Mathematics, Pseudomathematics, Artificial Intelligence
david petry
Part I Mathematics
The essence of science is the ability to make predictions
about the results of experiments. Mathematical statements
can be interpreted as making predictions about the
results of computational experiments, and hence,
mathematics is a science. One example should suffice.
The Riemann Hypothesis is an example of a statement which
makes predictions about the results of computational
experiments. The Riemann Hypothesis asserts that if we
were to compute the value of one of the complex zeros of
the zeta function, then to within the accuracy of the
computation, the real part of that value will be 1/2. So
once we have ascertained that the zeros of the zeta
function can actually be computed to arbitrary precision
with a sensible amount of computation, then we clearly
see that the Riemann Hypothesis is making predictions
about the results of computing those zeros, and actually
doing the computations can be thought of as performing an
experiment. It makes sense to say that we can "observe"
the results of the experiment. And we can think of the
experiments taking place in a world of computation.
The power and utility and meaning of mathematics comes
from its ability to predict the results of computational
experiments. All of the mathematics which has the
potential to be applied to the task of modeling phenomena
in the real world necessarily has the ability to make
such predictions. The notion of "truth" in mathematics
has no concrete meaning without a connection to the
predictions of experimental results.
The astute reader will point out that some "mathematical"
statements cannot be interpreted as making predictions
about the results of computational experiments. For
example, the Continuum Hypothesis makes no such
predictions. And that's a good point; mathematics has
been infected by a virus, namely, Cantor's Theory.
Part II Pseudomathematics
Cantor's Theory (classical set theory) is a conceptual
framework which encompasses both mathematics (the science
of phenomena observable in the world of computation) and
a fabricated world of the super-infinite. It merges the
two worlds so cleverly and so seamlessly that those who
accept that framework of thought often lose the ability
to distinguish between the two worlds. (Note: henceforth,
I'll refer only to the fabricated world of the
super-infinite as Cantor's Theory)
Cantor's Theory is a pseudomathematics. It has the form
but not the function of mathematics. It follows the
definition-theorem-proof format of mathematics, but in
the end, the theorems have no interpretation as
predictions about the results of computational
experiments.
Cantor's Theory has been aptly described as "formal
operations on meaningless symbols". Formally, Cantor's
Theory is impeccable, but ultimately, it is absurd. It
builds upon notions of infinite sets and power sets of
those infinite sets, which it does not define in terms of
the reality that we observe (i.e. the world of
computatioin), but rather, it merely asserts their
existence by decree. The result is that the existential
quantifier becomes a meaningless symbol. To assert that
the infinite objects in Cantor's Theory "exist" is a lie.
Formal logic has the "garbage in, garbage out" property;
it does not have the power to transform meaningless
assumptions and symbols into meaningful assertions. The
"meaning" of mathematics comes from its ability to make
predictions about the results of computational
experiments. The theorems in Cantor's Theory are not
truths in any meaningful sense.
The problem with Cantor's Theory is not the notion of
infinity, nor the notion of power sets. A theory of the
infinite can be designed such that every concept and
every object in the theory has corresponding
approximations in the world of computation. In such a
theory, the notions of infinite sets and power sets will
be defined as abstractions derived from the finite world
we actually observe. Such a theory can serve as a
conceptual aid in mathematics, and is part of the science
of mathematics.
Mathematics has tremendous power to enlighten us about
the nature of reality. Cantor's Theory has no such power.
The only "power" Cantor's Theory has is the power to
bamboozle. The Cantorians have parlayed this power into
enormous political power within the mathematics
community. To maintain their power, the Cantorians must
impose their "theory" on everyone in the community, and
weed out from the community anyone who questions the
reality of the world implied to exist by their theory. In
general, a community which has a power structure built
around knowledge of a mythology about a world beyond what
can be observed is thought of as either a religious
community or a mystical cult, and either could describe
the Cantorian community.
The world of the super-infinite - a world lying beyond
the world that is accessible in the scientific paradigm -
is essentially theological, and it has theological
origins. Cantor himself was hoping to unify mathematics
and theology with his theory.
I'm hardly the first person to notice that there is no
mathematical content to Cantor's Theory. Consider the
following quote from a contemporary of Cantor:
"I don't know what predominates in Cantor's theory -
philosophy or theology, but I am sure that there is no
mathematics there" (Kronecker)
It would have been quite difficult in Cantor's time to
make an airtight case that Cantor's Theory is not
mathematics. At that time, one might have argued that
mathematics is a science which makes predictions about
the results of computational experiments, but if the
question were asked, "where do these experiments take
place?", the likely answer would have been, "in the
mind". But then it could have been argued that objects in
the theory of the infinite live in the imagination, and
hence in the mind, and hence computational experiments
and objects in the theory of the infinite have the same
ontological status.
Furthermore, in Cantor's time, the ability to compute was
so limited that the mathematics of the time was making
predictions about computational experiments which were
far beyond the ability of then current technology to
carry out, and so it might have seemed to the
mathematicians that the mathematics they were doing was
no more connected to reality than Cantor's Theory.
The computer revolution really has changed the way we
look at the world. The computer revolution has improved
our ability to compute by some 15 orders of magnitude,
which is a truly stunning number to any practical minded
person. It outdoes any other revolution in history. It
has opened up the world of computation to us. It has
given us a new paradigm for mathematics. It's changed our
thinking about the ontological status of mathematical
objects.
Mathematics can now be clearly seen as a science. The
computer serves as the mathematicians' microscope. It
serves as the mathematician's test tube in which
computational experiments are performed. The world of
computation - the world viewed through the
mathematicians' microscope - has an objective existence
independent of the mind, and mathematics is the science
which studies that world.
It should be emphasized that formalisms themselves are
objects that can be viewed in the mathematicians'
microscope. They are objects that are studied in the
science of mathematics. Hence, Cantor's Theory, as a
formalism, is something to be studied as part of
mathematics. But the formal theorems in Cantor's Theory
have no interpretation as theorems in the science of
mathematics. They do not make predictions about the
results of computational experiments. The objects in
Cantor's world of the infinite are merely figments of the
imagination. Cantor's Theory most certainly is not part
of the foundation of mathematics.
Mathematics belongs in the public arena. As long as we
agree that religion belongs in the private arena and not
in the public arena, as we do in the United States, then
mathematics must not include the Cantorian religion.
Part III Artificial Intelligence
When I was in graduate school, I thought it would be a
really cool idea to build a foundation for mathematics
incorporating as one of the cornerstones, the idea that
mathematical statements must make predictions about the
results of computational experiments. I believe it would
be a straightforward task to teach computers to
understand mathematics when it is built on such a
foundation, and then it would be seen that the foundation
of mathematics is also the foundation of artificial
intelligence. After all, mathematics is the tool we use,
and the tool computers could be using, to understand the
world in a precise and quantitative way. I believe that
making available a solid theoretical foundation for
artificial intelligence could have profound implications
for the future of mankind.
Even without the connection to artificial intelligence, I
believe I had a good idea. All of the mathematics which
has the potential to be applied must have the capacity
for making predictions, so I would have been building a
theoretical foundation for applied mathematics. If
nothing else, it could be a good pedagogical tool. And
failing that, intellectual curiosity alone should justify
pursuing such a goal.
Alas, the Cantorians told me essentially that my ideas
are worthless crackpot ideas with no connection to real
mathematics, and that I really have no right to pursue
such ideas until I first prove that I am fully competent
in Cantorian mathematics. It was discouraging. I ended up
believing that as long as Cantor's Theory is part of the
canon of the mathematics community, I don't fit in.
Maybe it's too much to ask that the current generation of
mathematicians should abandon Cantor's Theory. But it
can't possibly be too much to ask that the mathematicians
be tolerant of those students who have been deeply
affected by the radical technological changes taking
place in the world today.
As I've already pointed out, the computer revolution has
opened up the world of computation to us. Many of today's
students want to explore that world, and they see
mathematics as the science which studies that world. They
want to be mathematicians. So I'm asking that
mathematicians tolerate those students.
But even more significant is the dream of artificial
intelligence. Right now, it's only a dream, but it's a
powerful and compelling dream, and it has captured the
imaginations of some of the best and brightest students.
What the dream of artificial intelligence leads us to
believe is that there is no magic ingredient in the mind
which in not available to a computer. That is, there's an
equivalence between artificial intelligence and the mind,
and they both live in the world of computation; all of
the tools we need to understand the world around us, live
in the world of computation; all of the tools we need to
understand the world of computation itself, live in the
world of computation; mathematics lives in the world of
computation; the idea that there exists a world of the
super-infinite which has no connection to the world of
computation, is a fantasy; the Cantorian idea that we can
"prove" that such a world exists is preposterous. Again,
what I'm asking is that mathematicians tolerate the
students who hold that world view; encourage them, allow
them to be part of the mathematics community, don't think
of them as second rate minds, and don't call them
crackpots.
Part IV Babble
It's not the kind of thing I can prove beyond doubt, but
I believe that the mathematicians' adherence to Cantor's
Theory has held back progress in artificial intelligence
by as much as thirty years and counting. If mathematics
were built on a foundation which emphasized the
connection between mathematics and computation, then all
of the mathematical knowledge accumulated over the past
few thousand years would be immediately transferable to
the computer, and the computer would be then intelligent;
it would have the tools it needs to understand the
physical world, and it would have the tools it needs to
understand its own mind; it would be self-aware.
Computers simply need not understand the complexities and
nuances of human social behavior in order to be deemed
intelligent; the Turing Test is bunk (another of my pet
peeves).
There are lots of reasons for debunking pseudoscience. I
shouldn't have to elaborate on that for the relatively
sophisticated audience likely to be reading this article.
Sure, it's likely that those who have devoted themselves
to the pseudoscience will experience a stressfull period
of readjustment when that pseudoscience is debunked, and
we should do what we can to minimize that stress, but
that is no reason to refrain from the debunking process.
Cantor's Theory is pseudomathematics. It is
pseudoscience. It has theological origins. It's not part
of the foundation of mathematics. It doesn't belong in
the universities. It needs to be debunked.
Cantor's Theory has been debunked.
Jesse F. Hughes
> The essence of science is the ability to make predictions
> about the results of experiments. Mathematical statements
> can be interpreted as making predictions about the
> results of computational experiments, and hence,
> mathematics is a science. One example should suffice.
Sure, why *wouldn't* a single example suffice for this broad, sweeping
and utterly stupid claim?
--
"Looking at their behavior I see them endangering not only their own
futures, but that of their families, and now, considering my latest
result, the future of people all over the world." -- James S. Harris,
on the shortsightedness of his mathematical critics
Daryl McCullough
>There are lots of reasons for debunking pseudoscience.
Physician, heal thyself.
--
Daryl McCullough
Ithaca, NY
|-|erc
> Part IV Babble
>
>
> It's not the kind of thing I can prove beyond doubt, but
> I believe that the mathematicians' adherence to Cantor's
> Theory has held back progress in artificial intelligence
> by as much as thirty years and counting. If mathematics
You'll cop a lot of slack for writing the truth, when you challenge the Cantorians
they scare easiily, but they'll be back and in greater numbers. UH HA HA
Most people here think godels proof proves that AI is impossible!
Humans can understand why G is true, but a computer can never do so!
Incompleteness is another pseudo maths, or just plain nuts wrong,
The statement "You can't prove me" debunks AI?? AI is a bit
more advanced than the mathematicians here, "you can't prove me"
would only be true in a Belief Revision system, not in Truth Maintencance
when only proven facts hold.
Also, why are all the Cantorian supporters afraid to answer this question.
oo coin flippers flip coins oo times each. how many flips of the antidiagonal
have been flipped one after the other? INFINITY
Its quite easy to find counterexamples to popular pseudo maths, but then
you encounter the ambiguous arguments they believe, and the lies they cling to.
----------------------undisputed proof of a single oo type ----------------------
A sentence is any mathematical expression, probably allowing some pseudo english for
sufficient descripteness of any real you can think of.
like "antidiag(utm(number,digit)mod10)"
with an alphabet of 100 characters this is real number
114200904090107302120133014211302051833040907092032131504515032
"utm(100,digit)mod10)" is
2120133051505033040907092032131504515032
"utm(101,digit)mod10)" is
2120133051505133040907092032131504515032
that's 1, 2, 1/3, sqrt(2), pi, e, sin(0.5), antidiag, tau, omega, .. all on the one countable list.
"YOU CAN'T PROVE ME" debunks a complete formalism?? WHAT A LAUGH!!
Herc
fishfry
"david petry" <david_lawr...@yahoo.com> wrote:
> Part I Mathematics
>
>
> The essence of science is the ability to make predictions
> about the results of experiments. Mathematical statements
> can be interpreted as making predictions about the
> results of computational experiments
One need read no further. Crank alert.
conesetter
To Hughes, McCullough, fishfry: If you can answer petry the fair
thing would be to do so.
Torkel Franzen
> To Hughes, McCullough, fishfry: If you can answer petry the fair
> thing would be to do so.
He's been answered many times. His present tirade adds nothing to
earlier ones.
Torkel Franzen
>To Hughes, McCullough, fishfry: If you can answer petry the fair
>thing would be to do so.
His been answered many times. His present tirade adds nothing to
earlier ones.
Herman Jurjus
[snip]
> ... And that's a good point; mathematics has
> been infected by a virus, namely, Cantor's Theory.
Just FYI: it's said that there have been German (nazi) mathematicians in
the 1930s, who said they could recognize jewish mathematicians from
aryan ones, by looking only at their attitude towards Cantor's set theory.
If you meet a lot of opposition and disgust about your opinion regarding
Cantor, this may be partly due to that remembrance.
If you have a point (and i think you have, although i don't agree
completely), please consider to reformulate it, and avoid the name Cantor.
"Never give unsolicited advice" - Erasmus's advice
--
Cheers,
Herman Jurjus
Jesse F. Hughes
I'll pass. I've had all the long, dull conversations with Petry's
deep philosophical theories (roughly: computers good, Cantor bad) that
I care for.
The claim that mathematics is an experimental science and that a
single example would suffice to prove this point was funny. That's
all I wanted to point out.
If you want to engage Petry's big brane in philosophical discussion,
be my guest. I find it somewhat less enjoyable and considerably less
educational than licking razor blades.
--
Jesse F. Hughes
"To be honest, I don't have enough interest in math to spend the time
it would take to clean up the mess that I believe has been created in
the past 100 or so years." -- Curt Welch lets the world down.
robert j. kolker
david petry wrote:
> Cantor's Theory is pseudomathematics. It is
> pseudoscience. It has theological origins. It's not part
> of the foundation of mathematics. It doesn't belong in
> the universities. It needs to be debunked.
>
> Cantor's Theory has been debunked.
>
The currently accepted formalisms of set theory have not been shown to
produce contradictions. Until such a time, there is no reason to reject
it. This objection to Cantorian set theory goes back to the end of the
19-th century. Most mathematicians have ignoried and produced perfectly
good (which is to say, consisternt) mathematics.
If you want to give up over half the theorem in analysis that depend in
some fashion on the axiom of choice, or non-constructive methods, go
right ahead. The loss is yours.
Bob Kolker
Larry Hammick
> > Part I Mathematics
> >
> >
> > The essence of science is the ability to make predictions
> > about the results of experiments. Mathematical statements
> > can be interpreted as making predictions about the
> > results of computational experiments
>
> One need read no further. Crank alert.
subject line. And then the use of the very famous RH as
the only necessary illustration of something.
For a long time, there were cranks who insisted they could
trisect an angle with a ruler and compass. Today the computer
had taken over the role of those instruments. If there's not
enough RAM in the universe to distinguish countable from
uncountable, or .9999... from 1, then they must the same, right?
LH
robert j. kolker
conesetter wrote:
>
> To Hughes, McCullough, fishfry: If you can answer petry the fair
> thing would be to do so.
Here is your answer. In its currently accepted form Cantorian set theory
produces no known contradictions. Consistency is the only requirement
for a mathematical theory. Not finite constructability. Not ease of
intuitive grasp. Just plain old logical consistency.
Bob Kolker
>
David C. Ullrich
<cones...@btopenworld.com> wrote:
The above _was_ an answer, albeit expressed a little tersely
and ironically. A more verbose and explicit version of the same
answer is this:
>> > The essence of science is the ability to make predictions
>> > about the results of experiments. Mathematical statements
>> > can be interpreted as making predictions about the
>> > results of computational experiments
Oh yeah? Sez you.
Or an even more verbose and explicit form:
It's simply not true that all mathematical statements can
be interpreted as making predictions about the results
of computational experiments. Ok, whether that's true
or not depends on the definition of the word "mathematics",
and giving such a definition that will make everybody
happy is an unsolved problem. But it's not true of
mathematics as the term is commonly understood by
mathematicians - Petry's proof that set theory is
pseudomathematics relies on a non-standard definition
of the term.
I mean really. If I define mathematics to be my
microwave oven then calculus is not part of mathematics.
So what? Mathematics is _not_ my microwave oven;
a dog has four legs even if you _call_ a tail a leg.
************************
David C. Ullrich
Larry Hammick
"Larry Hammick" <larryh...@OMIT-MEtelus.net> wrote in message
news:Q0oPd.27725$K54.2590@edtnps84...
show that .9999... equals 1, then they must be different. Ah, that's
better. Glad I got back to this error before DLP spotted it :)
Daryl McCullough
> To Hughes, McCullough, fishfry: If you can answer petry the fair
>thing would be to do so.
He's been answered many times. The summary:
1. No, Cantorian set theory is not theology.
2. No, the purpose of mathematics is not to predict the results of
computations.
3. No, the reason that AI hasn't made more progress is not because
of Cantorian set theory.
David's claims to the contrary are just bizarre.
robert j. kolker
Daryl McCullough wrote:
> 3. No, the reason that AI hasn't made more progress is not because
> of Cantorian set theory.
The reason for lack of progress in AI is simple and clear. We do not yet
know what Real Intelligence is. How can one make a similacarum if one
does not have the Real Thing at hand?
Bob Kolker
Mike Oliver
> david petry wrote:
> [snip]
>
>> ... And that's a good point; mathematics has
>> been infected by a virus, namely, Cantor's Theory.
>
> Just FYI: it's said that there have been German (nazi) mathematicians in
> the 1930s, who said they could recognize jewish mathematicians from
> aryan ones, by looking only at their attitude towards Cantor's set theory.
Is that true? It's a new one on me. A little surprising, given
that Cantor wasn't Jewish. Cantor was, I believe, Lutheran, though
with strong pro-Catholic leanings. Dauben suggests that one of
his main goals was to save the Catholic church from the errors that
Aquinas (I think) had introduced into its ideas about the infinite.
(A noble goal, I'd say, and I'm not Catholic either.)
He did have a very Jewish sounding *name*, and may have had
Jewish ancestors, so it's certainly possible that that was
more than enough for the Nazis.
robert j. kolker
Mike Oliver wrote:
>
> He did have a very Jewish sounding *name*, and may have had
> Jewish ancestors, so it's certainly possible that that was
> more than enough for the Nazis.
In 1937 one third of the Germans had some Jewish ancestory. Jews had
lived in what become Germany for nearly six hundred years, so this can
hardly be surprising.
Cantor might have had Jewish grandparents on his father's side (hence
the name) but he himself was Lutheran. Look at the composer Felix
Mendelson. His grandpa was the famous rabbi Moses Mendelson, but Felix
was a Christian.
Bob Kolker
LordBeotian
"david petry" <david_lawr...@yahoo.com> ha scritto
> The astute reader will point out that some "mathematical"
> statements cannot be interpreted as making predictions
> about the results of computational experiments. For
> example, the Continuum Hypothesis makes no such
> predictions. And that's a good point; mathematics has
> been infected by a virus, namely, Cantor's Theory.
CH (like AC) don't seems to make a prediction about a computation but could
be used to make a prediction that you could be unable to make without it.
Torkel Franzen
> CH (like AC) don't seems to make a prediction about a computation but could
> be used to make a prediction that you could be unable to make
> without it.
CH and AC have no arithmetical consequences not provable without them.
Mike Oliver
But that doesn't refute what LordBeotian said, which at least on
its face is not about what can be formally proved, but about the
predictions you would in fact be able to make.
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
> > CH (like AC) don't seems to make a prediction about a computation but
could
> > be used to make a prediction that you could be unable to make
> > without it.
>
> CH and AC have no arithmetical consequences not provable without them.
Why?
Does it come from Turing theorem obout ordinal logic?
However it could take much longer to find the proof without AC or CH,
couldn't it?
Suppose for example that the first integer that does not satisfy collatz
conjecture is bigger than the longest integer ever writable in the life of
the earth, you could in principle prove that in PA but maybe not in
practice. Maybe this could be prove in a reasonable time using some
nonconstructive principle.
JXStern
<david_lawr...@yahoo.com> wrote:
>The essence of science is the ability to make predictions
>about the results of experiments.
OK, science as instrumentalism.
> Mathematical statements
>can be interpreted as making predictions about the
>results of computational experiments, and hence,
>mathematics is a science.
NS. Mathematics is the tool, not the craft. A hammer is not
architecture.
Penmanship can be interpreted as making predictions about experiments,
therefore penmanship is science? No, I don't think so.
>What the dream of artificial intelligence leads us to
>believe is that there is no magic ingredient in the mind
>which in not available to a computer. That is, there's an
>equivalence between artificial intelligence and the mind,
>and they both live in the world of computation; all of
>the tools we need to understand the world around us, live
>in the world of computation; all of the tools we need to
>understand the world of computation itself, live in the
>world of computation; mathematics lives in the world of
>computation;
I agree with all that.
> the idea that there exists a world of the
>super-infinite which has no connection to the world of
>computation, is a fantasy; the Cantorian idea that we can
>"prove" that such a world exists is preposterous.
Hey, who cares if it exists? Work on AI within the bounds of
constructivism and computation and instrumentalism.
> Again,
>what I'm asking is that mathematicians tolerate the
>students who hold that world view; encourage them, allow
>them to be part of the mathematics community, don't think
>of them as second rate minds, and don't call them
>crackpots.
Well now, this debate has been going on since at least Brouwer around
1900, if not Lucretius 50 BCE.
>Cantor's Theory is pseudomathematics. It is
>pseudoscience. It has theological origins. It's not part
>of the foundation of mathematics. It doesn't belong in
>the universities. It needs to be debunked.
It's non-constructive. That's an issue.
I agree that only constructive mathematics has anything foundational
to say about AI.
J.
JXStern
Just to pick a nit, maybe making a simulacrum (sic) is not the point.
And I've come around to believe that we know even less about what
computation is, than we know about what real intelligence is.
Daryl pretty well does answer Petry's complaint, yet I have a lot of
sympathy for some of Petry's points, and for the relevance of the
constructivist viewpoint in regards to AI, which, in a charitable
reading, is what Petry is going on about.
J.
JXStern
<larryh...@OMIT-MEtelus.net> wrote:
>Oops, that didn't come out right :) If there's not enough RAM to
>show that .9999... equals 1, then they must be different. Ah, that's
>better. Glad I got back to this error before DLP spotted it :)
OK, I'm cancelling my order for more RAM.
J.
Babylonian
calculations through a technique called "Lazy evaluation". They are
functional languages like LISP, and might be useful in your work.
http://en.wikipedia.org/wiki/Lazy_evaluation
Whether or not they can ever share anything with cardinal theory, I
have no idea, but nobody regards cardinal thoery as "the foundation of
mathematics".
Neil W Rickert
>Daryl pretty well does answer Petry's complaint, yet I have a lot of
>sympathy for some of Petry's points, and for the relevance of the
>constructivist viewpoint in regards to AI, which, in a charitable
>reading, is what Petry is going on about.
You are far too charitable to Petry, who is a well known crank.
Petry asks that his world view of mathematics be respected. Yet he
does not himself respect views other than his own, as his use of
"pseudomathematics" demonstrates.
Petry's version of what constitutes mathematics would seem to leave
out geometry.
David C. Ullrich
wrote:
>david petry wrote:
>[snip]
>
> > ... And that's a good point; mathematics has
>> been infected by a virus, namely, Cantor's Theory.
>
>Just FYI: it's said that there have been German (nazi) mathematicians in
>the 1930s, who said they could recognize jewish mathematicians from
>aryan ones, by looking only at their attitude towards Cantor's set theory.
Note that the above is not what I'm referring to below:
>If you meet a lot of opposition and disgust about your opinion regarding
>Cantor, this may be partly due to that remembrance.
This is ridiculous - the reason his views meet with a lot of
opposition is because his views are mistaken.
And the idea that the people here are saying he's wrong for any reason
other than the fact that he's wrong is just a touch insulting.
>If you have a point (and i think you have, although i don't agree
>completely), please consider to reformulate it, and avoid the name Cantor.
>
> "Never give unsolicited advice" - Erasmus's advice
************************
David C. Ullrich
Daryl McCullough
>Daryl pretty well does answer Petry's complaint, yet I have a lot of
>sympathy for some of Petry's points, and for the relevance of the
>constructivist viewpoint in regards to AI, which, in a charitable
>reading, is what Petry is going on about.
There is nothing whatsoever getting in the way of using constructive
mathematics in the development of AI. That's what's so bizarre about
David Petry's diatribe. Nobody is *stopping* him from using only
computable mathematics.
I think maybe what he is saying is that he thinks AI could be solved
if it weren't for the fact that so many brilliant people are wasting
time doing something else. Maybe so, but it seems to me that Petry's
attitude is basically totalitarian---nobody should work on anything
except problems that Petry thinks are worth working on.
I really don't think that constructive mathematics has any relevance
to AI. The difficulties in achieving AI are not mathematical
difficulties.
Lester Zick
A little imagination might help. Basically you're indicating we can't
do it if it hasn't been done. Reminisces me of defining addition in
terms of incrementation and subtraction in terms of decrementation.
Bit circular, what?
Regards - Lester
JXStern
McCullough) wrote:
>>Daryl pretty well does answer Petry's complaint, yet I have a lot of
>>sympathy for some of Petry's points, and for the relevance of the
>>constructivist viewpoint in regards to AI, which, in a charitable
>>reading, is what Petry is going on about.
>
>There is nothing whatsoever getting in the way of using constructive
>mathematics in the development of AI. That's what's so bizarre about
>David Petry's diatribe. Nobody is *stopping* him from using only
>computable mathematics.
Agreed.
>I think maybe what he is saying is that he thinks AI could be solved
>if it weren't for the fact that so many brilliant people are wasting
>time doing something else. Maybe so, but it seems to me that Petry's
>attitude is basically totalitarian---nobody should work on anything
>except problems that Petry thinks are worth working on.
It's easy to get carried away in newsgroup postings, I'm not all that
familiar with the gentleman's positions, and it's the nature of
research that a lot of it goes off in the wrong direction, no use
getting excited about that.
>I really don't think that constructive mathematics has any relevance
>to AI. The difficulties in achieving AI are not mathematical
>difficulties.
Well, I did say a "constructivist viewpoint", there's no results of
constructivist mathematics that I have in mind when saying that, I
just want to suggest that the process of computation is a constructive
one, and that anything that we hope or expect or succeed in doing with
computation is therefore going to be described best in a
constructivist manner. So at the least, I agree that any problems
with AI are not mathematical ones, not even finite or constructivist
ones of combinatorics or scale.
J.
Zim Olson
>
>
>Part I Mathematics
>
>
>The essence of science is the ability to make predictions
>can be interpreted as making predictions about the
>results of computational experiments, and hence,
>
>The Riemann Hypothesis is an example of a statement which
>makes predictions about the results of computational
>experiments. The Riemann Hypothesis asserts that if we
>were to compute the value of one of the complex zeros of
>the zeta function, then to within the accuracy of the
>computation, the real part of that value will be 1/2. So
>once we have ascertained that the zeros of the zeta
>function can actually be computed to arbitrary precision
>with a sensible amount of computation, then we clearly
>see that the Riemann Hypothesis is making predictions
>about the results of computing those zeros, and actually
>doing the computations can be thought of as performing an
>experiment. It makes sense to say that we can "observe"
>the results of the experiment. And we can think of the
>experiments taking place in a world of computation.
>
>The power and utility and meaning of mathematics comes
>from its ability to predict the results of computational
>experiments. All of the mathematics which has the
>potential to be applied to the task of modeling phenomena
>in the real world necessarily has the ability to make
>such predictions. The notion of "truth" in mathematics
>has no concrete meaning without a connection to the
>predictions of experimental results.
>
>The astute reader will point out that some "mathematical"
>statements cannot be interpreted as making predictions
>about the results of computational experiments. For
>example, the Continuum Hypothesis makes no such
>been infected by a virus, namely, Cantor's Theory.
>
>
>
>
>Cantor's Theory (classical set theory) is a conceptual
>framework which encompasses both mathematics (the science
>of phenomena observable in the world of computation) and
>a fabricated world of the super-infinite. It merges the
>two worlds so cleverly and so seamlessly that those who
>accept that framework of thought often lose the ability
>to distinguish between the two worlds. (Note: henceforth,
>I'll refer only to the fabricated world of the
>super-infinite as Cantor's Theory)
>
>Cantor's Theory is a pseudomathematics. It has the form
>but not the function of mathematics. It follows the
>definition-theorem-proof format of mathematics, but in
>the end, the theorems have no interpretation as
>predictions about the results of computational
>
>Cantor's Theory has been aptly described as "formal
>operations on meaningless symbols". Formally, Cantor's
>Theory is impeccable, but ultimately, it is absurd. It
>builds upon notions of infinite sets and power sets of
>those infinite sets, which it does not define in terms of
>the reality that we observe (i.e. the world of
>computatioin), but rather, it merely asserts their
>existence by decree. The result is that the existential
>quantifier becomes a meaningless symbol. To assert that
>the infinite objects in Cantor's Theory "exist" is a lie.
>
>Formal logic has the "garbage in, garbage out" property;
>it does not have the power to transform meaningless
>assumptions and symbols into meaningful assertions. The
>"meaning" of mathematics comes from its ability to make
>predictions about the results of computational
>experiments. The theorems in Cantor's Theory are not
>truths in any meaningful sense.
>
>The problem with Cantor's Theory is not the notion of
>infinity, nor the notion of power sets. A theory of the
>infinite can be designed such that every concept and
>every object in the theory has corresponding
>approximations in the world of computation. In such a
>theory, the notions of infinite sets and power sets will
>be defined as abstractions derived from the finite world
>we actually observe. Such a theory can serve as a
>conceptual aid in mathematics, and is part of the science
>of mathematics.
>
>Mathematics has tremendous power to enlighten us about
>the nature of reality. Cantor's Theory has no such power.
>The only "power" Cantor's Theory has is the power to
>bamboozle. The Cantorians have parlayed this power into
>enormous political power within the mathematics
>community. To maintain their power, the Cantorians must
>impose their "theory" on everyone in the community, and
>weed out from the community anyone who questions the
>reality of the world implied to exist by their theory. In
>general, a community which has a power structure built
>around knowledge of a mythology about a world beyond what
>can be observed is thought of as either a religious
>community or a mystical cult, and either could describe
>the Cantorian community.
>
>The world of the super-infinite - a world lying beyond
>the world that is accessible in the scientific paradigm -
>is essentially theological, and it has theological
>origins. Cantor himself was hoping to unify mathematics
>and theology with his theory.
>
>I'm hardly the first person to notice that there is no
>mathematical content to Cantor's Theory. Consider the
>following quote from a contemporary of Cantor:
>
>"I don't know what predominates in Cantor's theory -
>philosophy or theology, but I am sure that there is no
>mathematics there" (Kronecker)
>
>It would have been quite difficult in Cantor's time to
>make an airtight case that Cantor's Theory is not
>mathematics. At that time, one might have argued that
>mathematics is a science which makes predictions about
>the results of computational experiments, but if the
>question were asked, "where do these experiments take
>place?", the likely answer would have been, "in the
>mind". But then it could have been argued that objects in
>the theory of the infinite live in the imagination, and
>hence in the mind, and hence computational experiments
>and objects in the theory of the infinite have the same
>ontological status.
>
>Furthermore, in Cantor's time, the ability to compute was
>so limited that the mathematics of the time was making
>predictions about computational experiments which were
>far beyond the ability of then current technology to
>carry out, and so it might have seemed to the
>mathematicians that the mathematics they were doing was
>no more connected to reality than Cantor's Theory.
>
>The computer revolution really has changed the way we
>look at the world. The computer revolution has improved
>our ability to compute by some 15 orders of magnitude,
>which is a truly stunning number to any practical minded
>person. It outdoes any other revolution in history. It
>has opened up the world of computation to us. It has
>given us a new paradigm for mathematics. It's changed our
>thinking about the ontological status of mathematical
>objects.
>
>Mathematics can now be clearly seen as a science. The
>computer serves as the mathematicians' microscope. It
>serves as the mathematician's test tube in which
>computational experiments are performed. The world of
>computation - the world viewed through the
>mathematicians' microscope - has an objective existence
>independent of the mind, and mathematics is the science
>which studies that world.
>
>It should be emphasized that formalisms themselves are
>objects that can be viewed in the mathematicians'
>microscope. They are objects that are studied in the
>science of mathematics. Hence, Cantor's Theory, as a
>formalism, is something to be studied as part of
>mathematics. But the formal theorems in Cantor's Theory
>have no interpretation as theorems in the science of
>mathematics. They do not make predictions about the
>results of computational experiments. The objects in
>Cantor's world of the infinite are merely figments of the
>imagination. Cantor's Theory most certainly is not part
>of the foundation of mathematics.
>
>agree that religion belongs in the private arena and not
>in the public arena, as we do in the United States, then
>mathematics must not include the Cantorian religion.
>
>
>Part III Artificial Intelligence
>
>
>When I was in graduate school, I thought it would be a
>really cool idea to build a foundation for mathematics
>incorporating as one of the cornerstones, the idea that
>mathematical statements must make predictions about the
>results of computational experiments. I believe it would
>be a straightforward task to teach computers to
>understand mathematics when it is built on such a
>foundation, and then it would be seen that the foundation
>of mathematics is also the foundation of artificial
>intelligence. After all, mathematics is the tool we use,
>and the tool computers could be using, to understand the
>world in a precise and quantitative way. I believe that
>making available a solid theoretical foundation for
>artificial intelligence could have profound implications
>for the future of mankind.
>
>Even without the connection to artificial intelligence, I
>believe I had a good idea. All of the mathematics which
>has the potential to be applied must have the capacity
>for making predictions, so I would have been building a
>theoretical foundation for applied mathematics. If
>nothing else, it could be a good pedagogical tool. And
>failing that, intellectual curiosity alone should justify
>pursuing such a goal.
>
>Alas, the Cantorians told me essentially that my ideas
>are worthless crackpot ideas with no connection to real
>mathematics, and that I really have no right to pursue
>such ideas until I first prove that I am fully competent
>in Cantorian mathematics. It was discouraging. I ended up
>believing that as long as Cantor's Theory is part of the
>canon of the mathematics community, I don't fit in.
>
>Maybe it's too much to ask that the current generation of
>mathematicians should abandon Cantor's Theory. But it
>can't possibly be too much to ask that the mathematicians
>be tolerant of those students who have been deeply
>affected by the radical technological changes taking
>place in the world today.
>
>As I've already pointed out, the computer revolution has
>opened up the world of computation to us. Many of today's
>students want to explore that world, and they see
>mathematics as the science which studies that world. They
>want to be mathematicians. So I'm asking that
>mathematicians tolerate those students.
>
>But even more significant is the dream of artificial
>intelligence. Right now, it's only a dream, but it's a
>powerful and compelling dream, and it has captured the
>imaginations of some of the best and brightest students.
>What the dream of artificial intelligence leads us to
>believe is that there is no magic ingredient in the mind
>which in not available to a computer. That is, there's an
>equivalence between artificial intelligence and the mind,
>and they both live in the world of computation; all of
>the tools we need to understand the world around us, live
>in the world of computation; all of the tools we need to
>understand the world of computation itself, live in the
>world of computation; mathematics lives in the world of
>super-infinite which has no connection to the world of
>computation, is a fantasy; the Cantorian idea that we can
>what I'm asking is that mathematicians tolerate the
>students who hold that world view; encourage them, allow
>them to be part of the mathematics community, don't think
>of them as second rate minds, and don't call them
>crackpots.
>
>
>
>
>It's not the kind of thing I can prove beyond doubt, but
>I believe that the mathematicians' adherence to Cantor's
>Theory has held back progress in artificial intelligence
>by as much as thirty years and counting. If mathematics
>were built on a foundation which emphasized the
>connection between mathematics and computation, then all
>of the mathematical knowledge accumulated over the past
>few thousand years would be immediately transferable to
>the computer, and the computer would be then intelligent;
>it would have the tools it needs to understand the
>physical world, and it would have the tools it needs to
>understand its own mind; it would be self-aware.
>Computers simply need not understand the complexities and
>nuances of human social behavior in order to be deemed
>intelligent; the Turing Test is bunk (another of my pet
>peeves).
>
>There are lots of reasons for debunking pseudoscience. I
>shouldn't have to elaborate on that for the relatively
>sophisticated audience likely to be reading this article.
>Sure, it's likely that those who have devoted themselves
>to the pseudoscience will experience a stressfull period
>of readjustment when that pseudoscience is debunked, and
>we should do what we can to minimize that stress, but
>that is no reason to refrain from the debunking process.
>
>Cantor's Theory is pseudomathematics. It is
>pseudoscience. It has theological origins. It's not part
>of the foundation of mathematics. It doesn't belong in
>the universities. It needs to be debunked.
>
Hello:
As far as Pseudo Mathematics goes. I think Modern Mathematics could be
also put in that category. See my thoughts on Math. Axioms at:
http://www.zimmathematics.com/htm/creati.htm
I think the whole notion of what an Axiom is or at least it's role in
Mathematics in not common knowledge even to Mathematicians. I guess my
wisdom/message for the moment to Mathematicians is "Chill Out".
Sometimes Axioms are Axioms and sometimes they aren't, even in
Mathematics. Imagine that! I don't think we are going to be able to
define ourselves to Mathematics Narvana.
All I do , is see what I can dream up, and see if it makes sense and
works. That is my general working philosophy in Mathematics for now.
I have a Creative Mathematics site at:
Zim Olson
|-|erc
"Daryl McCullough" ,
oo coin flippers flip coins oo times each. how many flips of the antidiagonal
have been flipped one after the other?
ANSWERS NOT EXCUSES
Herc
|-|erc
"Jesse F. Hughes" ,
|-|erc
"Zim Olson" ,
|-|erc
"JXStern" ,
|-|erc
"LordBeotian" ,
|-|erc
"Mike Oliver" ,
|-|erc
"robert j. kolker" ,
|-|erc
|-|erc
"David C. Ullrich" ,
|-|erc
"LordBeotian" ,
|-|erc
"Torkel Franzen" ,
|-|erc
"Mike Oliver" ,
|-|erc
"Larry Hammick" ,
|-|erc
|-|erc
"Herman Jurjus" ,
|-|erc
"JXStern" ,
|-|erc
"David C. Ullrich" ,
|-|erc
"JXStern" ,
|-|erc
"Lester Zick" ,
|-|erc
"Daryl McCullough" ,
Shmuel (Seymour J.) Metz
at 09:39 PM, zimmath...@yahoo.com (Zim Olson) said:
>As far as Pseudo Mathematics goes. I think Modern Mathematics could
>be also put in that category. See my thoughts
ITYM delusions.
>Sometimes Axioms are Axioms and sometimes they aren't,
That depends on what is is.
>All I do , is see what I can dream up, and see if it makes sense and
> works.
For purely idiosyncratic meanings of "makes sense" and "works".
--
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
Torkel Franzen
> But that doesn't refute what LordBeotian said, which at least on
> its face is not about what can be formally proved, but about the
> predictions you would in fact be able to make.
True. Results about formal derivability doesn't tell us what
mathematics is actually useful or indeed indispensable.
Stephen Harris
"Mike Oliver" <mike_...@verizon.net> wrote in message
news:376obrF...@individual.net...
> Herman Jurjus wrote:
>> david petry wrote:
>> [snip]
>>
>>> ... And that's a good point; mathematics has
>>> been infected by a virus, namely, Cantor's Theory.
>>
>> the 1930s, who said they could recognize jewish mathematicians from aryan
>> ones, by looking only at their attitude towards Cantor's set theory.
>
> Is that true? It's a new one on me. A little surprising, given
> that Cantor wasn't Jewish. Cantor was, I believe, Lutheran, though
> with strong pro-Catholic leanings. Dauben suggests that one of
> his main goals was to save the Catholic church from the errors that
> Aquinas (I think) had introduced into its ideas about the infinite.
> (A noble goal, I'd say, and I'm not Catholic either.)
>
> He did have a very Jewish sounding *name*, and may have had
> Jewish ancestors, so it's certainly possible that that was
> more than enough for the Nazis.
His father was Lutheran and his mother Catholic with some
racial Jew in his ancestry. Saying that he was Catholic doesn't
indicate much. Quite a few Jews changed their names and
became Catholics because they were discriminated against
for university positions. Tarski for instance whose father's
name was Ignacy Teitelbaum, became a Catholic. Sol
Feferman isn't very fond of Cantor's Set Theory so I don't
think ancestry much to do with attitudes towards Set Theory.
Torkel Franzen
> Sol Feferman isn't very fond of Cantor's Set Theory [...]
On what do you base this?
Torkel Franzen
> Does it come from Turing theorem obout ordinal logic?
No, from Gödel's work on the constructible hierarchy. The full story
is told by Platek in a paper in the Journal of Symbolic Logic,
vol 34 no 2 1969.
> However it could take much longer to find the proof without AC or CH,
> couldn't it?
In the particular case of AC or CH, there is no significant increase
in the length of proofs. But we know that extending a theory T by a
statement A undecidable in T will shorten the proofs of some
statements to any extent, in the sense that for any recursive function
f there is a statement B such that f(p)<p', where p is the length of
the shortest proof of B in T+A and p' the length of the shortest proof
of B in T. (Gödel was the first to prove a result of this kind.)
Friedman has given dramatic actual examples of how non-finitary
principles make it possible to prove statements that otherwise have no
feasible proof. And of course we also know that there are statements
that cannot be proved without such principles.
Jesse F. Hughes
> "He argues that the freedom provided by Cantorian set theory was
> purchased at a heavy philosophical price, namely adherence to a form
> of mathematical platonism that is difficult to support."
Feferman thinks that the mathematical theory ZF somehow requires
Platonism?
Bizarre. I'll have to put this book on my list, because I sure can't
guess what his argument might be.
--
Jesse F. Hughes
"Contrariwise," continued Tweedledee, "if it was so, it might be, and
if it were so, it would be; but as it isn't, it ain't. That's logic!"
-- Lewis Carroll
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
> Friedman has given dramatic actual examples of how non-finitary
> principles make it possible to prove statements that otherwise have no
> feasible proof. And of course we also know that there are statements
> that cannot be proved without such principles.
What infinitary principles are you referring to?
Is what you say true for arithmetical statements?
Thanks
Stephen Harris
"Torkel Franzen" <tor...@sm.luth.se> wrote in message
news:vcbzmy8...@beta19.sm.ltu.se...
> "Stephen Harris" <cyberguard...@yahoo.com> writes:
>
>> Sol Feferman isn't very fond of Cantor's Set Theory [...]
>
> On what do you base this?
>
>
Much briefer than the longer post I sent before:
http://math.stanford.edu/~feferman/book98.html
In the Light of Logic. Author: Solomon Feferman. (Oxford University Press,
1998, ISBN 0-19-508030-0, Logic and Computation in Philosophy series)
Description from the jacket flap:
"In this collection of essays written over a period of twenty years, Solomon
Feferman explains advanced results in modern logic and employs them to cast
light on significant problems in the foundations of mathematics. Most
troubling among these is the revolutionary way in which Georg Cantor
elaborated the nature of the infinite, and in doing so helped transform the
face of twentieth-century mathematics. Feferman details the development of
Cantorian concepts and the foundational difficulties they engendered. He
Stephen Harris
>
>> Sol Feferman isn't very fond of Cantor's Set Theory [...]
>
ad(ii) "The platonist position does not necessarily require the
existence of infinite sets ...(but of course that is essential to
Cantorian set theory.)"
"Is Cantor Necessary?," is answered with a resounding "no."
> On what do you base this?
>
>
SH: I've been reading a lot of Feferman lately and have a directory
for him as well as his book "In The Light Of Logic". I think Feferman
is not a platonist and he thinks Cantorian set theory requires platonism
when he quotes Cohen's reasons for rejecting the platonist realist
position and accepting a formalist position in the set-theoretical
foundations of mathematics.
Yes, it is in chapter 2 of "In The Light of Logic" (page 30)
"Infinity In Mathematics: Is Cantor Necessary?
"In order to explain the objections to Cantor's ideas in mathematics
that lead one to search for viable alternatives, one must first provide
some understanding of their nature and use. This will done here more
or less historically though necessarily in outline; we start with a rather
innocent looking problem about the existence of certain special kinds
of numbers 2."
*the actual infinite is not required for the mathematics of the physical
world*
Page 44:
ad(i). "Views of mathematical objects as independently existing abstract
entitities are generally called a form of _platonism_. In its particular
set-theoretical manifestation, this reveals itself most obviously in such
principles as the Axiom of Extensionality and the Axiom of Choice. *
ad(ii). The platonist position _per se_ does not necessarily require the
existence of infinite sets (or, for that matter, of sets at all, but of
course
that is essential to Cantorian set theory.
--------------------------------------
http://wwwpersonal.ksu.edu/~aarana/papers/FefermanReviewMathIntelligencer.pdf
Review of "In The Light Of Logic"
"In Part I, Feferman raises as a problem the role of transfinite
set theory in mathematics. Since transfinite sets are supposed
to be infinite objects about which facts are true independently of
our abilities to verify them, it seems that these abstract entities
must exist independently of human thoughts or constructions. This
family of beliefs about sets is frequently called platonism.
Feferman finds platonism philosophically unsatisfying, and thus
presents three projects aimed at avoiding platonism:
L.E.J. Brouwer's 'intuitionism', David Hilbert's 'finitism', and
Hermann Weyl's 'predicativism'. [SH: Feferman prefers Weyl.]
In feferman.unlimited.pdf "Some Formal Systems For The Unlimited
Theory of Structures and Categories"
On page 1 he lists some informal statements in the "naive theory
of structures and categories. On page 2 Feferman writes:
"These sorts of statements cannot be accounted for directly in
currently (generally) accepted mathematics, i.e. as formulated
in systems such as ZF or its immediate extensions. But they do not
have an unreasonable or "cooked-up" look. Each of them arises
as a natural continuation of ordinary mathematical talk about
structures (in particular, categories)."
* Franzen wrote: "CH and AC have no arithmetical consequences not provable
without them."
SH: I've read that they are independent.
http://math.stanford.edu/~feferman/book98.html
"In his concluding chapters, Feferman uses tools from the special
part of logic called proof theory to explain how the vast part, if
not all of scientifically applicable mathematics can be justified
on the basis of purely arithmetical principles. At least to that
extent, the question raised in two of the essays of the volume,
"Is Cantor Necessary?," is answered with a resounding "no."
SH: Also pages 68 and 69 of In the Light of Logic seem relevant.
Regards,
Stephen
Torkel Franzen
> Then the FAQ is mistaken?
No, the statements cited are not arithmetical.
Stephen Harris
> "LordBeotian" <poki...@CANCELLAMIyahoo.it> writes:
>
>> CH (like AC) don't seems to make a prediction about a computation but
>> could
>> be used to make a prediction that you could be unable to make
>> without it.
>
> CH and AC have no arithmetical consequences not provable without them.
>
>
http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/axiomchoice.html
Then the FAQ is mistaken?
And these statements depend on AC (i.e., they cannot be proved in ZF
without AC):
* The union of countably many countable sets is countable.
* Every infinite set has a denumerable subset.
* The Loewenheim-Skolem Theorem: Any first-order theory which has a
model has a denumerable model.
* The Baire Category Theorem: The reals are not the union of
countably many nowhere dense sets (i.e., the reals are not
meager).
* The Ultrafilter Theorem: Every Boolean algebra has an ultrafilter
on it.
Torkel Franzen
> What infinitary principles are you referring to?
> Is what you say true for arithmetical statements?
By the incompleteness theorem, statements asserting the existence of
large infinite sets have new arithmetical consequences.
Stephen Harris
> "Stephen Harris" <cyberguard...@yahoo.com> writes:
>
>> Sol Feferman isn't very fond of Cantor's Set Theory [...]
>
> On what do you base this?
>
"In The Light of Logic" by Sol Feferman page 248:
Infinity in Mathematics: Is Cantor Necessary? (conclusion)
"Independently of such detailed work which puts into question
the necessity of higher set theory for everyday mathematics*,
I am convinced that the platonism which underlies Cantorian
set theory is utterly unsatisfactory as a philosophy or our
subject, despite the apparent coherence of current set-theoretical
conceptions and methods. To echo Weyl, platonism is the
medieval metaphysics of mathematics; surely we can do better."
SH: It took me a few hours to find something compact and convincing.
Regards,
Stephen
Torkel Franzen
> SH: It took me a few hours to find something compact and convincing.
Yes, that certainly justifies describing Feferman as "not very fond
of" Cantor's set theory!
Mike Oliver
> [Cantor's] father was Lutheran and his mother Catholic with some
> racial Jew in his ancestry. Saying that he was Catholic doesn't
> indicate much.
Maybe that's why I didn't say it, then. Cantor was philo-Catholic,
not Catholic.
> Quite a few Jews changed their names and
> became Catholics because they were discriminated against
> for university positions.
Just about the opposite of Cantor's case in both senses. Rather
than becoming Catholic without genuine Catholic sympathies, he
*didn't* become Catholic *in*spite* of genuine Catholic sympathies.
And he *kept* a Jewish-sounding surname even though he wasn't
Jewish.
Stephen Harris
The point of my post was about ancestralism:
> Just FYI: it's said that there have been German (nazi) mathematicians in
> the 1930s, who said they could recognize jewish mathematicians from aryan
> ones, by looking only at their attitude towards Cantor's set theory.
Oliver wrote:
Is that true? It's a new one on me. A little surprising, given
that Cantor wasn't Jewish. Cantor was, I believe, Lutheran, ...
SH wrote: Saying that he was Catholic doesn't indicate much.
I meant that religion wasn't the key issue. Ancestry was the key issue.
I think about 1/3 of Germans had Jewish ancestry with the density
higher in cities. Cantor probably would have been included in the
pogrom because of his ancestry, not his current religious taste.
My post was actually about recognizing jewish mathematicians
from aryan due to their attitude toward Cantor's set theory. A
Nazi would primarily care about the bloodline, not the religion.
A Nazi might also expect Jewish mathematicains to react favorably
to Cantor's set theory, after all they ^^ or the Nazis might not know
what religion Cantor was or that he was say only 1/8 genetically Jewish.
I was pointing out that there might be some truth to this perception
of favoring a mathematical contribution because of ancestry or
perceived ancestry, it would hardly amount to accurate prediction;
but be boosted by Nazi perception of Jewish banking conspiracies.
I was at first disinclined to believe that Nazis would employ such an
illogical criteria. However, they also pursued religous relics as a
possible source of power, which is not logical ... Indiana Jones is
only an embellishment of this direction of thought, not totally fictional.
The practice of Jews changing their religion and last names in order
to secure university positions arose from the first world war and
the depression and financial penalties placed on Germany. Anti-
Semitism is hardly anything new, just starting with the Nazis.
Stephen Harris
> Stephen Harris wrote:
>
>> [Cantor's] father was Lutheran and his mother Catholic with some
>> racial Jew in his ancestry. Saying that he was Catholic doesn't
>> indicate much.
>
> Maybe that's why I didn't say it, then. Cantor was philo-Catholic,
> not Catholic.
>
There isn't much difference between Lutheranism and Catholicism
especially regarding the role of Jesus Christ. So I don't think it
matters which one I picked for Cantor, it isn't relevant.
Technically, I think the word Hebrew has more to do with
racial origins/bloodlines/ethnic land of origin. Jewish used to be
reserved for religion. I decided to add this note because internet has
some picky users who might complain about "jewish ancestry"
meaning genetics rather than continued commitment to Judaism.
robert j. kolker
Stephen Harris wrote:
>
> Technically, I think the word Hebrew has more to do with
> racial origins/bloodlines/ethnic land of origin. Jewish used to be
> reserved for religion. I decided to add this note because internet has
> some picky users who might complain about "jewish ancestry"
> meaning genetics rather than continued commitment to Judaism.
How do you explain black Jews? They are israelites in the biblical
sense. If they keep the Commandments they are no different from Moses or
Abraham.
Bob Kolker
Mike Oliver
> Oliver wrote:
> Is that true? It's a new one on me. A little surprising, given
> that Cantor wasn't Jewish. Cantor was, I believe, Lutheran, ...
>
> SH wrote: Saying that he was Catholic doesn't indicate much.
Well, you can sort of see how this would be confusing, since
on the face of it you wrote a complete non-sequitur.
Stephen Harris
"robert j. kolker" <now...@nowhere.net> wrote in message
news:379sv2F...@individual.net...
I don't know. People can share the same religion and obviously not be
members of the same race. The nation of Israel was divided into a
dozen or so tribes. Intermarriage with Philistines used to be forbidden.
Isolation causes a pattern in the phenotype.
I was pointing out that when somebody said, 'he or she is Jewish'
that technically they should have said 'he or she was of Hebrew descent'
if they meant a lineage rather than whether the individual practiced
Judaism (Jewish) as a religion. Many people made the common error
of saying Jewish to mean the lineage or genetic heritage.
But that differentiation has eroded with time. So Sammy Davis Jr.
converted to Judaism, he was Jewish, even though he was not Hebrew.
This distinction is standard if you are well-educated in this area.
The distinction is still made even if identifying features do not have a
high level of correspondence in the population or the lines are blurred.
I made this correction because I didn't want to be nitpicked by people
who knew the correct terms since I originally said 'jewish lineage' or
something close when I should have said hebrew lineage which would
be more precise. The purpose of my note was not to teach the distinction
to people who were never aware of it in the first place, but to write more
precisely for those who were aware of the distinction. You appear not
to be aware of the distinction, so read about it if you are curious as I
wasn't promoting an anthropological tangent to the Subject:
just a corrective note about imprecise language in an earlier post.
There are Semetic people with black skin who are grouped as Caucasoids.
They may share a common religion or various religions or be atheists.
Ancestralism, http://www.everything2.com/index.pl?node_id=1691674
Stephen
Jeffrey Ketland
> In the particular case of AC or CH, there is no significant increase
> in the length of proofs. But we know that extending a theory T by a
> statement A undecidable in T will shorten the proofs of some
> statements to any extent, in the sense that for any recursive function
> f there is a statement B such that f(p)<p', where p is the length of
> the shortest proof of B in T+A and p' the length of the shortest proof
> of B in T. (Gödel was the first to prove a result of this kind.)
How do we get this for a single undecidable? Are you thinking of Statman's
results, described in Sam Buss's paper on lengths of proof? In these cases,
the statements B are propositional tautologies.
> Friedman has given dramatic actual examples of how non-finitary
> principles make it possible to prove statements that otherwise have no
> feasible proof.
I didn't know that Friedman had detailed results related to speed-up. Is it
in one of his series of FOM postings?
--- Jeff
Torkel Franzen
> How do we get this for a single undecidable?
By a simple recursion theoretic argument! Suppose T is an
essentially undecidable theory, like PA. Suppose A is undecidable in
T, and prove that there is, for any recursive f, a B as indicated.
(This is another exercise from Shoenfield's _Mathematical Logic_.)
> I didn't know that Friedman had detailed results related to
> speed-up.
I learned about these results from the Harrington volume (_Harvey
Friedman's research on the foundations of mathematics_). His more
recent work should be available through his web page.
Keith Ramsay
robert j. kolker wrote:
| david petry wrote:
| > Cantor's Theory is pseudomathematics. It is
| > pseudoscience. It has theological origins. It's not part
| > of the foundation of mathematics. It doesn't belong in
| > the universities. It needs to be debunked.
[...]
| If you want to give up over half the theorem in analysis that depend
in
| some fashion on the axiom of choice, or non-constructive methods, go
| right ahead. The loss is yours.
It's not clear to me that constructivity has much to do
with the distinction between what David Petry wants us
to discard and what he wants us to keep.
Keith Ramsay
Torkel Franzen
> Are they "constructive" principles? (Like tranfinite induction up to some
> contructible ordinal?)
(Assuming you mean "constructive ordinal".) Such principles can have
new arithmetical consequences, but are not necessarily constructive.
LordBeotian
"Torkel Franzen" <tor...@sm.luth.se> ha scritto
> > What infinitary principles are you referring to?
> > Is what you say true for arithmetical statements?
>
> By the incompleteness theorem, statements asserting the existence of
> large infinite sets have new arithmetical consequences.
Are they "constructive" principles? (Like tranfinite induction up to some
contructible ordinal?)
Herman Jurjus
> Herman Jurjus wrote:
>
>> david petry wrote:
>> [snip]
>>
>>> ... And that's a good point; mathematics has
>>> been infected by a virus, namely, Cantor's Theory.
>>
>>
>> in the 1930s, who said they could recognize jewish mathematicians from
>> aryan ones, by looking only at their attitude towards Cantor's set
>> theory.
>
>
> that Cantor wasn't Jewish.
I didn't say he was. Actually, i don't know for sure whether there were
any nazi's who really held this opinion. The only thing i know for sure
is: a few years ago, i read this in a popular mathematics booklet,
pocket format, that i quickly browsed through in a second hand bookshop.
Unfortunately, i totally forgot author and title. I think it was a book
from the 1960s or so, and it looked much more well-informed and serious
than most popular books sold today.
Also, i've read in other (equally questionable) locations that
Bieberbach did hold such views. (But still, having nothing perse to do
with Cantor personally being Jewish or not. Perhaps he/they just noted
the enormous changes in mathematics between, say, 1870 and 1930?)
One more remark that's perhaps interesting (given that Petry claims
Cantor's motives were religious). Poincare reports that Hermite at one
time expressed anti-Cantoreanist leanings, exactly for religious
reasons. Hermite felt that 'the infinite' was something to be left to
God, and when handled by us humble earthlings could only lead to disaster.
So it looks that Hermite agreed with Petry on this, only... -his- was
the religious point of view, not Cantor's. There you have it: there's
always a second side to a story. <g>
--
Cheers,
Herman Jurjus
anon...@mathforum.org
>
>"Zim Olson" ,
>
>oo coin flippers flip coins oo times each. how many flips of the antidiagonal
>have been flipped one after the other?
>
>ANSWERS NOT EXCUSES
>
>Herc
>
>
Herc:
Instead of waiting for a reply to my request for an explanation (since
I may not get one) I will try to give an "Answer".
"oo coin flippers" Herc and " oo coin flips" Herc and "No Table or
flat surface" Zim, give "no basis for probability" Zim .
All these "things" need to be part of a "probability EQUATION ". Or no
basis for "probability".
Best I can come up with , without explanation of your question.
Zim Olson
http://www.zimmathematics.com
Jeffrey Ketland
> "Jeffrey Ketland" <ket...@ketland.fsnet.co.uk> writes:
>
>> How do we get this for a single undecidable?
>
> By a simple recursion theoretic argument! Suppose T is an
> essentially undecidable theory, like PA. Suppose A is undecidable in
> T, and prove that there is, for any recursive f, a B as indicated.
> (This is another exercise from Shoenfield's _Mathematical Logic_.)
Ah. Clever.
>> I didn't know that Friedman had detailed results related to
>> speed-up.
>
> I learned about these results from the Harrington volume (_Harvey
> Friedman's research on the foundations of mathematics_). His more
> recent work should be available through his web page.
Thanks. I'll take a look at both.
--- Jeff
exama...@gmail.com
fishfry wrote:
> In article <1108156809.7...@f14g2000cwb.googlegroups.com>,
> "david petry" <david_lawr...@yahoo.com> wrote:
>
> > Part I Mathematics
> >
> >
> > The essence of science is the ability to make predictions
> > about the results of experiments. Mathematical statements
> > can be interpreted as making predictions about the
> > results of computational experiments
>
> One need read no further. Crank alert.
Really. Care to explain why? Because the universe is fundamentally
non-mechanical, or because there are objective truths independent of
human mind that are not mechanical?
Which one is it?
--
Eray
exama...@gmail.com
I think your post is well written and to the point. It does seem that
phil. of mathematics has a very tight relationship to phil. of mind,
which seems to pass by many of those who have read in only either of
the subjects.
Please join our ai-philosophy group which has as members many
professional and amateur researchers who are interested in theoretical
AI.
I would be pleased if you could repost this article there.
http://groups.yahoo.com/group/ai-philosophy
Regards,
--
Eray ai-philosophy mod.
david petry wrote:
> Part I Mathematics
>
>
> The essence of science is the ability to make predictions
> about the results of experiments. Mathematical statements
> can be interpreted as making predictions about the
> results of computational experiments, and hence,
> mathematics is a science. One example should suffice.
>
> The Riemann Hypothesis is an example of a statement which
> makes predictions about the results of computational
> experiments. The Riemann Hypothesis asserts that if we
> were to compute the value of one of the complex zeros of
> the zeta function, then to within the accuracy of the
> computation, the real part of that value will be 1/2. So
> once we have ascertained that the zeros of the zeta
> function can actually be computed to arbitrary precision
> with a sensible amount of computation, then we clearly
> see that the Riemann Hypothesis is making predictions
> about the results of computing those zeros, and actually
> doing the computations can be thought of as performing an
> experiment. It makes sense to say that we can "observe"
> the results of the experiment. And we can think of the
> experiments taking place in a world of computation.
>
> The power and utility and meaning of mathematics comes
> from its ability to predict the results of computational
> experiments. All of the mathematics which has the
> potential to be applied to the task of modeling phenomena
> in the real world necessarily has the ability to make
> such predictions. The notion of "truth" in mathematics
> has no concrete meaning without a connection to the
> predictions of experimental results.
>
> The astute reader will point out that some "mathematical"
> statements cannot be interpreted as making predictions
> about the results of computational experiments. For
> example, the Continuum Hypothesis makes no such
> predictions. And that's a good point; mathematics has
> been infected by a virus, namely, Cantor's Theory.
>
>
> Part II Pseudomathematics
>
>
> Cantor's Theory (classical set theory) is a conceptual
> framework which encompasses both mathematics (the science
> of phenomena observable in the world of computation) and
> a fabricated world of the super-infinite. It merges the
> two worlds so cleverly and so seamlessly that those who
> accept that framework of thought often lose the ability
> to distinguish between the two worlds. (Note: henceforth,
> I'll refer only to the fabricated world of the
> super-infinite as Cantor's Theory)
>
> Cantor's Theory is a pseudomathematics. It has the form
> but not the function of mathematics. It follows the
> definition-theorem-proof format of mathematics, but in
> the end, the theorems have no interpretation as
> predictions about the results of computational
> experiments.
>
> Cantor's Theory has been aptly described as "formal
> operations on meaningless symbols". Formally, Cantor's
> Theory is impeccable, but ultimately, it is absurd. It
> builds upon notions of infinite sets and power sets of
> those infinite sets, which it does not define in terms of
> the reality that we observe (i.e. the world of
> computatioin), but rather, it merely asserts their
> existence by decree. The result is that the existential
> quantifier becomes a meaningless symbol. To assert that
> the infinite objects in Cantor's Theory "exist" is a lie.
>
> Formal logic has the "garbage in, garbage out" property;
> it does not have the power to transform meaningless
> assumptions and symbols into meaningful assertions. The
> "meaning" of mathematics comes from its ability to make
> predictions about the results of computational
> experiments. The theorems in Cantor's Theory are not
> truths in any meaningful sense.
>
> The problem with Cantor's Theory is not the notion of
> infinity, nor the notion of power sets. A theory of the
> infinite can be designed such that every concept and
> every object in the theory has corresponding
> approximations in the world of computation. In such a
> theory, the notions of infinite sets and power sets will
> be defined as abstractions derived from the finite world
> we actually observe. Such a theory can serve as a
> conceptual aid in mathematics, and is part of the science
> of mathematics.
>
> Mathematics has tremendous power to enlighten us about
> the nature of reality. Cantor's Theory has no such power.
> The only "power" Cantor's Theory has is the power to
> bamboozle. The Cantorians have parlayed this power into
> enormous political power within the mathematics
> community. To maintain their power, the Cantorians must
> impose their "theory" on everyone in the community, and
> weed out from the community anyone who questions the
> reality of the world implied to exist by their theory. In
> general, a community which has a power structure built
> around knowledge of a mythology about a world beyond what
> can be observed is thought of as either a religious
> community or a mystical cult, and either could describe
> the Cantorian community.
>
> The world of the super-infinite - a world lying beyond
> the world that is accessible in the scientific paradigm -
> is essentially theological, and it has theological
> origins. Cantor himself was hoping to unify mathematics
> and theology with his theory.
>
> I'm hardly the first person to notice that there is no
> mathematical content to Cantor's Theory. Consider the
> following quote from a contemporary of Cantor:
>
> "I don't know what predominates in Cantor's theory -
> philosophy or theology, but I am sure that there is no
> mathematics there" (Kronecker)
>
> It would have been quite difficult in Cantor's time to
> make an airtight case that Cantor's Theory is not
> mathematics. At that time, one might have argued that
> mathematics is a science which makes predictions about
> the results of computational experiments, but if the
> question were asked, "where do these experiments take
> place?", the likely answer would have been, "in the
> mind". But then it could have been argued that objects in
> the theory of the infinite live in the imagination, and
> hence in the mind, and hence computational experiments
> and objects in the theory of the infinite have the same
> ontological status.
>
> Furthermore, in Cantor's time, the ability to compute was
> so limited that the mathematics of the time was making
> predictions about computational experiments which were
> far beyond the ability of then current technology to
> carry out, and so it might have seemed to the
> mathematicians that the mathematics they were doing was
> no more connected to reality than Cantor's Theory.
>
> The computer revolution really has changed the way we
> look at the world. The computer revolution has improved
> our ability to compute by some 15 orders of magnitude,
> which is a truly stunning number to any practical minded
> person. It outdoes any other revolution in history. It
> has opened up the world of computation to us. It has
> given us a new paradigm for mathematics. It's changed our
> thinking about the ontological status of mathematical
> objects.
>
> Mathematics can now be clearly seen as a science. The
> computer serves as the mathematicians' microscope. It
> serves as the mathematician's test tube in which
> computational experiments are performed. The world of
> computation - the world viewed through the
> mathematicians' microscope - has an objective existence
> independent of the mind, and mathematics is the science
> which studies that world.
>
> It should be emphasized that formalisms themselves are
> objects that can be viewed in the mathematicians'
> microscope. They are objects that are studied in the
> science of mathematics. Hence, Cantor's Theory, as a
> formalism, is something to be studied as part of
> mathematics. But the formal theorems in Cantor's Theory
> have no interpretation as theorems in the science of
> mathematics. They do not make predictions about the
> results of computational experiments. The objects in
> Cantor's world of the infinite are merely figments of the
> imagination. Cantor's Theory most certainly is not part
> of the foundation of mathematics.
>
> Mathematics belongs in the public arena. As long as we
> agree that religion belongs in the private arena and not
> in the public arena, as we do in the United States, then
> mathematics must not include the Cantorian religion.
>
>
> Part III Artificial Intelligence
>
>
> When I was in graduate school, I thought it would be a
> really cool idea to build a foundation for mathematics
> incorporating as one of the cornerstones, the idea that
> mathematical statements must make predictions about the
> results of computational experiments. I believe it would
> be a straightforward task to teach computers to
> understand mathematics when it is built on such a
> foundation, and then it would be seen that the foundation
> of mathematics is also the foundation of artificial
> intelligence. After all, mathematics is the tool we use,
> and the tool computers could be using, to understand the
> world in a precise and quantitative way. I believe that
> making available a solid theoretical foundation for
> artificial intelligence could have profound implications
> for the future of mankind.
>
> Even without the connection to artificial intelligence, I
> believe I had a good idea. All of the mathematics which
> has the potential to be applied must have the capacity
> for making predictions, so I would have been building a
> theoretical foundation for applied mathematics. If
> nothing else, it could be a good pedagogical tool. And
> failing that, intellectual curiosity alone should justify
> pursuing such a goal.
>
> Alas, the Cantorians told me essentially that my ideas
> are worthless crackpot ideas with no connection to real
> mathematics, and that I really have no right to pursue
> such ideas until I first prove that I am fully competent
> in Cantorian mathematics. It was discouraging. I ended up
> believing that as long as Cantor's Theory is part of the
> canon of the mathematics community, I don't fit in.
>
> Maybe it's too much to ask that the current generation of
> mathematicians should abandon Cantor's Theory. But it
> can't possibly be too much to ask that the mathematicians
> be tolerant of those students who have been deeply
> affected by the radical technological changes taking
> place in the world today.
>
> As I've already pointed out, the computer revolution has
> opened up the world of computation to us. Many of today's
> students want to explore that world, and they see
> mathematics as the science which studies that world. They
> want to be mathematicians. So I'm asking that
> mathematicians tolerate those students.
>
> But even more significant is the dream of artificial
> intelligence. Right now, it's only a dream, but it's a
> powerful and compelling dream, and it has captured the
> imaginations of some of the best and brightest students.
> What the dream of artificial intelligence leads us to
> believe is that there is no magic ingredient in the mind
> which in not available to a computer. That is, there's an
> equivalence between artificial intelligence and the mind,
> and they both live in the world of computation; all of
> the tools we need to understand the world around us, live
> in the world of computation; all of the tools we need to
> understand the world of computation itself, live in the
> world of computation; mathematics lives in the world of
> computation; the idea that there exists a world of the
> super-infinite which has no connection to the world of
> computation, is a fantasy; the Cantorian idea that we can
> "prove" that such a world exists is preposterous. Again,
> what I'm asking is that mathematicians tolerate the
> students who hold that world view; encourage them, allow
> them to be part of the mathematics community, don't think
> of them as second rate minds, and don't call them
> crackpots.
>
>
> Part IV Babble
>
>
> It's not the kind of thing I can prove beyond doubt, but
> I believe that the mathematicians' adherence to Cantor's
> Theory has held back progress in artificial intelligence
> by as much as thirty years and counting. If mathematics
> were built on a foundation which emphasized the
> connection between mathematics and computation, then all
> of the mathematical knowledge accumulated over the past
> few thousand years would be immediately transferable to
> the computer, and the computer would be then intelligent;
> it would have the tools it needs to understand the
> physical world, and it would have the tools it needs to
> understand its own mind; it would be self-aware.
> Computers simply need not understand the complexities and
> nuances of human social behavior in order to be deemed
> intelligent; the Turing Test is bunk (another of my pet
> peeves).
>
> There are lots of reasons for debunking pseudoscience. I
> shouldn't have to elaborate on that for the relatively
> sophisticated audience likely to be reading this article.
> Sure, it's likely that those who have devoted themselves
> to the pseudoscience will experience a stressfull period
> of readjustment when that pseudoscience is debunked, and
> we should do what we can to minimize that stress, but
> that is no reason to refrain from the debunking process.
>
> Cantor's Theory is pseudomathematics. It is
> pseudoscience. It has theological origins. It's not part
> of the foundation of mathematics. It doesn't belong in
> the universities. It needs to be debunked.
>
> Cantor's Theory has been debunked.
Zim Olson
>
>"Zim Olson" ,
>
>oo coin flippers flip coins oo times each. how many flips of the antidiagonal
>have been flipped one after the other?
>
>ANSWERS NOT EXCUSES
>
>Herc
>
>
Herc:
Not sure what you mean. What do you mean by "flips of the
anitdiagonal"??
Zim Olson
http://www.zimmathematics.com
Mike Oliver
> One more remark that's perhaps interesting (given that Petry claims
> Cantor's motives were religious). Poincare reports that Hermite at one
> time expressed anti-Cantoreanist leanings, exactly for religious
> reasons. Hermite felt that 'the infinite' was something to be left to
> God, and when handled by us humble earthlings could only lead to disaster.
>
> So it looks that Hermite agreed with Petry on this, only... -his- was
> the religious point of view, not Cantor's. There you have it: there's
> always a second side to a story. <g>
I think there is no doubt that Cantor had a theological agenda. Not
that there's anything wrong with that. The view you attribute to
Hermite was a rather common one, and one that Cantor specifically
addressed, in theological terms.
Of course this is quite different from saying that set theory
as practiced today is theology.
george
> Torkel Franzen
>
> > In the particular case of AC or CH, there is no significant
increase
> > in the length of proofs. But we know that extending a theory T by a
> > statement A undecidable in T will shorten the proofs of some
> > statements to any extent, in the sense that for any recursive
function
> > f there is a statement B such that f(p)<p', where p is the length
of
> > the shortest proof of B in T+A and p' the length of the shortest
proof
> > of B in T. (Gödel was the first to prove a result of this kind.)
Jeffrey Ketland wrote:
> How do we get this for a single undecidable? Are you thinking of
Statman's
> results, described in Sam Buss's paper on lengths of proof? In these
cases,
> the statements B are propositional tautologies.
This is a point of which you seemed blissfully ignorant
when you and I were having our donnybrook over practical
feasibility of certain "first-order" proofs that, being
over finite domains, were actually propositional.
>
> > Friedman has given dramatic actual examples of how non-finitary
> > principles make it possible to prove statements that otherwise have
no
> > feasible proof.
>
> I didn't know that Friedman had detailed results related to speed-up.
You were not at all clear about what "speed-up"
even was, or the context in which it was meaningful, in that
argument. The world might be a better place if you would
just shut up for a LITTLE while longer and let THIS thread
take its course, BEFORE diverting it with your misconceptions
(yes, I KNOW that sounds like Projection on my part).
My point here was simply that the tack TF was taking here
shows that he actually knows what "speed-up" MEANS. The argument
you were having with me last year shows that you didn't.
david petry
Herman Jurjus wrote:
> david petry wrote:
> [snip]
>
> > been infected by a virus, namely, Cantor's Theory.
>
in
> the 1930s, who said they could recognize jewish mathematicians from
> aryan ones, by looking only at their attitude towards Cantor's set
theory.
regarding
> Cantor, this may be partly due to that remembrance.
FWIW, I'm quite certain the Kronecker was Jewish, so there can't be
any strong relationship between being against Cantor's Theory and
being against Jews.
robert j. kolker
david petry wrote:
> FWIW, I'm quite certain the Kronecker was Jewish, so there can't be
> any strong relationship between being against Cantor's Theory and
> being against Jews.
Nowadays most of the goyim in mathematics accept cantorian set theory.
Bob Kolker
Jeffrey Ketland
> This is a point of which you seemed blissfully ignorant ...
Liar.
> You were not at all clear about what "speed-up" ...
So why did I have to explain the concept to you, who had not the foggiest
clue what was going on? (But you continued to rant, scream, shout and hurl
expletives like a 3 year old child).
--- Jeff
Stephen Harris
After reading Sol Feferman I'm not sure it makes any practical difference.
He mentions his system W and Harvey Friedman's (Steven G. Simpson)
Reverse Mathematics as adequate to the task of scientific mathematics.
"In The Light Of Logic" by Sol Feferman, page 287
3. What's Wrong with Set-Theoretical Foundations?
"Philosophically, set theory -- even in its "moderate" form given by
Zermelo's axioms -- require for its justification a strong form of
platonic realism. This is not without its defenders, most notably
Godel (1944 and 1947/1964) (cd. also Maddy 1990). For its
critics, however, the following are highly problematic features of
this philosophy:
(i) abstract entities are assumed to exist independently of any
means of human definition or construction;
(ii) classical reasoning (leading to nonconstructive existence results)
is admitted, since the statements of set theory are supposed to
be about such an independently existing reality and thus have a
determinate truth value (true or false);
(iii) completed infinite totalities and, in particular, the totality of all
subsets of any infinite set are assumed to exist;
(iv) in consequence of (iii) and the Axiom of Separation,
impredicative definitions of sets are routinely admitted;
(v) the Axiom of Choice is assumed in order to carry through the
Cantorian theory of transfinite cardinals."
SH: I think the Law of the Excluded Middle is assumed which would
seem to have a consequence for probabilistic approaches to AI. I
don't think Feferman considers Category Theory an adequate substitute
for Cantorian set theory but he mentions it as a possibility.
exama...@gmail.com
Thank you for these extensive and informative notes Stephen.
The readership must note that (i) is prima facie wrong. To admit it
would be to also admit some obsolete form of dualism, how do we get in
touch with these guys? A physical possibility Godel wanted to say was:
they are out there in heaven, but *somehow* they put themselves in
implicit patterns of the nature, so we can detect them. (a "second
order" of existence, sure...) This is theology, not science. Why should
we assume such a complex schizophrenic explanation?
Besides, what if our physical world is finite (which by all appearances
is the case), then there are also independently existing abstract
objects that do not choose to manifest themselves in the nature. Great!
There are patterns in the nature, because nature is *lawful*, not
because nature is awash by the light of God! What do you say, is
"gravity" a person, an entity? Or is it just part of our description of
the physical world, a common property of the material?
To say that something like "*the* empty set actually exists" is a
confusion of the worst kind which led some previous philosophers to
believe in immaterial souls (Descartes), computations that exist
independently of matter (Putnam's first view), infinite minds (angels,
God), etc. That line of thought is a dead end. It explains nothing
about the world or mathematicians or minds in general!
Mathematicians (and computer scientists) are nothing more or less than
imaginative people who think abstract, efficiently and in disciplined
ways. There is absolutely no fundamental epistemic or ontic difference
between common sense concepts such as "space" and mathematical concepts
such as "set". That is why you can use exactly the same hardware to
study both concepts.
Regards,
--
Eray Ozkural
exama...@gmail.com
> One more remark that's perhaps interesting (given that Petry claims
> Cantor's motives were religious). Poincare reports that Hermite at
one
> time expressed anti-Cantoreanist leanings, exactly for religious
> reasons. Hermite felt that 'the infinite' was something to be left to
> God, and when handled by us humble earthlings could only lead to
disaster.
I think it is well known that Cantor had a theological motivation, or
at least that he was quite religious.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cantor.html
"In 1905 Cantor wrote a religious work after returning home from a
spell in hospital. He also corresponded with Jourdain on the history of
set theory and his religious tract."
No materialist in his right mind would write a religious work, or try
to associate a mathematical theory with religion.
We should try to look for the philosophical explanations to see how he
develops his philosophy of set theory.
"He turned from the mathematical development of set theory towards two
new directions, firstly discussing the philosophical aspects of his
theory with many philosophers (he published these letters in 1888)"
Are these letters available on the net?
Of course, if it is not obvious from his theory that he is trying to
reach up to heavens, what can I say?
Regards,
--
Eray
Barb Knox
exama...@gmail.com wrote:
[snip]
>Mathematicians (and computer scientists) are nothing more or less than
>imaginative people who think abstract, efficiently and in disciplined
>ways. There is absolutely no fundamental epistemic or ontic difference
>between common sense concepts such as "space" and mathematical concepts
>such as "set". That is why you can use exactly the same hardware to
>study both concepts.
That line of reasoning also implies that there is no fundamental epistemic
or ontic difference between those concepts and clearly non-physical ones
such as "Alice" (in Wonderland) and God (in heaven, or wherever), since
people "use exactly the same hardware" for these too. Oops.
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
exama...@gmail.com
Mike Oliver wrote:
> Herman Jurjus wrote:
>
> > One more remark that's perhaps interesting (given that Petry claims
> > Cantor's motives were religious). Poincare reports that Hermite at
one
> > time expressed anti-Cantoreanist leanings, exactly for religious
> > reasons. Hermite felt that 'the infinite' was something to be left
to
> > God, and when handled by us humble earthlings could only lead to
disaster.
> >
> > So it looks that Hermite agreed with Petry on this, only... -his-
was
> > the religious point of view, not Cantor's. There you have it:
there's
> > always a second side to a story. <g>
>
> I think there is no doubt that Cantor had a theological agenda. Not
> that there's anything wrong with that.
I don't think so.
Regards,
--
Eray
exama...@gmail.com
> In article <1108422348....@f14g2000cwb.googlegroups.com>,
> exama...@gmail.com wrote:
> [snip]
>
> >Mathematicians (and computer scientists) are nothing more or less
than
> >imaginative people who think abstract, efficiently and in
disciplined
> >ways. There is absolutely no fundamental epistemic or ontic
difference
> >between common sense concepts such as "space" and mathematical
concepts
> >such as "set". That is why you can use exactly the same hardware to
> >study both concepts.
>
> That line of reasoning also implies that there is no fundamental
epistemic
> or ontic difference between those concepts and clearly non-physical
ones
> such as "Alice" (in Wonderland) and God (in heaven, or wherever),
since
> people "use exactly the same hardware" for these too. Oops.
By fundamental ontic difference, I mean change in substance. By
fundamental epistemic difference, I mean change in the basic means to
attain knowledge. (To simplify the discussion). I don't specify which
concept/entity is more useful, has more meaning, is better, etc. Those
are details, self-evident, trivial details. Let's then delve into utter
triviality.
"Alice" is not a concept, it is a fictional character. It exists only
in the fictional sense (e.g. does not exist physically). It has meaning
only and only relative to a particular textual context.
Set is a concept. Space is a concept. They are general concepts that
are not tied to one narrow context. These are useful tools of
abstraction we have derived from our sensory surfaces.
God, either as the theological concept, or as a unique entity does not
exist. It has the same status as Alice. (If you redefine "God" such
that it is identical to the universe, it exists) Although it was
intended as an "explanatory" concept/entity, it is a failure, a bad
theory.
No, there is absolutely no *fundamental* ontic or epistemic difference
between any of these. Thinking in that vein would be dualism, which
some people in this thread seem to presuppse.
All of these "ideas" are human thoughts, admitting this does not
corrupt the computationalist view. The difference is that set and space
have primarily scientific uses, while Alice and God have, it seems,
only fictional uses. It would be absurd to try to use "Alice" or "God"
to predict the world, not a good way at least.
So, I think we can grant "set" and "space" the wisdom they deserve,
after all both these concepts can be used to explain/predict so many
things in the world.
The mathematical realists are completely resistant to the following
idea.
(*) If the concept of "set" had NECESSARILY no explanatory power in the
physical world, then it would be MEANINGLESS. It would have no use.
If you understand what this really means, then you've grown out of
shabby dualism into the 20th century doctrine of monism and
physicalism. A considerable improvement from useless scholastic
philosophy.
Regards,
--
Eray
exama...@gmail.com
> I really don't think that constructive mathematics has any relevance
> to AI. The difficulties in achieving AI are not mathematical
> difficulties.
Do you mean they are conceptual difficulties, then?
Regards,
--
Eray
Daryl McCullough
Yes. What does it mean to generalize from concrete examples? What does it mean
to learn by induction? How in thinking about a problem do we sort out which
details are relevant from those that are irrelevant? What does it mean to say
that two things look similar? What does it mean to be creative?
These aren't really mathematical questions (although the answers might very well
involve mathematics).
--
Daryl McCullough
Ithaca, NY
dave
robert j. kolker wrote:
> Consistency is the only requirement
> for a mathematical theory.
So if a theology is consistent, then it's mathematics? So if
I desire a stronger reality check than mere consistency, I can't
be part of the mathematics community?
Did you read the following paragraph in my article?
Ross A. Finlayson
> http://math.stanford.edu/~feferman/book98.html
>
> "In his concluding chapters, Feferman uses tools from the special
> part of logic called proof theory to explain how the vast part, if
> not all of scientifically applicable mathematics can be justified
> on the basis of purely arithmetical principles. At least to that
> extent, the question raised in two of the essays of the volume,
> "Is Cantor Necessary?," is answered with a resounding "no."
>
> SH: Also pages 68 and 69 of In the Light of Logic seem relevant.
>
> Regards,
> Stephen
Hi Stephen,
As usual your posts are florid and cogent.
http://mathforge.net/index.jsp?page=seeReplies&messageNum=1096
An article on yesterday's "MathForge" from some anonymous submitter
mentions Feferman about Goedel. Sci.math's Milo "Akhmim Wooden Tablet"
Gardner compliments Fowler. Is it so that "This means the great
monuments of mathematics are about to fall"? I think not. What is
this "natural independence"?
Cantor selected the Alef's, where Alef or Aleph is the first letter of
the Hebraic alphabetic script, for reasons that we don't know. Perhaps
one reason is that the Latin and Greek alphabets are completely
overloaded with symbols, and he wanted some novelty in his notation.
Perhaps another was to curry support for his theory among the Jewish
and those supportive or sympathetic of or to Jews, as they're
influential in academic mathematics. Perhaps another was to associate
it with the mystical qabalah and gematria, which is an axiom-free set
theory with ubiquitous ordinals. Perhaps another was as a nod to his
own Jewish heritage.
Georg Cantor, the mathematician, is not the cantor from the synagogue.
Cantor, the man, was a Christian and also basically a Jew, and he was
circumspect about his beliefs, which probably included faith in a
higher power.
Criticizing mathematics for secular reasons is polemic and
unproductive. While that is so, addressing that some people harbor
notions of defense of a mathematical school as defense of a school of
faith can help understand some of the strong emotions associated with
the issue(s).
Some people don't _care_ about mathematics. Some people, as would
probably be evidenced by mathematicians and contributors to this forum
on mathematics, very much do.
Anyways, people who address and argue about the foundations of
mathematics are not necessarily cranks, trolls, lunatics, nor idiots,
nor anti-semitic. Where here we have been discussing perceived
inadequacies of contemporary treatments of the foundations of
mathematics, blanket denial of those accusations is a form of
discrimination itself, and is basically representative of fear of
change.
That's not to say mathematicians shouldn't be discriminating with
regards to rigor and mathematical proof. A useful 14'th century column
circumference proof doesn't see the light of day today not because it's
wrong, but because there are more powerful, general, and useful
mathematical tools to answer the same questions. Some people think
current treatments of foundations of mathematics are wrong or
unjustified, or even just not useful. With their interests in
mathematics they seek better ways.
A logical paradox is just a sticky-note saying "fix me". They're not
done yet, doc. "No one expects the anti-prerogative."
Infinite sets are equivalent. I think the actual infinite _is_
necessary for the explanation of the physical.
Ross
--
A Pertti Lounesto Shrine:
http://www.tiki-lounge.com/~raf/lounesto/LounestoShrine.html
robert j. kolker
dave wrote:
> robert j. kolker wrote:
>
>
>>Consistency is the only requirement
>>for a mathematical theory.
>
>
> So if a theology is consistent, then it's mathematics? So if
No. Because theology is not a mathematical theory.
I intended to say, that if a mathematical theory is not internally
consistent it is useless and should be discarded.
Theology is another thing entirely. It is not based on definite axioms
or axiom schemata nor are its conclusion derived logicallly from its
axioms. So it isn't mathematics.
I am aware that Spinoza tried to establish theology in the style of
Euclidean geometry but he kept sneaking in additional assumptions.
Euclid did this too, and Hilbert had to clean up his act.
Bob Kolker