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Set Theory: Should you believe?

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Norman Wildberger

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Jul 13, 2006, 11:37:51 AM7/13/06
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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


malb...@yahoo.com

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Jul 12, 2006, 6:56:13 PM7/12/06
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I've added sci.logic to your posting. karl m

Gonçalo Rodrigues

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Jul 12, 2006, 9:43:48 PM7/12/06
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On Thu, 13 Jul 2006 08:37:51 -0700, "Norman Wildberger"
<wildb...@pacific.net.au> fed this fish to the penguins:

I have read it all and I have survived. I will not offer any reasoned
response because you have not written a reasoned article. While at
some points your position is tenable and understandable, there are
just too many blunders, confusions and errors to try to answer it.

With my best regards,
G. Rodrigues

Peter Webb

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Jul 12, 2006, 10:11:34 PM7/12/06
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<malb...@yahoo.com> wrote in message
news:1152744973.1...@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. Much of it admittedly
from me.

>> 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.

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.


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?

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?

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?

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.

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?

Gene Ward Smith

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Jul 12, 2006, 10:16:43 PM7/12/06
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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.

Your paper starts out saying a lot of things that are blatanly false.
This is not a good way to make a reasoned argument.

You say, for instance, that "the Academy" has consistently refused to
get serious about foundational questions, whereas doing that has been a
major theme of twentieth century mathematics, and many great
mathematicians have made it their life's work. You claim physicists
have trouble with string theory because they make use of set theory,
which is simply nonsense. I suspect you don't know much about string
theory. You claim the axioms of set theory have not been questioned,
which is drivel. You sneer at people who do what you claim you think
should be done, namely study the foundations of mathematics, and put
the word "supposedly" before "difficult", indicating that you think set
theory and logic are dead easy. But do you know any serious set theory?
You claim that most mathematicians could not define a vector or a
function, as if you possessed the secret decoder ring which made you
smarter than the rest of us.

In short, you sound, just in your opening few paragraphs, altogether
too much like the kind of people we constantly come across here on
sci.math. You sound like a crank. I think you *are* a crank. I suggest
you stick to subjects you've studied, and not sound off on topics you
don't understand.

david petry

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Jul 12, 2006, 10:54:44 PM7/12/06
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Norman Wildberger wrote:

> 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.

Here's two relevant quotes:

"ordinary mathematical practice does not require an enigmatic
metaphysical universe of sets" (Nik Weaver)

"the actual infinite is not required for the mathematics of the
physical world" (Feferman)

Most mathematicians working in the field of foundations understand and
accept those quotes, but the average mathematician may not.

What I have suggested is that mathematics needs a reality check;
mathematics can and should be treated as a science in which testable
consequences are required. I suspect that is what you are getting at,
although I don't think you've said it especially clearly.


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

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

Gene Ward Smith

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Jul 12, 2006, 11:07:11 PM7/12/06
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david petry wrote:

> Most mathematicians working in the field of foundations understand and
> accept those quotes, but the average mathematician may not.

Most mathematicians are not enthusiastic about being forced to work
only in terms of first-order arithmetic, or even second-order
arithmetic, either. Because it would be a big, fat pain and accomplish
nothing they see as necessary. On the other hand, they might very well
be interested in the question of whether, and to what extent, what they
are doing can be done using weaker assumptions--so long as you don't
try to *make* them do it that way.

leste...@cableone.net

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Jul 12, 2006, 11:15:06 PM7/12/06
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Well technically of course all solutions you point out define a
perfect sphere not a perfect circle.

~v~~

Virgil

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Jul 12, 2006, 11:37:30 PM7/12/06
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In article <ggebb2dj42j589uhh...@4ax.com>,
leste...@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.

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

Gerry Myerson

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Jul 13, 2006, 12:02:25 AM7/13/06
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In article <44b5abd7$0$1207$afc3...@news.optusnet.com.au>,
"Peter Webb" <webbfamily...@optusnet.com.au> wrote:

> <malb...@yahoo.com> wrote in message
> news:1152744973.1...@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.

> >> 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.

> 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.

> 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.

> 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. 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.

> 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.

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.

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

Gene Ward Smith

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Jul 13, 2006, 2:04:39 AM7/13/06
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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?

Gene Ward Smith

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Jul 13, 2006, 2:34:03 AM7/13/06
to

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?

> 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?

Gerry Myerson

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Jul 13, 2006, 3:06:23 AM7/13/06
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In article <1152772442.9...@m73g2000cwd.googlegroups.com>,

"Gene Ward Smith" <genewa...@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.

As for measureable cardinals, I'm guessing the question of their
existence doesn't interest Norm one way or the other, on the
grounds that the stability or otherwise of the Sydney Harbor
Bridge is unlikely to depend on the outcome. The real world
questions mathematics sprang from do not depend on these
abstractions, nor on the axiom systems in which they are debated.

> > 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?

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."

Gene Ward Smith

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Jul 13, 2006, 3:07:25 AM7/13/06
to

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.

Here's a quote from the paper, which I think shows how confused
Wildberger is about axioms:

"In ordinary mathematics, statements are either true, false, or they
don't make sense. If you have an elaborate theory of 'hierarchies upon
hierarchies of infinite sets', in which you cannot even in principle
decide if there is something between the first and second 'infinity' on
your list, there's a time to admit you are no longer doing
mathematics."

Of course, the statement about not being able to decide if there is a
cardinal between aleph_0 and aleph_1 is absurd, but leave that aside
and look instead at the statement that there is a cardinal between
aleph_0 and 2^aleph_0. How does this differ from the statement that the
sum of the angles of a triangle cannot be determined in absolute
geometry?

Suppose, when faced with the fact that not everyone found the parallel
postulate to be intuituively true, the Greek geometers had simply
removed it from consideration. They could then be attacked with the
sneer that they were not doing mathematics at all, because they could
not answer so basic a question as whether the sum of the angles of a
triangle was less than, equal to, or greater than two right angles. Yet
I think it is clear they *would* be doing geometry. For that matter a
group theorist who cannot tell you if a generic group is abelian or
nonabelian, since it might be either, is not failing to do mathematics.

The difference is that we've given up on the idea that there is a
single correct geometry, but still feel (and that's nothing but
intuition speaking) that there is a single true set theory, just as we
think there is a single true number theory. But our intution is not
strong enough to settle all the questions as to what this true set
theory actually is. This is really a meta problem for mathematics, and
not a question which can allow a person to conclude that set theorists
are not mathematicians, any more than the existence of nonstandard
models for first-order arithmetic would prove number theorists are not
mathematicians. As to rigor, the mechanically verified proofs of the
Mizar project are more rigorous than mere mortals like you, me, or
Wildberger do, and they are based on set theory with (if needed)
inacessible cardinals. The rigor argument is clearly therefore baloney,
and Wildberger cannot seem to separate mathematics from metaphysics for
the rest of it.

Gerry Myerson

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Jul 13, 2006, 3:13:19 AM7/13/06
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In article <1152770679....@p79g2000cwp.googlegroups.com>,

"Gene Ward Smith" <genewa...@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. Maybe Norm does. I don't know.

Russell Easterly

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Jul 13, 2006, 3:14:39 AM7/13/06
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"Gene Ward Smith" <genewa...@gmail.com> wrote in message
news:1152770679....@p79g2000cwp.googlegroups.com...

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


Gene Ward Smith

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Jul 13, 2006, 3:18:30 AM7/13/06
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Gerry Myerson wrote:

> 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?

Gene Ward Smith

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Jul 13, 2006, 3:19:52 AM7/13/06
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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?

Stephen Montgomery-Smith

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Jul 13, 2006, 3:48:33 AM7/13/06
to

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

Kevin Karn

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Jul 13, 2006, 3:58:25 AM7/13/06
to
Gene Ward Smith wrote:
> 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.

Virgil

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Jul 13, 2006, 4:01:43 AM7/13/06
to
In article <UbWdnZXmzLV8byjZ...@comcast.com>,
"Russell Easterly" <logi...@comcast.net> wrote:

> "Gene Ward Smith" <genewa...@gmail.com> wrote in message
> news:1152770679....@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.


>
> 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.

> 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.


> 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.

Gene Ward Smith

unread,
Jul 13, 2006, 4:40:53 AM7/13/06
to

Stephen Montgomery-Smith wrote:

> 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
> 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.

> 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.

Gene Ward Smith

unread,
Jul 13, 2006, 4:42:21 AM7/13/06
to

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.

Peter Webb

unread,
Jul 13, 2006, 4:42:56 AM7/13/06
to

"Gerry Myerson" <ge...@maths.mq.edi.ai.i2u4email> wrote in message
news:gerry-2088DC....@sunb.ocs.mq.edu.au...

> In article <44b5abd7$0$1207$afc3...@news.optusnet.com.au>,
> "Peter Webb" <webbfamily...@optusnet.com.au> wrote:
>
>> <malb...@yahoo.com> wrote in message
>> news:1152744973.1...@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.

>
>> >> 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.


>> 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.

>> 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?

>> 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.

>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.

>> 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.

> 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?

> --
> Gerry Myerson (ge...@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.


R. Srinivasan

unread,
Jul 13, 2006, 5:26:25 AM7/13/06
to
> [...]


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

Peter Webb

unread,
Jul 13, 2006, 5:46:02 AM7/13/06
to

"Gerry Myerson" <ge...@maths.mq.edi.ai.i2u4email> wrote in message
news:gerry-D682F9....@sunb.ocs.mq.edu.au...

> In article <1152770679....@p79g2000cwp.googlegroups.com>,
> "Gene Ward Smith" <genewa...@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?

>Maybe Norm does. I don't know.
>
> --
> Gerry Myerson (ge...@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" ?

Hatto von Aquitanien

unread,
Jul 13, 2006, 6:20:38 AM7/13/06
to
Norman Wildberger wrote:

> 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

Daryl McCullough

unread,
Jul 13, 2006, 6:10:02 AM7/13/06
to
Gerry Myerson says...

>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

Tez

unread,
Jul 13, 2006, 7:18:28 AM7/13/06
to
Gerry Myerson wrote:
> In article <44b5abd7$0$1207$afc3...@news.optusnet.com.au>,
> "Peter Webb" <webbfamily...@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]


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

-Tez

guenther vonKnakspot

unread,
Jul 13, 2006, 7:22:50 AM7/13/06
to

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.

Hatto von Aquitanien

unread,
Jul 13, 2006, 9:00:11 AM7/13/06
to
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 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

Richard Herring

unread,
Jul 13, 2006, 9:11:00 AM7/13/06
to
In message <1152759284....@h48g2000cwc.googlegroups.com>, david
petry <david_lawr...@yahoo.com> writes
>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

Hatto von Aquitanien

unread,
Jul 13, 2006, 9:33:58 AM7/13/06
to
Virgil wrote:

> 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

Robert J. Kolker

unread,
Jul 13, 2006, 10:43:42 AM7/13/06
to
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. We still need Inspiration. If you regard all, so-called "mental"
processes, as really physical then your assertion may have some basis.

Bob Kolker

guenther vonKnakspot

unread,
Jul 13, 2006, 10:29:07 AM7/13/06
to
Hatto von Aquitanien wrote:
> 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.

> 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).

> 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.

Dave L. Renfro

unread,
Jul 13, 2006, 10:29:49 AM7/13/06
to
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.

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

Hatto von Aquitanien

unread,
Jul 13, 2006, 10:31:00 AM7/13/06
to
Robert J. Kolker wrote:

> 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?

> 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

ste...@nomail.com

unread,
Jul 13, 2006, 11:04:53 AM7/13/06
to

> Russell Easterly wrote:

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

Stephen

kunzmilan

unread,
Jul 13, 2006, 11:06:18 AM7/13/06
to

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

Hatto von Aquitanien

unread,
Jul 13, 2006, 11:18:22 AM7/13/06
to
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.
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.



>> 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.

>> 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

guenther vonKnakspot

unread,
Jul 13, 2006, 11:39:58 AM7/13/06
to
Hatto von Aquitanien wrote:
> 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.

> 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.


> 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.


> >> 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.


> 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 ?

Lee Rudolph

unread,
Jul 13, 2006, 11:44:58 AM7/13/06
to

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

Aatu Koskensilta

unread,
Jul 13, 2006, 11:52:47 AM7/13/06
to
Lee Rudolph wrote:
> 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.kos...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

leste...@cableone.net

unread,
Jul 13, 2006, 12:42:07 PM7/13/06
to
On Wed, 12 Jul 2006 21:37:30 -0600, Virgil <vir...@comcast.net> wrote:

>In article <ggebb2dj42j589uhh...@4ax.com>,


> leste...@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.

>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~~

lDont...@nowhere.net

unread,
Jul 13, 2006, 1:15:04 PM7/13/06
to
On 12 Jul 2006 23:34:03 -0700, "Gene Ward Smith"
<genewa...@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.

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

> 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~~

Stephen Montgomery-Smith

unread,
Jul 13, 2006, 1:18:23 PM7/13/06
to

I have (tried) working in fluid dynamics for a number of years, and
certainly some sort of naive set theory is indispensible. I already
indicated this in another post where I talked about problems in showing
which differential equations have unique solutions and which don't.
Basically anything beyond numerical simulations or solving simple
laminar flows requires sophisticated techniques that involve objects
much more abstract than the real numbers (namely things like measure
spaces and Besov spaces and things like that). A good example is the
result of Caffarelli, Kohn and Nirenberg which looks at the maximal
possible Hausdorff dimension of the set of singularities in the solution
to the Navier-Stokes equation.

Stephen

Aatu Koskensilta

unread,
Jul 13, 2006, 1:19:13 PM7/13/06
to
lDont...@nowhere.net wrote:
> On 12 Jul 2006 23:34:03 -0700, "Gene Ward Smith"
> <genewa...@gmail.com> wrote:
>
>> 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.

How do you demonstrate that?

george

unread,
Jul 13, 2006, 1:21:44 PM7/13/06
to

Gerry Myerson wrote:

> More ad hominem.

Shut up. You are not a rhetoric major
and You Don't Know what "ad hominem" means.
The literal translation is "against the man" but that
does NOT mean that EVERY time somebody insults
a person, he is committing a sin of type "ad hominem".
What is going on here is an ARGUMENT. Arguments
are characterized by mutual attempts at LOGICAL rebuttal.
If you have logically demonstrated that somebody is flaunting
his ignorance and being a dipshit then it is NOT "ad hominem"
Or Any Other fallacy to point this out.

> Norm accepts the axioms of group theory as the
> definition of what a group is,

Shut up. You don't know Norm, either.
Norm attacks "logicians" and the axiomatic method
generally. IF he accepted axioms as legitimate for
defining things then he wouldn't have anything to argue about.

> and has no problem with them

Wrong.

> because he can construct (finite) models of them.

Obviously, he AND EVERYBODY ELSE can ALSO construct
INFINITE models of them -- AND HE DEALS with infinite
groups ALL THE TIME in his work.
So this is simply not a legitimate distinction.

> He argues that the axioms
> of set theory (in particular, ZFC) don't define what a set is

That is completely incoherent.
Since, in ZFC, EVERYthing is a set, the axioms of ZFC
could NOT POSSIBLY DO ANYTHING BUT define what
a set is.

> and don't lead to sensible constructions of infinite sets.

That is ridiculous too. He is not even saying that.
You need to quote him. What he DOES say is that
textbooks aren't much on defining what an infinite set is,
and he has looked. That just makes HIM look stupid.
Arthur Rubin and others have posted 5 different definitions
of infinite set, in case he was too stupid to find one.
More to the point, classical FOL IN GENERAL is NOT
constructive and one NEVER looks to the axioms themselves
to construct ANYthing -- the model construction language is
ALWAYS something DIFFERENT.


> 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.

Well, now you are a lot closer to reading him rightly,
but you do need to understand that your 2nd point here (about
math not needing axioms in the first place) contradicts your point
above about Norm respecting the axioms of group theory because
he can construct finite models of them. Group theory is in fact
defined by its axioms. Norm canNOT simultaneously appreciate that
AND think that math doesn't need axioms. Neither can you.

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

Well, you shouldn't be.
That's what you get for farting in church. NW is presuming
to pontificate about something he has not studied.

> 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.

He IS NOT, you IDIOT! IF he were doing that, he would've
actually GOOGLED "constructive mathematics" before presuming
to pontificate, and would've SAID something about it in his article!
NW * doesn't know SHIT * about constructive mathematics!

> It's an unpopular view,

And one that he has never heard of; the only thing that is even
MOTIVATING him to LEAN in that direction are the things
he DOES ACTUALLY TALK about in his article, things you
would've been WISE TO QUOTE, IF you were going to be
stupid enough to be attributing views to another person,
things like the failure of the profession at large to logically
ground its concepts. If this was what was actually bothering
him then (obviously) he should've been supporting axioms,
not attacking them.

> it doesn't particularly appeal to me, but I don't see the need
> to go ballistic in response.

The going ballistic is NOT in response to constructivism.
NW didn't cite Bishop or anybody who is trying to go that
way. He HIMSELF FIRST went ballistic vs. "logicians" by
calling them a priesthood. He embarrassed himself by sounding
like James Harris.

lDont...@nowhere.net

unread,
Jul 13, 2006, 1:22:02 PM7/13/06
to
On Thu, 13 Jul 2006 09:33:58 -0400, Hatto von Aquitanien
<ab...@AugiaDives.hre> wrote:

>Virgil wrote:
>
>> 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 believe set theory is essential to modern math.

> 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?

Not exactly the same question. The historical or anthropological
foundations of math are certainly interesting. But the more
interesting question is why people are using intuitive assumptions in
math at all?
~v~~

Hatto von Aquitanien

unread,
Jul 13, 2006, 1:27:07 PM7/13/06
to
leste...@cableone.net wrote:

>
> 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~~

My suspicion is that the objection is to the idea of applying operations
which are valid for finite sets to infinite sets and arriving at
conclusions which challenge conventional wisdom. For example it was
recently pointed out that permutations of N result in a demonstration that
there exist uncountable bijections. Another example is Cantor's theory of
transfinite induction. One can reasonably ask if any of these findings are
meaningful.

I'm told by some that the Löwenheim-Skolem theorem will do all kinds of
wonderful things for me. The theorem depends upon Cantor's findings. Now,
if I take the time to truly understand this stuff, will I simply be led
down the primrose path?
--
Nil conscire sibi

Aatu Koskensilta

unread,
Jul 13, 2006, 1:33:48 PM7/13/06
to
Hatto von Aquitanien wrote:
> For example it was recently pointed out that permutations of N result in a demonstration that
> there exist uncountable bijections.

Really? Where can I find out more about this startling discovery?

ste...@nomail.com

unread,
Jul 13, 2006, 1:41:13 PM7/13/06
to
In sci.math Aatu Koskensilta <aatu.kos...@xortec.fi> wrote:
> Hatto von Aquitanien wrote:
>> For example it was recently pointed out that permutations of N result in a demonstration that
>> there exist uncountable bijections.

> Really? Where can I find out more about this startling discovery?

The above is Aquitanien's garbled interpretation of the fact
that there exist countable sets that are not recursively
enumerable.

Stephen

Stephen Montgomery-Smith

unread,
Jul 13, 2006, 1:49:04 PM7/13/06
to

Well let me start by saying that I am only stating an opinion so there
is no need to get upset or mad about it. I do admit that I didn't
deeply scrutinize his writings, but I didn't feel that he said anything
mathematically false. So it really all boils down to feelings about the
current state of mathematics.

Next, while he was definitely ranting, and perhaps venting a certain
anger he also feels, I think it is important to listen to the message
rather than the messenger or the style in which the message is communicated.

There is a clear distinction between someone who expresses discomfort
with the present way we do mathematics which is at worst an unpopular
view, and saying that Cantor's diagonal argument is logically flawed
which at best is a crackpot position to hold. This web site is clearly
in the former category.

The view that mathematics should try to constrain itself to physical
reality is, in my opinion, not a crackpot position. I think this is
what Morris Kline was getting at in his books "Mathematics: Loss of
Certainty." Attempts to place axiomatic number theory and axiomatic set
theory on firmer ground by proving its consistency starting with a much
weaker system are doomed to failure by Goedel's Theorem, and thus the
possibility that the whole thing is built on a house of cards becomes
ever more reasonable. I agree with Kline that the strongest evidence
for the consistency of these theories, or at least some weaker version
which would still contain most of our important discoveries, is just how
very well it seems to work in practice.

Let me use an example. The mathematicians Wiener and Banach lived at
about the same time. Banach chose a much more abstract direction to
take his mathematics, looking at general properties of infinite
dimensional normed vector spaces beyond Hilbert spaces and spaces of
continuous or measurable function. Wiener looked more to real life
applications, and building on the work of Einstein put the phenominum of
Brownian motion onto a rigourous mathematical foundation. Many very
talented mathematicians have studied Banach spaces, but it has proven to
be a remarkably barren field. On the other hand Wiener's approach has
led to the rich theory of stochastic calculus, which has proved
extremely useful in studing the heat equation, harmonic functions, and
even economics.

Stephen

DontB...@nowhere.net

unread,
Jul 13, 2006, 1:53:03 PM7/13/06
to
On Thu, 13 Jul 2006 20:19:13 +0300, Aatu Koskensilta
<aatu.kos...@xortec.fi> wrote:

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

>How do you demonstrate that?

That people assume their assumptions? Axioms are assumed. Assumption
is the alternative to demonstration. Assumption is not demonstration.
If people are "not too lazy or stupid" to demonstrate theorems - since
many theorems have been demonstrated - then the inference is they are
too lazy or stupid to demonstrate axioms by conversion of "not not too
lazy or stupid".

~v~~

Hatto von Aquitanien

unread,
Jul 13, 2006, 1:52:31 PM7/13/06
to
ste...@nomail.com wrote:

If you cannot recursively enumerate it, you can't count it.
--
Nil conscire sibi

Aatu Koskensilta

unread,
Jul 13, 2006, 1:57:24 PM7/13/06
to
DontB...@nowhere.net wrote:
> On Thu, 13 Jul 2006 20:19:13 +0300, Aatu Koskensilta
> <aatu.kos...@xortec.fi> wrote:
>
>> lDont...@nowhere.net wrote:
>>> Isn't everyone? The only reason for axioms is that people are too lazy
>>> or stupid to demonstrate the truth of their assumptions
>> How do you demonstrate that?
>
> That people assume their assumptions? Axioms are assumed.

I'm not convinced. Please demonstrate that.

Hatto von Aquitanien

unread,
Jul 13, 2006, 2:03:12 PM7/13/06
to
Hatto von Aquitanien wrote:

I guess that may not be completely accurate. As long as I know that every
location has a unique element I can count the elements and not know which
one I am counting. I stand corrected.
--
Nil conscire sibi

Dave L. Renfro

unread,
Jul 13, 2006, 2:05:23 PM7/13/06
to
Hatto von Aquitanien wrote:

>> For example it was recently pointed out that
>> permutations of N result in a demonstration that
>> there exist uncountable bijections.

Aatu Koskensilta wrote:

> Really? Where can I find out more about this
> startling discovery?

The identity map on the set of all permutations
of N is an example of an uncountable permutation,
but I'm not sure if this is what Hatto von Aquitanien
had in mind.

Dave L. Renfro

Hatto von Aquitanien

unread,
Jul 13, 2006, 2:20:01 PM7/13/06
to
lDont...@nowhere.net wrote:

> On Thu, 13 Jul 2006 09:33:58 -0400, Hatto von Aquitanien
> <ab...@AugiaDives.hre> wrote:
>

>>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 believe set theory is essential to modern math.

What can you do with set theory which you can't do with symbolic logic
and/or some kind of BNF-like grammar?

> Not exactly the same question. The historical or anthropological
> foundations of math are certainly interesting. But the more
> interesting question is why people are using intuitive assumptions in
> math at all?
> ~v~~

Because without intuitive assumptions they would have absolutely not concept
of existence. That would make doing mathematics rather difficult. My
point is that if one were able to identify and codify these assumptions, it
may be possible to establish a good formal foundation that way.
--
Nil conscire sibi

Gene Ward Smith

unread,
Jul 13, 2006, 2:49:00 PM7/13/06
to

Stephen Montgomery-Smith wrote:

> The view that mathematics should try to constrain itself to physical
> reality is, in my opinion, not a crackpot position.

It's a meaningless position unless you can give "confine itself to
physical reality" in connection to a subject which is not *about*
physical reality a meaning.

> Let me use an example. The mathematicians Wiener and Banach lived at
> about the same time. Banach chose a much more abstract direction to
> take his mathematics, looking at general properties of infinite
> dimensional normed vector spaces beyond Hilbert spaces and spaces of
> continuous or measurable function. Wiener looked more to real life
> applications, and building on the work of Einstein put the phenominum of
> Brownian motion onto a rigourous mathematical foundation. Many very
> talented mathematicians have studied Banach spaces, but it has proven to
> be a remarkably barren field. On the other hand Wiener's approach has
> led to the rich theory of stochastic calculus, which has proved
> extremely useful in studing the heat equation, harmonic functions, and
> even economics.

Stephen, I suggest you confine yourself to topics you know something
about, and cease spouting ignorant horseshit. A very famous
incident--so famous it makes it dead obvious you don't know what the
hell you are talking about--is Gelfand's proof of Wiener's theorem
using Banach algebras. Wiener had proven that if Fourier series on a
closed interval which converges absolutely to a function f such that f
has no zero on the interval, then 1/f also has an absolutely convergent
Fourier series on that interval. The proof was not easy, yet Gelfand
proves it in a few lines using the theory of Banach algebras.

Gene Ward Smith

unread,
Jul 13, 2006, 2:55:30 PM7/13/06
to

george wrote:

> Shut up. You don't know Norm, either.

They are both live in Sydney, so he probably does know him or has at
least met him. I suspect his views have been influenced by something
more than the feeble contents of Wildberger's diatribe.

Gene Ward Smith

unread,
Jul 13, 2006, 2:59:24 PM7/13/06
to

Stephen Montgomery-Smith wrote:

> Basically anything beyond numerical simulations or solving simple
> laminar flows requires sophisticated techniques that involve objects
> much more abstract than the real numbers (namely things like measure
> spaces and Besov spaces and things like that).

If you want a really scary thought, try this one--the large cardinal
axioms set theorists like to ponder have implications for measure
theory and descriptive set theory.

Stephen Montgomery-Smith

unread,
Jul 13, 2006, 3:17:11 PM7/13/06
to
Gene Ward Smith wrote:
> Stephen Montgomery-Smith wrote:
>
>
>>The view that mathematics should try to constrain itself to physical
>>reality is, in my opinion, not a crackpot position.
>
>
> It's a meaningless position unless you can give "confine itself to
> physical reality" in connection to a subject which is not *about*
> physical reality a meaning.

I disagree, but while I would otherwise be glad to continue this
interesting discussion...


> Stephen, I suggest you confine yourself to topics you know something
> about, and cease spouting ignorant horseshit.

...it simply isn't worth this kind of abuse. If you disagree with me
that's fine, but do so in a civilized manner.

> A very famous
> incident--so famous it makes it dead obvious you don't know what the
> hell you are talking about--is Gelfand's proof of Wiener's theorem
> using Banach algebras.

I do know what I am talking about. I have written papers in Banach
spaces and Banach algebras. For the purposes of this discussion (which
is not about the hard facts of math) these two fields are essentially
different.

Frankly I don't mind you thinking I am wrong, and engaging in a
civilized discussion about how I am wrong, but don't throw abuse at me
like this.

This is a discussion about the philosophy of math. Fankly I don't see
this discussion of worthy of anything more than having fun, and if it
ceases to be fun I don't want to be part of it.

Stephen

Hatto von Aquitanien

unread,
Jul 13, 2006, 3:16:41 PM7/13/06
to
Stephen Montgomery-Smith wrote:

> Hatto von Aquitanien wrote:
>> Virgil wrote:
>>
>>
>>>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?
>
> I have (tried) working in fluid dynamics for a number of years, and
> certainly some sort of naive set theory is indispensible.

I believe that it is vital to have an understanding of sets to the extent of
knowing what a Ven diagram is, as well as what intersections, unions,
complements, etc., are. But do I really need

# 0 = { }
# 1 = {0} = {{ }}
# 2 = {0,1} = {0, {0}} = {{ }, {{ }}}
# 3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, {{ }}, {{ }, {{ }}}}
?

1+1=2, 2+1=3, 3+1=4,... or even 1,2,3,... seems more useful to me.

> I already
> indicated this in another post where I talked about problems in showing
> which differential equations have unique solutions and which don't.
> Basically anything beyond numerical simulations or solving simple
> laminar flows requires sophisticated techniques that involve objects
> much more abstract than the real numbers (namely things like measure
> spaces and Besov spaces and things like that). A good example is the
> result of Caffarelli, Kohn and Nirenberg which looks at the maximal
> possible Hausdorff dimension of the set of singularities in the solution
> to the Navier-Stokes equation.

Does any of that involve transfinite induction?
--
Nil conscire sibi

Gene Ward Smith

unread,
Jul 13, 2006, 3:21:38 PM7/13/06
to

Peter Webb wrote:

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

Wildberger is the guy who is promoting the idea that we should start
off in geometry by doing it over vector spaces of the rationals, and
then replacing all formulas which lead out of the rationals by things
which do some of the same work but confine themselves to rational
values. Instead of the norm, you use the associated bilinear and
quadratic forms; he calls the square of the distance between two points
"quadrance". Instead of angles, we can use "spread", which is
sin^2(theta). Then multiple angle formulas involve a transformed
version of the Chebychev polynomials, transferring from the [-1,1]^2
square to the [0,1]^2 square, which he calls "spread polynomials".

This is a cute idea but he wrecks it somewhat by rants about the
"wrong" trigonometry and how terribly, terribly revolutionary this
fairly obvious idea is.

Gene Ward Smith

unread,
Jul 13, 2006, 3:27:47 PM7/13/06
to

Stephen Montgomery-Smith wrote:

> > A very famous
> > incident--so famous it makes it dead obvious you don't know what the
> > hell you are talking about--is Gelfand's proof of Wiener's theorem
> > using Banach algebras.
>
> I do know what I am talking about. I have written papers in Banach
> spaces and Banach algebras.

In that case, you were deliberately misleading.

For the purposes of this discussion (which
> is not about the hard facts of math) these two fields are essentially
> different.

What the hell does this mean--for the purposes of this discussion we
are going to pretend that Banach algebras have nothing whatever to do
with Banach spaces, so we can continue to insult Stefan Banach? Tough.
I won't play such a silly game.

> Frankly I don't mind you thinking I am wrong, and engaging in a
> civilized discussion about how I am wrong, but don't throw abuse at me
> like this.

You were not being honest by your own admission here. You *knew* that
Banach's work had applications, and deliberately tried to make it
appear otherwise.

> This is a discussion about the philosophy of math. Fankly I don't see
> this discussion of worthy of anything more than having fun, and if it
> ceases to be fun I don't want to be part of it.

It's more fun if you play by the rules.

imagin...@despammed.com

unread,
Jul 13, 2006, 3:50:07 PM7/13/06
to
Stephen Montgomery-Smith wrote:

<snip>

> The view that mathematics should try to constrain itself to physical
> reality is, in my opinion, not a crackpot position.

It does seem to me that - as stated - this is precisely a crackpot
position. The problem is none of the words "physical", "reality", or
"constrain", the problem is the word "should". In practice mathematics,
as referred to by the overwhelming population of mathematicians, means
the abstract study of formal patterns, and that's all. The mystery of
it all is the way in which the patterns investigated by abstract
mathematicians turn out to apply so directly to aspects of the real
world. But it's going to be hard to develop these abstract theories if
you have to be consulting one of these "real world characters" [do I
have to call them other than "cranks"?] to see if _they_ happen to
think what you're doing is OK.

What about geometry? Do we really have an absolutely clear answer to
what the geometry of the physical world is? If I can assume not, then
what is anyone to do? Can't study Euclidean geometry, because it might
turn out not to be maths; similarly can't study non-Euclidean geometry
either. In fact, the maths department is at the mercy of the physic
department to inform it (eventually, perhaps) what it may or may not
include in the curriculum. Clearly this is insane.

I am no expert here, but I looked at the OP's rant, and it appeared to
me to be classic crankery. How else can you interpret it but that he
thinks "mathematics" means "calculating things (in the real world)"? I
see he has also written a book - can anyone make any sense of it?
Again, there are all the standard crank signs: cranks do not
investigate some new idea, they "replace" the bit of maths that's
beyond them with something usually half-baked - and even if it could be
formalised into something coherent, it's very far from clear that it
achieves anything other than applying a patch to the crank's pet peeve.
I mean, OP "replaces" trigonometrical functions with something
"better": can that ever be coherent? If the "replacement" is different,
it has not replaced anything, since the original functions still exist;
if it's just a different description of the same thing, this hardly
counts as "replacement" either.

I'm also puzzled by a quite simple question: if you reject the axiom of
infinity, then you have no infinite sets. Can you still say - for
example - that the integers form a group under addition? (How? By some
grotesque circumlocution? I don't know...) Do you not simply lose all
algebra relating to infinite sets? When people like Tony Orlow babble
on about "numbers", it's easy to understand why it doesn't bother them
that their creations have more or less no well-defined algebraic
properties at all (and I assume that they have simply no grasp of what
they are missing, in their headlong flight to create their
"anti-Cantor" patch), but how can someone apparently involved in
teaching maths resolve this? (This is partly a genuine question: I know
it's perfectly legitimate to investigate set theory with an axiom of
anti-infinity, but I just can't quite imagine how you proceed.
Presumably you can define the naturals using more or less Peano's
axioms; you can show therefore that there is no maximum natural; you
can see that it cannot be possible to count the naturals against a
ditty that stops anywhere, yet somehow you can't actually talk formally
about "all the naturals". ??)

<snip>

> Let me use an example. The mathematicians Wiener and Banach lived at
> about the same time. Banach chose a much more abstract direction to
> take his mathematics, looking at general properties of infinite
> dimensional normed vector spaces beyond Hilbert spaces and spaces of
> continuous or measurable function. Wiener looked more to real life
> applications, and building on the work of Einstein put the phenominum of
> Brownian motion onto a rigourous mathematical foundation. Many very
> talented mathematicians have studied Banach spaces, but it has proven to
> be a remarkably barren field. On the other hand Wiener's approach has
> led to the rich theory of stochastic calculus, which has proved
> extremely useful in studing the heat equation, harmonic functions, and
> even economics.

That's an example (or rather two examples) about which I know next to
nothing. But another example is the use of finite fields in encryption.
Surely (in Hardy's time, say) if a Petry or any other of these "Should"
freaks had come along, finite fields would have been a prime candidate
for banning, since they "had no relation to the physical world". The
history of maths is all about people wandering off and investigating
something just because they wanted to, constrained only by the need for
mathematics to be, um, mathematical. That is, to proceed by axiomatic,
logical argument (exactly the sort of thing that the cranks never seem
to manage).

Actually, I'm not sure that your examples above are fair anyway.
Consider the relative contributions of fortune-telling and astronomy to
the everyday happiness of the proverbial rider of the Clapham omnibus.
Fortune-telling provides lots of entertainment, keeps proles out of gin
palaces, provides employment for considerable numbers of people (at
least, around here it does), and generally forms a part of the social
fabric. What has astronomy done? Sell a few telescopes to schoolboys?
Of course I have no doubt which has the greater fundamental value, but
I do not think this can be assessed by the sort of naive economic
argument you seem to be relying on.

Brian Chandler
http://imaginatorium.org

>
> Stephen

Stephen Montgomery-Smith

unread,
Jul 13, 2006, 3:51:30 PM7/13/06
to

Let me take this up later.

First, this result does use measure theory, and in this field the
notions of countability and uncountability play a very crucial role.
Without Cantor's diagonal argument, which is essential if one wishes to
see that a countable set has measure zero, you really cannot construct a
decent workable theory. The notion of Hausdorff measure in this context
really is something quite sophisticated, and unless you feel very easy
manipulating sets as if they were real objects, you are not going to
reach it.

Now as to transfinite induction - most non-set theoriest use this in the
form of Zorn's Lemma (transfinite induction and axiom of choice joined
together). Zorn's Lemma is the source of many important theorems,
including for example the Hahn-Banach Theorem, a result that is crucial
in the kind of analysis that is the background to much of the modern
study of fluid dynamics. Now its not to say that maybe each use of the
Hahn-Banach Theorem that is used by someone studying fluids might in its
context be provable without the full generality of the Hahn-Banach
Theorem, and might be provable without transfinite induction. But its
nice to know that even if one could prove it without transfinite
induction, that still we know that it is true in advance because we have
a proof that does use transfinite induction. (And in any case, if you
are deeply involved in fluid dynamics you simply don't have the time to
question every use of Hahn-Banach in this manner.)

Now I'm not saying that transfinite induction is correct. But I am
saying that so far we don't know of any flaws in it (or rather in the
set theory needed to support it), and in the meantime it provides such a
useful framework in which to do mathematics, even practical mathematics,
that it would be foolish to throw it away before we know what, if
anything, is wrong with it.

Stephen

Russell Easterly

unread,
Jul 13, 2006, 6:39:46 PM7/13/06
to

"Kevin Karn" <kk...@mail.com> wrote in message
news:1152777505.5...@m79g2000cwm.googlegroups.com...

> 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?

$1 million in grants over 15 years is pretty pitiful for someone
who is one of the top in his field.
Even relatively minor cancer researchers make more than
that a year.

Pure mathematics doesn't get much funding and I doubt
that set theory gets more than a fraction of that.


Russell
- 2 many 2 count


Gerry Myerson

unread,
Jul 13, 2006, 7:52:53 PM7/13/06
to
In article <1152816930.8...@m73g2000cwd.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

For what it's worth, I have met Norm. In the past, we have shared
an interest in Gaussian periods. We have never discussed set theory.

To the best of my knowledge, I haven't ever met george,
and, if he is as repulsive in person as he is on Usenet,
I hope I never do. I certainly have no intention of responding
to his foul posts.

My experience on Usenet has been that, by and large, it is
the cranks and trolls who resort to shouting, name-calling,
and other rude behavior. Between george and Norm, who is
displaying more crankish traits?

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

Lester Zick

unread,
Jul 13, 2006, 8:02:59 PM7/13/06
to
On Thu, 13 Jul 2006 14:20:01 -0400, Hatto von Aquitanien
<ab...@AugiaDives.hre> wrote:

>lDont...@nowhere.net wrote:
>
>> On Thu, 13 Jul 2006 09:33:58 -0400, Hatto von Aquitanien
>> <ab...@AugiaDives.hre> wrote:
>>
>
>>>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 believe set theory is essential to modern math.
>
>What can you do with set theory which you can't do with symbolic logic
>and/or some kind of BNF-like grammar?
>
>> Not exactly the same question. The historical or anthropological
>> foundations of math are certainly interesting. But the more
>> interesting question is why people are using intuitive assumptions in
>> math at all?
>> ~v~~
>
>Because without intuitive assumptions they would have absolutely not concept
>of existence.

Which in itself is an axiomatic assumption used to validate other
axiomatic assumptions.

> That would make doing mathematics rather difficult.

Unless axioms are demonstrated instead of assumed.

> My
>point is that if one were able to identify and codify these assumptions, it
>may be possible to establish a good formal foundation that way.

Isn't that what modern math does? The only real difference seems to be
that it codifies these axiomatic assumptions in various set terms
instead of the geometric terms used in classical math.

~v~~

cbr...@cbrownsystems.com

unread,
Jul 13, 2006, 8:10:08 PM7/13/06
to
imagin...@despammed.com wrote:
> Stephen Montgomery-Smith wrote:
>
> <snip>
>
> > The view that mathematics should try to constrain itself to physical
> > reality is, in my opinion, not a crackpot position.
>
> It does seem to me that - as stated - this is precisely a crackpot
> position. The problem is none of the words "physical", "reality", or
> "constrain", the problem is the word "should". In practice mathematics,
> as referred to by the overwhelming population of mathematicians, means
> the abstract study of formal patterns, and that's all. The mystery of
> it all is the way in which the patterns investigated by abstract
> mathematicians turn out to apply so directly to aspects of the real
> world. But it's going to be hard to develop these abstract theories if
> you have to be consulting one of these "real world characters" [do I
> have to call them other than "cranks"?] to see if _they_ happen to
> think what you're doing is OK.

And furthermore, such claims of "should" seem to be moving targets.

For instance: the claim that "there is no such thing as a physical
infinity" is a relatively recent "physical truth". As late as the
1960's, a cosmology consisting of an infinite, steady state universe
was still being debated.

That isn't to say that I do or don't find the steady state type
theories compelling. What I find is that those who claim that the
concept of mathematical infinity is suspect because "there can be no
such physical thing as infinity" tend to be myopic regarding the
history of what has been accepted as reasonable physical models of our
universe; and close-minded regarding the possibility that further data
may change scientific opinion once again.

Cheers - Chas

Gerry Myerson

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Jul 13, 2006, 8:13:55 PM7/13/06
to
In article <1152780141.6...@35g2000cwc.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

> 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.

For what it's worth, Norm has written papers in number theory,
e.g.,

MR1314396 (96a:11029) Wildberger, N. J. Row-reduction and invariants of
Diophantine equations. Proc. Indian Acad. Sci. Math. Sci. 104 (1994),
no. 3, 549--555.

Whether this was before he came to his current views on set theory,
I do not know.

Gerry Myerson

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Jul 13, 2006, 8:47:33 PM7/13/06
to
In article <44b6165b$0$1210$afc3...@news.optusnet.com.au>,
"Peter Webb" <webbfamily...@optusnet.com.au> wrote:

> "Gerry Myerson" <ge...@maths.mq.edi.ai.i2u4email> wrote in message
> news:gerry-D682F9....@sunb.ocs.mq.edu.au...
> > In article <1152770679....@p79g2000cwp.googlegroups.com>,


> > "Gene Ward Smith" <genewa...@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?

In its day, phlogiston was a well-accepted theory. Alchemy was orthodox.
So was spontaneous generation. Bright people, not cranks, spent
a lot of time and effort trying to prove the parallel postulate.
It's not unknown in the history of science, even of mathematics,
for very good people to do a lot of work that makes later generations
scratch their heads and say, why did they go down that blind alley?
why did they even bother to think about those things?

It may be wrong to say that today's set theorists are yesterday's
phlogiston theorists - but is it crankish?

Evolution - I've read statements along the lines of, "nothing in modern
biology makes sense, without evolution." I've never read anything like,
"nothing in modern mathematics makes sense, without ZFC." I read (and
write) papers of mathematical research all the time that go on for pages
and never mention ZFC, or large cardinals, or anything else that I
recognize as coming from modern set theory. I think the analogy between
evolution and set theory is overstated.

> 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" ?

If you haven't already guessed it, I know very little about modern set
theory myself. It shouldn't be surprising that I can't tell whether
someone else does.

Kevin Karn

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Jul 13, 2006, 8:54:16 PM7/13/06
to
Russell Easterly wrote:
> "Kevin Karn" <kk...@mail.com> wrote in message
> news:1152777505.5...@m79g2000cwm.googlegroups.com...
>
> > 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?
>
> $1 million in grants over 15 years is pretty pitiful for someone
> who is one of the top in his field.
> Even relatively minor cancer researchers make more than
> that a year.

Cancer researchers are doing something useful. Hugh Woodin isn't. In
terms of making a useful contribution to society, Hugh Woodin stands
exactly on a par with that Japanese hotdog eating champion. They're
both at the top of their fields, and their fields are totally useless
in both cases.

What did Hugh use the $1 million for? He's already making a
$100,000/year salary. So what's the $1 million for? Pencils? Chalk?
Pepsi? Cancer researchers need money for equipment/supplies etc. What
do set theory researchers need money for? Nothing, so set theory is
obviously a scam.

> Pure mathematics doesn't get much funding and I doubt
> that set theory gets more than a fraction of that.

It's probably best to make judgments like that after looking into the
facts. Who's getting the money in math today? Who decides where the
money goes? It's time somebody took a good hard look at the "business"
of math.

Stephen Montgomery-Smith

unread,
Jul 13, 2006, 9:11:50 PM7/13/06
to

This affirms what I felt about the OP, that while his views are
unpopular, he is not a crank. (I also just checked him out on
MathSciNet, and it is obvious that he is an active research
mathematician.) In my opinion, his remarks about the natural numbers
are spot on. The only way I disagree with him is that I don't think
that now is the time to work on these problems.

This whole thread has left a bad taste in my mouth because of the speed
at which people were willing to spew abuse on him. Just because he has
a superficial resemblence to crankpot anti-Cantorians doesn't make him
one. His views are controversial, but that doesn't mean we respond
rudely, rather it gives us an opportunity to engage in civilized
conversation. Even if we end up still disagreeing, we might have
learned something, or at least honed our arguments better.

Now, one of the respondents said that Norm had rejected the axiom of
infinity, and so how is he supposed to do number theory. But nowhere in
his article has he rejected the axiom of infinity. He has rejected the
whole notion that axioms are the way to go. I would paraphrase what he
said slightly differently - axioms (might) describe the natural numbers,
but they don't define them. There is clearly a problem that there are
numbers between 1 and googolplex that we can never write down or
meaningfully describe. What makes us think that they are really there?
The evidence is at best empirical.

The responses have led me to think that some people here believe in the
modern axiomatic system like a religion.

Stephen

mariano.su...@gmail.com

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Jul 13, 2006, 9:46:19 PM7/13/06
to
Stephen Montgomery-Smith wrote:
> [snip]

> The responses have led me to think that some people here believe in the
> modern axiomatic system like a religion.

An axiomatic system is just a method of singling out a set of
propositions (those that follow from the axioms). For example,
the Peano axioms for natural numbers select a very definite
subset of all the posible propositions (those that follow from
the axioms).

Axiomatic systems are thus just tools. In particular, the idea
of "believing in the modern axiomatic system" makes as
much sense as that of "believing in hammers".

-- m

Gerry Myerson

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Jul 13, 2006, 9:52:01 PM7/13/06
to
In article <44b60791$0$22363$afc3...@news.optusnet.com.au>,
"Peter Webb" <webbfamily...@optusnet.com.au> wrote:

> 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?

My guess is that Norm would be just as uncomfortable with decimal
expansions that are not generated by a rule as he is with (other)
Cauchy sequences that are not generated by a rule, although there
is the difference that when you know the first trillion decimals
you know rather more than when you know the first trillion terms
in a Cauchy sequence. I may be missing the point of your
true-crank-misleading trichotomy here. If Wildberger is rejecting
sequences not given by rules, he isn't the first serious mathematician
to do so, to do so honestly and while in full possession of his
faculties.

> 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?

This bothered me, too, as I have vague memories of having seen
associativity and the rest proved from the Cauchy sequence definition
in some course I attended decades ago. I don't know what to make of
this paragraph. I hope he elaborates on it sometime.

> 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?

Is he saying anything that differs from what Bishop had to say
about examples where you couldn't tell whether a real number was zero
or not because the definition of the real number depended on knowing
whether the decimal expansion of pi does or doesn't contain 17
consecutive zeros? Was Bishop a 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?

Let's break it into pieces. "A set of rational numbers is essentially
a sequence of zeros and ones." You could quibble about the word,
"essentially," but I suppose it's true that anything you can do with
a set of rational numbers, you can do with a sequence of zeros and
ones. "A sequence of zeros and ones is specified properly [only] when
you have a finite function or computer program which generates it."
I'd class this as "opinion," an option you've omitted from your
trichotomy, and an opinion that is unpopular but not unknown in the
mathematical research community. "Otherwise, the sequence is not
accessible in a finite universe." I think he makes it clear that what
he means is that if the sequence is not generated by a rule the only
way you could know enough about the sequence to answer all quesitons
about it is to have it all written out, which you can't do in a finite
universe.

> --------------------------
> 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?

This one threw me for a loop, too. I know about the trigonometry book,
although I haven't seen it. I don't know whether that book contains
his argument that the computable numbers are complete and not countable.
It would seem to be an odd thing to include in a book that purports to
do high school trig.

Anyway, I haven't taken a few days to think clearly about the subject,
and even if I did, I don't know that I'd see what he says about the
computable reals.

I'd still reject your trichotomy, and label this paragraph, "unclear."

> 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?

You can give a rule to construct an irrational number which you can
then prove to be unequal to any algebraic number someone else hands you.
You can do this without thinking about the set of all irrational
numbers, just as you can prove the square root of two is irrational
without thinking about the set of all rational numbers (heck, Euclid
pulled that one off).

You can (Norm says) work with functions by knowing what kind of thing
is an input, what kind of thing is an output, and what the rule is,
without ever thinking of the set of all inputs. If you can show that
any rule that takes an integer as an input and gives a computable
number as an output must omit some computable number, then you have
a proof that computable numbers are not countable, and you've done it
without using set-theoretic constructions.

Don't ask me how to do it (but isn't all this stuff in Bishop?).
Again, I'd reject your alternatives, and say, "needs elaboration."

> ------------------------------------
> 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?

Well, now I think you're being misleading. I think it's clear
that he means he has never had to ponder which model of set theory
he is working in.

> -----------------------------
> 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?

Way beyond my area of expertise. I won't go there.

To sum up my reaction to your examples, I'd say Norm has a lot of
explaining to do if he wants to gain acceptance for his ideas.
But I'm not comfortable labelling him a crank, or accusing him
of deceit.

Stephen Montgomery-Smith

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Jul 13, 2006, 9:53:12 PM7/13/06
to

I disagree.

Gerry Myerson

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Jul 13, 2006, 9:56:40 PM7/13/06
to
In article <1152789508.2...@35g2000cwc.googlegroups.com>,
"Tez" <terence...@gmail.com> wrote:

> 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.

I accept that my use of the phrase, "ad hominem," was inaccurate.
I don't know what the correct term is for the fallacy of rebutting
someone's argument by calling it "bullshit" and "crank babble."

Gerry Myerson

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Jul 13, 2006, 10:03:37 PM7/13/06
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In article <1152774445.6...@m79g2000cwm.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

> The difference is that we've given up on the idea that there is a
> single correct geometry, but still feel (and that's nothing but
> intuition speaking) that there is a single true set theory, just as we
> think there is a single true number theory.

The other difference is that we never said that all mathematics
is based on geometry, can be expressed in terms of geometry,
can be justified on the basis of geometry. We do hear that all
of mathematics can be expressed in ZFC (or at any rate all the
mathematics that isn't specifically designed to be done outside
ZFC), and this makes the independence results of set theory
a bit more worrisome than those of geometry.

Stephen Montgomery-Smith

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Jul 13, 2006, 10:11:50 PM7/13/06
to

I cannot claim to have a great knowledge of what modern set theory is
about myself. They just don't talk about it much at the conferences I
attend. The odd talk on the subject that I have seen has subjects like
what kind of cardinals we might expect between aleph-null and the
continuum, given that the continuum is not well ordered. I have to say
it really does come across like the old theological debates about how
many angels could fit on a pinhead.

When, in another post, I said that mathematics should try to get its
ideas about reality, I certainly did not mean that every mathematician
should call the physics or economics department before embarking on a
course of research. But it does require using a bit of common sense and
taste in deciding whether a course of research is worth pursuing, and
from time to time reassessing these decisions.

For example, it seems absolutely clear to me that investigation of
polynomials on finite fields is well grounded in reality. Maybe someone
else disagrees, but there is going to be some concensus. Another
example - funding agencies might feel that there is not much future in
investigating Banach spaces that are not isomorphic to their hyperplanes.

We have no idea whether the axioms of set theory are going to hold up.
It makes sense to investigate mathematical pursuits that have some
connection with the world around us, even if that connection is
philosophical or metaphorical in its nature. This does not preclude
that a few people should be allowed to explore the weird stuff (or
whoever wants to in their spare time) because you never know what is
going to be useful (and useful in this context does not mean how many
dollars it is going to make). But some math areas are clearly drying
up, and set theory is one of them.

Stephen

Lester Zick

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Jul 13, 2006, 10:19:43 PM7/13/06
to

It occurs to me that beliefs are beliefs whatever direction they take.
The relevant difference seems to be the value and utility of that
direction. Belief systems like religion seem comparatively ad hoc in
this respect and scientific/mathematical systems considerably more
rigorous.

~v~~

Gene Ward Smith

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Jul 13, 2006, 10:21:02 PM7/13/06
to

Gerry Myerson wrote:

> The other difference is that we never said that all mathematics
> is based on geometry, can be expressed in terms of geometry,
> can be justified on the basis of geometry.

That's pretty well Euclid's approach. Numbers, including number theory,
are grounded in geometry. Had Euclid been shown the proof of the
independence of the parallel postulate, my guess is that he's still
assume there was one correct geometry, and hyperbolic and elliptic
geometries are just things we can define in terms of that geometry.


We do hear that all
> of mathematics can be expressed in ZFC (or at any rate all the
> mathematics that isn't specifically designed to be done outside
> ZFC), and this makes the independence results of set theory
> a bit more worrisome than those of geometry.

Isn't this a contradiction? The independence results show that
mathematics can't all be reduced to ZFC. Of course, the independence
results themselves involve inner models and forcing models, but it
seems to me the obvious way to interpret it all is that there are
reasonable ways to add axioms to ZFC.

Lester Zick

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Jul 13, 2006, 10:23:56 PM7/13/06
to
On 13 Jul 2006 18:46:19 -0700, "mariano.su...@gmail.com"
<mariano.su...@gmail.com> wrote:

I suspect people do believe in hammers. They believe in their senses
for probably no better reason than they have no better alternatives to
believe in as long as they have to believe in something. And that
would seem to be the only alternative Aristotle's reliance on
syllogistic inference for the establishment of truth left them.

~v~~

Gene Ward Smith

unread,
Jul 13, 2006, 10:29:52 PM7/13/06
to

Gerry Myerson wrote:

> This bothered me, too, as I have vague memories of having seen
> associativity and the rest proved from the Cauchy sequence definition
> in some course I attended decades ago. I don't know what to make of
> this paragraph. I hope he elaborates on it sometime.

It's not difficult to show Cauchy sequences define a commutative ring,
in fact an algebra.
When you show null sequences are a maximal ideal, you are set. Of
course if you object on principle to things like maximal ideals, you
aren't set, but that is my objection to the whole enterprise--it's just
a version of Mathematics Made Difficult. It's like having someone tell
you you will go to hell if you don't wear the right kind of underwear;
there's just nothing much attractive about making proofs harder, though
finding what is the minimal assumptions you are required to make is
interesting mathematics in itself.

Lester Zick

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Jul 13, 2006, 10:30:13 PM7/13/06
to

Oh you might be surprised. I've seen atrocious behavior exhibited by
very conventional thinkers driven to reactionary extremes. Of course
they're not cranks because they're conventional thinkers and cranks
aren't.

~v~~

Lester Zick

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Jul 13, 2006, 10:32:18 PM7/13/06
to

Probably more along the lines of reactionary and irrelvant.

~v~~

Gene Ward Smith

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Jul 13, 2006, 10:35:42 PM7/13/06
to

Gerry Myerson wrote:

> My experience on Usenet has been that, by and large, it is
> the cranks and trolls who resort to shouting, name-calling,
> and other rude behavior. Between george and Norm, who is
> displaying more crankish traits?

I'm hardly in a position to play Miss Manners, but I think one could
reasonably complain in both cases. One thing I am very tired of reading
on this list are the constant attempts to denigrate the mathematical
work of one distinguished mathematician or another, or one school of
mathematicians or another. I think ripping in to the positions people
take on the philosophy of mathematics is one thing, but treating the
mathematical work of the people involved is quite something else.

Gene Ward Smith

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Jul 13, 2006, 10:39:19 PM7/13/06
to

Stephen Montgomery-Smith wrote:

> This whole thread has left a bad taste in my mouth because of the speed
> at which people were willing to spew abuse on him.

If Norm's article had not been very rude, which it was, I think the
response would have been different.

> There is clearly a problem that there are
> numbers between 1 and googolplex that we can never write down or
> meaningfully describe. What makes us think that they are really there?
> The evidence is at best empirical.

Saying you are not allowed to prove the fundamental theorem of number
theory strikes me as a really, really, really bad way to try to do
number theory, and speaking of religion, it seems almost like a
religious objection here.

Lester Zick

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Jul 13, 2006, 10:44:11 PM7/13/06
to

It occurs to me that physical arguments related to mathematical
infinities are the rather ugly stepchildren of a finite universe and
empirical utilitarian justifications for science and mathematics.

~v~~

Stephen Montgomery-Smith

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Jul 13, 2006, 11:20:53 PM7/13/06
to
Gene Ward Smith wrote:
> Stephen Montgomery-Smith wrote:
>
>
>>This whole thread has left a bad taste in my mouth because of the speed
>>at which people were willing to spew abuse on him.
>
>
> If Norm's article had not been very rude, which it was, I think the
> response would have been different.

I would describe it as provocative rather than rude. He had a very
forceful style, but his language stayed professional.

>>There is clearly a problem that there are
>>numbers between 1 and googolplex that we can never write down or
>>meaningfully describe. What makes us think that they are really there?
>> The evidence is at best empirical.
>
>
> Saying you are not allowed to prove the fundamental theorem of number
> theory strikes me as a really, really, really bad way to try to do
> number theory,

In context, his remark that his number "in effect" has no prime
factorization is rather sharp. I *think* I disagree with him, but I
greatly respect his point of view.

> and speaking of religion, it seems almost like a
> religious objection here.
>

I don't deny it.

Stephen

Gerry Myerson

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Jul 14, 2006, 12:01:13 AM7/14/06
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In article <1152843662....@h48g2000cwc.googlegroups.com>,

"Gene Ward Smith" <genewa...@gmail.com> wrote:

> Gerry Myerson wrote:
>
> > We do hear that all of mathematics can be expressed in ZFC (or at
> > any rate all the mathematics that isn't specifically designed to be
> > done outside ZFC), and this makes the independence results of set
> > theory a bit more worrisome than those of geometry.
>
> Isn't this a contradiction? The independence results show that
> mathematics can't all be reduced to ZFC. Of course, the independence
> results themselves involve inner models and forcing models, but it
> seems to me the obvious way to interpret it all is that there are
> reasonable ways to add axioms to ZFC.

Let me try to get at what the worry is.

How do we know the Wiles-Taylor solution of the Fermat problem
is correct? If you push a mathematician hard enough on this,
or any similar question, never accepting an answer as final
until you come down to bedrock, the eventual answer will be
that it can all be formalized and validated in ZFC. But ZFC
doesn't even decide a simple question like, is there an infinite
subset of the reals that can be put into one-one correspondence
with neither the integers nor the reals? If ZFC doesn't capture
such a basic part of our mathematical world, what reason is there
to think it's a good framework for our mathematical work? and what
good does it do to say that you can carry out the Wiles-Taylor
proof in it?

The independence of the parallel postulate doesn't make anyone
worry, because no one says you can carry out the proof of Fermat
in Euclidean geometry. The independence of the continuum
hypothesis says we really don't understand sets, and that's a worry.

mariano.su...@gmail.com

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Jul 14, 2006, 12:10:56 AM7/14/06
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I honestly do not see what are you disagreeing with.

I cannot imagine you are disagreeing with my statement
that axiomatic systems are descriptions of sets of propositions;
this statement is completely equivalent to the statement that
Chomski grammars are descriptions of sets of words.

I could understand if your point was that axiomatic systems
make lousy descriptions of sets of propositions and/or that
what the sets of propositions described by the "usually
acepted" axiomatic systems is not the set one would be
most interested in describing (be it because they include
propositions which "should" be false, be it because they do
not include propositions which "should" be true, and so on)

To the first point I would probably respond that in the
whole history of mankind the number of effective alternatives
to the axiomatic method that we have devised is zero.

The second point is not a problem with the axiomatic method,
but with the fact that the usually used axiom systems may do not
model what you (or I) believe that should be true or false.

Yet most people end up agreeing that Banach-Tarski's paradox
is a fair price to pay for the existence of maximal ideals in
arbitrary commutative rings, say. And no one lost any sleep
when Shelah proved that the answer to Whitehead's conjecture
depends on the set theory chosen: my guess for this is that
essentially no one who has thought deeply about the issue has
any real intuition as to what the answer should be, so it is not
a problem if changing the set theory used changes the answer.

And that puts to the fore what it is that people believe in:
their idea of what things should be, the set of propositions
they believe/feel/intuit should hold. Now that set of propositons
is very hard to communicate and it will certainly not work to base
mathematics on "what Mr. X believes sets are" because eventually
Mr. X will die and the decision procedure which consists in
"asking Mr. X on the truth of propositions relative to sets" will
stop working.

Axiom systems are a tool used when trying to describe the set
of propositions people believe should hold. What people believe
in precedes the axiom system. If some one came up with an
alternative way of describing the kind of sets of propositions that
people have in mind when they think about the integers, say, then
people will try to describe the set of propositions about the
integers that they believe should hold using that new method.

In short, what I am saying is: what people believe in precedes
axiom systems (which are just an imperfect way of conveying it)


-- m

Stephen Montgomery-Smith

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Jul 14, 2006, 12:41:02 AM7/14/06
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I read what you wrote, and I admit I am not quite getting it. So let me
describe my position and hope that it is a worthy response to what you
wrote. I'm not quite sure whether I am disagreeing with you or not.

There is a way in which people try to persuade other people of their
belief system. I think that our current scheme (axioms, tautologies,
modus ponens etc) we use in math is a very good description of what we
feel are really effective ways of persuading other people. (Of course
we know that using pure reason doesn't work with most people, but we
have this sense that there is a Platonic reality as to how an ideal
argument should work.) However, I believe that our ultimate sense of
what we think is true and not true comes down to simply "what do I
believe." And the methods of logical deduction are no more than one
other set of things that I simply "believe."

On a completely different tack, while methods of logical deduction seem
to be a good description of how to do mathematics, I don't think it is
how mathematicians really think. The example I would cite is Euler's
(rather brilliant) discovery that sum 1/n^2 = p^2/6 (which basically
involves pretending that the Taylor series of sin(x)/x is a polynomial).
His original proof certainly was not rigorous. Rather it was a flash
of insight, and certainly thinking out of the box.

Stephen

abo

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Jul 14, 2006, 1:47:13 AM7/14/06
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First, I think Norm is incorrect in saying the fundamental theorem of
number theory cannot be proven. He believes he has a counterexample,
and this is what he says:

"For example c+23 is almost surely such a number---I claim it has no
prime factorization. Neither you nor I nor anyone ever living in this
universe will ever be able to factor this number, since most of its
`prime factors' are almost surely so huge as to be inexpressible, which
means they don't exist."

Suppose the usual prime factors don't exist. Then that would simply
mean that other numbers become prime - and a prime factorization still
exist. That is, suppose in standard arithmetic (c+23) = a*b. And
suppose that, unlike (c+23), a and b are too big to write down, so in
Norm's world they do not exist. Well if they do not exist, then (c+23)
is a prime number, because it cannot be factored into anything other
than itself and 1. Prime factorization holds.

Here's a system which captures, I think, Norm's intutition. Consider
second-order PA without the successor axiom (that every number has a
successor) - call this F. F does not assume there is a maximum number
(i.e. a number which does not have a successor) - it is simply agnostic
about the matter, neither assuming nor not assuming it. Note in
passing that in F you can prove the existence and uniqueness of prime
factorization. This is not Norm's world, because in Norm's world the
natural numbers have gaps, and in F the natural numbers do not have
any gaps. That is, in F, if n exists, then all numbers less than n (to
speak loosely) also exist.

To get gaps, consider this system: define a *second-order natural
number* to be a function from a set {x : x <= n} to N union
{"(",")","+","-","*","/","^"} where certain conditions have to hold.
These conditions are precisely the ones that the function maps to a
term. For instance, f below is an example of a second-order natural
number

f(0) = "("
f(1) = "("
f(2) = 3
f(3) = "^"
f(4) = 2
f(5) = ")"
f(6) = "+"
f(7) = 1
f(8) = ")"

So f represents ((3 ^ 2) + 1). Now it cannot be shown in F that
second-order natural numbers do not have gaps. (And if one added to F,
contra the Successor Axiom, that there is a maximum number, then one
would be able to prove that there are indeed gaps.) Still, it seems
to me very straight-foward to prove the existence and uniqueness of
Prime Factorization for second-order natural numbers.

Anyone who is interested in "Arithmetic without the Successor Axiom"
can consult my text here: www.andrewboucher.com/papers/arith-succ.pdf

mariano.su...@gmail.com

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Jul 14, 2006, 1:49:09 AM7/14/06
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Stephen Montgomery-Smith wrote:
> I read what you wrote, and I admit I am not quite getting it. So let me
> describe my position and hope that it is a worthy response to what you
> wrote. I'm not quite sure whether I am disagreeing with you or not.

You asserted that "the responses have led [you] to think that


some people here believe in the modern axiomatic system

like a religion". I replied that "believing in the axiomatic system"
does not make sense, because axiomatic systems are just a way
to describe a set of propositions, and you said "[you] disagree".
I assumed you meant that.

In any case, there is no obligation to agree or disagree (despite
what one might think after reading certain newsgroups ;-) ) One
can simply be interested in trying to understand what others think.

> There is a way in which people try to persuade other people of their
> belief system. I think that our current scheme (axioms, tautologies,
> modus ponens etc) we use in math is a very good description of what we
> feel are really effective ways of persuading other people. (Of course
> we know that using pure reason doesn't work with most people, but we
> have this sense that there is a Platonic reality as to how an ideal
> argument should work.) However, I believe that our ultimate sense of
> what we think is true and not true comes down to simply "what do I
> believe." And the methods of logical deduction are no more than one
> other set of things that I simply "believe."

IME, it is not only that axiom systems and formal theories are good
at persuading other people. And it is not only that they are good in
modeling our intuitions (what we believe). They are actually good at
modeling reality. Say, ZFC and its deduction means can be used to
model the physics of elementary particles so well that at the moment
we cannot carry on experiments fine enough to confirm the predicted
mass of the electron with the precision we are able to compute.

I wouldn't say I believe in logical deduction. I'd be symtomatically
more verbose: I believe in there is a "reality", external from myself
and from by beliefs. I do not know if we can know that reality (I do
not even know what that means, in fact). But I know that we can attempt

to build models of it. And so far, the models we have built, using
modus
ponens, and logical deduction in general, have been quite good.

> On a completely different tack, while methods of logical deduction seem
> to be a good description of how to do mathematics, I don't think it is
> how mathematicians really think. The example I would cite is Euler's
> (rather brilliant) discovery that sum 1/n^2 = p^2/6 (which basically
> involves pretending that the Taylor series of sin(x)/x is a polynomial).
> His original proof certainly was not rigorous. Rather it was a flash
> of insight, and certainly thinking out of the box.

No one who knows how a mathematician works, and least of
all a working mathematician, would ever say that mathematicians
work by logic deduction. Insights do not come in the form of
chains of applications of modus ponens.

If you think about it, the fact that such a limited tool as the
axiomatic system, and formal theories in general, with not much
more than modus ponens, provides a good model for genial
intuitions such as Euler's, is one of the things that validates it.

-- m

Gene Ward Smith

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Jul 14, 2006, 1:57:55 AM7/14/06
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Gerry Myerson wrote:

> The independence of the parallel postulate doesn't make anyone
> worry, because no one says you can carry out the proof of Fermat
> in Euclidean geometry. The independence of the continuum
> hypothesis says we really don't understand sets, and that's a worry.

It says we don't completely understand sets, but it also says most of
the time, we don't need to. ZFC, which cannot settle the the continuum
question, is actually stronger than we need most of the time. Most of
the time, second order arithmetic would be fine. While in fact FLT was
not proven in a system that weak, it's extremely likely that it could
be, and we know that for a huge bunch of mathematics, it suffices.

On the other hand, sometimes painful issues do arise. The Whitehead
problem, whether or not Ext^1(A, Z)=0 for A an abelian group entails
that A is free, was a shocker to algebraists, you may recall. We would
like to think what seem like basic algebraic questions of that nature
are independent of set theory considerations--and usually, they are.
But not this time, and in fact the whole question is bound up with the
continuum hypothesis.

Cabal-type intutions, that the continuum hypothesis is false and
Martin's axiom true, lead to one answer, whereas the minimalist
hypothesis that all sets are constructible leads to another. You get
the exact same division over Suslin lines; there are two very different
flavors of set theory extensions which these reflect.

Gene Ward Smith

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Jul 14, 2006, 2:07:35 AM7/14/06
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abo wrote:

> Here's a system which captures, I think, Norm's intutition.

This does exactly what I complained Norm doesn't do, which is to write
down some axioms. As a result, you can get mathematical results instead
of just engaging in diatribes. Needless to say, I think this is a
better approach, but it's not clear to me that this is Norm's intution.

How strong a consistency result can you prove in "F"?

Rupert

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Jul 14, 2006, 2:19:24 AM7/14/06
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Gerry Myerson wrote:
> I'm not sure how you think your method of not ignoring Norm's post
> will prevent his students from being taught his ideas.
>

Norm isn't teaching his ideas about foundations to his students in
classes. We're not very big on foundational studies at UNSW. (I offered
to teach an Honours course about the independence of the continuum
hypothesis, but nothing came of that). Obviously if Norm was going to
present his ideas in lectures he'd have to give a fair amount of time
to competing ideas as well.

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