpoints and developments. The articles are well illustrated and
supplied with references to the literature, both current and ?classical.??

The outstanding pedagogical quality of this work was made possible only by
the unique method by which it was written. There are, in general, two
authors for each chapter: one a university researcher, the other a teacher
of long experience in the German educational system. (In a few cases, more
than two authors have collaborated.) And the whole book has been
coordinated in repeated conferences, involving altogether about 150 authors
and coordinators.

Volume I opens with a section on mathematical foundations. It covers such
topics as axiomatization, the concept of an algorithm, proofs, the theory
of sets, the theory of relations, Boolean algebra, and antinomies. The
closing section, on the real number system and algebra, takes up natural
numbers, groups, linear algebra, polynomials, rings and ideals, the theory
of numbers, algebraic extensions of a fields, complex numbers and
quaternions, lattices, the theory of structure, and Zorn?s lemma.

Volume II begins with eight chapters on the foundations of geometry,
followed by eight others on its analytic treatment. The latter include
discussions of affine and Euclidean geometry, algebraic geometry, the
Erlanger Program and higher geometry, group theory approaches, differential
geometry, convex figures, and aspects of topology.

Volume III, on analysis, covers convergence, functions, integral and
measure, fundamental concepts of probability theory, alternating
differential forms, complex numbers and variables, points at infinity,
ordinary and partial differential equations, difference equations and
definite integrals, functional analysis, real functions, and analytic
number theory. An important concluding chapter examines ?The Changing
Structure of Modern Mathematics.?
</quote>

The part of the first volume specifically dedicated to foundational
mathematics strikes me as fairly agnostic regarding which of several
possible approaches are ideal in establishing the proper foundations of
mathematics. It is so condensed as to be cryptic. Each subsection could
fill a chapter, each section could fill a book, and each chapter could fill
a book shelf if they were elaborated upon to the point of covering their
topics comprehensively.


--
Nil conscire sibi
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tez_h@nospam.yahoo.com
science forum beginner


Joined: 27 Jun 2006
Posts: 9

PostPosted: Thu Jul 13, 2006 11:18 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Gerry Myerson wrote:
Quote:
In article <44b5abd7$0$1207$afc38c87@news.optusnet.com.au>,
"Peter Webb" <webbfamily-diespamdie@optusnet.com.au> wrote:
[snip]

Well, the discussion that I have seen - on this newsgroup (sci.math or
sci.logic, I can't remember) - is that it is bullshit.

Ad hominem.
[snip]

I don't know about the "mathematical establishment" (as a whole) not
understanding "elementary mathematics", but your own writings on set theory
and the axiomatic method don't fill me with confidence.

Ad hominem.
[snip]

This is pure crank stuff. Describing a huge and extremely rigorously defined
area such as the construction of the Reals as a "joke" without any
mathematical justification is flakey at best; the phrase "computational
specification of the continuum" (a phrase that gets exactly zero matches on
Google) is crank babble.

The mathematical justification for describing the current theory of
real numbers as a joke is given in the paper. You may not find it
convincing - I may not find it convincing - but it's there.

"Crank babble" is ad hominem.

[snip]

Your paper has no mathematical content, and is pure crank stuff. The stuff
about Axioms somehow being irrelevant to mathematics is just your own
philosophical ramblings. Surprising, since you seem to accept the axioms of
group theory, but not set theory (because they are too complicated and too
abstract for your liking)? Do groups with an infinite number of elements
exist, by the way?

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. He argues that the axioms
of set theory (in particular, ZFC) don't define what a set is and
don't lead to sensible constructions of infinite sets.


An ad hominem fallacy is of the form:

Person P made argument A

Person P is ignorant/poor/gay/female/stupid/handicapped/etc

Therefore argument A is invalid.


It is *not* ad hominem to say:

Person Q made argument B

Argument B is incoherent/lacks reasoning/lacks evidence/is purely
assertion/is muddled/makes categorical errors/equivocates/is
nonsense/is irrelevant/etc

Therefore person Q is ignorant/poorly
read/uninformed/unreasonable/stupid/irrelevant/etc


Sure, such a statement may be insulting. But it isn't invalid.

[snip]
Quote:

--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

-Tez
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Thu Jul 13, 2006 11:22 am    Post subject: Re: Set Theory: Should you believe? Reply with quote

Stephen Montgomery-Smith wrote:
Quote:
Norman Wildberger wrote:
I have posted an article at http://web.maths.unsw.edu.au/~norman/views.htm
that has caused a bit of discussion in some logic circles.

My claims in short: 1) most of `elementary mathematics' is not sufficiently
well understood by the mathematical establishment, leading to weaknesses in
K12 and college curriculum, 2) the current theory of `real numbers' is a
joke, and sidesteps the crucial issue of understanding the computational
specification of the continuum, and 3) `infinite sets' are a metaphysical
concept, and unnecessary for correct mathematics.

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

Assoc Prof N J Wildberger
School of Maths
UNSW

After casually reading his notes, I think that he is saying this. The
axioms of modern set theory are too burdensome for actual
mathematicians, who in practice take a somewhat Platonic view of their
subject. But he is unsatisfied with the pragmatic Platonist approach
that we take, particular in the manner in which mathematics is taught at
pre-college levels. He thinks we need to roll up our sleeves and
rethink foundations so that we get something that really is usuable, so
that we can finally truly rid ourselves of Platonism, superstition,
instinct, gut reaction, religion, etc, etc.

Personally I really like the Platonic approach, using set theory as a
highly convenient crutch. I'm not going to preclude the possibility
that one day a set of foundations for mathematics will be found, that
will greatly simplify and advance the extent that we will be able to
think about mathematics, but I think it will take a great genius, and
also some crisis of cicumstance (perhaps the discovery of some horribly
unresolvable contradiction).

But on the whole, even if his tone was not exactly politic, I liked a
lot of what he said. But I don't think it is going to change the world.

Stephen

Hi, without any intention of antagonizing you, let me ask you this
question: What of what he said did you like and for what reasons?
I would like to understand what the appeal of such a rant as
Wildberger's is. In my view, it is evidently wrong in many parts, wrong
under light scrutiny in many others and falacious for the most part of
the rest. The only statement I can think of which is undebatable is the
fact that education in mathematics is ever more deficient. But this is
also strongly suggested by a lot of other factors, as the ever growing
number of crackpots, 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.
Thanks in advance for any effort you may put into your answer.
Regards.
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