Cantorian pseudomathematics
david petry
It's pretty clear that there is at least a small handful of prominent
mathematicians, applied mathematicians, and logicians who agree with me
that Georg Cantor introduced an element of make-believe into
mathematics, and that this element should be expunged. Look at what
they say:
"[Cantor's paradise] is a paradise of fools, and besides feels more
like hell" (Doron Zeilberger)
"those of us who work in probability theory or any other area of
applied mathematics have a right to demand that this disease [Cantorian
set theory] be quarantined and kept out of our field" (From
"Probability Theory: The Logic of Science" by E. T. Jaynes)
"I am convinced that the platonism which underlies Cantorian set theory
is utterly unsatisfactory as a philosophy of our subject, despite the
apparent coherence of current set-theoretical conceptions and methods
... platonism is the medieval metaphysics of mathematics; surely we can
do better" (From "Infinity in Mathematics: Is Cantor Necessary?" by
Soloman Feferman)
As Feferman and others have suggested, Cantorian set theory has more in
common with Medieval mysticism (e.g. Kabbalah) than with the modern
scientific world view; it is simply not essential to scientifically
applicable mathematics, and if mathematics is to be a serious
discipline having the purpose of helping us understand the world in
which we live, rather than merely being a game, then Cantorian set
theory must not be regarded as lying at the foundation of mathematics.
In the modern scientific world view, the "real" world is the world we
can observe. It's the world with which we can interact, and in which we
can perform experiments. Assertions about the real world are meaningful
if and only if they make testable (and falsifiable) predictions about
the results of experiments.
Mathematics is not only a tool we use to create models of the real
world, but is itself the study of a part of the real world - the world
of computation. As a conceptual aid, we can think of the computer as
being analogous to both a microscope and a test tube: it helps us peer
deeply into the world of computation, and it gives us a way to perform
experiments within that world of computation - it gives us a way to
interact with the world of computation - and hence, this world of
computation has the essential characteristics of being "real". (We use
the computer as a conceptual aid to drive home the fact that the world
of computation is not merely something that lives in our imaginations)
Then "real" mathematics is the science which studies the phenomena
observed in that world of computation, and "real" mathematical
statements are meaningful if and only if they make testable predictions
about the results of computational experiments. All of the mathematics
which is scientifically applicable fits within this paradigm.
Cantorian set theory implies the "existence" of a world far beyond the
world of computation. Even if we imagine ourselves having access to
computers that could take a countably infinite amount of data as the
input, then perform an infinite amount of computation on that data, and
then give an output, that would only push us up one very small step in
the cumulative hierarchy of Cantorian transfinite sets. The Cantorian
world of infinite sets is invisible, and mystical.
Cantorian set theory is a virus. It's a disease. It not only consumes
resources without giving anything back to its hosts, but it turns
mathematicians into inhuman monsters. Let me elaborate on this.
When I was a mathematics student, I had a dream. I thought it would be
really cool to teach computers how to really and truly understand
mathematics. It looked to me like that would be a very promising route
to artificial intelligence; mathematics is, after all, the most
powerful tool we have to understand the world we live in, so if a
computer could understand mathematics, it would have the tools it needs
to understand the real world, and it would be intelligent. I believed
that my pursuit of this idea could lead to a fascinating career.
So the first question I tried to answer was, what is the "meaning" of a
mathematical statement, and how could I explain it to an artificial
intelligence? And the answer is fairly straightforward: an artificial
intelligence (A.I.) lives in the world of computation, and hence, if a
mathematical statement can be interpreted as making predictions about
the results of an experiment within the world of computation - an
experiment which the A.I. itself could perform - then all the A.I.
needs to understand the meaning of the statement is to know how to
convert the statement into a computational experiment which could test
(potentially falsify) the statement. Then mathematics itself would be
seen to be essentially a very high level programming language, and an
A.I. would be built up as a collection of mathematical statements.
So anyway, I wanted to pursue this idea from a purely theoretical point
of view, so I decided I could do it from within the mathematics
department. My plan was to build a foundation for mathematics
incorporating as one of the cornerstones, the idea that mathematical
statements must have an interpretation as making predictions about the
results of computational experiments. I argued that it not only leads
to a foundation of mathematics which can be used as a foundation for
artificial intelligence, but it also gives a foundation for mathematics
which would be of practical value to those who apply mathematics. And
to be sure, I did my homework on this; although at first glance it may
look to some people like I'm just doing what the constructivists have
already done, it really isn't; there's something very new and original
about it. For one thing, I argue that probabilistic reasoning is the
key to the development of the notion of a continuum, which makes my
ideas distinctly different from any form of constructivism that I'm
aware of.
Things really didn't go very well for me. The Cantorians told me
essentially that I'm not even doing real mathematics. They claimed I'm
doing philosophy. They accused me of attempting to deprive
mathematicians of their intellectual freedom and trying to imposed my
religion on others. They claimed that I'm trying to turn back the
clock, and that I'm clinging to stale philosophical ideas. They claimed
that by doing what I was trying to do, I would be destroying some
beautiful thing that the mathematicians have created. They told me I
had no right to pursue my own ideas until I first prove beyond doubt
that I fully understand Cantorian mathematics. They hinted that my
ideas were crackpot ideas. They made it clear that I really wasn't very
welcome in the mathematics community. And I ended up convinced that I
wouldn't be able to publish in the mathematical literature, for
instance. I ended up discouraged, and I quit.
So that's the situation. I enthusiastically argue that the study of the
foundations of mathematics has tremendous potential practical value,
and the "mathematicians" attack me; they crush my dreams and tell me
I'm not even doing mathematics. It may seem bizarre, but it's really
pretty clear what's going on: the Cantorian pseudomathematicians are
defending a religion, and they really can't see what monsters they have
become.
What the Cantorians are doing is nothing less than a crime against
humanity. They are imposing their religion on the world, and they are
discriminating against and excluding from their community, those who
question the axioms of that religion; I'm pretty sure I'm not the only
one who has had his dreams and ambitions crushed by the Cantorians.
What they are doing is evil. What they are doing has social and
political implications, as well as theoretical and technological
implications. And sadly, it's just the tip of the iceberg of a very
large problem facing academia and the world at large. Do I not have a
moral obligation to speak out about his evil?
mensa...@aol.com
That's the difference between you and Cantor.
Cantor had his problems, but he wasn't a quitter.
>
> So that's the situation. I enthusiastically argue that the study of the
> foundations of mathematics has tremendous potential practical value,
> and the "mathematicians" attack me; they crush my dreams and tell me
> I'm not even doing mathematics. It may seem bizarre, but it's really
> pretty clear what's going on: the Cantorian pseudomathematicians are
> defending a religion, and they really can't see what monsters they have
> become.
So why don't you just go ahead and build your AI machine and
show them a thing or two. The world will beat a path to your door.
Oh wait, you're a quitter, aren't you?
> What the Cantorians are doing is nothing less than a crime against
> humanity. They are imposing their religion on the world, and they are
> discriminating against and excluding from their community, those who
> question the axioms of that religion; I'm pretty sure I'm not the only
> one who has had his dreams and ambitions crushed by the Cantorians.
> What they are doing is evil. What they are doing has social and
> political implications, as well as theoretical and technological
> implications. And sadly, it's just the tip of the iceberg of a very
> large problem facing academia and the world at large. Do I not have a
> moral obligation to speak out about his evil?
Not if you're a quitter. Prove yourself right first and from that
will come both the right and the obligation to speak out.
Robert J. Kolker
> become.
> What the Cantorians are doing is nothing less than a crime against
> humanity.
Really? Sir, you are barking mad. If you don't like purely constructive
math then don't do it.
Bishop Berkely issued a similar diatribe against Newton and his fluxions
and infinitesimal quantities. Totally ridiculous. True, but it produced
differential equations which are at the gut of the mathematical
methodology of physics.
You want you math computational and finitary. Fine. But be prepared to
give up some good stuff.
Bob Kolker
Chip Eastham
Hi, David:
It's a pretty nice essay as these things go. Good spelling,
punctuation, and a fairly consistent development of a theme.
As far as labelling some group of people and "speak[ing] out
about [t]his evil", I can't advise you whether you have a moral
obligation or not. That sounds like a personal decision. We
often make the greater contribution by working on improving
ourselves, rather than directly changing those around us.
To me the best part of your note by far was the paragraph on
teaching math to computers. It's not _doing_ math, but I can
respect your having a dream and pursuing it. Teaching (and
programming) are honorable professions.
If you're any good at programming, you might google for Douglas
Lenat and the Cyc project.
I suspect you can find a more constructive use of your talents
than stereotyping folks who don't share your vision.
best wishes, chip
John Savard
<david_lawr...@yahoo.com> wrote, in part:
>As Feferman and others have suggested, Cantorian set theory has more in
>common with Medieval mysticism (e.g. Kabbalah)
Jewish mathematics!
Ah, yes, there's the problem!
Let's stick to good old Aryan mathematics, and we'll all get along just
fine!
Some people may be uncomfortable with the transfinite, and it isn't
needed for the pedestrian parts of mathematics, but you have just taken
this particular debate to a new low - even if unintentionally.
John Savard
http://www.quadibloc.com/index.html
_________________________________________
Usenet Zone Free Binaries Usenet Server
More than 140,000 groups
Unlimited download
http://www.usenetzone.com to open account
fishfry
"Robert J. Kolker" <now...@nowhere.com> wrote:
> david petry wrote:
>
> > become.
> > What the Cantorians are doing is nothing less than a crime against
> > humanity.
>
> Really? Sir, you are barking mad. If you don't like purely constructive
> math then don't do it.
>
> Bishop Berkely issued a similar diatribe against Newton and his fluxions
> and infinitesimal quantities. Totally ridiculous. True, but it produced
> differential equations which are at the gut of the mathematical
> methodology of physics.
>
I would not want the reputation of George Berkeley (rhymes with Charles
Barkley) to suffer by comparison with a Usenet crank.
Berkeley was a contemporary of Newton. He wrote incisive and accurate
criticism of the careless and logically unsupportable use of
infinitesimals in the work of Newton.
Berkeley's criticisms were logically correct. He was not a crank; and he
did not go around calling anyone's work a "crime against humanity." He
simply pointed out the logical deficiencies in Newton's concept of
differentials: technical difficulties that would not be fully resolved
(by the work of Weirstrass, Cauchy, and Cantor) for another 150 years
after the publication of Newton's Principia.
The World Wide Wade
> david petry wrote:
>
> > become.
> > What the Cantorians are doing is nothing less than a crime against
> > humanity.
>
> Really? Sir, you are barking mad. If you don't like purely constructive
> math then don't do it.
>
> Bishop Berkely issued a similar diatribe against Newton and his fluxions
> and infinitesimal quantities. Totally ridiculous.
You're quite misinformed. Berkeley's critique of the calculus was
well founded and led to the attempt on the part of many eminent
mathematicians of the day in bringing the rigor of Euclidean
geometry to the new mathematics of Newton and Leibnitz.
Daryl McCullough
>So that's the situation. I enthusiastically argue that the study of the
>foundations of mathematics has tremendous potential practical value,
>and the "mathematicians" attack me; they crush my dreams and tell me
>I'm not even doing mathematics. It may seem bizarre, but it's really
>pretty clear what's going on: the Cantorian pseudomathematicians are
>defending a religion, and they really can't see what monsters they have
>become.
This is an unbelievable case of projection. You are the religious
fanatic who accuses others of not doing "real" mathematics. No
real mathematician has ever said that finitistic mathematics isn't
"real" mathematics.
>What the Cantorians are doing is nothing less than a crime against
>humanity. They are imposing their religion on the world, and they are
>discriminating against and excluding from their community, those who
>question the axioms of that religion;
David, a general pattern in your statements is that you attribute
to others what is true about yourself. You are the one who has
made a religion out of your particular brand of mathematics. You
are the one who wants to impose his own particular type of religion
on the world.
Nobody is attacking you for your finitistic mathematics. People
are attacking you for saying insane things like "What the Cantorians
are doing is nothing less than a crime against humanity." That
is *lunacy*. That judgement does not have anything to do with
your mathematics. If a "Cantorian" described non-Cantorian mathematics
as a "crime against humanity" *that* would be lunacy, as well.
I think you are not well. And it has nothing to do with
mathematics.
--
Daryl McCullough
Ithaca, NY
Robert J. Kolker
> Barkley) to suffer by comparison with a Usenet crank.
>
> Berkeley was a contemporary of Newton. He wrote incisive and accurate
> criticism of the careless and logically unsupportable use of
> infinitesimals in the work of Newton.
Careless? Yes. Insupportable? At the time. Did it work? You bet it did!
Bob Kolker
Robert J. Kolker
>
> You're quite misinformed. Berkeley's critique of the calculus was
> well founded and led to the attempt on the part of many eminent
> mathematicians of the day in bringing the rigor of Euclidean
> geometry to the new mathematics of Newton and Leibnitz.
Yes. And about the middle of the 19-th century the limit concept was
sufficiently refined. But for 150 years the logically incorrect,
insupportable mathematics of Newton and Leibniz carried physics to
triumph. Eventually the infinitesimal was made respectable at last circa
1960.
Bob Kolker
Tony Orlow
>
>
> It's pretty clear that there is at least a small handful of prominent
> mathematicians, applied mathematicians, and logicians who agree with me
> that Georg Cantor introduced an element of make-believe into
> mathematics, and that this element should be expunged. Look at what
> they say:
>
> "[Cantor's paradise] is a paradise of fools, and besides feels more
> like hell" (Doron Zeilberger)
>
> "those of us who work in probability theory or any other area of
> applied mathematics have a right to demand that this disease [Cantorian
> set theory] be quarantined and kept out of our field" (From
> "Probability Theory: The Logic of Science" by E. T. Jaynes)
>
> "I am convinced that the platonism which underlies Cantorian set theory
> is utterly unsatisfactory as a philosophy of our subject, despite the
> apparent coherence of current set-theoretical conceptions and methods
> ... platonism is the medieval metaphysics of mathematics; surely we can
> do better" (From "Infinity in Mathematics: Is Cantor Necessary?" by
> Soloman Feferman)
Iteresting quotes. Of the three, who made more progress? My guess is Feferman,
who is coincidentally the most polite in his criticism. While I share your
distaste for transfiniute set theory and its methods and conclusions, I am not
sure it helps to go around calling people's religion a "disease", even if it
may seem so to you. I usually find that convincing someone of the opposite of
what they already think cannot be achieved with bold statements, but only with
pointed questions.
To some extent, this is true. Computers are math machines. However, modern
computers are all based on a similar underlying computational structure,
strings of binary digits. Some areas of mathemtics seem to be less accessible
than others in this context. For instance, I am trying to figure out the most
efficient way to calculate any arbitrary binary fractional power of 2, in order
to implement my H-riffic numbers, and there doesn't seem to be any good way, so
far as I can tell.
Uh, I have to go pick up a sick kid at school. More later. In general, I agree,
but I don't think it pays to get too vitriolic about it. By the way, I also
went into computer science rather than math, largely because the "foundations"
as they currently stand are unacceptable. So, you're not alone. Later.
Tony
--
Smiles,
Tony
fishfry
"Robert J. Kolker" <now...@nowhere.com> wrote:
If I'm understanding correctly, it would be wrong to criticize the logic
in any theory as long as the details will be filled in 150 years later.
I'm sure that will save peer reviewers a lot of trouble, since nothing
could be criticized as long as it apparently gave approximately correct
results.
The World Wide Wade
<fishfry-3E9ED7...@comcast.dca.giganews.com>,
fishfry <fis...@your-mailbox.com> wrote:
> Berkeley was a contemporary of Newton. He wrote incisive and accurate
> criticism of the careless and logically unsupportable use of
> infinitesimals in the work of Newton.
I think you want "fluxions" in place of "infinitesimals".
The World Wide Wade
You wrote that "Berkely issued a similar diatribe against Newton
and his fluxions and infinitesimal quantities. Totally
ridiculous." It was not a diatribe, and it was far from
ridiculous. It was trenchant criticism. Berkeley was of course
well aware of the power of the calculus in physics.
Torkel Franzen
> You wrote that "Berkely issued a similar diatribe against Newton
> and his fluxions and infinitesimal quantities. Totally
> ridiculous." It was not a diatribe, and it was far from
> ridiculous. It was trenchant criticism. Berkeley was of course
> well aware of the power of the calculus in physics.
Berkeley had a very good head on his shoulders, and _The Analyst_
is a brilliant piece of work.
Robert J. Kolker
>
>
>
> If I'm understanding correctly, it would be wrong to criticize the logic
> in any theory as long as the details will be filled in 150 years later.
No. The point is that a theory that gets so much right, cannot be easily
dismissed even if its foundations are a bit rickety.
Bob Kolker
Robert J. Kolker
>
>
> and his fluxions and infinitesimal quantities. Totally
> ridiculous." It was not a diatribe, and it was far from
> ridiculous. It was trenchant criticism. Berkeley was of course
> well aware of the power of the calculus in physics.
"Of course"? Where does the Bishop showed that he grasped Newton's
system of physics? In what writings? Can you give a citation?
Bob Kolker
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> david petry said:
> >
> >
> > Mathematics is not only a tool we use to create models of the real
> > world, but is itself the study of a part of the real world - the
> > world of computation. As a conceptual aid, we can think of the
> > computer as being analogous to both a microscope and a test tube:
> > it helps us peer deeply into the world of computation, and it gives
> > us a way to perform experiments within that world of computation -
> > it gives us a way to interact with the world of computation - and
> > hence, this world of computation has the essential characteristics
> > of being "real". (We use the computer as a conceptual aid to drive
> > home the fact that the world of computation is not merely something
> > that lives in our imaginations) Then "real" mathematics is the
> > science which studies the phenomena observed in that world of
> > computation, and "real" mathematical statements are meaningful if
> > and only if they make testable predictions about the results of
> > computational experiments. All of the mathematics which is
> > scientifically applicable fits within this paradigm.
But one can find suddenly that the mathematics despised by those for
whom application is the only justification has suddenly become
applicable.
A fairly recent instance of this is the application of number theory to
the security of electronic transmission of information.
>
> To some extent, this is true. Computers are math machines. However,
> modern computers are all based on a similar underlying computational
> structure, strings of binary digits.
Finite strings of binary digits. and in all machines to date, the
strings have a fixed word size, usually of some fixed power of 2.
> Some areas of mathemtics seem to
> be less accessible than others in this context. For instance, I am
> trying to figure out the most efficient way to calculate any
> arbitrary binary fractional power of 2, in order to implement my
> H-riffic numbers, and there doesn't seem to be any good way, so far
> as I can tell.
Attempts to deal with the the unending with a finite machines is to
likely to produce much of any use.
Robert J. Kolker
>
>
> Attempts to deal with the the unending with a finite machines is to
> likely to produce much of any use.
I assume you meant unlikely.
Bob Kolker
David R Tribble
>> [...] Computers are math machines. However,
>> modern computers are all based on a similar underlying computational
>> structure, strings of binary digits.
>
Virgil wrote:
> Finite strings of binary digits. and in all machines to date, the
> strings have a fixed word size, usually of some fixed power of 2.
> Attempts to deal with the the unending with a finite machines is to
> [un]likely to produce much of any use.
Indeed. IEEE floating-point supports +/-infinity values, but these
are represented as special bit patterns, and are only 32 or 64 bits
wide.
Even the "unlimited precision" BIGNUMs supported by LISP, Java,
and other languages are not actally unlimited in any real sense, but
are constrained by the number of bits available in a physical machine.
Large numbers such as googolplex cannot be represented exactly on
any matter-based machine, and such numbers are extremely small
(infinitely so) compared to infinite values.
The World Wide Wade
"Robert J. Kolker" <now...@nowhere.com> wrote:
You are obviously not very familiar with Berkeley (which hasn't
prevented you from declaring his critiques "diatribes" or
"ridiculous"). You could get acquainted with some of his thinking
by looking at _The Analyst_, which is here:
http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analy
st.html#Sect36
Robert J. Kolker
>
>
> You are obviously not very familiar with Berkeley (which hasn't
> prevented you from declaring his critiques "diatribes" or
> "ridiculous"). You could get acquainted with some of his thinking
> by looking at _The Analyst_, which is here:
I sit corrected. Apparently Berkley was right on top of things.
Which leads to
Qu 68. Why does mathematics so ill founded give the right answers?
Apparently one does not need rigorous foundations to get correct
results. That should give one pause. Rigour is overrated and empirical
correctness is underrated.
Bob Kolker
michaeld
Calculus, even in its non-rigorous state, is of very clear demonstrable
value and that would be true even if there were no known way of making
it rigorous.
However putting calculus on a rigorous footing was also of very clear
value. Not just because the results are "more certain", but the
rigorous version opens up many new avenues for exploration and allows
many applications that simply don't exist with the original
non-rigorous version.
Michael
Dik T. Winter
...
> Qu 68. Why does mathematics so ill founded give the right answers?
>
> Apparently one does not need rigorous foundations to get correct
> results. That should give one pause. Rigour is overrated and empirical
> correctness is underrated.
I do not think so. When it appears to be the case that some methods do
give good results, when the method has no foundation, it is the task of
the mathematician to try to find out *why* the results are so good, and
if it is not a fluke, to try to find the foundation behind it all.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Virgil
"Robert J. Kolker" <now...@nowhere.com> wrote:
Actually I meant 'not' likely, but when I missed the 'n' in 'not', my
overactive spell checker auto corrected 'ot' to 'to'.
imagin...@despammed.com
> In article <MPG.1e2ee410e...@newsstand.cit.cornell.edu>,
> Tony Orlow <ae...@cornell.edu> wrote:
> > To some extent, this is true. Computers are math machines. However,
> > modern computers are all based on a similar underlying computational
> > structure, strings of binary digits.
>
> Finite strings of binary digits. and in all machines to date, the
> strings have a fixed word size, usually of some fixed power of 2.
Not "usually", I think. I've used computers with word lengths of 48,
18, 60 bits, and (8-bit) byte base machines. Only one of those is a
power of 2. 36 was really the best number, because it's easy to
represent a draughts (checkers) board in one word.
I don't see any obvious advantage in the word size being a power of 2 -
it just needs to be a multiple of the character size in the native
character set.
Brian Chandler
http://imaginatorium.org
Robert J. Kolker
> I do not think so. When it appears to be the case that some methods do
> give good results, when the method has no foundation, it is the task of
> the mathematician to try to find out *why* the results are so good, and
> if it is not a fluke, to try to find the foundation behind it all.
Admirably correct. It is the mathematicians burden. Meanwhile physicists
will go on doing good physics. Rigor is highly overrated.
One good result after another is sufficient proof that the heuristic is
not a fluke.
Bob Kolker
Robert J. Kolker
>
> Actually I meant 'not' likely, but when I missed the 'n' in 'not', my
> overactive spell checker auto corrected 'ot' to 'to'.
Transposition and omission will be the death of all of us.
Bob Kolker
Robert J. Kolker
>
> Not "usually", I think. I've used computers with word lengths of 48,
> 18, 60 bits, and (8-bit) byte base machines. Only one of those is a
> power of 2. 36 was really the best number, because it's easy to
> represent a draughts (checkers) board in one word.
You have given away your age. Oh for the days of the IBM 7000 series and
the PDP-10.
Bob Kolker
Han de Bruijn
>
> But one can find suddenly that the mathematics despised by those for
> whom application is the only justification has suddenly become
> applicable.
Any mathematics that is unapplicable will remain unapplicable forever.
Any mathematics that is despised but applicable might become _applied_.
People should not confuse "applicable" and "applied".
Han de Bruijn
Han de Bruijn
>
> Apparently one does not need rigorous foundations to get correct
> results. That should give one pause. Rigour is overrated and empirical
> correctness is underrated.
Hmm ... Is it the same Bob Kolker who is talking here?
Han de Bruijn
Han de Bruijn
> In article <42lm0fF...@individual.net> "Robert J. Kolker" <now...@nowhere.com> writes:
> ...
> > Qu 68. Why does mathematics so ill founded give the right answers?
> >
> > Apparently one does not need rigorous foundations to get correct
> > results. That should give one pause. Rigour is overrated and empirical
> > correctness is underrated.
>
> I do not think so. When it appears to be the case that some methods do
> give good results, when the method has no foundation, it is the task of
> the mathematician to try to find out *why* the results are so good, and
> if it is not a fluke, to try to find the foundation behind it all.
If this is the mathematicians task, why don't they _do_ it then?
(Sorry, I have too many examples where they just didn't keep up)
Han de Bruijn
Han de Bruijn
> "those of us who work in probability theory or any other area of
> applied mathematics have a right to demand that this disease [Cantorian
> set theory] be quarantined and kept out of our field" (From
> "Probability Theory: The Logic of Science" by E. T. Jaynes)
Jaynes work, that is: extremely interesting. Starting at:
Han de Bruijn
Tony Orlow
positive and negative powers of 2, which is what I am talking about
implementing, for the sake of exploring the behavior of thes enumbers. I
believe several people were asking about arithmetic on such numbers?. Second,
the machine does not have to be infinite to support a system which can be
arbitrarily large in theory. A T-riffic application might be useful where we
need precision in fractions that canot be handled by the floating-point
registers. Nobody here claims to be able to store an actually infinite string
of bits in a finite machine, but processes can be implemented and tested to
high degrees of accuracy, given the right algorithms. So, you can stop
pretending I am trying to put infinity in a finite box.
--
Smiles,
Tony
Tony Orlow
In mathemtics, I think this is true. Intuition is also entirely discounted. You
ever seen any empirical evidence for the vase-ball solution, Bob?
>
> Bob Kolker
>
>
--
Smiles,
Tony
Han.de...@dto.tudelft.nl
>
> "those of us who work in probability theory or any other area of
> applied mathematics have a right to demand that this disease [Cantorian
> set theory] be quarantined and kept out of our field" (From
> "Probability Theory: The Logic of Science" by E. T. Jaynes)
http://bayes.wustl.edu/etj/prob/book.pdf
Quoted from this last and unfinished symphony by E.T. Jaynes:
In our view, an infinite set cannot be said to possess any "existence"
and mathematical properties at all - at least, in probability theory -
until we have specified the limiting process that is to generate it
from
a finite set. In other words, we sail under the banner of Gauss,
Kronecker, and Poincare rather than Cantor, Hilbert, and Bourbaki.
We hope that readers who are shocked by this will study the indictment
of Bourbakism by the mathematician Morris Kline (1980), and then bear
with us long enough to see the advantages of our approach.
Very much the same approach has been taken by myself and
subsequently has been refuted by the mathematical community.
See e.g. this message and subsequent ones in the thread:
http://groups.google.nl/group/sci.math/msg/fe0e86eaa0139db2?hl=en&
The basic statement is that the probability of a natural being
divisible by another natural a is just P(natural | a) = 1/a .
Mainstream mathematics says that such a probability does
not exist / cannot be defined. They use "densities" instead.
Han de Bruijn
Jesse F. Hughes
> The basic statement is that the probability of a natural being
> divisible by another natural a is just P(natural | a) = 1/a .
> Mainstream mathematics says that such a probability does
> not exist / cannot be defined. They use "densities" instead.
Your syntax seems odd to me. You write:
P(natural | a)
which may mean:
The probability of a natural, given a (????)
The probability that natural divides a (????)
Something else (????)
Did you mean the probability that a natural n is divisible by a? That
*might* be written as P( a | n ), although the pipe symbol is used for
other meaning in probability theory. And in any case, how could this
make any sense at all? How could this probability depend on a and not
n?
Boy, I can't figure what you mean.
--
"No feeling sympathy for mathematicians who start marching with signs
like 'Will work for food' in the future... I will not show mercy
going forward. I was trained as a soldier in the United States Army
after all... We play to win." --James Harris, feel his wrath!
Han.de...@dto.tudelft.nl
>
> to be sure, I did my homework on this; although at first glance it may
> look to some people like I'm just doing what the constructivists have
> already done, it really isn't; there's something very new and original
> about it. For one thing, I argue that probabilistic reasoning is the
> key to the development of the notion of a continuum, which makes my
> ideas distinctly different from any form of constructivism that I'm
> aware of.
I'm not so sure. L.E.J. Brouwer devised his "choice sequences"
for the purpose of constructing the intuitionistic continuum.
See i.e.
http://www.univ-nancy2.fr/poincare/colloques/symp02/abstracts/niekus.pdf
Han de Bruijn
Han.de...@dto.tudelft.nl
> ideas were crackpot ideas. They made it clear that I really wasn't very
> welcome in the mathematics community. And I ended up convinced that I
> wouldn't be able to publish in the mathematical literature, for
> instance. I ended up discouraged, and I quit.
Come on, David ! What's the beef in being "able to publish in the
mathematical literature", if you have free access to the Internet.
As a matter of fact, you didn't quit at all. I'm glad you didn't !
And nothing prevents you from setting up a web site as well.
If you have trouble with finding resources, contact us.
Han de Bruijn
Han.de...@dto.tudelft.nl
> What the Cantorians are doing is nothing less than a crime against
> humanity. They are imposing their religion on the world, and they are
> discriminating against and excluding from their community, those who
> question the axioms of that religion; I'm pretty sure I'm not the only
> one who has had his dreams and ambitions crushed by the Cantorians.
As I have experienced, crushing someone's dreams and ambitions
is not typical for Cantorians, but for human beings in general. There
is nothing "anti-Cantorian" in the following work of mine:
http://hdebruijn.soo.dto.tudelft.nl/jaar2004/nlrlsfem.pdf
But if you read it, you can understand very well why it is not accepted
by the (numerical) mathematics community. It undermines tradition,
authority, in short: establishment. That's all. And it's nothing new.
Han de Bruijn
Dik T. Winter
> Dik T. Winter wrote:
> > In article <42lm0fF...@individual.net> "Robert J. Kolker" <now...@nowhere.com> writes:
...
> > give good results, when the method has no foundation, it is the task of
> > the mathematician to try to find out *why* the results are so good, and
> > if it is not a fluke, to try to find the foundation behind it all.
>
> If this is the mathematicians task, why don't they _do_ it then?
> (Sorry, I have too many examples where they just didn't keep up)
They do, hence Cantor, which gives a foundation to Newton's work. But
you do not like it.
BTW, it is already some months ago that you would look at the way I
did numerical differentation. Have you already done so?
Han de Bruijn
> In article <560a7$43c63cdf$82a1e2b0$12...@news2.tudelft.nl> Han de Bruijn
<Han.de...@DTO.TUDelft.NL> writes:
>
> > (Sorry, I have too many examples where they just didn't keep up)
>
> They do, hence Cantor, which gives a foundation to Newton's work. But
> you do not like it.
WHAT ??!! Did Cantor give a foundation to Newton's work ??!!
Unbelievable ! I didn't expect such a silly remark from you, Dik.
> BTW, it is already some months ago that you would look at the way I
> did numerical differentation. Have you already done so?
Yes. I've seen it. And decided to let it rest for a while.
Han de Bruijn
Han de Bruijn
> Han.de...@DTO.TUDelft.NL writes:
>
>>The basic statement is that the probability of a natural being
>>divisible by another natural a is just P(natural | a) = 1/a .
>>Mainstream mathematics says that such a probability does
>>not exist / cannot be defined. They use "densities" instead.
>
> Your syntax seems odd to me. You write:
>
> P(natural | a)
>
> which may mean:
>
> The probability of a natural, given a (????)
> The probability that natural divides a (????)
> Something else (????)
The probability of finding the right meaning here is 1/3.
But, in the context given, the conditional probability is 1 :-)
> Did you mean the probability that a natural n is divisible by a? That
> *might* be written as P( a | n ), although the pipe symbol is used for
> other meaning in probability theory.
Yeah. Conditional probability. That is not meant here. And it's quite
clear that you have already understood.
> And in any case, how could this
> make any sense at all? How could this probability depend on a and not
> n?
Did I mention "n" somewhere in _this_ article?
> Boy, I can't figure what you mean.
Yes you can. Just think a little bit harder.
Han de Bruijn
Jesse F. Hughes
You wrote "natural | a", which would mean "natural divides a", but I
don't know what that means. The probability that a fixed natural
divides a? The probability that some natural divides a? The
probability that a "random" natural divides a? If you mean the last,
what does that *mean*?
Finally, why did you write "natural | a" but speak of "the probability
of a natural being divisible by another natural a"? That would be
written "a | natural" -- but again, fixing this wouldn't make your
meaning clear to me.
>
>> Boy, I can't figure what you mean.
>
> Yes you can. Just think a little bit harder.
No I can't. Why make things deliberately obtuse?
--
Jesse F. Hughes
"And hey, if you're moping and miserable because mathematics tests you,
then maybe, if you think you're a mathematician, you might want to try
a different field." -- Another James S. Harris self-diagnosis.
Shmuel (Seymour J.) Metz
01/12/2006
at 11:50 AM, Han.de...@DTO.TUDelft.NL said:
>Quoted from this last and unfinished symphony by E.T. Jaynes:
>In other words, we sail under the banner of Gauss,
>Kronecker, and Poincare
An interesting quote, given that they[1] dealt with the Real line,
which is infinite.
In <1137101977.2...@f14g2000cwb.googlegroups.com>, on
01/12/2006
at 01:39 PM, Han.de...@DTO.TUDelft.NL said:
>As I have experienced, crushing someone's dreams and ambitions is not
>typical for Cantorians,
In fact, the obvious example of someone doing that is Kronecker, who
persecuted Kantor.
>http://hdebruijn.soo.dto.tudelft.nl/jaar2004/nlrlsfem.pdf
>But if you read it, you can understand very well why it is not
>accepted by the (numerical) mathematics community.
Indeed, but our understanding of why might be very different from what
you wanted.
>It undermines tradition, authority, in short: establishment.
>That's all.
Is it? Or is that all that you are able to understand?
>And it's nothing new.
It is nothing new for an outsider who does not understand a field to
produce material of no value. No doubt I could buy some oils, hire a
model and mark up a sheet of canvass. Getting the Met to display it
would be a more difficult proposition. Would that mean that there was
a conspiracy against outsiders, or simply that my talents lay in other
areas?
[1] Yes, even Kronecker.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Shmuel (Seymour J.) Metz
01/11/2006
at 08:02 PM, imagin...@despammed.com said:
>Not "usually", I think.
Are you talking about old machines or only current ones? CDC is dead
and so are their 12, 24, 48 and 60 bit machines. Philco is gone.
Sylvania is dead, and so are their 36 bit machines. RCA stopped
shipping the 601 and CDP long since. The 7090 et all are no longer in
production. DEC is gone and Hewlett PackPaq does not manufacture their
PDP-5/8, PDP-6/10 or PDP-7/9/15 lines. Unisys still sells 36 and 48
bit machines, but at a low volume. Essentially everything with a
viable market these days is a twos complement machine whose word size
is a power of 2.
"Silly wabbit, trits are for kids."
Han de Bruijn
OK. Fixing first things first: P(a | n) , meaning: the probability that
a fixed natural number a divides an arbitrary natural number n .
>>>Boy, I can't figure what you mean.
>>
>>Yes you can. Just think a little bit harder.
>
> No I can't. Why make things deliberately obtuse?
Not deliberately, but I've been quite sloppy here, admittedly.
My excuse is that there have been references to other threads, where
everything is explained the best I can. The "best I can" meaning that
I can't do much better. Satisfied? Wild guess: not.
Han de Bruijn
Han de Bruijn
> In <1137095418.8...@g44g2000cwa.googlegroups.com>, on
> 01/12/2006 at 11:50 AM, Han.de...@DTO.TUDelft.NL said:
>
>>Quoted from this last and unfinished symphony by E.T. Jaynes:
>
>> In other words, we sail under the banner of Gauss,
>> Kronecker, and Poincare
>
> An interesting quote, given that they[1] dealt with the Real line,
> which is infinite.
That's true. But the question is: infinite in what sense?
There is a significant difference between Cantor's infinite
and infinity in mathematics as a science.
> [1] Yes, even Kronecker.
Han de Bruijn
Han de Bruijn
> In <1137095418.8...@g44g2000cwa.googlegroups.com>, on
> 01/12/2006 at 11:50 AM, Han.de...@DTO.TUDelft.NL said:
>
>> http://hdebruijn.soo.dto.tudelft.nl/jaar2004/nlrlsfem.pdf
>> But if you read it, you can understand very well why it is not
>> accepted by the (numerical) mathematics community.
>
> Indeed, but our understanding of why might be very different from what
> you wanted.
>
>> It undermines tradition, authority, in short: establishment.
>> That's all.
>
> Is it? Or is that all that you are able to understand?
>
>>And it's nothing new.
>
> It is nothing new for an outsider who does not understand a field to
> produce material of no value. [ ... snip ... ]
If you want to prove that I'm rather ignorant about numerical analysis
then I wish you good luck. Search the web for "Han de Bruijn" and this
subject and note that Google is my friend.
Han de Bruijn
Jesse F. Hughes
> OK. Fixing first things first: P(a | n) , meaning: the probability that
> a fixed natural number a divides an arbitrary natural number n .
Great. But I still haven't a clue what that might mean.
What does "a divides an arbitrary natural n" *mean*? What sort of
event is that and how am I supposed to understand the probability of
this event?
--
Jesse F. Hughes
"And I'm one of my own biggest skeptics as I had *YEARS* of wrong
ideas, and attempts that failed. Worse, for some of them it took
*MONTHS* before I figured out where I screwed up." -- James Harris
Tony Orlow
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
> > OK. Fixing first things first: P(a | n) , meaning: the probability that
> > a fixed natural number a divides an arbitrary natural number n .
>
> Great. But I still haven't a clue what that might mean.
>
> What does "a divides an arbitrary natural n" *mean*? What sort of
> event is that and how am I supposed to understand the probability of
> this event?
>
Oh, come on Jesse. Han is talking about the probability that a randomly chosen
natural n is divisible by any given fixed natural a. If you consider an initial
segment of the naturals, and the limit of this probability as that initial
segment approaches oo, the probability approaches 1/a, as Han states. You put
all the natural numbers on balls in your vase and pick one at random. The
chances that it is an integral multiple of your pre-chosen a is 1/a. As you
continue to pick random balls, the ratio of them that are integral multiples of
a will approach 1/a. You get this. It is obvious. Why play dumb?
--
Smiles,
Tony
Jesse F. Hughes
I understand the probability a divides n when we have a uniform
distribution on initial segments of N (although my terminology may be
off).
It seems you're right that the limit of these values is 1/a.
But there is no uniform distribution on N, so I don't see what Han
means there and your explanation doesn't clarify it.
--
"I arrest anybody I think needs arresting, Mr. Carter, and I'm not in
the habit of explaining why."
"There's a law about that ---"
"You're in Dodge, Mr. Carter." -- Gunsmoke radio show / John Ashcroft
Randy Poe
Tony Orlow wrote:
> Jesse F. Hughes said:
> > Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
> >
> > > OK. Fixing first things first: P(a | n) , meaning: the probability that
> > > a fixed natural number a divides an arbitrary natural number n .
> >
> > Great. But I still haven't a clue what that might mean.
> >
> > What does "a divides an arbitrary natural n" *mean*? What sort of
> > event is that and how am I supposed to understand the probability of
> > this event?
> >
>
> Oh, come on Jesse. Han is talking about the probability that a randomly chosen
> natural n
The concept of "randomly chosen natural n" is precisely where
the problem is. Intuitively we'd like there to be such a thing
as a distribution which could produce any natural with equal
probability. But there's no such thing.
One can cook up non-uniform distributions over all of N, but
not a uniform one.
- Randy
Tony Orlow
> Tony Orlow <ae...@cornell.edu> writes:
>
> > Jesse F. Hughes said:
> >> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
> >>
> >> > OK. Fixing first things first: P(a | n) , meaning: the probability that
> >> > a fixed natural number a divides an arbitrary natural number n .
> >>
> >> Great. But I still haven't a clue what that might mean.
> >>
> >> What does "a divides an arbitrary natural n" *mean*? What sort of
> >> event is that and how am I supposed to understand the probability of
> >> this event?
> >>
> >
> > Oh, come on Jesse. Han is talking about the probability that a randomly chosen
> > natural n is divisible by any given fixed natural a. If you consider an initial
> > segment of the naturals, and the limit of this probability as that initial
> > segment approaches oo, the probability approaches 1/a, as Han states. You put
> > all the natural numbers on balls in your vase and pick one at random. The
> > chances that it is an integral multiple of your pre-chosen a is 1/a. As you
> > continue to pick random balls, the ratio of them that are integral multiples of
> > a will approach 1/a. You get this. It is obvious. Why play dumb?
>
> I understand the probability a divides n when we have a uniform
> distribution on initial segments of N (although my terminology may be
> off).
>
> It seems you're right that the limit of these values is 1/a.
>
> But there is no uniform distribution on N, so I don't see what Han
> means there and your explanation doesn't clarify it.
>
>
It seems to me that the distribution of such integral multiples of a is quite
uniform, since you get one such multiple exactly once every a steps as you step
through the naturals. Otherwise, what do you mean by a uniform distribution?
Han is speaking probabilistically anyway, so the distribution need not be
uniform if the selection is random. It just needs to approach that ration
asymptotically (more or less, with irregularities).
I mean, it sounds like you are agruing against the notion of set density, or
frequency, or probability of distribution, but all those concepts, if they
differ, make sense to me.
__
Smiles,
Tony
Tony Orlow
n. That seems like a pretty uniform distribution to me. It doesn't get any more
uniformly distributed than that. When you are talking about a random pick, then
it doesn't have to be uniform, does it? It just has to gradually get closer to
1/n overall.
--
Smiles,
Tony
Jesse F. Hughes
> Jesse F. Hughes said:
>> I understand the probability a divides n when we have a uniform
>> distribution on initial segments of N (although my terminology may be
>> off).
>>
>> It seems you're right that the limit of these values is 1/a.
>>
>> But there is no uniform distribution on N, so I don't see what Han
>> means there and your explanation doesn't clarify it.
>>
>>
>
> It seems to me that the distribution of such integral multiples of a
> is quite uniform, since you get one such multiple exactly once every
> a steps as you step through the naturals. Otherwise, what do you
> mean by a uniform distribution?
It is the part about "a randomly chosen natural n" that makes no
particular sense to me.
> Han is speaking probabilistically anyway, so the distribution need not be
> uniform if the selection is random. It just needs to approach that ration
> asymptotically (more or less, with irregularities).
--
Jesse F. Hughes
"Casting [Demi] Moore as a woman who has come to the New World so that
she can 'worship without fear or persecution' in _The_Scarlet_Letter_
is like casting Bruce Willis as Young Rene Descartes." -Joe Queenan
Randy Poe
Tony Orlow wrote:
> Randy Poe said:
> >
> > Tony Orlow wrote:
> > > Jesse F. Hughes said:
> > > > Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
> > > >
> > > > > OK. Fixing first things first: P(a | n) , meaning: the probability that
> > > > > a fixed natural number a divides an arbitrary natural number n .
> > > >
> > > > Great. But I still haven't a clue what that might mean.
> > > >
> > > > What does "a divides an arbitrary natural n" *mean*? What sort of
> > > > event is that and how am I supposed to understand the probability of
> > > > this event?
> > > >
> > >
> > > Oh, come on Jesse. Han is talking about the probability that a randomly chosen
> > > natural n
> >
> > The concept of "randomly chosen natural n" is precisely where
> > the problem is. Intuitively we'd like there to be such a thing
> > as a distribution which could produce any natural with equal
> > probability. But there's no such thing.
> >
> > One can cook up non-uniform distributions over all of N, but
> > not a uniform one.
> >
> >
> n. That seems like a pretty uniform distribution to me. It doesn't get any more
> uniformly distributed than that. When you are talking about a random pick, then
> it doesn't have to be uniform, does it? It just has to gradually get closer to
> 1/n overall.
This is really garbled, apparently due to your misinterpretation
of the phrase "uniform distribution".
When you talk about a random pick of a value n, that means
that you are generating a value from a probability distribution
of potential values. A uniform distribution has equal probability
of generating any of the possible values. I can generate
random picks from {1,2,..., 100} easily enough, with the
distribution which says that any particular value will arise
with probability 0.01.
The term "uniform distribution" refers to the picking process,
not the density of divisors of n. There is no such thing
as a uniform distribution over all natural numbers. So
when you want to talk about "pick a random n", then
you need to define the distribution, i.e. the probability
that your choice = n for any particular n.
- Randy
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> Jesse F. Hughes said:
> > Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
> >
> > > OK. Fixing first things first: P(a | n) , meaning: the probability that
> > > a fixed natural number a divides an arbitrary natural number n .
> >
> > Great. But I still haven't a clue what that might mean.
> >
> > What does "a divides an arbitrary natural n" *mean*? What sort of
> > event is that and how am I supposed to understand the probability of
> > this event?
> >
>
> Oh, come on Jesse. Han is talking about the probability that a randomly
> chosen
> natural n is divisible by any given fixed natural a. If you consider an
> initial
> segment of the naturals, and the limit of this probability as that initial
> segment approaches oo, the probability approaches 1/a, as Han states. You put
> all the natural numbers on balls in your vase and pick one at random.
To pick one of a finite set of n objects at random, probability theory
says that the probability of any one object being picked in any one
trial is 1/n, and that the sum of these individual probabilities must
add up to 1.
But the limit of theis probability as n increases without bound is 0,
and no countable number of 0's add up to anything other than 0 (adding
up a countably infinite number of them is only defined as the limit of
their partial sums, and all those partial sums are 0's).
The
> chances that it is an integral multiple of your pre-chosen a is 1/a. As you
> continue to pick random balls, the ratio of them that are integral multiples
> of
> a will approach 1/a. You get this. It is obvious. Why play dumb?
So that TO must be insisting that LIM_{n -> +oo} [sum_{1<=k<=n} 0] = 1
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
You can define the "limiting" density of some subset of the naturals,
but this does not require a probability for each member of the naturals.
The limiting density of S in N, where /\ indicates intersection
and N_n indicates the set {k in N: k <= n}. is
"limiting density" = LIM{n -> oo} card(S /\ N_n)/card(N_n)
> Han is speaking probabilistically anyway,
> so the distribution need not be uniform if the selection is random.
That betrays a profound ignorance of probability theory.
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> If you step through the naturals, then every nth element is evenly
> divisible by n. That seems like a pretty uniform distribution to me.
> It doesn't get any more uniformly distributed than that. When you are
> talking about a random pick, then it doesn't have to be uniform, does
> it? It just has to gradually get closer to 1/n overall.
To pick randomly from the Dedekind infinite set of finite naturals i
impossible.
To pick at random from the residue classes mod n, for any positive
finite naturals n, is what is meant here, and is quite possible.
There is no probability that can be assigned to the Dedekind infinite
set of finite naturals, N, or any other countably infinite set, such
that every subset (every event) has a probability compatible with the
requirements of probability theory.
apoorv
> In the modern scientific world view, the "real" world
> is the world we
> can observe. It's the world with which we can
> interact, and in which we
> can perform experiments. Assertions about the real
> world are meaningful
> if and only if they make testable (and falsifiable)
> predictions about
> the results of experiments.
and
> Cantorian set theory implies the "existence" of a
> world far beyond the
> world of computation. Even if we imagine ourselves
> having access to
> computers that could take a countably infinite amount
> of data as the
> input, then perform an infinite amount of computation
> on that data, and
> then give an output, that would only push us up one
> very small step in
> the cumulative hierarchy of Cantorian transfinite
> sets. The Cantorian
> world of infinite sets is invisible, and mystical.
In addition to verifiability, there is the equally (perhaps more) important issue of logical self-consistency.
Further,if the recognition of numbers is a cognitive state of the brain,we must accept set theory to be a model of reality and not the reality itself. I think that the absolute model of computation (Peano axioms) that we work with prevents us from looking at other models and more importantly, from fully exploring the links between computation and cognition. See for example,
http://www.sciencenews.org/articles/20020622/bob9.asp
-Apoorv
apoorv
apoorv
Han.de...@dto.tudelft.nl
>
> I understand the probability a divides n when we have a uniform
> distribution on initial segments of N (although my terminology may be
> off).
Your terminology is clear enough.
> It seems you're right that the limit of these values is 1/a.
>
> But there is no uniform distribution on N, so I don't see what Han
> means there and your explanation doesn't clarify it.
Who the hell has decided that "there is no uniform distribution on N"
!?
Your logic and mine are mutually incompatible if you can't agree
upon the fact that an initial segment (1..n) of N is just N as n -> oo
.
Oh, and for the sake of "clarity", the limit of a segment (1..n) for
n -> oo "means": if you have a finite segment (1..m), then replace
it by a larger finite segment (1..n) where n > m. Recursively.
Han de Bruijn
Han.de...@dto.tudelft.nl
The fact that I am challenging common notions in mathematics
doesn't mean that I'm quite ignorant about how to employ them
according to the "official" rules. It's a common misunderstanding
in 'sci.math' that those who try to present alternative approaches
are always ignorant about mathematics "as it should be done".
Han de Bruijn
Han.de...@dto.tudelft.nl
>
> The concept of "randomly chosen natural n" is precisely where
> the problem is. Intuitively we'd like there to be such a thing
> as a distribution which could produce any natural with equal
> probability. But there's no such thing.
Duh! There is no such thing because you have not _defined_ it.
There *is* such a thing if you define it on initial segments (1..n)
and then take the limit for n -> oo . There is no guarantee that
this procedure always results in meaningful outcomes (which
is just typical for limits on functions).
But for the case P(arbitrary natural divisible by fixed natural)
the outcome is sensible and equals 1/(fixed natural).
You want a counter example? The probability that a natural
begins with 1 in decimal notation does not exist.
> One can cook up non-uniform distributions over all of N, but
> not a uniform one.
Both. Non-uniform distributions as well as uniform distributions.
Han de Bruijn
Han.de...@dto.tudelft.nl
> When you talk about a random pick of a value n, that means
> that you are generating a value from a probability distribution
> of potential values. A uniform distribution has equal probability
> of generating any of the possible values. I can generate
> random picks from {1,2,..., 100} easily enough, with the
> distribution which says that any particular value will arise
> with probability 0.01.
So you understand very well what we are going to do next.
> The term "uniform distribution" refers to the picking process,
> not the density of divisors of n. There is no such thing
> as a uniform distribution over all natural numbers. So
> when you want to talk about "pick a random n", then
> you need to define the distribution, i.e. the probability
> that your choice = n for any particular n.
There is no such thing as a uniform distribution over all natural
numbers, as long as "the" natural numbers are considered as
a _completed infinity_. Then we will all agree about the "fact"
that the probability to select an arbitrary natural is exactly 0.
And therefore the sum of all those probabilities cannot be 1.
Which means that they do not exist. Right?
But that is not what _we_ are doing, because, in the constructive
approach, completed infinity is rejected. Therefore we start with
a finite initial segment (1..n) , define a function (probability) at
this
segment and see if the limit of that function exists for n -> oo .
It has been demonstrated that such limits can exist and this proves
that the outcome of such a constructive approach can be different
from the traditional outcome. Obviously, that is a _contradiction_
between constructive and mainstream mathematics. So make up
your mind.
Han de Bruijn
Han.de...@dto.tudelft.nl
>
> But the limit of theis probability as n increases without bound is 0,
> and no countable number of 0's add up to anything other than 0 (adding
> up a countably infinite number of them is only defined as the limit of
> their partial sums, and all those partial sums are 0's).
>
>
> So that TO must be insisting that LIM_{n -> +oo} [sum_{1<=k<=n} 0] = 1
You are making the same mistake as some first year students do
when taking limits. It's essentially the same as in:
lim(n->oo) Sum_n 1/n = lim(n->oo) Sum_n [lim(n->oo) 1/n] =
lim(n->oo) Sum_n [0] = lim(n->oo) 0 = 0
While it should be:
lim(n->oo) [Sum_n 1/n] = lim(n->oo) [1] = 1 .
Han de Bruijn
Randy Poe
Han.de...@DTO.TUDelft.NL wrote:
> Jesse F. Hughes wrote:
> >
> > I understand the probability a divides n when we have a uniform
> > distribution on initial segments of N (although my terminology may be
> > off).
>
> Your terminology is clear enough.
>
> > It seems you're right that the limit of these values is 1/a.
> >
> > But there is no uniform distribution on N, so I don't see what Han
> > means there and your explanation doesn't clarify it.
>
> Who the hell has decided that "there is no uniform distribution on N"
> !?
It isn't "decided". It is "proven". The fact that people like you,
Ross, and Tony think theorems are arbitrary "decisions" and
alternate decisions could be made from the same axioms,
does not make those theorems false.
You can define a uniform distribution over a finite set of
n discrete points. The probability that X = any particular
point from the set is 1/n.
You can define a uniform distribution over an interval
containing uncountably many points, such as [a,b].
There is a probability density 1/(b-a), defined such
that the probability of picking a point in any sub-interval
of length s is s/(b-a). If you divide [a,b] into n intervals,
the probability of each is 1/n.
A probability distribution must satisfy the condition
that the total probability over all events is 1. There is
no way to do that with a countable set of discrete
points.
Feel free to prove us wrong. Define the uniform
distribution over N. Show us a procedure to generate
a random integer.
- Randy
Jesse F. Hughes
> Jesse F. Hughes wrote:
>>
>> I understand the probability a divides n when we have a uniform
>> distribution on initial segments of N (although my terminology may be
>> off).
>
> Your terminology is clear enough.
>
>> It seems you're right that the limit of these values is 1/a.
>>
>> But there is no uniform distribution on N, so I don't see what Han
>> means there and your explanation doesn't clarify it.
>
> Who the hell has decided that "there is no uniform distribution on N"
> !?
>
> Your logic and mine are mutually incompatible if you can't agree
> upon the fact that an initial segment (1..n) of N is just N as n -> oo
> .
Okay. We'll use your logic for determining P( {n} ) for a uniform
distribution. My approach will mirror your method for determining
P( {na | n in N} ) where a is fixed. I take it that we agree that
{na | n in N} is the right way to define the event "a divides n"?
Let's see, you did something like this: Let P_k be the uniform
distribution on the set {1,...,k} and calculate
lim_{k -> oo} P_k( {na | n in N} n {1,...,k} )
This came out to be 1/a, so you conclude P(a divides x) = 1/a.
Now let's try that again. P_k( {n} n {1,...,k} ) = 1/k if n <= k, 0
else. Thus,
P( {n} ) = lim_{k -> oo}P_k( {n} n {1,...,k} ) = lim_{k-> oo} 1/k = 0.
Uh oh. By definition of distribution, P is not a distribution since
it fails to satisfy countable additivity.
Now maybe you have some other definition of distribution in mind, but
you haven't told me what it is.
Note: I don't deny that lim_{k->oo} {1,...,k} = N (given an
appropriate topology on PN). But that's not at issue.
> Oh, and for the sake of "clarity", the limit of a segment (1..n) for
> n -> oo "means": if you have a finite segment (1..m), then replace
> it by a larger finite segment (1..n) where n > m. Recursively.
That's not clearer at all. That isn't what limit means at all.
You need to give a topology on PN that allows one to prove
lim_{k->oo} {1,...,k} = N.
It seems to me that the topology with basis {(n,oo) | n in N} would
do, so we needn't dispute this point. But it's apparent you don't
know what mathematical clarity means.
--
"Just because you're ... in a Ph.d program it does not mean that
you're up to the challenge of being a real mathematician. Only those
who have a purity of mind and dedication to the truth as the highest
ideal have a chance." --James Harris, as Sir Galahad the Pure.
Han.de...@dto.tudelft.nl
> Han.de...@DTO.TUDelft.NL wrote:
> >
> > Who the hell has decided that "there is no uniform distribution on N" !?
>
> It isn't "decided". It is "proven". The fact that people like you,
> Ross, and Tony think theorems are arbitrary "decisions" and
> alternate decisions could be made from the same axioms,
> does not make those theorems false.
There you say it ! But *I* don't think that "alternate decisions
could be made from the same axioms". And *I* dont think
that those theorems are "false", in the sense that they do
not follow from the axioms. What *I* think is that the axioms
themselves are false. Worse, what *I* think is that the whole
of mathematics cannot be founded on just axioms and logic.
> You can define a uniform distribution over a finite set of
> n discrete points. The probability that X = any particular
> point from the set is 1/n.
>
> You can define a uniform distribution over an interval
> containing uncountably many points, such as [a,b].
> There is a probability density 1/(b-a), defined such
> that the probability of picking a point in any sub-interval
> of length s is s/(b-a). If you divide [a,b] into n intervals,
> the probability of each is 1/n.
We all agree on this.
> A probability distribution must satisfy the condition
> that the total probability over all events is 1. There is
> no way to do that with a countable set of discrete
> points.
I have repeatedly pointed out that there _is_ a way.
But I also understand why that way is unacceptable
for mainstream mathematics, if its set of all naturals
is to be considered as a completed infinity. Which it
is _not_.
> Feel free to prove us wrong. Define the uniform
> distribution over N. Show us a procedure to generate
> a random integer.
I cannot "prove" that you are wrong because that would
imply that I should step into your vicious circle and be
happy to stay there. I won't do this, because I find that
Cantor's paradise is not like heaven, but rather like hell,
for some good reasons.
Han de Bruijn
Han.de...@dto.tudelft.nl
>
> Okay. We'll use your logic for determining P( {n} ) for a uniform
> distribution. My approach will mirror your method for determining
> P( {na | n in N} ) where a is fixed. I take it that we agree that
> {na | n in N} is the right way to define the event "a divides n"?
>
> Let's see, you did something like this: Let P_k be the uniform
> distribution on the set {1,...,k} and calculate
>
> lim_{k -> oo} P_k( {na | n in N} n {1,...,k} )
>
> This came out to be 1/a, so you conclude P(a divides x) = 1/a.
>
> Now let's try that again. P_k( {n} n {1,...,k} ) = 1/k if n <= k, 0
> else. Thus,
>
> P( {n} ) = lim_{k -> oo}P_k( {n} n {1,...,k} ) = lim_{k-> oo} 1/k = 0.
>
> Uh oh. By definition of distribution, P is not a distribution since
> it fails to satisfy countable additivity.
Yes. I cannot help if you define elementary probabilities the wrong
way around. I cannot help if you find that the probability a die will
throw up the numbers 1,2,3 with a chance of 1/6 and the numbers
4,5,6 with a chance of 0 and that all chances do not sum up to 1.
Duuhh ...
> Now maybe you have some other definition of distribution in mind, but
> you haven't told me what it is.
I _have_ told you what it is: P(a fixed natural in (1..n)) = 1/n .
[ ... snip ... ]
> It seems to me that the topology with basis {(n,oo) | n in N} would
> do, so we needn't dispute this point. But it's apparent you don't
> know what mathematical clarity means.
Uh, uh, isn't that a bit of a generalization? I do not consider myself
as a mathematical genious, but I have done some decent work in
some areas which might be called sort of mathematical. With so
much clarity that some decent computer code could be developed.
Han de Bruijn
Virgil
Han.de...@DTO.TUDelft.NL wrote:
>
> There you say it ! But *I* don't think that "alternate decisions
> could be made from the same axioms". And *I* dont think
> that those theorems are "false", in the sense that they do
> not follow from the axioms. What *I* think is that the axioms
> themselves are false.
And just what do you propose to base your proof of that falsity upon?
Are you claiming there must be some sort of "ultimate truth" accessible
to mankind?
Until we find it, and we don't seem to have found it yet, axiomatics
works better than intuition.
> Worse, what *I* think is that the whole
> of mathematics cannot be founded on just axioms and logic.
Better a mathematics built on axioms than one built on nothing at all.
If you don't like our axiom systems, you are quite free in present day
mathematics to develop your own. have you done so?
>
> > You can define a uniform distribution over a finite set of
> > n discrete points. The probability that X = any particular
> > point from the set is 1/n.
> >
> > You can define a uniform distribution over an interval
> > containing uncountably many points, such as [a,b].
> > There is a probability density 1/(b-a), defined such
> > that the probability of picking a point in any sub-interval
> > of length s is s/(b-a). If you divide [a,b] into n intervals,
> > the probability of each is 1/n.
>
> We all agree on this.
>
> > A probability distribution must satisfy the condition
> > that the total probability over all events is 1. There is
> > no way to do that with a countable set of discrete
> > points.
>
> I have repeatedly pointed out that there _is_ a way.
> But I also understand why that way is unacceptable
> for mainstream mathematics, if its set of all naturals
> is to be considered as a completed infinity. Which it
> is _not_.
At least not in your philosophy. But our axiom system sasys otherwise.
Can you build a better axiom system?
>
> > Feel free to prove us wrong. Define the uniform
> > distribution over N. Show us a procedure to generate
> > a random integer.
>
> I cannot "prove" that you are wrong because that would
> imply that I should step into your vicious circle and be
> happy to stay there. I won't do this, because I find that
> Cantor's paradise is not like heaven, but rather like hell,
> for some good reasons.
Until you can come up with an alternate axiom system that will deliver
the mathematics that sciences depend on but not keep you in your self
made hell, you will remain there. So get busy!
Han.de...@dto.tudelft.nl
>
> There is no probability that can be assigned to the Dedekind infinite
> set of finite naturals, N, or any other countably infinite set, such
> that every subset (every event) has a probability compatible with the
> requirements of probability theory.
And that is *true*. As long as you don't step out of the world of
Cantor
and Dedekind, there doesn't exist a counter argument to this. I repeat:
as long as you don't step out.
Han de Bruijn
Virgil
Han.de...@DTO.TUDelft.NL wrote:
> Jesse F. Hughes wrote:
> >
> > Okay. We'll use your logic for determining P( {n} ) for a uniform
> > distribution. My approach will mirror your method for determining
> > P( {na | n in N} ) where a is fixed. I take it that we agree that
> > {na | n in N} is the right way to define the event "a divides n"?
> >
> > Let's see, you did something like this: Let P_k be the uniform
> > distribution on the set {1,...,k} and calculate
> >
> > lim_{k -> oo} P_k( {na | n in N} n {1,...,k} )
> >
> > This came out to be 1/a, so you conclude P(a divides x) = 1/a.
> >
> > Now let's try that again. P_k( {n} n {1,...,k} ) = 1/k if n <= k, 0
> > else. Thus,
> >
> > P( {n} ) = lim_{k -> oo}P_k( {n} n {1,...,k} ) = lim_{k-> oo} 1/k = 0.
> >
> > Uh oh. By definition of distribution, P is not a distribution since
> > it fails to satisfy countable additivity.
>
> Yes. I cannot help if you define elementary probabilities the wrong
> way around. I cannot help if you find that the probability a die will
> throw up the numbers 1,2,3 with a chance of 1/6 and the numbers
> 4,5,6 with a chance of 0 and that all chances do not sum up to 1.
So what is the HdB definition of probability, and what are its
properties?
> > Now maybe you have some other definition of distribution in mind, but
> > you haven't told me what it is.
>
> I _have_ told you what it is: P(a fixed natural in (1..n)) = 1/n .
As stated, that is nonsense. For any fixed natural <=n it is 1 and for
all others it is 0.
if you meaqn that the probability of a randomly selected member of
{1,2,3,...,n} will match some fixed member of that set, then the
probability should be 1/n, but that does not apply to countably infinite
sets.
>
> [ ... snip ... ]
>
> > It seems to me that the topology with basis {(n,oo) | n in N} would
> > do, so we needn't dispute this point. But it's apparent you don't
> > know what mathematical clarity means.
>
> Uh, uh, isn't that a bit of a generalization? I do not consider myself
> as a mathematical genious, but I have done some decent work in
> some areas which might be called sort of mathematical. With so
> much clarity that some decent computer code could be developed.
>
> Han de Bruijn
Those who confuse computer science with mathematics are no more right
than those who confuse physics with mathematics.
MoeBlee
> As long as you don't step out of the world of
> Cantor
> and Dedekind
In what logisitic system, with what primitives, and from what axioms do
you derive the conclusions that describe the world you've stepped out
into?
MoeBlee
Jesse F. Hughes
> Jesse F. Hughes wrote:
>>
>> Okay. We'll use your logic for determining P( {n} ) for a uniform
>> distribution. My approach will mirror your method for determining
>> P( {na | n in N} ) where a is fixed. I take it that we agree that
>> {na | n in N} is the right way to define the event "a divides n"?
>>
>> Let's see, you did something like this: Let P_k be the uniform
>> distribution on the set {1,...,k} and calculate
>>
>> lim_{k -> oo} P_k( {na | n in N} n {1,...,k} )
>>
>> This came out to be 1/a, so you conclude P(a divides x) = 1/a.
>>
>> Now let's try that again. P_k( {n} n {1,...,k} ) = 1/k if n <= k, 0
>> else. Thus,
>>
>> P( {n} ) = lim_{k -> oo}P_k( {n} n {1,...,k} ) = lim_{k-> oo} 1/k = 0.
>>
>> Uh oh. By definition of distribution, P is not a distribution since
>> it fails to satisfy countable additivity.
>
> Yes. I cannot help if you define elementary probabilities the wrong
> way around. I cannot help if you find that the probability a die will
> throw up the numbers 1,2,3 with a chance of 1/6 and the numbers
> 4,5,6 with a chance of 0 and that all chances do not sum up to 1.
That's not a distribution. You're confused. You should be arguing
against the axioms of probability theory, but instead you give an
example that is not a distribution in *anyone's* sense at all.
Or are you claiming that I misapplied your method in calculating
P( {n} )? Well, show me then. If you claim there's some sense
to be made in randomly choosing a natural number (with equal
probability for each n), by all means show me how to calculate
P( {n} ). 'Cause I tried you use your method, but maybe I got it
wrong.
>> Now maybe you have some other definition of distribution in mind, but
>> you haven't told me what it is.
>
> I _have_ told you what it is: P(a fixed natural in (1..n)) = 1/n .
That is not a definition of distribution. Maybe you mean that it is
an example of a distribution, but I don't understand it. There are
two things you should tell me:
(1) If you don't accept the standard definition of "distribution",
what definition do you prefer. Note: your answer should say something
like, "X is a distribution iff X ...."
(2) If you also want to show me that there *is* a uniform definition
on N, give it. You didn't do so above, since no initial segment of N
is equal to N.
If you mean that for a fixed set {1,...,n}, P( {k} ) = 1/n for k <= n
is a uniform distribution, well, no kidding. That wasn't a question I
asked.
>> It seems to me that the topology with basis {(n,oo) | n in N} would
>> do, so we needn't dispute this point. But it's apparent you don't
>> know what mathematical clarity means.
>
> Uh, uh, isn't that a bit of a generalization?
Not much. You wrote this:
Oh, and for the sake of "clarity", the limit of a segment (1..n)
for n -> oo "means": if you have a finite segment (1..m), then
replace it by a larger finite segment (1..n) where n >
m. Recursively.
That clarified not a bit what your "limit" means. Not even with the
magic word, "Recursively".
> I do not consider myself as a mathematical genious, but I have done
> some decent work in some areas which might be called sort of
> mathematical. With so much clarity that some decent computer code
> could be developed.
Yeah, whatever. Good for you.
--
"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
Virgil
Han.de...@DTO.TUDelft.NL wrote:
What HdB means by "stepping out" is denial of the existence of any
infinite sets at all. Which denial also scuttles the ring of integers,
the field of rational numbers, the field of real numbers, and a whole
lot more of standard mahematics.
Randy Poe
Han.de...@DTO.TUDelft.NL wrote:
> Randy Poe wrote:
> > Han.de...@DTO.TUDelft.NL wrote:
> > >
> > > Who the hell has decided that "there is no uniform distribution on N" !?
> >
> > It isn't "decided". It is "proven". The fact that people like you,
> > Ross, and Tony think theorems are arbitrary "decisions" and
> > alternate decisions could be made from the same axioms,
> > does not make those theorems false.
>
> There you say it ! But *I* don't think that "alternate decisions
> could be made from the same axioms". And *I* dont think
> that those theorems are "false", in the sense that they do
> not follow from the axioms. What *I* think is that the axioms
> themselves are false.
What does it mean for an axiom to be false?
> Worse, what *I* think is that the whole
> of mathematics cannot be founded on just axioms and logic.
Well, you can feel free to build up illogical systems, but
don't use the word "mathematics" for it.
> > A probability distribution must satisfy the condition
> > that the total probability over all events is 1. There is
> > no way to do that with a countable set of discrete
> > points.
>
> I have repeatedly pointed out that there _is_ a way.
I suppose what that means (I haven't seen it) is that
you have repeatedly *claimed* there is a way. But when
I ask you to actually that way, you decline, saying that
to do so would be to lower yourself in some way.
So pardon me if I don't believe you.
> > Feel free to prove us wrong. Define the uniform
> > distribution over N. Show us a procedure to generate
> > a random integer.
>
> I cannot "prove" that you are wrong
You can "prove" me wrong about the non-existence of the
uniform distribution over N by showing me the existence
of the uniform distribution over N.
> because that would
> imply that I should step into your vicious circle
Backing up your claims is not "stepping into a vicious
circle". Unbackable claims are just hot air.
- Randy
Han.de...@dto.tudelft.nl
>
> if you meaqn that the probability of a randomly selected member of
> {1,2,3,...,n} will match some fixed member of that set, then the
> probability should be 1/n, but that does not apply to countably infinite
> sets.
Yes, that's what I mean. And it _does_ apply to countably infinite
sets,
provided that these sets are potential infinite and not actual
infinite. For
the simple reason that, in constructive mathematics, an infinite set
can
only be reached through (a sequence of) finite sets.
> Those who confuse computer science with mathematics are no more right
> than those who confuse physics with mathematics.
My purpose is not to "confuse" things, rather than to see things from a
viewpoint that _unifies_ mathematics with the other sciences, like i.e.
computer science and physics, indeed, as you say.
Han de Bruijn
Han.de...@dto.tudelft.nl
You just invited me to step _in_, instead of to step _out_. A world
that
is "logistic", with "primitives" and "axioms" is too incomplete to form
a
foundation for constructive mathematics. (If I have understood what you
are saying here. English is not my native language)
Han de Bruijn
Shmuel (Seymour J.) Metz
01/14/2006
at 04:50 AM, Han.de...@DTO.TUDelft.NL said:
>Who the hell has decided that "there is no uniform distribution on N"
>!?
Everybody who understands what N is.
>Your logic and mine are mutually incompatible if you can't agree
>upon the fact that an initial segment (1..n) of N is just N as n ->
>oo
You have neither fact not logic until you define what you mean by
limit and probe that it has the requisite properties.
>Oh, and for the sake of "clarity", the limit of a segment (1..n) for
>n -> oo "means": if you have a finite segment (1..m), then replace
>it by a larger finite segment (1..n) where n > m. Recursively.
There is no clarity where there is no definition. Trplacing a finite
set with a finite set recursively gives you a finite set.
In <1137243963.9...@z14g2000cwz.googlegroups.com>, on
01/14/2006
at 05:06 AM, Han.de...@DTO.TUDelft.NL said:
>The fact that I am challenging common notions in mathematics doesn't
>mean that I'm quite ignorant about how to employ them according to
>the "official" rules.
No. But that fact that you are unable to cherently express your
challenge does demonstrate your ignorance.
>It's a common misunderstanding in 'sci.math' that those who try to
>present alternative approaches are always ignorant about mathematics
>"as it should be done".
Au contraire, it is a common observation of fact that those claiming
to present alternative approaches often fail to either understand the
conventional approaches or to express a coherent alternative. It is a
common observation that those making such claims often fail to either
understand conventional definitions of standard nomenclature or to
coherently present their private definitions. It is a common
observation that those claiming that their original methods are being
capriciously rejected fail to realize just how common it is for the
Mathematical community to welcome new methods with open arms.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Han.de...@dto.tudelft.nl
> Han.de...@DTO.TUDelft.NL writes:
>
> > Jesse F. Hughes wrote:
> >>
> >> Okay. We'll use your logic for determining P( {n} ) for a uniform
> >> distribution. My approach will mirror your method for determining
> >> P( {na | n in N} ) where a is fixed. I take it that we agree that
> >> {na | n in N} is the right way to define the event "a divides n"?
> >>
> >> Let's see, you did something like this: Let P_k be the uniform
> >> distribution on the set {1,...,k} and calculate
> >>
> >> lim_{k -> oo} P_k( {na | n in N} n {1,...,k} )
> >>
> >> This came out to be 1/a, so you conclude P(a divides x) = 1/a.
> >>
> >> Now let's try that again. P_k( {n} n {1,...,k} ) = 1/k if n <= k, 0
> >> else. Thus,
> >>
> >> P( {n} ) = lim_{k -> oo}P_k( {n} n {1,...,k} ) = lim_{k-> oo} 1/k = 0.
> >>
> >> Uh oh. By definition of distribution, P is not a distribution since
> >> it fails to satisfy countable additivity.
> >
> > Yes. I cannot help if you define elementary probabilities the wrong
> > way around. I cannot help if you find that the probability a die will
> > throw up the numbers 1,2,3 with a chance of 1/6 and the numbers
> > 4,5,6 with a chance of 0 and that all chances do not sum up to 1.
>
> That's not a distribution. You're confused. You should be arguing
> against the axioms of probability theory, but instead you give an
> example that is not a distribution in *anyone's* sense at all.
Uhm, right. Now I see where the confusion comes from: it's from
using "n" for the variable "n" as well as the symbol for intersection.
But my counter argument still stands. What you are saying is this:
P_k( {n} /\ {1,...,k} ) = 1/k if n <= k, 0 else.
The error is in "else". You are already assuming a completed infinity
of naturals _outside_ the interval {1,...,k} and you are assigning to
these a probability 0 . In my approach, there is no "else", because
the set of all naturals is growing out of the finite interval {1,...,k}
and
there are no other naturals yet. This is basically the same sillyness
as with throwing dice and assuming that there exist other numbers
than already contained in the set {1,2,3,4,5,6}.
The other possibility is that I still don't understand your argument.
Han de Bruijn
Han.de...@dto.tudelft.nl
>
> What HdB means by "stepping out" is denial of the existence of any
> infinite sets at all. Which denial also scuttles the ring of integers,
> the field of rational numbers, the field of real numbers, and a whole
> lot more of standard mahematics.
Nope. It is the denial of any _completed_ infinite set. Which denial
does _not_ scuttle the ring of integers, the field of rational numbers,
the field of real numbers. But it _does_ scuttle some other parts of
standard mathematics, indeed.
Han de Bruijn
Han.de...@dto.tudelft.nl
> Han.de...@DTO.TUDelft.NL wrote:
>
> What does it mean for an axiom to be false?
An axiom which is void of any possible real world content is false.
> > Worse, what *I* think is that the whole
> > of mathematics cannot be founded on just axioms and logic.
>
> Well, you can feel free to build up illogical systems, but
> don't use the word "mathematics" for it.
I didn't say "void of any logic". I said "solely based on logic and
axioms" and nothing else allowed (i.e. constructive means).
[ Ignoring the usual whining about "lack of clarity" and the like ]
Han de Bruijn
Han.de...@dto.tudelft.nl
> In <1137243049.1...@o13g2000cwo.googlegroups.com>, on
> 01/14/2006 at 04:50 AM, Han.de...@DTO.TUDelft.NL said:
>
> >Who the hell has decided that "there is no uniform distribution on N"
> >!?
>
> Everybody who understands what N is.
That is: everybody who _accepts_ what N is, according to mainstream
mathematics ideology. If want to you call that "understanding", fine.
> >Your logic and mine are mutually incompatible if you can't agree
> >upon the fact that an initial segment (1..n) of N is just N as n ->
> >oo
>
> You have neither fact not logic until you define what you mean by
> limit and probe that it has the requisite properties.
Ah, the usual trick that I haven't defined anything. Well, let me just
say that I might expect some intelligence from my readers. I have
defined already much more than some of my opponents are willing
to absorbe.
> >Oh, and for the sake of "clarity", the limit of a segment (1..n) for
> >n -> oo "means": if you have a finite segment (1..m), then replace
> >it by a larger finite segment (1..n) where n > m. Recursively.
>
> There is no clarity where there is no definition. Trplacing a finite
> set with a finite set recursively gives you a finite set.
And so on and so forth. That is what some people call infinity.
> In <1137243963.9...@z14g2000cwz.googlegroups.com>, on
> 01/14/2006 at 05:06 AM, Han.de...@DTO.TUDelft.NL said:
>
> >The fact that I am challenging common notions in mathematics doesn't
> >mean that I'm quite ignorant about how to employ them according to
> >the "official" rules.
>
> No. But that fact that you are unable to cherently express your
> challenge does demonstrate your ignorance.
Again, the usual trick. I'm not impressed by this kind of "argument".
> >It's a common misunderstanding in 'sci.math' that those who try to
> >present alternative approaches are always ignorant about mathematics
> >"as it should be done".
>
> Au contraire, it is a common observation of fact that those claiming
> to present alternative approaches often fail to either understand the
> conventional approaches or to express a coherent alternative. It is a
> common observation that those making such claims often fail to either
> understand conventional definitions of standard nomenclature or to
> coherently present their private definitions. It is a common
> observation that those claiming that their original methods are being
> capriciously rejected fail to realize just how common it is for the
> Mathematical community to welcome new methods with open arms.
If "common observation" is interpreted as "common observation by
the community of mainstream mathematicians", then you are right,
quite obviously. I suggest you read the original poster by David Petry,
which describes exactly the kind of behaviour you are exposing here.
Han de Bruijn
Virgil
Han.de...@DTO.TUDelft.NL wrote:
Then your purpose is not acheived, nor, indeed, desireable. If
mathematics is so "unified" with science's needs, there will be no new
math created until some need for it transpires. But history has shown us
that much of the math that turns out to be useful to science was created
before science was aware of the need for it.
Virgil
Han.de...@DTO.TUDelft.NL wrote:
A world that is constructed on no logically coherent basis is too
ambiguous to be reliable.
Virgil
Han.de...@DTO.TUDelft.NL wrote:
If you cannot axiomatize your "constructive means", how can you ever be
sure of what your system will, or will not, allow, or or even have any
faith that it is self-consistent?
Shmuel (Seymour J.) Metz
01/15/2006
at 06:16 AM, Han.de...@DTO.TUDelft.NL said:
>Yes, that's what I mean. And it _does_ apply to countably infinite
>sets,
No.
>provided that these sets are potential infinite and not actual
>infinite.
Mathematics is not Philosophy. Neither "potential infinite" nor
"actual infinite" are Mathematical terms. If you want to use them
without being laughed at, provide real definitions.
In <1137338538.7...@o13g2000cwo.googlegroups.com>, on
01/15/2006
at 07:22 AM, Han.de...@DTO.TUDelft.NL said:
>I didn't say "void of any logic". I said "solely based on logic and
>axioms" and nothing else allowed (i.e. constructive means).
A formal proof *is* a constructive mean.
cbr...@cbrownsystems.com
> Virgil wrote:
> >
> > if you meaqn that the probability of a randomly selected member of
> > {1,2,3,...,n} will match some fixed member of that set, then the
> > probability should be 1/n, but that does not apply to countably infinite
> > sets.
>
> Yes, that's what I mean. And it _does_ apply to countably infinite
> sets, provided that these sets are potential infinite and not actual
> infinite.
In that case, can you define what /you/ mean by "a randomly selected
member of a countably infinite set", potential or otherwise?
It might help if you first define what /you/ mean by "a randomly
selected member of {1,2,3,..., n}"?
Just to anticipate - if I have an n-sided die which I claim is
"random", what would you accept as evidence that my claim is true? It
doesn't seem sufficient to only require that in m rolls I find "nearly"
m/n of them come up each of 1,2,3,.. etc.; for if I rolled the
sequence:
(1,2,3,...,n,1,2,3,..,n,1,2,3,...n,...)
this would hardly be evidence of a "random" die to me.
Cheers - Chas
david petry
> Neither "potential infinite" nor
> "actual infinite" are Mathematical terms. If you want to use them
> without being laughed at, provide real definitions.
They are meta-mathematical terms. When we declare that all properties
of infinitary objects come from properties of finite approximations to
those infinitary objects, then we are talking about the potential
infinite. If we assert that infinitary objects can have properties that
don't correspond to properties of finite approximations to those
objects, then we are talking about an actual infinite. The problem with
the actual infinite is that it is not observable. That is, we have no
way to test (falsify) statements about the actual infinite.
cbr...@cbrownsystems.com
> Shmuel (Seymour J.) Metz wrote:
>
> > In <1137243049.1...@o13g2000cwo.googlegroups.com>, on
> > 01/14/2006 at 04:50 AM, Han.de...@DTO.TUDelft.NL said:
> >
<snip>
> > >Oh, and for the sake of "clarity", the limit of a segment (1..n) for
> > >n -> oo "means": if you have a finite segment (1..m), then replace
> > >it by a larger finite segment (1..n) where n > m. Recursively.
> >
> > There is no clarity where there is no definition. Trplacing a finite
> > set with a finite set recursively gives you a finite set.
>
> And so on and so forth. That is what some people call infinity.
>
Assuming that you also mean that that's what you call "infinity",
define the function P on an interval [1,n] as P([1,n]) = 0 if n is even
and 1 if n is odd. In your definition of "in the limit" what is the
limit of P([1,n]) as n->oo?
If you accept that P may not be definable in the limit, is it so odd
that "selecting a random member" may also not be definable "in the
limit" - even though it makes sense for every finite interval?
<snip>
> > >It's a common misunderstanding in 'sci.math' that those who try to
> > >present alternative approaches are always ignorant about mathematics
> > >"as it should be done".
> >
> > Au contraire, it is a common observation of fact that those claiming
> > to present alternative approaches often fail to either understand the
> > conventional approaches or to express a coherent alternative. It is a
> > common observation that those making such claims often fail to either
> > understand conventional definitions of standard nomenclature or to
> > coherently present their private definitions. It is a common
> > observation that those claiming that their original methods are being
> > capriciously rejected fail to realize just how common it is for the
> > Mathematical community to welcome new methods with open arms.
>
> If "common observation" is interpreted as "common observation by
> the community of mainstream mathematicians", then you are right,
> quite obviously.
If "mainstream mathematicians" are those who require an axiomatic
approach involving precise definitions, clear premises and logical
deductions, then yes, he is quite obviously right.
> I suggest you read the original poster by David Petry,
> which describes exactly the kind of behaviour you are exposing here.
>
David seems to have his own, separate axe to grind.
Cheers - Chas
Ross A. Finlayson
I'll disagree with that.
As you learn more about infinity, you'll find it ubiquitous, and
exhibited in, for example, parastatistics. The more closely subatomic
particles are measured, the more precisely, their measurements increase
in accuracy _and_ decrease in value. The universe is rather large, the
macroscale to the meso- and microscale.
It takes a rethinking of sorts to begin to comprehend the infinite, and
infinitesimal, in nature. Consider Zeno's arrow. The sum over the
positive naturals of 1/2^n _is_ one. Even the ultrafinitist must
infinitely approach that value.
It's the Ouroboros, Yggdrasil, turtles all the way around.
Ross
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
As even the finitest of mathematical objects has no physical reality, we
have no way of testing our conclusions about such finite mathematical
objects either, except against our expectations for them.
Those who rely on purely axiomatic systems, whether they allow or do not
allow infiniteness, restrict their expectations to what can be derived
only by logical deduction, and have excluded intuition based
expectations from their mathematics.
The process of axiomatic mathematics is like investigating a family of
games in which are all ruled by logic particular games are also ruled by
particular axiom sets.
One looks for the more interesting games and what one can get out of
playing with them.
Those who choose not to take part in some of the games should not have
the power, which those vociferous "anti-Cantorians" seem to demand, to
prevent others from playing them.
If history has told us anything about the development of mathematics, it
is that we cannot predict what part of it will next suddenly become
important. SO those who would cut off parts of it may easily be,
mathematically speaking, cutting off their noses to spite their faces.
cbr...@cbrownsystems.com
> Shmuel (Seymour J.) Metz wrote:
>
> > Neither "potential infinite" nor
> > "actual infinite" are Mathematical terms. If you want to use them
> > without being laughed at, provide real definitions.
>
> They are meta-mathematical terms. When we declare that all properties
> of infinitary objects come from properties of finite approximations to
> those infinitary objects, then we are talking about the potential
> infinite. If we assert that infinitary objects can have properties that
> don't correspond to properties of finite approximations to those
> objects, then we are talking about an actual infinite.
Not every mathematical object is numerical, and thus amenable to the
idea of a "finite approximation". In what way could the collection of
all non-abelian, finite groups form a "finite approximation" of an
infinite, non-abelian group (e.g., the free group on {a,b})?
I think rather one either accepts the existence of an infinite set, or
one doesn't. If one doesn't, there is no such thing as "the free group
on {a,b}" - which, I might add, is a perfectly reasonable stance to
take.
> The problem with
> the actual infinite is that it is not observable.
This is a reasonable philosophical stance; although not one I share. It
depends on what one considers "observable"; and that is a
philosophical, not a mathematical question. De gustibus non
disputandum!
> That is, we have no
> way to test (falsify) statements about the actual infinite.
This justification is the illogical part of your argument to me.
Is it possible to "observe" that there is no largest prime number? And
yet, it's easy to prove, using a finite process; thus we have falsified
the statement "the (actual) infinite set of naturals contains a largest
prime". Euclid did it; why can't we?
I always feel that the only way you will accept a statement such as
"(2^30,402,457) - 1 is prime" [1] is if I actually demonstrate that I
fail to divide it, in order, by each of 2, 3, 4, 5, 6, 7, etc. up to
2^30,402,457. This would surely be impossible in our universe; so in
your sense could not be "observed" as a computation, and therefore it
is unfalsifiable.
But there are other ways of proving that 2^30,402,457-1 is prime (or
not). Similarly, there are perfectly finite ways of drawing conclusions
about infinite sets - given that you accept their existence to start
with.
Cheers - Chas
[1] The recently discovered Mersenne prime.
MoeBlee
So you don't consider any of the intuitionistic logistic systems with
primitives and axioms to be a foundation for constructive mathematics?
How do you conclude that a constructive mathematics cannot be
axiomatized in a logistic system? And, 'logistic', 'primitive' and
'axiom' are well understand words; I don't what you mean to indicate by
using them with irony quote marks.
MoeBlee
Dik T. Winter
> Dik T. Winter wrote:
>
> > In article <560a7$43c63cdf$82a1e2b0$12...@news2.tudelft.nl> Han de Bruijn
> <Han.de...@DTO.TUDelft.NL> writes:
> >
> > > (Sorry, I have too many examples where they just didn't keep up)
> >
> > They do, hence Cantor, which gives a foundation to Newton's work. But
> > you do not like it.
>
> WHAT ??!! Did Cantor give a foundation to Newton's work ??!!
> Unbelievable ! I didn't expect such a silly remark from you, Dik.
Oh. I think there are three ways to get Newton's work on a serious
footing. The first is limits, and for that to work you ultimately
need set theory, I think (and here it is Cantor who started the basics).
A second way is Robinson's hyperreals that give a proper definition of
the infinitesimals. And the third way is Kock's method that uses
Brouwers non-exclusion of the middle, and that also gives a sound
basis for infinitesimals. But the latter two date from the 1960s and
1970s.
If you have another method in mind, pray state where I can find it.
And do not come up with Leibniz who even explained that he could not
give a proper definition of infinitesimals.
> > BTW, it is already some months ago that you would look at the way I
> > did numerical differentation. Have you already done so?
>
> Yes. I've seen it. And decided to let it rest for a while.
A sound method to resolve a dispute. Person A states that something
can be done. Person B states that that is false. Person A retains
his view. Person B gives evidence that A is wrong. Person A states
that he will let it rest for a while.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
david petry
cbr...@cbrownsystems.com wrote:
> david petry wrote:
> > The problem with
> > the actual infinite is that it is not observable.
>
> This is a reasonable philosophical stance; although not one I share. It
> depends on what one considers "observable"; and that is a
> philosophical, not a mathematical question. De gustibus non
> disputandum!
A statement has observable content if it makes predictions about the
results of a computational experiment. That's not merely a
philosophical notion.
> > That is, we have no
> > way to test (falsify) statements about the actual infinite.
>
> This justification is the illogical part of your argument to me.
>
> Is it possible to "observe" that there is no largest prime number? And
> yet, it's easy to prove, using a finite process; thus we have falsified
> the statement "the (actual) infinite set of naturals contains a largest
> prime". Euclid did it; why can't we?
In fact, an easy extension of Euclid's argument gives a bound on how
far we have to look for the next prime, given any prime. That extension
is falsifiable.
Randy Poe
Han.de...@DTO.TUDelft.NL wrote:
> Shmuel (Seymour J.) Metz wrote:
>
> > In <1137243049.1...@o13g2000cwo.googlegroups.com>, on
> > 01/14/2006 at 04:50 AM, Han.de...@DTO.TUDelft.NL said:
> >
> > >Who the hell has decided that "there is no uniform distribution on N"
> > >!?
> >
> > Everybody who understands what N is.
>
> That is: everybody who _accepts_ what N is, according to mainstream
> mathematics ideology.
Ah, I see. So in contrast, your position is that you believe there
is a uniform distribution on N, but you don't believe there is
such a thing as N.
I begin to see what you mean about building your philosophical
system on things other than logic.
- Randy