Update: Objections to Cantor's Theory
david petry
I'm starting a new thread because the other one got
out of control.
I want to thank those of you who gave me some useful
feedback on my article (you know who you are :-).
Here's the updated version of my article. I hope it
answers some of the questions people have asked.
********
In this article, "Cantor's Theory" refers to the pre-formal
ideas about set theory introduced by Cantor in the latter
part of the nineteenth century. The "anti-Cantorians" are
people who claim that Cantor created a fantasy world.
The anti-Cantorians claim that while infinite sets and
power sets of those infinite sets are undeniably useful
abstractions, what Cantor did was to take an argument
(the diagonalization argument), which is perfectly valid
in concrete mathematics, and wrecklessly apply it to
the abstractions of infinity, ultimately producing garbage.
The Cantorians (i.e. almost all pure mathematicians) claim
that since Cantor's Theory can be formalized in a logically
consistent way (e.g. ZFC), and since the study of
formalizations is certainly a part of pure mathematics, there
is absolutely no room for debate about Cantor's Theory.
When Cantor introduced his ideas, there was a heated debate
about whether they should be accepted as mathematics. For
reasons which are not entirely clear, the ideas were
accepted, and the debate has fallen silent within the
mathematical literature. However, the debate has flaired
up again on the internet.
Most of the debate on the internet about Cantor's Theory
is junk. The topic is a crank magnet. Most of the people
who participate in the debate, have no deep understanding of
the issues. However, hidden within all the noise, there does
seem to be some signal.
While the pure mathematicians almost unanimously accept
Cantor's Theory (with the exception of a small group of
constructivists), there are lots of intelligent people who
believe it to be an absurdity. Typically, these people
are non-experts in pure mathematics, but they are people
who have who have found mathematics to be of great practical
value in science and technology, and who like to view
mathematics itself as a science.
These "anti-Cantorians" see an underlying reality to
mathematics, namely, computation. They tend to accept the
idea that the computer can be thought of as a microscope
into the world of computation, and mathematics is the
science which studies the phenomena observed through that
microscope. They claim that that paradigm encompasses all
of the mathematics which has the potential to be applied to
the task of understanding phenomena in the real world (e.g.
in science and engineering).
Cantor's Theory, if taken seriously, would lead us to believe
that while the collection of all objects in the world of
computation is a countable set, and while the collection of all
identifiable abstractions derived from the world of computation
is a countable set, there nevertheless "exist" uncountable sets,
implying (again, according to Cantor's logic) the "existence"
of a super-infinite fantasy world having no connection to the
underlying reality of mathematics. The anti-Cantorians see
such a belief as an absurdity (in the sense of being
disconnected from reality, rather than merely counter-intuitive).
The mathematicians claim that they can "prove" the existence
of uncountable sets, and hence there's nothing to be debated.
But that merely calls into question the nature of "proof".
Certainly infinite sets and power sets exist as absractions.
But, abstractions don't necessarily obey exactly that same
laws of logic as directly observable objects. Assuming
otherwise can turn abstractions into fantasies, and proofs
into absurdities, and that's the crux of the anti-Cantorian's
argument.
The pure mathematicians tend to view mathematics as an art
form. They seek to create beautiful theories, which may happen
to be connected to reality, but only by accident. Those who apply
mathematics, tend to view mathematics as a science which explores
an objective reality (the world of computation). In science, truth
must have observable implications, and such a "reality check"
would reveal Cantor's Theory to be a pseudoscience; many of the
formal theorems in Cantor's Theory have no observable implications.
The artists see the requirement that mathematical statements must
have observable implications as a restriction on their intellectual
freedom.
The "anti-Cantorian" view has been around ever since Cantor
introduced his ideas. Witness the following quotes from
contemporaries of Cantor:
"I don't know what predominates in Cantor's
theory - philosophy or theology, but I am sure
that there is no mathematics there" (Kronecker)
"Set theory is a disease from which mathematics
will one day recover" (Poincare)
In the contemporary mainstream mathematical literature, there
is almost no debate over the validity of Cantor's Theory.
However, some mathematicians still drop hints that they see
the absurdity of Cantor's Theory. Consider:
"Set theory is based on polite lies, things we agree
on even though we know they're not true. In some ways,
the foundations of mathematics has an air of unreality."
(William P. Thurston)
It was the advent of the internet which revealed just how
prevalent the anti-Cantorian view still is; there seems to be a
never-ending heated debate in the Usenet newsgroups sci.math and
sci.logic over the validity of Cantor's Theory. Typically, the
anti-Cantorians accuse the pure mathematicians of living in a
dream world, and the mathematicians respond by accusing the
anti-Cantorians of being imbeciles, idiots and crackpots.
It is plausible that in the future, mathematics will be split
into two disciplines - scientific mathematics (i.e. the science
of phenomena observable in the world of computation), and
philosophical mathematics, wherein Cantor's Theory is merely
one of many possible formal "theories" of the infinite.
G. Frege
<david_lawr...@yahoo.com> wrote:
I's suggest that you delete the following paragraph from your article.
Rationale:
(1) It's definitely n o t plausible. You might f i r s t ask
professionals of mathematics before claiming such ridiculous things.
(2) It obviously reflects *your own* opinion concerning this topic -
hardly something that should show up in an article of an encyclopedia.
(3) It's complete nonsense.
>
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation), and
> philosophical mathematics, wherein Cantor's Theory is merely
> one of many possible formal "theories" of the infinite.
>
F.
Stephen J. Herschkorn
To which axioms of ZF do anti-Cantorians object, exactly?
According to anti-Cantorians, what, exactly, is the flaw in the proof of
the fact that there is no surjection from the natural numbers to the set
of subsets of natural numbers?
According to anti-Cantorians, what, exactly, is the flaw in the proof
that there is no surjection from the set of of natural numbers to the
set of real numbers?
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor in Central New Jersey and Manhattan
tor...@sm.luth.se
david petry wrote:
> "Set theory is a disease from which mathematics
> will one day recover" (Poincare)
This quote has been questioned in the past. What is the
actual reference?
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> [...] the mathematicians respond by accusing the
> anti-Cantorians of being imbeciles, idiots and[/or] crackpots.
And that's indeed true in most (if not all) cases.
F.
G. Frege
> >
> > [...] the mathematicians respond by accusing the
> > anti-Cantorians of being imbeciles, idiots and[/or] crackpots.
> And that's indeed true in most (if not all) cases.
> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Could y o u, David, show me a single exception?
F.
david petry
For all I know, it's just folklore, but it has
appeared in various reputable works.
Daryl McCullough
>
>
>I'm starting a new thread because the other one got
>out of control.
...starting with the first post.
--
Daryl McCullough
Ithaca, NY
Stephen Montgomery-Smith
> I'm starting a new thread because the other one got
> out of control.
>
> I want to thank those of you who gave me some useful
> feedback on my article (you know who you are :-).
>
> Here's the updated version of my article. I hope it
> answers some of the questions people have asked.
I think that I would object to the use of the word "anti-Cantorian" and
I would prefer that you use the word "intuitionist."
To the intuitionists, who believe that set theory is polite lie, I have
great respect and admiration. I myself use set theory, but only because
no-one has come up with anything better.
To the anti-Cantorians, who assert that Cantor's diagonal argument is
logically wrong, I can only give what the messenger related from King
Henry V to the prince regent of France:
Scorn and defiance. Slight regard, contempt,
And anything that may not misbecome
The mighty sender, doth he prize you at.
ste...@nomail.com
<snip>
This paragraph
> It was the advent of the internet which revealed just how
> prevalent the anti-Cantorian view still is; there seems to be a
> never-ending heated debate in the Usenet newsgroups sci.math and
> sci.logic over the validity of Cantor's Theory. Typically, the
> anti-Cantorians accuse the pure mathematicians of living in a
> dream world, and the mathematicians respond by accusing the
> anti-Cantorians of being imbeciles, idiots and crackpots.
seems to be undermined by this paragraph.
> Most of the debate on the internet about Cantor's Theory
> is junk. The topic is a crank magnet. Most of the people
> who participate in the debate, have no deep understanding of
> the issues. However, hidden within all the noise, there does
> seem to be some signal.
If mose of the debate on the internet is junk, then
it seems strange to use the internet as evidence
for the prevalence of the anti-Cantorian view.
Stephen
george
david petry wrote:
> I'm starting a new thread because the other one got
> out of control.
All threads about this are inherently out of control.
There is no there there.
> I want to thank those of you who gave me some useful
> feedback on my article (you know who you are :-).
Of course we do, but it's not clear that YOU do, since
you persist. I in particular replied late and it is not
clear that you were still paying attention.
> Here's the updated version of my article. I hope it
> answers some of the questions people have asked.
It doesn't answer the only question *I* asked, so I will
have to repeat myself.
> ********
> In this article, "Cantor's Theory" refers to the pre-formal
> ideas about set theory introduced by Cantor in the latter
> part of the nineteenth century.
NO, IT DOESN'T.
Cantror's Theorem is provable in perfectly standard formal
first-order set theories like ZFC and NBG. It is proof of
how ignorant the objectors are that they can't even recognize
THESE proofs AS proofs.
> The "anti-Cantorians" are
> people who claim that Cantor created a fantasy world.
Absolutely ALL of first-order logic (to the extent that
it is infinitary and symbolic) is a fantasy world!
Have you ever seen or observed the letter "a", or the digit
"0"? OF COURSE NOT! You have seen particular individual
physical tokens of characters; you have seen MANY DIFFERENT
a's and many different 0's, but you have NEVER ONCE seen
THE symbol 0 or THE symbol "a", or even THE word "David"!
THESE THINGS ARE ALL *ABSTRACT*! Saying about anything that
it is "a fantasy world" is just MEANINGLESS in this context.
OF COURSE it's fantasy, to the extent that it is not concrete.
Neither Zero nor One nor COUNTABLE infinity is ANY more concrete
than these higher infinities that the idiots are objecting to!
> The anti-Cantorians claim that while infinite sets and
> power sets of those infinite sets are undeniably useful
> abstractions,
They claim nothing of the kind. If this were the case then
the argument would simply be over and they would have simply
lost.
> what Cantor did was to take an argument
> (the diagonalization argument), which is perfectly valid
> in concrete mathematics,
It is a LOGICAL argument and it is valid in the context
of FIRST-ORDER LOGIC. This has absolutely NOTHING to do with
any sort of distinction between "concrete" and "abstract"
mathematics, a distinction which YOUR ignorant ass is FAR
from competent to define IN ANY case! ALL math is abstract
BY DEFINITION!
> and wrecklessly apply it to
> the abstractions of infinity,
> ultimately producing garbage.
This is ridiculous. There are an infinite number of natural numbers.
If you alleged that every last one of them had a double, or a square,
would you be doing something "wreckless"? That's spelled "reckless",
by the way. If you are doing a finitary simple operation (such as
taking
half of a rational number) and you can do it to ANY member of some
class,
then the fact that the class is infinite SIMPLY DOES NOT MATTER. There
ARE times when it matters but THIS IS NOT one of them.
Diagonalization IS A TURING-MACHINE argument, in ADDITION to a
Cantorian
one. TMs are infinitary BY DEFINITION: They MUST have infinitely long
tapes and they MUST admit NO upper bound on the length of their input
beyond that it be finite. So EVERY TM has AN INFINITE number of
possible
inputs and can represent a potentially infinite class of the
corresponding
outputs, if it halts on enough of them. Given that TM programs can
themselves
be encoded as inputs to TMs, ANTI-DIAGONALIZATION IS UNAVOIDABLE. YOU
CAN WRITE A SHORT TM program to anti-diagonalize ANY AND EVERY TM.
The fact that the idiots think we are applying this TM "wrecklessly"
does
NOT in ANY way compromise its VALIDITY as a TM. And no TM produces
"garbage",
no matter HOW it is applied: it either halts or it doesn't: THAT IS
ALL.
> The Cantorians (i.e. almost all pure mathematicians) claim
> that since Cantor's Theory can be formalized in a logically
> consistent way (e.g. ZFC), and since the study of
> formalizations is certainly a part of pure mathematics, there
> is absolutely no room for debate about Cantor's Theory.
This is backwards. ZFC does not formalize "Cantor's Theory".
ZFC, precisely because IT IS a formal first-order theory,
is bigger, better, and badder than Cantor's Informal Theory
COULD EVER HAVE HOPED to be. Moreover, this simple fact alone
entitles ALL of ZFC's theorems to the utmost respect UNTIL AFTER
somebody PROVES A CONTRADICTION from ZFC! UNTIL THEN, ALL the idiots
are PERFECTLY welcome to SIT DOWN AND SHUT UP, not only about Cantor's
theorem, but about EVERY theorem of ZFC that is not itself a deep
axiom (like Choice).
> When Cantor introduced his ideas, there was a heated debate
> about whether they should be accepted as mathematics. For
> reasons which are not entirely clear, the ideas were
> accepted,
Please. They were clear to Zermelo. They were clear to Hilbert.
They were accepted because nobody has ever derived any contradiction
from the axioms positing them. In math, that is ALWAYS *enough*!
The fact that it is not enough for the idiots just proves they're
idiots.
> and the debate has fallen silent within the
> mathematical literature. However, the debate has flaired
> up again on the internet.
>
> Most of the debate on the internet about Cantor's Theory
> is junk. The topic is a crank magnet. Most of the people
> who participate in the debate, have no deep understanding of
> the issues. However, hidden within all the noise, there does
> seem to be some signal.
No, there doesn't, and you're flaunting your own ferrosity
by suggesting otherwise.
> While the pure mathematicians almost unanimously accept
> Cantor's Theory (with the exception of a small group of
> constructivists), there are lots of intelligent people who
> believe it to be an absurdity.
No, there are not.
> Typically, these people
> are non-experts in pure mathematics, but they are people
> who have who have found mathematics to be of great practical
> value in science and technology,
Possibly.
> and who like to view
> mathematics itself as a science.
No, I'm sorry, they do nothing of the kind.
They do not know enough philosophy to even know
HOW to do THAT. They view math as something completely
other than what it actually is, possibly, as you said before,
as computation. But computability is every bit as diagonalizable
as powersetting, so that doesn't help.
> These "anti-Cantorians" see an underlying reality to
> mathematics, namely, computation. They tend to accept the
> idea that the computer can be thought of as a microscope
> into the world of computation, and mathematics is the
> science which studies the phenomena observed through that
> microscope. They claim that that paradigm
WHAT paradigm??
Last I heard, around here, we, being logicians, were
tempted to view the computational paradigm in terms of
Turing Machines. In that case, the idiots get no help,
because first-order reasoning CAN be formalized via the
TM paradigm. Moreover, the TM paradigm itself supports
(anti-)diagonalization! So saying "it's all about computation"
affords the idiots NO HELP WHATSOEVER.
> encompasses all
> of the mathematics which has the potential to be applied to
> the task of understanding phenomena in the real world (e.g.
> in science and engineering).
OK, fine. That still doesn't help them.
That paradigm shows that if every"thing" is computable, then
some"thing" different from all those"things", is, by diagonalization,
ALSO COMPUTABLE. THIS IS A CONTRADICTION. Therefore there
exist NON-computable "things". There is NOTHING they can do about
this.
> Cantor's Theory, if taken seriously, would lead us to believe
> that while the collection of all objects in the world of
> computation is a countable set, and while the collection of all
> identifiable abstractions derived from the world of computation
> is a countable set, there nevertheless "exist" uncountable sets,
Replace "uncountable" with "uncomputable" and you can just prove this,
both *about* TMs AND *with* TMs.
> implying (again, according to Cantor's logic)
Please. The only LOGIC relevant here is STANDARD
CLASSICAL FIRST-ORDER LOGIC! There is NOTHING
Cantorian about THAT! This is just logic, PERIOD!
THAT is what the idiots don't understand!
That's WHY we're justified in dismissing them as idiots!
> the "existence"
> of a super-infinite fantasy world having no connection to the
> underlying reality of mathematics.
There IS NO "underlying" reality of mathematics, unless you mean
precisely this fantasy world. Mathematics is ALL science fiction.
It's about WHAT IF *this* were axiomatically true?
> The anti-Cantorians see
> such a belief as an absurdity (in the sense of being
> disconnected from reality, rather than merely counter-intuitive).
Again, ALL axioms are like that.
0<1
is just 3 symbols. You cannot point to a physical 0 or 1
that they could be about. "<" does NOT have to connote being
to the left of something on a number line. It does not have
to be ANYthing concrete. ALL of this is abstract.
ALL of it is disconnected from "reality". JEEZUS.
> The mathematicians claim that they can "prove" the existence
> of uncountable sets, and hence there's nothing to be debated.
> But that merely calls into question the nature of "proof".
No, actually, it doesn't. We can say very unambiguously and
finitarily EXACTLY what we mean by "proof" (proof-predicates are
PRIMITIVE-recursive). At best it calls into question the
EXISTENTIAL IMPORT of proof. This is where you need to say something
about the Lowenheim-Skolem theorem. All these theories that prove
the existence of uncountable sets can (at first-order) be modeled
in a countable universe. This really does mean in some sense that
these proofs DON'T (by themselves) necessitate the existence of
uncountable sets. They do, however, given any alleged counting,
necessitate the existence of something that it leaves out. The
problem is that that's only ONE (as opposed to uncountably infinitely
many)thing left out.
> Certainly infinite sets and power sets exist as absractions.
I repeat, in that case, the cranks SIMPLY LOSE. EVERYthing being
talked about here is "abstract". YOUR NAME is abstract.
> But, abstractions don't necessarily obey exactly that same
> laws of logic as directly observable objects.
The level of ignorance you are flaunting here is breathtaking.
First of all, "directly observable objects" don't obey ANY LAWS
OF LOGIC, PERIOD. Logic IS abstract and it is ABOUT abstractions.
ONLY abstractions obey ANY laws of logic at all!
> Assuming
> otherwise can turn abstractions into fantasies, and proofs
> into absurdities, and that's the crux of the anti-Cantorian's
> argument.
Then they simply have no argument. There is no difference
between an "abstraction" and a "fantasy" TO BEGIN WITH, so
there is nothing to turn. Moreover, EVERYthing "concrete"
or non-abstract is UTTERLY AND COMPLETELY IRRELEVANT to this
WHOLE ENTERPRISE ANYWAY! The fact that the cranks surmise otherwise
is one thing that makes them cranks.
> The pure mathematicians tend to view mathematics as an art
> form. They seek to create beautiful theories, which may happen
> to be connected to reality, but only by accident. Those who apply
> mathematics, tend to view mathematics as a science which explores
> an objective reality (the world of computation).
I already rebutted all this in my previous reply in the other
thread, which you probably haven't read. You need to.
> In science, truth
> must have observable implications, and such a "reality check"
> would reveal Cantor's Theory to be a pseudoscience;
Quite the contrary: it reveals that it has a model, so the
cranks may sit down and shut up, at least until after
they have understood the Lowenheim-Skolem theorem.
> many of the
> formal theorems in Cantor's Theory have no observable implications.
This is simply a lie.
All theorems following from an axiom set have the OBSERVABLE
implication
that NO model of the axioms can decide the theorem falsely.
> The artists see the requirement that mathematical statements must
> have observable implications as a restriction on their intellectual
> freedom.
Here, you are just flaunting your personal ignorance.
"Observable implications" is not just oxymoronic, it's moronic.
In my previous reply, I gave you the difference between an observation
and an implication. I repeat, go look it up.
> The "anti-Cantorian" view has been around ever since Cantor
> introduced his ideas. Witness the following quotes from
> contemporaries of Cantor:
This is the most intellectually dishonest thing I have ever
seen here. Those contemporaries were certainly in a completely
different mindset from any modern thinkers, and Kronecker in
particular utterly lacks credibility in this whole context.
> "I don't know what predominates in Cantor's
> theory - philosophy or theology, but I am sure
> that there is no mathematics there" (Kronecker)
THAT is NOT a derivation of a contradiction from any of
Cantor's axioms.
> "Set theory is a disease from which mathematics
> will one day recover" (Poincare)
And THAT is not specific to Cantor.
> In the contemporary mainstream mathematical literature, there
> is almost no debate over the validity of Cantor's Theory.
Nor is there even any acknowledgment OF THE EXISTENCE of such
a thing. INSTEAD OF "Cantor's Theory", WE NOW have all the
THEOREMS following from AXIOM-SETS like ZFC and NBG. It is THESE
that don't need debating (because they HAVE been PROVEN)!
Cantor's TheorEM, NOT "Cantor's Theory", is what is relevant NOW.
> However, some mathematicians still drop hints that they see
> the absurdity of Cantor's Theory. Consider:
> "Set theory is based on polite lies, things we agree
> on even though we know they're not true. In some ways,
> the foundations of mathematics has an air of unreality."
> (William P. Thurston)
That's a much better quote. But it's
ridiculously hubristic. William P. Thurston in point of
actual fact doesn't know shit, let alone that the axiom of
choice or the continuum hypothesis "is not true".
At best he can say there are some models in which they are not
true. But there are others in which they are. He's got no
more clue than anyone else which side to pick, or should I say,
which side "the truth" has picked.
> It was the advent of the internet which revealed just how
> prevalent the anti-Cantorian view still is;
It has revealed no such thing. Have you never heard of "spam"?
A VERY SMALL number of people can create a VERY LARGE percentage
of the traffic. The percentage of the messages here that are about
this topic IS NOT in any relevant proportion to the fraction of
thinkers in the field who take that viewpoint seriously.
> there seems to be a
> never-ending heated debate in the Usenet newsgroups sci.math and
> sci.logic over the validity of Cantor's Theory.
No, there doesn't. There seem to be perennially recurring individuals
who bring it up. They come and they go. Herc is momentarily gone,
thank God.
> Typically, the
> anti-Cantorians accuse the pure mathematicians of living in a
> dream world, and the mathematicians respond by accusing the
> anti-Cantorians of being imbeciles, idiots and crackpots.
>
>
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation),
Well, I hate to break this to you, but (anti-)diagonalization
IS EVERY BIT AS OBSERVABLE "in the world of computation" as it is
in the proof of Cantor's Theorem.
> and philosophical mathematics, wherein Cantor's Theory is merely
> one of many possible formal "theories" of the infinite.
Even that is over-claiming; as far as math knows, there simply IS NO
"THE" Infinite for ANYthing to be a formal theory OF. Every theory
is different. Each one is going to be about a DIFFERENT infinity.
Stephen Montgomery-Smith
> To the intuitionists, who believe that set theory is polite lie, I have
> great respect and admiration. I myself use set theory, but only because
> no-one has come up with anything better.
Quite likely there is some great Platonic reality to mathematics,
towards which we are seeking. Set theory is merely a recent attempt at
groping in this direction. I do respect those who remind us that we
haven't yet arrived, and obviously this list includes some of the greats
like Poincare and Kronecker.
But this has nothing to do with anti-Cantorianism, which is
pseudo-mathematical babble. Really the problem with your proposed
article is that it only serves to obscure the real issues by identifying
them with the arguments of crackpots.
G. Frege
<ste...@math.missouri.edu> wrote:
>
> [...] I do respect those who remind us that we
> haven't yet arrived, and obviously this list includes some of the greats
> like Poincare and Kronecker.
>
> But this has nothing to do with anti-Cantorianism, which is pseudo-
> mathematical babble. Really the problem with your proposed article
> is that it only serves to obscure the real issues by identifying
> them with the ["]arguments["] of crackpots.
>
Agree!
F.
Stephen Montgomery-Smith
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation), and
> philosophical mathematics, wherein Cantor's Theory is merely
> one of many possible formal "theories" of the infinite.
Let me comment on this portion of your article. I think that you are
really describing more of a social phenominum in mathematics, where
there has been a tendency in the 20th century towards more abstract ways
of thinking. My sense is that this manner of thinking is currently
going out of vogue. For example, hardly anyone, it seems, studies set
theory for its own sake - I would say that the majority of
mathematicians use it simply because it is a convenient framework in
which to place their ideas.
One area which I am close to is the study of Banach spaces. This is a
difficult and technical and highly abstract area, but it is really
beginning to split into different directions. Some of them are going
into the theory of convex sets, and this is finding very genuine
applications in computer science (e.g. search algorithms in high
dimensional spaces, transmission of data down noisy lines, etc). Some
of them are moving into more concrete subjects like probability or
harmonic analysis. Harmonic analysis in turn is going back to its
roots, the study of partial differential equations, and the study of
partial differential equations is slowly moving back to its foundations
like engineering type applications.
In part, this trend away from the more abstract mathematics is being
encouraged by the funding agencies. For example, when applying for a
grant from the National Science Foundation they require you to explain
how you think that your work will improve the world. (The addition of
this requirement is fairly recent, maybe 10 years ago.)
However, while there is some recognition that maybe some mathematicians
have become perhaps a bit too abstract, it should be recognised that
much of the abstract stuff (particularly the better abstract stuff) is
going to find its way into other areas. For example, Cantor's diagonal
argument is clearly the forrunner of Turing's halting theorem and
Goedel's incompleteness theorem. These results, in turn, have helped
develop a field called computer science. Knowing that there were
uncomputable problems clearly helps people to look in the right
direction. In this way, it could be argued that Cantor played a vital
role in defeating the Nazi's in the Second World War (since so many of
these "abstract" computer scientists help crack the German secret code).
There is a split between pure mathematicians and applied mathematicians,
but I would contend that it is more along the lines of personality types
rather than any philosophical differences. I would agree that this
tension is particularly tight right now, but it is not a crisis, and in
time these two subjects will draw closer to each other again.
Stephen
Jesse Alama
> Quite likely there is some great Platonic reality to mathematics,
> towards which we are seeking. Set theory is merely a recent attempt at
> groping in this direction.
How does platonism follow from set theory? Or: how is set theory a
"groping in" the direction of "some great Platonic reality"?
Jesse
--
Jesse Alama (al...@stanford.edu)
Stephen Montgomery-Smith
> Stephen Montgomery-Smith <ste...@math.missouri.edu> writes:
>
>
>>Quite likely there is some great Platonic reality to mathematics,
>>towards which we are seeking. Set theory is merely a recent attempt at
>>groping in this direction.
>
>
> How does platonism follow from set theory? Or: how is set theory a
> "groping in" the direction of "some great Platonic reality"?
I think that you are taking me too literally.
Perhaps there is some reality of what mathematics is, beyond the grubby
reality in which we live. (This could also be compared to "justice" -
this might be a device used by society to keep us in order, or there
might be some absolute justice which we mortals try to approximate with
our laws.)
Set theory has the appearence to some of being this great math reality,
so that people think that there really are real numbers and power sets
of ordinals. I think that many mathematicians today take this for granted.
In any case, in our attempt to find out what this universal reality is,
we currently have, as a guess, set theory. I personally think that its
wrong, but I think that if one day we really do find out what this
reality is, that we will see that set theory has captured a lot of its
fundamentals.
Thus, for example, I can see that people might be uncomfortable with
notions like "the set of real numbers". But I think that the best thing
to do right now is to just "go along" with it. It is fine to have our
own personal misgivings about it, but to try to reconstruct our own
theories with limitations like, for example, no excluded middle, or no
infinite sets, or only computable or constructable ideas allowed, or
even no axiom of choice, is, I think, wrong headed. If we ever do find
out what mathematics really is, we will quite likely find that all the
effort we put into developing these limited theories was a complete
waste of time.
I think that in order to decide what is "good mathematics", our
intuition is a much better guide than any rules we might invent. For
example, it seems clear to me that Riesz's theorem about the boundedness
of the Hilbert transform is clearly an important part of mathematics,
whereas esetoric properties of non-separable Banach spaces is probably
not so important.
In actuality, I think that many mathematicians, and certainly the best
mathematicians, display this "good taste."
Incidently, when I use big words like "platonic" I am essentially
showing off, because I only have a vague idea what this word means.
Stephen
G. Frege
wrote:
> >
> > Quite likely there is some great Platonic reality to mathematics,
> > towards which we are seeking. Set theory is merely a recent attempt at
> > groping in this direction.
> >
>
> How does platonism follow from set theory? Or: how is set theory a
> "groping in" the direction of "some great Platonic reality"?
>
"[...] classes and concepts can be considered as real
objects [...] that exist independently from our definitions and
constructions. It seems to me that the assumption of such objects is as
justified as the assumption of physical objects, and there is similar
reason to believe in their existence." (Kurt Gödel) // Ad hoc
translation.
"Even recognizing the fruitfulness of my objectivism for my work,
people might choose not to adopt the objectivistic position but merely
do their work a s i f the position were true--provided they are able to
produce such an attitude. But then they only take this as-if point of
view toward this position after it has been shown to be fruitful.
Moreover, it is doubtful whether one can pretend so well as to yield the
desired effect of getting good scientific results."
[Kurt Gödel--in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: The MIT Press. (p. 239)]
"Abraham Robinson is a representative of an as-if position, according to
which it is fruitful to behave as if there were mathematical objects,
and in this way you achieve success by a false picture. This requires a
special art of pretending well. But such pretending can never reach the
same degree of imagination as one who believes objectivism to be true.
The success in the application of a belief in the existence of something
is the usual and most effective way of proving existence."
[Kurt Gödel--in: Wang, Hao (1996). /A logical journey:
From Gödel to philosophy/. Cambridge, MA: The MIT Press. (p. 240)]
F.
"The axioms [of set theory] force themselves upon us
as being true." (Kurt Gödel)
G. Frege
<ste...@math.missouri.edu> wrote:
>
> Incidently, when I use big words like "platonic" I am essentially
> showing off, because I only have a vague idea what this word means.
>
Two quotes... ;-)
»The working mathematician is a Platonist on weekdays, a formalist on
weekends. On weekdays, when doing mathematics, he's a Platonist,
convinced he's dealing with an objective reality whose properties he's
trying to determine. On weekends, if challenged to give a philosophical
account of the reality, it's easiest to pretend he doesn't believe it.
He plays formalist, and pretends mathematics is a meaningless game.«
(R. Hersh)
»On foundations we believe in the reality of mathematics,
but of course when philosophers attack us with their
paradoxes we rush to hide behind formalism and say
"Mathematics is just a combination of meaningless symbols,"
and then we bring out Chapters 1 and 2 on set theory.
Finally we are left in peace to go back to our mathematics
and do it as we have always done, with the feeling each
mathematician has that he is working with something real.
This sensation is probably an illusion, but is very convenient.
That is Bourbaki's attitude toward foundations.«
(Jean Dieudonné)
F.
Stephen Montgomery-Smith
Those quotes are brilliant!
Stephen
Dave Rusin
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
>Set theory has the appearence to some of being this great math reality,
>so that people think that there really are real numbers and power sets
>of ordinals. I think that many mathematicians today take this for granted.
I think that of the "people [who] think that there really are real numbers"
it is a minority who think this _because_ of set theory some how. Sure, sure,
we all go through the exercise of "constructing" the real numbers from
sets. But that's really just a way to affirm that the axioms we write down
which we hope capture the key properties of R are not patently incorrect,
that is, by making the construction of R from sets, we convince ourselves
that the axioms are _consistent_ (well, as consistent as ZF anyway). But
we chose those axioms in the first place because they seemed to distill
the essence of the what the real numbers _really are_. (And what they
"really are" ain't sets!)
You can liken this to the role of Choice in ZFC. Some people believe there
really are "sets" and that one of the properties they have is the axiom of
choice. But they concede that maybe their intuition is clouded (unlike
some of our cranks who won't grant this possibility...) and so they ask,
well, is it at least _possible_ that ZFC mirrors the reality I see? Or
is Choice a comforting intuition but patently impossible (in conjunction
with the other axioms)? Well, we know now that ZFC is consistent (assuming
ZF is) so sure, we can safely use Choice when we prove things about the
"sets" which exist in our universe. Other people may have other objects in
mind when they hear the word "sets", and there's no guarantee that their
"sets" satisfy ZFC, but that's not necessarily important.
In this way we might _think_ that, say, the Archimedean property holds for
the reals, but there's no way to go out and test this property, really;
at least we can ask, would it at least be _consistent_ to assume the
property holds, together with the ordered-field axioms? Here set theory
rewards us with a proof that yes, there is a model of this set of axioms
and so the whole kit and kaboodle of them are consistent. Having established
that, we are comforted that we are not misreading the nature of the
"real numbers" in our heads -- but those real numbers are not, to me anyway,
"sets" at all. The real numbers were there all along, and the only
thing set theory has given is some assurance that I've got a clear picture
of the key properties of R .
dave
quasi
wrote:
>...l
>the essence of the what the real numbers _really are_. (And what they
>"really are" ain't sets!)
Haha.
That reminds me of when I wanted to understand the concept of a
non-deterministic machine. I asked someone, supposedly an expert, to
explain it to me. He started out by saying a non-deterministic machine
was a 7-tuple ... and I stopped him right there. Even though I really
didn't know what they were, I knew that a formal definition as some
kind of 7-tuple was not the way to understand the essence.
quasi,
imagin...@despammed.com
> I'm starting a new thread because the other one got
> out of control.
>
> I want to thank those of you who gave me some useful
> feedback on my article (you know who you are :-).
>
> Here's the updated version of my article. I hope it
> answers some of the questions people have asked.
>
> ********
>
>
> In this article, "Cantor's Theory" refers to the pre-formal
> ideas about set theory introduced by Cantor in the latter
> part of the nineteenth century. The "anti-Cantorians" are
> people who claim that Cantor created a fantasy world.
This implies that non-anti-Cantorians think that mathematics is other
than a fantasy world. Can you explain exactly what you mean by
"fantasy", and perhaps give references to some of these
non-anti-Cantorians describing mathematics as, er, what, a model of the
physical world?
> The anti-Cantorians claim that while infinite sets and
> power sets of those infinite sets are undeniably useful
> abstractions, what Cantor did was to take an argument
> (the diagonalization argument), which is perfectly valid
> in concrete mathematics, and wrecklessly apply it to
> the abstractions of infinity, ultimately producing garbage.
"Garbage"? How is that defined, mathematically speaking?
<snip>
> These "anti-Cantorians" see an underlying reality to
> mathematics, namely, computation. They tend to accept the
> idea that the computer can be thought of as a microscope
> into the world of computation, and mathematics is the
> science which studies the phenomena observed through that
> microscope. They claim that that paradigm encompasses all
> of the mathematics which has the potential to be applied to
> the task of understanding phenomena in the real world (e.g.
> in science and engineering).
Ah, so these "anti-Cantorians" simply wish to define the word
"mathematics" as meaning a subset of what (normal) mathematicians mean
by mathematics. Why not have the simple decency to define a new term -
"computational mathematics", or whatever. Then write an article about
it (except that it's fairly clearly "original research", not wanted on
Wikipedia).
Brian Chandler
http://imaginatorium.org
Proginoskes
quasi wrote:
> [...]
> That reminds me of when I wanted to understand the concept of a
> non-deterministic machine. I asked someone, supposedly an expert, to
> explain it to me. He started out by saying a non-deterministic machine
> was a 7-tuple ... and I stopped him right there. Even though I really
> didn't know what they were, I knew that a formal definition as some
> kind of 7-tuple was not the way to understand the essence.
There's a big gap between understanding something and being able to
explain it. This is why there are so many good researchers in colleges
who are such horrible teachers.
Someone with more of the mind of a teacher would have probably said
something like: "A non-deterministic machine is an algorithm that
reaches points where a decision has to be made. It will choose the
correct decision, provided that there is a success somewhere further
down the road, among the future choices." I hope so, anyway, because I
wrote it. 8-)
--- Christopher Heckman
David Kastrup
It is one way to understand the essence.
Ask a player and a piano builder to teach you the essence of a piano.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Chan-Ho Suh
david petry <david_lawr...@yahoo.com> wrote:
> "Set theory is a disease from which mathematics
> will one day recover" (Poincare)
>
>
Poincare never said that. You will not be able to give a valid
reference for this.
> In the contemporary mainstream mathematical literature, there
> is almost no debate over the validity of Cantor's Theory.
> However, some mathematicians still drop hints that they see
> the absurdity of Cantor's Theory. Consider:
>
>
> "Set theory is based on polite lies, things we agree
> on even though we know they're not true. In some ways,
> the foundations of mathematics has an air of unreality."
> (William P. Thurston)
>
>
This quote is out of context and cannot be considered to be support for
the "anti-Cantorian" viewpoint. You are greatly distorting Thurston's
views, which are not really so different from those of many pure
mathematicians.
Keith Ramsay
There was a discussion of this on the Historia Mathematica
mailing list. One message mentioned an article about the
issue in the Mathematical Intelligencer, vol. 13 (1991)
p. 19-22. Allegedly it concludes he didn't say it.
On 7 Sept. 1998 Michael Detlefsen commented that the original
of the remark appears in Poincare's 1908 essay, "The Future
of Mathematics" (appearing as chapter 2 of _Science and
Method_). Here's how he quotes it (I word-wrapped it):
#"... it has come to pass that we have encountered certain
#paradoxes, certain apparent contradictions that would have
#delighted Zeno the Eleatic and the school of Megara. And
#then each must seek the remedy. For my part, I think, and
#I am not the only one, that the important thing is never to
#introduce entities not completely definable in a finite
#number of words. Whatever be the cure adopted, we may
#promise ourselves the joy of the doctor called in to follow
#a beautiful pathologic case."
So it seems that rather than describing set theory as a
disease, he is describing set theory as a patient to be
cured of a disease, i.e. paradoxes. His idea of a proper
cure wasn't the same as Cantor's, nor was it the same as
yours.
Keith Ramsay
muec...@rz.fh-augsburg.de
Chan-Ho Suh wrote:
> In article <1121888825.5...@g44g2000cwa.googlegroups.com>,
> david petry <david_lawr...@yahoo.com> wrote:
>
> > "Set theory is a disease from which mathematics
> > will one day recover" (Poincare)
> >
> >
>
> Poincare never said that. You will not be able to give a valid
> reference for this.
There is no actual infinity. The Catorians have forgotten that and have
fallen into contradictons. [H. Poincaré, Les mathématiques et la
logique III, Rev. métaphys. morale 14, p. 316, (1906).]
Regards, WM
muec...@rz.fh-augsburg.de
david petry wrote:
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation), and
> philosophical mathematics, wherein Cantor's Theory is merely
> one of many possible formal "theories" of the infinite.
Call this one relegious mathematics or matheology. The Cantorians close
their eyes in order to avoid obvious contradictions and to maintain
their useless pet, simply insisting that logic is not valid in the
"infinite".
One example: Take the set Q of all rationals q, in normal order.
Multiply each one by pi. This gives an equinumerous (by bijection) set
X of irrationals q*pi. The union of both sets consists of rationals and
irrationals. By symmetry reasons there is never a pair of irrationals
without a rational between them in this set. Now add one further
irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot
exist other than that two irrationals, namely sqrt(2) and one of the
elements q*pi of X have no rational q of Q between them. This is a
contradiction because in the whole set of real numbers in normal order,
there are never wo irrationals without a rational between them, as is
easy to prove.
Alone the latter fact is proof enough to see that there are not more
irrationals than rationals.
Probable solution: There are no irrational numbers existing at all, as
Kronecker already knew.
Regards, WM
Dik T. Winter
...
> The anti-Cantorians claim that while infinite sets and
> power sets of those infinite sets are undeniably useful
> abstractions,
Not all anti-Cantorians claim that.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Han de Bruijn
> On 20 Jul 2005 12:47:05 -0700, "david petry"
> <david_lawr...@yahoo.com> wrote:
>
> I's suggest that you delete the following paragraph from your article.
>
> Rationale:
>
> (1) It's definitely n o t plausible. You might f i r s t ask
> professionals of mathematics before claiming such ridiculous things.
>
> (2) It obviously reflects *your own* opinion concerning this topic -
> hardly something that should show up in an article of an encyclopedia.
>
> (3) It's complete nonsense.
>
>>It is plausible that in the future, mathematics will be split
>>into two disciplines - scientific mathematics (i.e. the science
>>of phenomena observable in the world of computation), and
>>philosophical mathematics, wherein Cantor's Theory is merely
>>one of many possible formal "theories" of the infinite.
As I've said in the other thread:
> Worse. It _has_ already split into two disciplines:
>
> http://huizen.dto.tudelft.nl/deBruijn/nag.htm
Han de Bruijn
Han de Bruijn
> I still await the inclusion of the answers to the following questions.
>
> To which axioms of ZF do anti-Cantorians object, exactly?
The last five or so. :-)
Han de Bruijn
Han de Bruijn
> I think that I would object to the use of the word "anti-Cantorian" and
> I would prefer that you use the word "intuitionist."
Though I feel much sympathy for intuitionism, I would object to that.
For the reason that intuitionism has already a well-defined meaning,
which conflicts on many issues with David Petry's description of an
"anti-Cantorian".
Han de Bruijn
Han de Bruijn
> Well, we know now that ZFC is consistent (assuming ZF is)
I didn't know that ZFC is consistent. And why should I assume that ZF
is consistent ? Isn't that just wishful thinking ?
Han de Bruijn
georgie
What exactly do you mean by the term "infinite
number?" Are you one of those morons who
apply a concept from one domain to another
domain and then expect to be taken seriously?
You do realize there are no infinite natural numbers
(counting numbers), right?
muec...@rz.fh-augsburg.de
muec...@rz.fh-augsburg.de wrote:
> david petry wrote:
>
> > It is plausible that in the future, mathematics will be split
> > into two disciplines - scientific mathematics (i.e. the science
> > of phenomena observable in the world of computation), and
> > philosophical mathematics, wherein Cantor's Theory is merely
> > one of many possible formal "theories" of the infinite.
>
> Call this one relegious mathematics or matheology. The Cantorians close
> their eyes in order to avoid obvious contradictions and to maintain
> their useless pet, simply insisting that logic is not valid in the
> "infinite".
>
CORRECTION:
> One example: Take the set Q of all rationals q, in normal order.
> ADD pi TO EACH ONE. This gives an equinumerous (by bijection) set
> X of irrationals q+pi. The union of both sets consists of rationals and
> irrationals. By symmetry reasons there is never a pair of irrationals
> without a rational between them in this set. Now add one further
> irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot
> exist other than that two irrationals, namely sqrt(2) and one of the
> elements q+pi of X have no rational q of Q between them. This is a
> contradiction because in the whole set of real numbers in normal order,
> there are never two irrationals without a rational between them, as is
Dave Rusin
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
>Dave Rusin wrote:
>
>> Well, we know now that ZFC is consistent (assuming ZF is)
>
>I didn't know that ZFC is consistent.
I didn't say it was known to be consistent. Read my whole sentence again.
Or read the work of Godel and of Cohen.
>And why should I assume that ZF
>is consistent ? Isn't that just wishful thinking ?
Yes, it is "just wishful thinking".
But isn't everything _you_ espouse also wishful thinking?
That is, your particular point of view seems to be to put greater
faith in the conclusions which are drawn from _inductive_ reasoning;
rather than conclusions drawn _deductively_ from axioms, you want
to work with statements which are "borne out in reality". This
is an eminently reasonable way to do some things (physical science,
for example), but of course it's just "wishful thinking" to suppose
that there are no exceptions nor contradictions out there somewhere,
simply because you haven't encountered any yet. Was Newtonian
mechanics correct? Experience said "yes" and so with "wishful
thinking" we believed it. Eventually we found it had to be tweaked.
So too with ZF. We actually know we can't prove it consistent
(unless it's inconsistent) but based on our experience we believe
it to be consistent. The "experiments" testing its consistency
consist of the zillions of proofs which have been constructed
of propositions phrased, ultimately, on the ZF axioms of set theory.
No contradictions have yet been found. Some day a contradiction
will emerge, and we'll have to tweak things just as the physicists did.
I wouldn't lose any sleep over it; the "experimental evidence"
is really strong that our "wishful thinking" is in fact correct.
dave
Chris Menzel
Excellent detective work. I've wondered for years about the legitimacy
of that quote.
Chris Menzel
Well, it might not be the way to *introduce* the concept to a beginner
-- one needs to start with an informal account of the underlying
intuitions. But that doesn't mean that it isn't, at the end of the day,
the right way to understand the concept's essence.
Virgil
muec...@rz.fh-augsburg.de wrote:
> Chan-Ho Suh wrote:
> > In article <1121888825.5...@g44g2000cwa.googlegroups.com>,
> > david petry <david lawrence pe...@yahoo.com> wrote:
> >
> > > "Set theory is a disease from which mathematics
> > > will one day recover" (Poincare)
> > >
> > >
> >
> > Poincare never said that. You will not be able to give a valid
> > reference for this.
>
> There is no actual infinity. The Catorians have forgotten that and have
> fallen into contradictons. [H. Poincaré, Les mathématiques et la
> logique III, Rev. métaphys. morale 14, p. 316, (1906).]
Poincare never said that.
G. Frege
<Han.de...@DTO.TUDelft.NL> wrote:
> >
> > Worse. It _has_ already split into two disciplines:
> >
> > http://huizen.dto.tudelft.nl/deBruijn/nag.htm
>
Nonsense.
Virgil
muec...@rz.fh-augsburg.de wrote:
> david petry wrote:
>
> > It is plausible that in the future, mathematics will be split
> > into two disciplines - scientific mathematics (i.e. the science
> > of phenomena observable in the world of computation), and
> > philosophical mathematics, wherein Cantor's Theory is merely
> > one of many possible formal "theories" of the infinite.
>
> Call this one relegious mathematics or matheology.
It is WM's beeleifs that are a matter of religion, as they require
simulteneous belief in the mutually exclusive.
> The Cantorians close their eyes in order to avoid obvious
> contradictions and to maintain their useless pet, simply insisting
> that logic is not valid in the "infinite".
Actually, it is the "Cantorian" insistence that logic IS valid in the
infinite that the the anti-Cantorian rabble objects to so fervently.
>
> One example: Take the set Q of all rationals q, in normal order.
> Multiply each one by pi. This gives an equinumerous (by bijection) set
> X of irrationals q*pi. The union of both sets consists of rationals and
> irrationals. By symmetry reasons there is never a pair of irrationals
> without a rational between them in this set. Now add one further
> irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot
> exist other than that two irrationals, namely sqrt(2) and one of the
> elements q*pi of X have no rational q of Q between them. This is a
> contradiction because in the whole set of real numbers in normal order,
> there are never wo irrationals without a rational between them, as is
> easy to prove.
The only "contradiction" here is produced by WM assuming, as usual, what
is patently false.
Between EVERY two reals there is a rational, in fact, between any two
distinct reals there are infinitely many rationals.
>
> Alone the latter fact is proof enough to see that there are not more
> irrationals than rationals.
The latter "fact" referred to is a falsehood which only someone as
mathematically incompetent as WM would be guilty of foisting on sci.math.
>
> Probable solution: There are no irrational numbers existing at all, as
> Kronecker already knew.
Solution: WM is too incompetent to discuss any mathematical issue, as
sci.math already knew.
>
> Regards, WM
Virgil
muec...@rz.fh-augsburg.de wrote:
> muec...@rz.fh-augsburg.de wrote:
> > david petry wrote:
> >
> > > It is plausible that in the future, mathematics will be split
> > > into two disciplines - scientific mathematics (i.e. the science
> > > of phenomena observable in the world of computation), and
> > > philosophical mathematics, wherein Cantor's Theory is merely
> > > one of many possible formal "theories" of the infinite.
> >
> > Call this one relegious mathematics or matheology. The Cantorians close
> > their eyes in order to avoid obvious contradictions and to maintain
> > their useless pet, simply insisting that logic is not valid in the
> > "infinite".
> >
> CORRECTION:
> > One example: Take the set Q of all rationals q, in normal order.
> > ADD pi TO EACH ONE. This gives an equinumerous (by bijection) set
> > X of irrationals q+pi. The union of both sets consists of rationals and
> > irrationals. By symmetry reasons there is never a pair of irrationals
> > without a rational between them in this set. Now add one further
> > irrational, for instance sqrt(2). The union Q u X u {sqrt(2)} cannot
> > exist other than that two irrationals, namely sqrt(2) and one of the
> > elements q+pi of X have no rational q of Q between them.
WM is deliberately trying to promulgate a falsehood here.
Between ANY two distinct reals there are as many rationals as in Q
itself. Since the union Q u X u {sqrt(2)} contians Q, there must be
infinitely many rationals between any two distinct members of it,
including those mentioned by WM.
> > This is a contradiction because in the whole set of real numbers in
> > normal order, there are never two irrationals without a rational
> > between them, as is easy to prove.
The only contradiction here is that WM contradicting the truth.
> >
> > Alone the latter fact is proof enough to see that there are not more
> > irrationals than rationals.
Alone, the latter fact is proof enough that WM is too mathematically
incompetent to assess any mathematical statements beyond the level of
2+2=4.
> >
> > Probable solution: There are no irrational numbers existing at all, as
> > Kronecker already knew.
Probable solution: WM, get thee to a shrink posthaste!
david petry
G. Frege wrote:
> On Wed, 20 Jul 2005 22:04:58 +0200, G. Frege <nomail@invalid> wrote:
>
>
> > >
> > > [...] the mathematicians respond by accusing the
> > > anti-Cantorians of being imbeciles, idiots and[/or] crackpots.
> > And that's indeed true in most (if not all) cases.
> Could y o u, David, show me a single exception?
Here's an article which was posted anonymously about 6 years ago,
which is an example of an anti-Cantorian who is far from being a
Crackpot.
************
5. dav2222 Jul 28 1999, 3:00 am show options
Newsgroups: sci.math
From: dav2...@my-deja.com - Find messages by this author
Date: 1999/07/28
Subject: More Cantor
Reply to Author | Forward | Print | Individual Message | Show original
| Report Abuse
Hello sci.math participants. My question relates
to a topic that has created a lot of heated
discussion in this group. I have done some
searches ondejanews and the world wide web and
have not found a convincing answer yet.
Like many persons, I am troubled by Cantor's
diagonalization proof.I feel that my math
teachers gave me a little "bait and switch."
Here's what I mean:
They taught me about whole numbers, then
integers, then fractions.I learned that fractions
are "rational" numbers. (by fraction I mean
something like A/B where A and B are integers)
Then I was taught that "rationals" are not
enough. For example, thesquare root of two can't
be represented as a fraction. Ditto for pi.
So then they introduce real numbers -- so far so
good.
Then comes the diagonalization proof, which seems
pretty cool, but a few weeks after learning about
it, it occurred to me that thereal numbers in the
proof are qualitatively different from numbers
like pi - this is the "bait and switch"
What I mean is, numbers like pi can be described
in English. The Enlish description can be mapped
to a whole number -- for example by letting each
letter be a "digit" in a base 26 numbering
system
. (Maybe
a higher base would be useful to get spaces,
symbols, etc. in)
By the way, I realize that this is not an
original thought. Please bear with me and I will
reach my question.
Anyways, it seems like one could make a
distinction between "meaningful" real numbers and
"meaningless" or "undescribable" real numbers.
I
don't have the mathematical skill to formally
define "meaningful" numbers, but informally, I
would offer the following definition:
A meaningful number is one that can be described
unambiguously, to a person of reasonable
intelligence, in a finite string of English
letters
, Arabic numerals, and commonly used
symbols. Ellipses are only permissible when
reasonable people would agree on the natural
continuation of the preceding sequence or series
of characters, numerals, and/or symbols.
Thus "THE SQUARE ROOT OF TWO" is meaningful.
Also, "3.1415926 . . . " is meaningful.
But, "0.97364380 . . . ", a number that might be
found in a diagonalization proof, is meaningless
by my definition. In fact, I am troubled even
calling "0.97364390 . . ." a "number" --
arguably it is just an ambiguous string of
numerals and symbols.
"THE SMALLEST NUMBER NOT DEFINABLE IN LESS THAN
19 SYLLABLES" is possibly meaningless. At any
rate, my instinct is that the question is not
relevant the the question I ask below.
So here's my question: What happens to
mathematics if we get rid of meaningless
numbers
? Note that I'm not trying to argue that
the diagonalization proof is wrong -
My
understanding is that mathematics is no longer
tied down by the real world and that many
mathematicians consider themselves free to make
whatever assumptions lead to interesting results.
e.g., non-Euclidean geometries
. Do we "need"
meaninglessnumbers in the same way that we "need"
sqrt (2)? What uses are there for meaningless
numbers? (besides proofs about aleph-0,
cardinality, and the like)
I also believe that without meaningless numbers
many of Cantor's results fall. Iam comfortable
with this - I'm just wondering what else we
lose. Can one construct a coherent mathematics
without all that stuff? My feeling is yes, but I
don't really know.
By the way, I have given the above some thought
and realize that arguably there is something more
fundamental at stake than meaningless numbers.
I would propose getting rid of "meaningless
sets." By analogy to the above definition, the
following are examples of meaningful sets:
{1, 2 , 9, 23}
{2, 3, 5, 7, 11, 13, 17, 19, 23 . . . }
{ALL EVEN INTEGERS} (You get the point).
{ALL SETS THAT ARE NOT MEMBERS OF THEMSELVES} is
possibly meaningless.
{9, 97, 973, 9736, 97364, 973643, 9736438,
97364380 . . .} is meaningless.
It seems to me that if you define sets in such a
way -- i.e. get rid of meaningless sets -- then
's proof about power sets always being of
greater cardinaliy does not work anymore.
As above, my question is -- what else do you
lose? Is mathematics still coherent? Do we
"need" meaningless sets? What uses are there for
them?
Thanks so much for bearing with me. I really
would appreciate an answer.
I apologise for my ignorance of mathematics -
please don't flame me for getting something
wrong. I am a lawyer and have not studied
mathematics in many years. This is just
something that has always troubled me. I have
done many searches and have not come across a
satisfactory answer.
P.S. It would be great if someone would give me
somelingo to describe this problem. e.g.: "Aha -
the problem you are describing is known
as Kubo's conundrum
david petry
Keith Ramsay wrote:
> |> > "Set theory is a disease from which mathematics
> |> > will one day recover" (Poincare)
> There was a discussion of this on the Historia Mathematica
> mailing list. One message mentioned an article about the
> issue in the Mathematical Intelligencer, vol. 13 (1991)
> p. 19-22. Allegedly it concludes he didn't say it.
>
> On 7 Sept. 1998 Michael Detlefsen commented that the original
> of the remark appears in Poincare's 1908 essay, "The Future
> of Mathematics" (appearing as chapter 2 of _Science and
> Method_). Here's how he quotes it (I word-wrapped it):
>
> #"... it has come to pass that we have encountered certain
> #paradoxes, certain apparent contradictions that would have
> #delighted Zeno the Eleatic and the school of Megara. And
> #then each must seek the remedy. For my part, I think, and
> #I am not the only one, that the important thing is never to
> #introduce entities not completely definable in a finite
> #number of words. Whatever be the cure adopted, we may
> #promise ourselves the joy of the doctor called in to follow
> #a beautiful pathologic case."
>
> So it seems that rather than describing set theory as a
> disease, he is describing set theory as a patient to be
> cured of a disease, i.e. paradoxes. His idea of a proper
> cure wasn't the same as Cantor's, nor was it the same as
> yours.
First, thanks for finding that reference for us.
However, I don't think you are interpreting it correctly. Look at
his proposed cure:
"For my part, I think, and I am not the only one,
that the important thing is never to introduce entities not completely
definable in a finite number of words."
That is almost exactly the "anti-Cantorian" view. There don't exist
more than a countable number of entities, given Poincare's cure.
He is claiming that it is Cantor's ideas about set theory that is the
disease. I don't think anyone is arguing that there is something
wrong with sets per se.
So I'm sticking with the quote.
>
> Keith Ramsay
david petry
Dik T. Winter wrote:
>> The anti-Cantorians claim that while infinite sets and
>> power sets of those infinite sets are undeniably useful
>> abstractions,
>Not all anti-Cantorians claim that.
As I suggested in my article, I'm trying to separate the signal
from the noise.
david petry
> In article <1121888825.518358.308...@g44g2000cwa.googlegroups.com>,
> david petry <david_lawrence_pe...@yahoo.com> wrote:
> > "Set theory is a disease from which mathematics
> > will one day recover" (Poincare)
> Poincare never said that. You will not be able to give a valid
> reference for this.
I can't, but Keith Ramsay already has. :-)
David Kastrup
Where "signal" is defined as what you happen to believe in...
david petry
>Mathematics is ALL science fiction.
>It's about WHAT IF *this* were axiomatically true?
That's not the way it is viewed by people who apply mathematics.
That's not the way it is viewed by the anti-Cantorians. It's not the
way it was viewed before formal set theory came on the scene.
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> Here's an article which was posted anonymously about 6 years ago,
> which is an example of an anti-Cantorian who is far from being a
> Crackpot.
>
*lol* You really don't know what you are talking about, do you?
Just a quote from that post:
"I don't have the mathematical skill to formally
define "meaningful" numbers, but [bla and bla]"
It's always the same storry..., here's another one:
"I was asked that before, and never got around to fully
analyzing the axioms for lack of time, but [bla and bla]"
(Tony Orlow)
--------------------
I am the very model of a modern non-Cantorian,
With insights mathematical as good as any saurian.
I rattle the Establishment foundations with prodigious ease,
And populate the counting numbers with some new infinities.
I've never studied axioms of sets all theoretical,
But that's just ted'ous detail; whereas MY thoughts are heretical
And cause the so-called experts rather quickly to exasperate,
While I sit back and mentally continue just to ....
(Barb Knox)
F.
david petry
>If mose of the debate on the internet is junk, then
>it seems strange to use the internet as evidence
>for the prevalence of the anti-Cantorian view.
For many of the anti-Cantorians, it is most obvious that
there's something absurd about Cantor's Theory, but they
really haven't a clue about how to argue the point to the
mathematicians.
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> I'm trying to separate the signal from the noise.
>
There is no "signal", only noise (produced by mathematical crackpots).
F.
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> For many of the anti-Cantorians, it is most obvious that
> there's something absurd about Cantor's Theory, but they
> really haven't a clue about how to argue the point to the
> mathematicians.
>
That's because the lack the necessary mathematical abilities.
Btw: If the h a d them, the wouldn't be "anti-Cantorians".
(It's r e a l l y that simple!)
F.
ste...@nomail.com
What is absurd about there not existing a bijection between
a set and its powerset?
In any case, as you said, most of the anti-Cantorian
arguments are nonsense. It seems strange to then
conclude that there is a serious and prevalent
anti-Cantorian view.
Stephen
malb...@yahoo.com
> In sci.math david petry <david_lawr...@yahoo.com> wrote:
>
> >>If mose of the debate on the internet is junk, then
> >>it seems strange to use the internet as evidence
> >>for the prevalence of the anti-Cantorian view.
>
> > For many of the anti-Cantorians, it is most obvious that
> > there's something absurd about Cantor's Theory, but they
> > really haven't a clue about how to argue the point to the
> > mathematicians.
>
> What is absurd about there not existing a bijection between
> a set and its powerset?
The view is that one can be constructed. karl m
Stephen Montgomery-Smith
>
> G. Frege wrote:
>
>>On Wed, 20 Jul 2005 22:04:58 +0200, G. Frege <nomail@invalid> wrote:
>>
>>
>>
>>>> [...] the mathematicians respond by accusing the
>>>>anti-Cantorians of being imbeciles, idiots and[/or] crackpots.
>
>
>>> And that's indeed true in most (if not all) cases.
>
>
>>Could y o u, David, show me a single exception?
>
>
>
> Here's an article which was posted anonymously about 6 years ago,
> which is an example of an anti-Cantorian who is far from being a
> Crackpot.
He's not an anti-Cantorian. He is a person who is just learning the
subject, and is asking some interesting questions. His ideas about
"meaningless" sets and "meaningless" real numbers are early attempts to
try to get to notions of recursively enumerable sets and computable real
numbers. If he had opportunity to study these very proper subjects, he
would come to find out that they don't really solve his problems, and
indeed diagonal arguments can be used with those to find "uncomputable"
or "non-primitive recursive" real numbers. He will come up against
problems like Turing's halting theorem (again effectively Cantor's
diagonal argument).
I was in a similar place 25 years ago. I read some article in a
magazine called "Wireless world" which debunked Einstein's theory of
relativity. I got very interested, and started asking the same
questions as the article. After a year or two of deep thinking I
figured out that Einstein really had got it right. That deep thinking
really helped me understand a lot of the issues.
I think that this kid has potential to go in the same direction. Just
give him time (and hopefully good advice).
Stephen Montgomery-Smith
Somebody else emailed me exactly that objection, and I agree with them
and you. They suggested the word "constructivist."
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> When Cantor introduced his ideas, there was a heated debate
> about whether they should be accepted as mathematics. For
> reasons which are not entirely clear, the ideas were
> accepted [...]
>
Actually, the reasons are rather clear. David Hilbert described Cantor's
work as
"the finest product of mathematical genius and one of the
supreme achievements of purely intellectual human activity."
F.
malb...@yahoo.com
> david petry wrote:
(...)
> > Here's an article which was posted anonymously about 6 years ago,
> > which is an example of an anti-Cantorian who is far from being a
> > Crackpot.
> I think that this kid has potential to go in the same direction. Just
> give him time (and hopefully good advice).
> > I apologise for my ignorance of mathematics -
> > please don't flame me for getting something
> > wrong. I am a lawyer and have not studied
> > mathematics in many years. This is just
> > something that has always troubled me. I have
> > done many searches and have not come across a
> > satisfactory answer.
>From what did you conclude that dav2222 is a kid? karl m
Stephen Montgomery-Smith
> As I've said in the other thread:
>
>> Worse. It _has_ already split into two disciplines:
>>
>> http://huizen.dto.tudelft.nl/deBruijn/nag.htm
>
>
> Han de Bruijn
I think that this article makes interesting points, although I would
venture to say that it is describing social problems rather than
philosophical problems. I also share with you a sense that mathematics
has recently tended to be a little overboard on the abstract. However,
I do believe that the trend is reversing. You don't want it to reverse
too quickly otherwise you will get a "French Revolution" rather than an
"American War of Independence" (I hope you get my meaning). Also, I
think that some tension between the "applied mathematicians" and "pure
mathematicians" is always going to be healthy.
I don't really think that objections to the axioms of ZF are at the root
of this problem. Well the emphasis on axiomization might have had a
psychological effect upon mathematicians. But these effects are
reversable. And in the meantime, people are trying to develop automatic
proof techniques, which will certainly prove to be useful in the future.
Anyway I have another post on this thread on this subject. I would be
interested in your comments on it.
Stephen
quasi
<progi...@email.msn.com> wrote:
>
>quasi wrote:
>> [...]
>> That reminds me of when I wanted to understand the concept of a
>> non-deterministic machine. I asked someone, supposedly an expert, to
>> explain it to me. He started out by saying a non-deterministic machine
>> was a 7-tuple ... and I stopped him right there. Even though I really
>> didn't know what they were, I knew that a formal definition as some
>> kind of 7-tuple was not the way to understand the essence.
>
>There's a big gap between understanding something and being able to
>explain it. This is why there are so many good researchers in colleges
>who are such horrible teachers.
>
>Someone with more of the mind of a teacher would have probably said
>something like: "A non-deterministic machine is an algorithm that
>reaches points where a decision has to be made. It will choose the
>correct decision, provided that there is a success somewhere further
>down the road, among the future choices." I hope so, anyway, because I
>wrote it. 8-)
>
> --- Christopher Heckman
Yes, I like that explanation. It's similar to the way I think about
it, and it does capture the essence. Once the basic concept is
communicated, then it's reasonable to ask the question "Ok, now how
can we formalize the concept?". After all, the definers must have had
the essence in mind when they devised the definition, so it's unfair
to just hit someone with an abstract, formal definition without first
motivating it with the concept of what the definition is trying to
model.
quasi
Jesse F. Hughes
By what principle?
Your article mentions the "world of computation" nonsense, introduced
by you and supported (you say) by Han de Bruijn[1].
There are other self-proclaimed anti-Cantorians on this group and a
number of them are committed to arguments involving infinite natural
numbers as a way of refuting Cantor[2]. I'd wager I can find more
people that support this tactic than that support the view that
infinity is a useful abstraction but not bound by normal logic.
So by what principle have you determined that your view is signal and
this other view noise? What test have you applied? Is it all just a
matter of whether you agree with the claims?
Footnotes:
[1] Sorry if I get the name wrong. Maybe you mentioned someone else.
[2] Orlow is currently the loudest, but I can try to find other names
if necessary.
--
"Now I realize that he got away with all of that because sci.math is
not important, and the rest of the world doesn't pay attention.
Like, no one is worried about football players reading sci.math
postings!" -- James S. Harris on jock reading habits
Stephen Montgomery-Smith
Well, you got me. I guess that invalidates everything else I said.
:-)
Daryl McCullough
>david petry wrote:
>
>> It is plausible that in the future, mathematics will be split
>> into two disciplines - scientific mathematics (i.e. the science
>> of phenomena observable in the world of computation), and
>> philosophical mathematics, wherein Cantor's Theory is merely
>> one of many possible formal "theories" of the infinite.
>
>Call this one relegious mathematics or matheology. The Cantorians close
>their eyes in order to avoid obvious contradictions and to maintain
>their useless pet, simply insisting that logic is not valid in the
>"infinite".
You've got that completely backwards. The one thing that (classical)
mathematicians insist on is that logic works the same regardless
of whether the domain is naturals, reals, infinite sets, or whatever.
"Cantorians" would never claim that logic is not valid for infinite
sets.
--
Daryl McCullough
Ithaca, NY
george
david petry wrote:
> George wrote:
>
> >Mathematics is ALL science fiction.
> >It's about WHAT IF *this* were axiomatically true?
>
> That's not the way it is
> viewed by people who apply mathematics.
How would you know?
I apply mathematics just as much as you do and
I am NOT confused by this. Your over-generalizations
about this group of "people who apply mathematics" are
even more insulting to them than to yourself.
> That's not the way it is viewed by the anti-Cantorians.
That's ridiculous.
Even anti-Cantorians are not so stupid
as to think that axioms are irrelevant.
They just tend to get distracted from them,
NOT by math, but by physics.
> It's not the
> way it was viewed before formal set
> theory came on the scene.
That's true, but So What?
Do you really think there is some past
truth you can go back to, before we took
this wrong turn? Logic has a validity completely
independent of its use in math. Math does NOT
enjoy a similar independence from logic, NOT EVEN
given Godel's theorem. Prominent among the things
that suggest that it does are things that are inexpressible
with classical first-order languages&axioms, and prominent
among those is Finitude. But there could nothing TO express
there if INfinity weren't "already" there also.
G. Frege
>
> Even anti-Cantorians are not so stupid
> as to think that axioms are irrelevant.
>
You didn't talk to Mückenheim, did you?
F.
Keith Ramsay
Stephen Montgomery-Smith wrote:
|I also share with you a sense that mathematics
|has recently tended to be a little overboard on the abstract.
However,
|I do believe that the trend is reversing.
Could you be a little more specific about both of these?
I don't know what sort of thing you have in mind. I don't
even really know what time scale you're talking about.
It's not clear to me that such a trend has existed in the
past 30 years, say, nor a recent reversal.
My impression is that a lot of these apparent trends are
just cyclical, and also localized to particular fields.
But I would also say that it's difficult to get a good
sense of the large trends, since even very broadly
knowledgable people tend to be aware of current research
in only a minority of fields.
Keith Ramsay
Stephen Montgomery-Smith
You could be right. On one of my other posts in this thread, I talked
in more detail in terms of analysis.
Stephen
Keith Ramsay
david petry wrote:
|Keith Ramsay wrote:
[...]
|> #"... it has come to pass that we have encountered certain
|> #paradoxes, certain apparent contradictions that would have
|> #delighted Zeno the Eleatic and the school of Megara. And
|> #then each must seek the remedy. For my part, I think, and
|> #I am not the only one, that the important thing is never to
|> #introduce entities not completely definable in a finite
|> #number of words. Whatever be the cure adopted, we may
|> #promise ourselves the joy of the doctor called in to follow
|> #a beautiful pathologic case."
|>
|> So it seems that rather than describing set theory as a
|> disease, he is describing set theory as a patient to be
|> cured of a disease, i.e. paradoxes. His idea of a proper
|> cure wasn't the same as Cantor's, nor was it the same as
|> yours.
|
|First, thanks for finding that reference for us.
|
|However, I don't think you are interpreting it correctly. Look at
|his proposed cure:
|
| "For my part, I think, and I am not the only one,
|that the important thing is never to introduce entities not completely
|definable in a finite number of words."
|
|That is almost exactly the "anti-Cantorian" view. There don't exist
|more than a countable number of entities, given Poincare's cure.
Poincare is famous for being in favor of predictive definitions.
(Often this also means that one doesn't believe in the existence
of a set containing all real numbers, although I don't remember
whether he ever said he believed that. One can believe in a
hierarchy of different kinds of real numbers, each kind being
countable, the diagonal proof showing that there are reals of
a higher type that aren't of a lower type.)
There's a big difference between predicativism and believing
in the kind of essential relationship between mathematics and
computation that you do, however.
|He is claiming that it is Cantor's ideas about set theory that is the
|disease.
That's not in the quotation. The quote doesn't even
mention Cantor.
|I don't think anyone is arguing that there is something
|wrong with sets per se.
Hence, he probably wouldn't actually describe set theory,
as such, as a disease, would he? Just particular verions
of it. Which is what I took him to be writing.
|So I'm sticking with the quote.
I hope you mean you're going to quote what he actually
wrote? (And probably you should look it up to be sure.)
Please don't present this poor paraphrase as if it were
a direct quotation.
Keith Ramsay
Keith Ramsay
david petry wrote:
|In this article, "Cantor's Theory" refers to the pre-formal
|ideas about set theory introduced by Cantor in the latter
|part of the nineteenth century.
You might want to think a little about how to place this
relative to the other existing articles. The Wikipedia
article "naive set theory" claims to be about nearly the
same thing as what you're describing as "Cantor's Theory".
[...]
|in concrete mathematics, and wrecklessly apply it to
"recklessly".
[...]
|In science, truth
|must have observable implications, and such a "reality check"
|would reveal Cantor's Theory to be a pseudoscience; many of the
|formal theorems in Cantor's Theory have no observable implications.
I think an example would make it clearer to the reader
what you have in mind.
Keith Ramsay
Jesse F. Hughes
> The anti-Cantorians claim that while infinite sets and
> power sets of those infinite sets are undeniably useful
> abstractions,
[...]
> But, abstractions don't necessarily obey exactly that same
> laws of logic as directly observable objects.
I wonder how any abstractions can be "undeniably useful" if we are
incapable of reasoning about them with our standard deductive logic
(and Petry doesn't specify any fragment of logic that abstractions
obey).
--
Jesse F. Hughes
"You see 300 of something, anything, and you go `[Man], that's a lot of
stuff.'" -- Jim Bigler, quoted in the Pittsburgh Post-Gazette.
Chan-Ho Suh
david petry <david_lawr...@yahoo.com> wrote:
If you consider Keith's reference to a different quote of Poincare's
that says something quite different as supporting your alleged quote,
then I guess I can't stop you.
muec...@rz.fh-augsburg.de
malb...@yahoo.com wrote:
> > What is absurd about there not existing a bijection between
> > a set and its powerset?
>
> The view is that one can be constructed. karl m
But the view is not justified! The bijection as a meaningful tool to
measure set-sizes is no even laid down in the axioms. It is nothing but
a thoughtless extrapolation from the finite into the infinite.
Regards, WM
muec...@rz.fh-augsburg.de
Daryl McCullough wrote:
> muec...@rz.fh-augsburg.de says...
>
> >david petry wrote:
> >
> >> It is plausible that in the future, mathematics will be split
> >> into two disciplines - scientific mathematics (i.e. the science
> >> of phenomena observable in the world of computation), and
> >> philosophical mathematics, wherein Cantor's Theory is merely
> >> one of many possible formal "theories" of the infinite.
> >
> >Call this one relegious mathematics or matheology. The Cantorians close
> >their eyes in order to avoid obvious contradictions and to maintain
> >their useless pet, simply insisting that logic is not valid in the
> >"infinite".
>
> You've got that completely backwards. The one thing that (classical)
> mathematicians insist on is that logic works the same regardless
> of whether the domain is naturals, reals, infinite sets, or whatever.
Then they should agree that there cannot exist more irrationals than
rationals in the real continuum, because, in normal order <, there does
never exist a pair of irrational numbers without a rational number
between them. There is no logic available to circumvent this fact.
Regards, WM
muec...@rz.fh-augsburg.de
Chan-Ho Suh wrote:
> > > > "Set theory is a disease from which mathematics
> > > > will one day recover" (Poincare)
> >
> >
> > > Poincare never said that. You will not be able to give a valid
> > > reference for this.
> >
> > I can't, but Keith Ramsay already has. :-)
> >
>
> If you consider Keith's reference to a different quote of Poincare's
> that says something quite different as supporting your alleged quote,
> then I guess I can't stop you.
Poincare's opinion is quite clear:
There is no actual infinity. The Catorians have forgotten that and have
fallen into contradictons. [H. Poincaré, Les mathématiques et la
logique III, Rev. métaphys. morale 14, p. 316, (1906).]
Here is another opinion, somewhat newer:
Abraham Robinson (1964): (i) Infinite totalities do not exist in any
sense of the word (i.e., either really or ideally). More precisely, any
mention, or purported mention, of infinite totalities is, literally,
meaningless. (ii) Nevertheless, we should continue the business of
Mathematics 'as usual', i.e., we should act as if infinite totalities
really existed." (In: Formalism 64, auch abgedruckt in Robinson 1979,
p. 507.)
Regards, WM
Han de Bruijn
Closer, but not quite. "Constructivist" is already a "reserved word" as
well: Google it up. The main problem with this terminology is that some
constructivists - and most certainly all intuitionists - don't want to
be involved with physical reality. The laboratory of mathematics is one
of David Petry's main issues in defining the anti-Cantorian, though.
Han de Bruijn
G. Frege
<Han.de...@DTO.TUDelft.NL> wrote:
> > >
> > > "anti-Cantorian"
> > >
Read: mathematical crackpot
There are no other "anti-Cantorians". On the other hand there are
constructivists, intuitionists and finitists; but they are certainly
not the type of "anti-Cantorians" showing up in math newsgroups (i.e.
unscholared crackpots).
F.
Martin Shobe
Sure there is. The fact that Q is dense in R does not entail that
Card(R) <= Card(Q). There is no logic that can show that it does.
(Note: we are using the standard definition of Card() here, not your
attempted definition).
Martin
David C. Ullrich
What _I_ wonder about here is exactly what "introduce" means.
The set of real numbers _can_ be defined in a finite number
of words ("the set of all sets of rationals such that...").
I don't know what Poincare meant by the word, but given _my_
interpretation of the word "introduce" we can introduce the
set of all reals without having introduced each element of
that set.
>There's a big difference between predicativism and believing
>in the kind of essential relationship between mathematics and
>computation that you do, however.
>
>|He is claiming that it is Cantor's ideas about set theory that is the
>|disease.
>
>That's not in the quotation. The quote doesn't even
>mention Cantor.
>
>|I don't think anyone is arguing that there is something
>|wrong with sets per se.
>
>Hence, he probably wouldn't actually describe set theory,
>as such, as a disease, would he? Just particular verions
>of it. Which is what I took him to be writing.
>
>|So I'm sticking with the quote.
>
>I hope you mean you're going to quote what he actually
>wrote? (And probably you should look it up to be sure.)
>Please don't present this poor paraphrase as if it were
>a direct quotation.
>
>Keith Ramsay
************************
David C. Ullrich
David C. Ullrich
<david_lawr...@yahoo.com> wrote:
>
>>If mose of the debate on the internet is junk, then
>>it seems strange to use the internet as evidence
>>for the prevalence of the anti-Cantorian view.
>
>For many of the anti-Cantorians, it is most obvious that
>there's something absurd about Cantor's Theory, but they
>really haven't a clue about how to argue the point to the
>mathematicians.
Yup. Similarly it's clear to most people (or to most
people who've heard about the things relativity says)
that the predictions of relativity are absurd -
measuring rods don't contract just because they're
moving, etc. But they haven't got a clue how to
argue the point to the physicists. There's a reason
for that: relativity actually _is_ how things are.
The fact that something is obiously absurd really
doesn't prove it's not so.
************************
David C. Ullrich
Dik T. Winter
> On Fri, 22 Jul 2005 11:10:03 +0200, Han de Bruijn
> <Han.de...@DTO.TUDelft.NL> wrote:
> > > > "anti-Cantorian"
>
> Read: mathematical crackpot
>
> There are no other "anti-Cantorians".
I do not know. I would say that Kronecker was as anti-Cantorian as you
can get. On the other hand, he did not reject Cantorianism on the
grounds that it was invalid, inconsistent, or whatever, but on the
grounds that in his opinion mathematicians should only operate with
finite numbers and with a finite numbers of operations. So in his
opinion transcendental numbers did not exist. (Note the contrast with
Mueckenheim who is of the opinion that irrational numbers do not exist.
For Kronecker, algebraic numbers did exist, that is a field were his
major contributions are.) So when Lindemann presented his proof that
pi was transcendental, Kronecker complimented him for a beautiful proof,
but nevertheless worthless, as transcendental numbers did not exist.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
David Kastrup
> In article <k4f1e153e6lv9qjob...@4ax.com> G. Frege <nomail@invalid> writes:
> > On Fri, 22 Jul 2005 11:10:03 +0200, Han de Bruijn
> > <Han.de...@DTO.TUDelft.NL> wrote:
> > > > > "anti-Cantorian"
> >
> > Read: mathematical crackpot
> >
> > There are no other "anti-Cantorians".
>
> I do not know. I would say that Kronecker was as anti-Cantorian as you
> can get. On the other hand, he did not reject Cantorianism on the
> grounds that it was invalid, inconsistent, or whatever, but on the
> grounds that in his opinion mathematicians should only operate with
> finite numbers and with a finite numbers of operations. So in his
> opinion transcendental numbers did not exist.
"Did not exist" does not jibe with an axiomatic approach. "don't
apply to reality" could.
> (Note the contrast with Mueckenheim who is of the opinion that
> irrational numbers do not exist. For Kronecker, algebraic numbers
> did exist, that is a field were his major contributions are.) So
> when Lindemann presented his proof that pi was transcendental,
> Kronecker complimented him for a beautiful proof, but nevertheless
> worthless, as transcendental numbers did not exist.
Weird.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Robert Low
> The bijection as a meaningful tool to
> measure set-sizes is no even laid down in the axioms. It is nothing but
> a thoughtless extrapolation from the finite into the infinite.
You mis-spelled 'sensible'.
Of course it's an extrapolation. We're trying to develop something
beyond what our intuition readily handles. The only strategy
we have is to take something that works where we know what
the answer ought to be, and try to generalise it. This
generalisation also has many properties that we'd like to
see: the Schroeder-Bernstein theorem, for example.
Now, you can say 'I don't like the axiom of infinity',
and work in what's left. That's your choice. But including
the axiom of infinity and then using the existence of
a bijection between two sets to say that they have the
same number of elements works well: for those of us
who are happy with the axiom of infinity and wish to
be able to use infinite sets, there is no competitor
that I know of. At least, none that makes as much sense.
Han de Bruijn
> On 21 Jul 2005 11:40:06 -0700, "david petry"
> <david_lawr...@yahoo.com> wrote:
>
>>>If mose of the debate on the internet is junk, then
>>>it seems strange to use the internet as evidence
>>>for the prevalence of the anti-Cantorian view.
>>
>>For many of the anti-Cantorians, it is most obvious that
>>there's something absurd about Cantor's Theory, but they
>>really haven't a clue about how to argue the point to the
>>mathematicians.
>
> Yup. Similarly it's clear to most people (or to most
> people who've heard about the things relativity says)
> that the predictions of relativity are absurd -
> measuring rods don't contract just because they're
> moving, etc. But they haven't got a clue how to
> argue the point to the physicists. There's a reason
> for that: relativity actually _is_ how things are.
Any such comparison between mathematical and physical "reality" is
completely besides the point. One of our problems with mathematics
is that some of us (: me) find physics the leading discipline and
that "A little bit of Physics would be NO Idleness in Mathematics".
BTW, mathematics has not such "convincing arguments" for its truth
as relativity has with E = mc^2 and the atomic bomb.
Han de Bruijn
Han de Bruijn
> Now, you can say 'I don't like the axiom of infinity',
> and work in what's left. That's your choice. But including
> the axiom of infinity and then using the existence of
> a bijection between two sets to say that they have the
> same number of elements works well: for those of us
> who are happy with the axiom of infinity and wish to
> be able to use infinite sets, there is no competitor
> that I know of. At least, none that makes as much sense.
Oh yeah ? Let's see.
Theorem
-------
The number of even naturals is half the number of all naturals.
Proof
-----
Consider the finite sequence of natural numbers from 1 to N:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N
Then the total number of naturals is N and, depending on whether N is
odd or even, the number of even naturals in that sequence is (N-1)/2
or N/2 respectively. The quotient of the two counts is 1/2.(1-1/N) or
just 1/2, respectively.
Now let the sequence grow to infinity, that is: take the limit of the
above quotient for N -> oo . In both cases, the limit is 1/2.
This proves the theorem.
See? Nothing else is needed than the classical, pre-Cantorian concept
of a limit, which is accepted by all kind of mathematicians I know of.
Note: it is assumed here that the sequence of all naturals is the same
as the _set_ of all naturals, which seems crucial for the argument.
Han de Bruijn
Aatu Koskensilta
> Well, it might not be the way to *introduce* the concept to a beginner
> -- one needs to start with an informal account of the underlying
> intuitions. But that doesn't mean that it isn't, at the end of the day,
> the right way to understand the concept's essence.
I don't think the essence of "a non-deterministic machine" has anything
to do with 7-tuples.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Daryl McCullough
>Daryl McCullough wrote:
>> You've got that completely backwards. The one thing that (classical)
>> mathematicians insist on is that logic works the same regardless
>> of whether the domain is naturals, reals, infinite sets, or whatever.
>
>Then they should agree that there cannot exist more irrationals than
>rationals in the real continuum, because, in normal order <, there does
>never exist a pair of irrational numbers without a rational number
>between them. There is no logic available to circumvent this fact.
What *logical* claim are you saying that mathematicians are violating?
"Betweenness", "order", "rational", "irrational", are *mathematical*
concepts, not logical concepts. Yes, of course, the *mathematics*
of infinite sets is different from the mathematics of finite sets.
But the *logic* is the same.
The logical operators, to refresh your memory are:
and
or
implies
not
exists
forall
What *logical* statement do you think mathematicians are
saying is true for finite sets but not infinite sets?
ste...@nomail.com
> Robert Low wrote:
>> Now, you can say 'I don't like the axiom of infinity',
>> and work in what's left. That's your choice. But including
>> the axiom of infinity and then using the existence of
>> a bijection between two sets to say that they have the
>> same number of elements works well: for those of us
>> who are happy with the axiom of infinity and wish to
>> be able to use infinite sets, there is no competitor
>> that I know of. At least, none that makes as much sense.
> Oh yeah ? Let's see.
> Theorem
> -------
> The number of even naturals is half the number of all naturals.
Theorem
The number of perfects squares is 0 times the number of all naturals.
> Proof
> -----
> Consider the finite sequence of natural numbers from 1 to N:
> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N
Proof
Consider the finite sequence of natural numbers from 1 to N:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... N
> Then the total number of naturals is N and, depending on whether N is
> odd or even, the number of even naturals in that sequence is (N-1)/2
> or N/2 respectively. The quotient of the two counts is 1/2.(1-1/N) or
> just 1/2, respectively.
Then the the total number of naturals is N and the number
of perfect squares in that sequence is floor(sqrt(N)). The
quotient of the two counts is floor(sqrt(N))/N.
> Now let the sequence grow to infinity, that is: take the limit of the
> above quotient for N -> oo . In both cases, the limit is 1/2.
Now let the sequence grow to infinity, that is: take the limit of the
above quotient for N -> oo. The limit is 0.
> This proves the theorem.
This proves the theorem.
> See? Nothing else is needed than the classical, pre-Cantorian concept
> of a limit, which is accepted by all kind of mathematicians I know of.
See all the neat stuff you can prove? You can also prove
that the number of perfect cubes and the number of primes both
also equal 0 times the number of naturals, and so are
presumably equal to each other.
Of course this idea fails miserably when considering
non numerical sets.
Stephen
Dik T. Winter
> "Dik T. Winter" <Dik.W...@cwi.nl> writes:
> > I do not know. I would say that Kronecker was as anti-Cantorian as you
> > can get. On the other hand, he did not reject Cantorianism on the
> > grounds that it was invalid, inconsistent, or whatever, but on the
> > grounds that in his opinion mathematicians should only operate with
> > finite numbers and with a finite numbers of operations. So in his
> > opinion transcendental numbers did not exist.
>
> "Did not exist" does not jibe with an axiomatic approach. "don't
> apply to reality" could.
Apparently (I have read a bit in his works) he thought that being roots
of polynomials with integer coefficients was sufficient reason for a
number to exist, and thought that was constructive. On the other hand,
he clearly thought that the various constructions to get the reals from
the rationals were not constructive. But even in his works he is himself
not entirely consistent. See also the biography at
<http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html>.
G. Frege
wrote:
> >
> > There a r e no other "anti-Cantorians".
> >
> [...] I would say that Kronecker w a s as anti-Cantorian as
> you can get.
>
Agree. (Same may be true for the late Poincare.)
>
> On the other hand, he did not reject Cantorianism on the
> grounds that it was invalid, inconsistent, or whatever, but on the
> grounds that in his opinion mathematicians should only operate with
> finite numbers and with a finite numbers of operations.
>
Right.
Actually, he was a forerunner of constructivism, that's well known.
F.
Alan Morgan
Sure. However, it leaves you completely unable to compare the cardinality
of these two sets
{1, 2, 3, 4, .... } and { one, two, three, four, ..... }
or, for that matter
{1, 2, 3, 4, .... } and { 1, 2, watermelon, 4, ..... }
without adding more axioms.
Alan
--
Defendit numerus
Virgil
muec...@rz.fh-augsburg.de wrote:
> malb...@yahoo.com wrote:
>
> > > What is absurd about there not existing a bijection between
> > > a set and its powerset?
> >
> > The view is that one can be constructed. karl m
>
> But the view is not justified! The bijection as a meaningful tool to
> measure set-sizes is no even laid down in the axioms.
The C in ZFC specifically allows construction of injections/bijections
between sets.
So that WM is wrong! Again! Consistent, isn't he?
Virgil
muec...@rz.fh-augsburg.de wrote:
There is nothing in standard logic or any standard axiom systemthat
reqeuires assumption of anything like WM's "relative cardinality", and
unless it is assumed as an axiom or otherwise required, WM's
non-standard conclusions are unsupported by anything except WM's
non-standard assumptions.
Virgil
muec...@rz.fh-augsburg.de wrote:
>
> Here is another opinion, somewhat newer:
> Abraham Robinson (1964):
.... Nevertheless, we should continue the business of
> Mathematics 'as usual', i.e., we should act as if infinite totalities
> really existed." (In: Formalism 64, auch abgedruckt in Robinson 1979,
> p. 507.)
>
> Regards, WM
WM conveniently forgets, when he quotes Robinson, that Robinson say that
even if WM were right, mathematicians should act as if WM were wrong.
So that when we act like WM is wrong, we are only following advice
posted by WM himself.
What Robinson is saying is that even if we are wrong about "existence",
we are right to reject WM.
Dik T. Winter
> On Fri, 22 Jul 2005 13:34:43 GMT, "Dik T. Winter" <Dik.W...@cwi.nl>
> wrote:
> > > There a r e no other "anti-Cantorians".
> > >
> > [...] I would say that Kronecker w a s as anti-Cantorian as
> > you can get.
> >
> Agree. (Same may be true for the late Poincare.)
> > On the other hand, he did not reject Cantorianism on the
> > grounds that it was invalid, inconsistent, or whatever, but on the
> > grounds that in his opinion mathematicians should only operate with
> > finite numbers and with a finite numbers of operations.
> >
> Right.
>
> Actually, he was a forerunner of constructivism, that's well known.
Yes, so I doubt your statement that there are no other "anti-Cantorians"
than cranks. I would not put Kronecker in that category.
G. Frege
wrote:
>
> Yes, so I doubt your statement that there are no other "anti-Cantorians"
> than cranks. I would not put Kronecker in that category.
>
Kronecker i s d e a d. (Actually, he died long ago.)
And I wouldn't call modern intuitionists, konstructivists or finitist
"anti-Cantorians". They are "non-Cantorians", but not "anti-Cantorians".
That's a certain difference, imho.
F.
P.S.
I agree, Kronecker certainly w a s an "anti-Cantorian", as is well
known, but that's _history_!
Martin Shobe
You need quite a bit more, since you need some sort of arithmetic with
infinite numbers, which, I believe, wasn't developed until Cantor.
>Note: it is assumed here that the sequence of all naturals is the same
>as the _set_ of all naturals, which seems crucial for the argument.
No, you don't use that at all. The assumtion you make is the ratio of
evens to odds in N is the limit of your sequence.
Martin
G. Frege
wrote:
>
> ...but those real numbers are not, to me anyway, "sets" at all.
>
Given this is true, there still might exist a set of the (real) real
numbers, though not in ZFC (since t h e r e we only have sets, but no
urelements). No?
F.
G. Frege
<david_lawr...@yahoo.com> wrote:
>
> In this article, "Cantor's Theory" refers to the pre-formal
> ideas about set theory...
>
Who cares about that anyway? Mathematicians and logicians are
concerned with /axiomatic set theory/ these days. (Another thing
"anti-Cantorians", i.e. mathematical crackpots, won't realize.)
F.
Han.de...@dto.tudelft.nl
> See all the neat stuff you can prove? You can also prove
> that the number of perfect cubes and the number of primes both
> also equal 0 times the number of naturals, and so are
> presumably equal to each other.
Did I suggest somewhere that I would consider what I have proved
here as "neat stuff" ? Have you got the message anyway ?
Han de Bruijn
Han.de...@dto.tudelft.nl
> >See? Nothing else is needed than the classical, pre-Cantorian concept
> >of a limit, which is accepted by all kind of mathematicians I know of.
>
> Sure. However, it leaves you completely unable to compare the cardinality
> of these two sets
>
> {1, 2, 3, 4, .... } and { one, two, three, four, ..... }
>
> or, for that matter
>
> {1, 2, 3, 4, .... } and { 1, 2, watermelon, 4, ..... }
>
> without adding more axioms.
Sure. But that's quite another question.
Han de Bruijn