Skepticism, mysticism, Jewish mathematics
david petry
Is there any reason to be skeptical about modern mathematics? Is it
possible that modern mathematics is culturally biased? Is it possible
that there is an element of fraud in modern mathematics? Has
mathematics become clever argumentation with no concrete content?
I'm like the reader to consider the possibility that the answer to all
those questions is yes.
First of all, it would be absurd to say that the modern academic system
based on peer review would preclude fraud and cultural bias. The "Sokal
Affair" seems to have proven that when academic writing becomes
indistinguishable to the non-expert from buzzword salad, then it's
likely that even experts can be fooled. (see
http://en.wikipedia.org/wiki/Sokal_Affair) Even physicists can be
fooled (see http://math.ucr.edu/home/baez/bogdanoff/ )
But even ignoring the possibility of outright fraud, cultural bias can
creep into academia slowly and progressively so that it's hardly
recognized by the majority of people involved. The best example might
be the takeover of the humanities by the Marxists (or extreme left
wing).
The Marxists produce beautiful theories. They produce complex, clever,
precise, and apparently logically consistent arguments, which must pass
a rigorous peer review process. The Marxists believe themselves to be
open minded, unbiased, compassionate, independent thinkers. But, of
course, the skeptics don't see it that way at all. According to the
skeptics, market forces are simply part of reality, and responding to
those forces is a natural and compassionate thing for humans to do, and
hence the implementation of Marxism (which tries to replace market
forces with governmental planning) requires a brutally oppressive and
intrusive government willing to criminalize human nature, so that no
matter how beautiful the Marxist theories may be, there is something
fundamentally very very wrong with the Marxist world view. Many
skeptics in academia believe that they are severely discriminated
against by the Marxists.
In other words, the skeptics will claim that despite the cleverness and
consistency of the Marxists' arguments, those arguments are built upon
a defective (or at least, a culturally biased) model of reality. So the
question becomes, is it possible that modern mathematics is built upon
a flawed model of reality which gives it a cultural bias? Does this
bias have any connection to the left wing bias in the humanities?
For the sake of this argument, let's consider an idealized skeptic. Our
skeptic will be intelligent and honest to a fault. She will have a
technical background, and will be fully aware of the power of
mathematics in technology. She will have no axe to grind, and she will
have no philosophical, religious or political biases, other than a
propensity for skepticism. That is, she looks for concrete evidence,
observable implications, and testable consequences. She is skeptical of
mere clever argumentation. And she refuses to be intimidated by appeals
to authority or ad hominem attack.
So let's say that our skeptic has been reading popular books about
mathematics. For example, books by Hofstadter, Penrose, Rucker,
Smullyan, Kline, Singh, Aczel, and maybe others. She is troubled that a
lot of the modern mathematics she has been reading about seems to be
nothing more than clever argumentation with no concrete content and no
testable consequences. She wants to know how the ideas she has been
reading about can help us to understand the world in which we live.
Let's look at specific examples of the mathematical ideas she is
skeptical about.
1) Set theory and Cantor's Theorem. It seems obvious to our skeptic
that the mathematical constructs we actually deal with must be
identifiable, and that we can only identify a countable number of such
constructs (since our language is countable). So Cantor's Theorem
asserting the existence of uncountable sets (and hence the existence of
objects which cannot be identified) cannot have any concrete content.
The idea that there must exist more unidentifiable objects than
identifiable objects appears to be silly word play. Clearly, to our
skeptic, set theory includes an element of make-believe. So she
concludes that much of what she has been reading is nothing more than
clever argumentation with no concrete content; what possible testable
consequences are there to the assertion that unidentifiable objects
exist? And why don't the books she has been reading address the obvious
skeptical objections to such ideas?
2) Godel's Theorem (loosely, no consistent formalism can prove its own
consistency) Informally, our skeptic claims, a proof is a compelling
argument. It seems clear to our skeptic that if we are to believe that
the formal theorems in our formalism should be accepted as compelling
arguments, then at the very least it must be the case that we already
believe that our formalism is consistent, and hence, no possible formal
proof within that formalism could be considered to be the evidence that
compels us to believe that the formalism is consistent. And our skeptic
asks, is that not already the essential content of Godel's theorem?
Even if you argue that Godel's proof is superior because it is actually
formal, you still have to deal with the informal notion of proof: does
Godel's proof compel us to believe that Godel's theorem is actually
true? So, our skeptic asks, what is the concrete content to Godel's
theorem? What does it tell us that is not implicit in the definition of
"proof"? How can it be tested? Is it anything more than clever
argumentation? How can such a theorem be regarded as one of the most
important theorems in all of mathematics? Why don't mathematicians
raise these kind of questions? Why aren't they at least a little bit
skeptical?
3) Self-reference and paradox. (note that some of the popular books our
skeptic has been reading do suggest that this is of great importance in
mathematics, and essential for understanding Godel's theorem). First of
all, our skeptic notes, the assertion that paradox is in some sense
"real" (i.e. something more than an illusion or a game or a joke),
would appear to be almost equivalent to the assertion that logical
reasoning can be used to prove that logic is flawed, which is
immediately highly suspicious. But it can be analyzed further: one of
the "ground rules" in communication is that we should always intend to
tell the truth. That is, when we speak, we are implicitly claiming to
be telling the truth, and we need to explicitly comment on the truth
value of our assertions (e.g. with modifiers such as 'probably',
'possibly', or 'not') only when we do not feel certain about the truth
of what we are saying. Hence, an utterance such as "I am lying" (i.e.
the Liar paradox) must be analyzed as if it contained its implicit
claim to truth, i.e., it must be deemed logically equivalent to
"(implicitly) I am telling the truth; (explicitly) I am lying", which
is nothing more than a simple contradiction, with nothing paradoxical
about it. So our skeptic wonders, how can the study of paradox can be
anything more than a game; how can the contemplation of paradox
possibly help us understand the world in which we live; how can it
possibly have testable consequences? And yet, whole books have been
devoted to its study -- why? Why don't mathematicians ask and address
these questions? And as far as self-reference goes, clearly humans can
talk about themselves, but to claim that sentences can talk about
themselves would seem to be a bizarre anthropomorphization of abstract
symbols; natural language gives us no way to create sentences which
unambiguously refer to themselves.
4) Fermat's Last Theorem. Our skeptic notes that while FLT itself has
clear meaning and a concrete content, there's nevertheless something
fishy about the idea that it has been proved. There is something that
is immediately clear to anyone who has dared to search for a
counterexample to FLT: just due to chance alone, it seems unlikely that
there is a counterexample. That is, for an exponent 'p' of modest size
or larger, the set of integers which are p'th powers is a very very
sparse set of integers, and for an arbitrary set of integers that is
that sparse, straightforward probabilistic reasoning tells us that it
is very unlikely that the sum of two of its elements will turn out to
be another element of the set. In fact, for example, a
back-of-the-envelope calculation suggests that for a set as sparse as
the set of 50'th powers, the probability that two of its elements will
sum to a third element of the set is about 1/10^200, and this can be
loosely interpreted as giving a probabilistic proof that FLT is almost
certainly true for exponent 50. Going further, given that FLT had been
proven for all exponents up to 10^6 before Wiles came along, using the
same heuristic argument, the probability (in the Bayesian sense where a
probability is a degree of belief) that there could be a counterexample
to FLT could be taken to be about 1/10^10^7. So, in other words, Wiles
spent seven years locked in his attic (so the story goes) to do nothing
more than remove that last little bit (1 part in 10^10^7) of
uncertainty that FLT is true, assuming that we generously assign a
value of less than 1/10^10^7 to the probability that his proof is
flawed! Since the proof tells us nothing that we do not already believe
to be true with very very high probability, searching for
counterexamples to the theorem in no way can be deemed a test of the
proof. So our skeptic has to wonder whether a man who is willing to
devote so much energy to such an insignificant task, for no apparent
reason other than to seek fame, would he not be willing to pull off a
hoax? How could we know? And furthermore, the proof itself is
presumably accessible to only the top one tenth of one percent of
mathematicians, so our skeptic notes that she has no realistic hope of
ever determining for herself whether the proof is consistent. But why
should she trust the "experts"? Why should the proof of FLT qualify as
headline news? Why don't the books our skeptic has been reading address
these kinds of questions?
So, does Cantor's proof compel us to believe that there exist
mathematical objects that cannot be identified? Does it compel us to
believe that there are more unidentifiable objects than identifiable
objects? Of course not! For example, we could simply assert that as
part of the definition, mathematics only studies identifiable objects,
and then with less magic than was used to prove the existence of
unidentifiable objects in the first place, all of the unidentifiable
objects would vanish from the mathematical universe! And to be sure,
the mathematics that does have testable consequences would hardly be
affected at all by such a change of definition.
So what's going on? Our skeptic will note that somehow mathematicians
are cheating. When they use words like "proof", "truth", "exists",
"logic", and even the word "mathematics" itself, they are not using
them in the way the rest of the world uses them. The mathematicians
have chosen convenient definitions and convenient axioms which let the
mathematicians formally "prove" what they want to prove; they have
completely abandoned the idea that mathematics should have testable
consequences; they are playing word games; they have insulated
themselves from reality.
So now our skeptic asks, given that we see what games the
mathematicians play, is it even remotely plausible that the
mathematicians could be capable of coming up with important insights
into the nature of proof, truth, existence, and logic? For one thing,
the mathematicians seem to be totally clueless about what is
"important" to anyone but themselves, given that they think testable
consequences are not important.
As far as the proof of Fermat's Last Theorem goes, our skeptic admits
that she has no special insight. But she has to wonder why
mathematicians apparently refuse to even think about such things from a
probabilistic point of view. Probabilistic reasoning does produce
results with testable consequences, and if we regard mathematics as a
science with the purpose of explaining the phenomena we observe within
the world of computation (a view completely compatible with the views
of those who apply mathematics), then probabilistic reasoning should be
accepted as part of mathematics. And furthermore, once we recognize
that formal reasoning and probabilistic reasoning complement each other
(i.e. probabilistic reasoning works especially well where formal
reasoning fails, and vice versa), we have to reexamine the content of
Godel's theorem (in this case, the assertion that there exist true
statements not provable in a given formalism). Clearly we can come up
with an unlimited number of statements which can be shown by
probabilistic reasoning to be almost certainly true but for which we
have no reason whatsoever to believe that they can be proven true in
any particular formalism. So Godel's theorem is true with a vengeance,
but it's not Godel's proof which compels us to believe that. And the
important question -- are there true mathematical statements having
testable consequences which cannot be understood and explained with
some combination of formal and probabilistic reasoning -- is most
certainly not answered by Godel's theorem.
Our skeptic notes that consistency is the concern of the liar. Those
who are devoted to truth get consistency for free. The argument for
believing that mathematics is consistent is compelling, but it
necessarily comes from outside mathematics itself, and here's the
argument: we simply have to believe that we are capable of consistent
reasoning (clearly we could not "reason" about the possibility that we
lack the ability to reason consistently), and the best model we have of
our own minds is that our minds are equivalent to computers. And, the
basic laws of mathematics are implicit in our best models of
computation. Together, those three assertions compel us to believe that
the basic laws of mathematics must be consistent.
To our skeptic, the mathematicians are playing a very twisted game when
they try to "prove" that mathematics (e.g. PA) is consistent. They
start with the basic principles of mathematics, and then they add on a
mythology about a world of the infinite, and then they claim that
within this bigger theory they can construct a "model" for the more
basic mathematics, and then they claim that that constitutes a proof
that the basic mathematics is consistent. Our skeptic notes that the
mathematicians' "proof" does not compel us to believe anything that we
do not already believe.
Our skeptic notes that "real" mathematics (i.e. the mathematics which
has the potential to help us understand the observable world in which
we live; the mathematics used by physicists, computer scientists,
statisticians, economists, and applied mathematicians) has testable
consequences (see appendix). But how can we test statements about a
world of the infinite lying beyond what we can observe? What test could
we perform to compel us to believe that Godel's proof tells us more
than what a simple and immediate informal argument compels us to
believe? How can we test the assertion that paradox is something other
than pure nonsense? How can we test the proof of a theorem which tells
us nothing more than what we should expect from simple probabilistic
reasoning? Should we not be skeptical of modern mathematics? Is modern
mathematics anything more than clever argumentation with no content? Is
it possible that modern mathematics is built on a flawed model of
reality? Are mathematicians lost? Given that modern mathematics has so
little content, and that it is almost completely inaccessible to the
average person, is it not plausible that the mathematicians have
created an environment in which outright fraud is possible?
So now, let's pretend that our skeptic ventures into sci.math and
sci.logic to explain her reasons for being skeptical of modern
mathematics. What will happen? It's not a pretty picture; the
mathematicians will go on the offensive. They'll call her a crackpot.
They'll claim that she is unqualified to even have skeptical thoughts
about mathematics, and that she is unreasonably demanding that the
experts come down to her level of understanding. They'll claim that she
is trying to impose her religion on others, and that she is trying to
take away the mathematicians' freedom. They'll borrow vocabulary from
the liberals, and accuse her of being a closed minded, ignorant nut
case. They'll try to dismiss all of the skeptical objections as a
result of an inability to deal with abstract thought. Ultimately our
skeptic will be told that mathematics is rightly defined by the
experts, and by definition, mathematics is what expert mathematicians
do, and that skepticism is simply not part of what mathematicians do,
and hence, by definition, mathematicians cannot be doing anything
wrong, and they have no obligation to respond to skepticism. And our
skeptic notes that this last argument is an almost perfect example of
the mathematicians' clever but vacuous, circular argumentation methods.
>From our skeptic's point of view, those brilliant mathematicians simply
cannot respond to skepticism honestly and intelligently, or even
civilly and coherently. They just play games. The skeptic is claiming
that all statements must have testable consequence, and hence, it's
possible for both 'A' and 'not A' to be meaningless if neither has
testable consequences. The mathematicians will insist that if 'A' is a
grammatically correct sentence, then either 'A' or 'not A' must be
true, and they will try to force the skeptic to commit to one or the
other. They willfully refuse to even understand the skeptic's position.
Our skeptic notes that the problem with modern mathematics lies in the
language itself; in order to accommodate Cantor's world of the
infinite, the mathematicians had to expunge the notion of testable
consequences from their language, and now there is essentially no way
to express that idea using their language, and communication between
the skeptics and mathematicians has become impossible.
So what's going on? It may not be entirely clear, but one of the books
that our skeptic has been reading offers an interesting perspective on
the situation. A book by Amir Aczel, "The Mystery of the Aleph :
Mathematics, the Kabbalah, and the Search for Infinity" draws parallels
between modern mathematics and the Kabbalah. The book itself speaks
positively of both modern mathematics and the Kabbalah; it seems to
suggest that we should marvel that the Medieval mystics (Kabbalists)
were able to anticipate the important results of twentieth century
mathematics. But there's another way of looking at what that book is
saying: maybe mathematics has been corrupted by Medieval mysticism.
Does the mathematicians' belief in a world of the infinite lying beyond
what we can observe have its roots in Kabbalism? There is certainly
evidence pointing in that direction. For one thing, Cantor was strongly
influenced by his religious beliefs, and those were mystical beliefs.
And other mystical influences can be found in the historical record.
Some evidence suggests that the mathematicians' belief that important
insights are to be gained from the contemplation of paradox has
Kabbalistic origins. Where does the mathematicians apparent belief
that the knowledge with the very least significant implications for the
world we observe is somehow the most important knowledge, come from?
Why do mathematicians place so much emphasis on clever argumentation
and attach no importance to the testable consequences of their
theories? Why are mathematicians unable to respond rationally to
skepticism? Such questions suggest that mathematics has been corrupted
by mysticism.
If we accept the notion that truth necessarily has observable
implications, and some cultures do emphasize that idea, then we must
admit that modern mathematics is built on a lie. Modern mathematics is
not culturally neutral. Although there is little evidence that anyone
is discriminated against on the basis of race or gender or nationality
or religious affiliation, there is an extreme bias against those who
accept the idea that truth necessarily has observable implications;
those people are regarded as untermenschen (crackpots) by the
mathematics community.
At the start of this article, I posed the question of whether
mathematics has been influenced by the liberalism (humanism, Marxism)
which has taken over the humanties and the social sciences. I believe
it has. Both Cantorian mathematics and Godel's results, if taken
seriously, would appear to validate the liberal view that truth and
reality and logic are merely social constructs, and I suggest that that
is a big part of the reason why those "theories" are accepted as part
of mathematics. As I see it, there is a really big problem here which
needs to be addressed; those who question the liberal dogma are
severely discriminated against in our universities. I believe that the
preferred solution to the humanism (the religion behind liberalism)
problem is to recognize humanism as a religion and apply the laws that
keep religion separate from government. Likewise, the Cantorian
religion (i.e. the belief in the existence of a world of the infinite
lying beyond what we can observe) doesn't belong in the publicly funded
universities.
Appendix Testable consequences of mathematics
Mathematics that is applied necessarily has testable consequences. For
one thing, bridges would fall down, airplanes wouldn't fly, computers
wouldn't work, and weather reports would be wrong half the time if
mathematics were flawed, and those things could be considered to be
tests of mathematics. But even in a more abstract sense, mathematics
has testable consequences.
We can think of mathematics as a science which studies the phenomena
observed in the world of computation. All of the mathematics that has
the potential to be applied can be thought of that way. As a conceptual
aid, we can think of the (abstract) computer as both a microscope and a
test tube: it helps us peer deeply into the world of computation, and
it gives us a way to perform experiments within the world of
computation. Mathematics studies what we observe when we look through
that microscope. Then, roughly speaking, a statement may be said to
have testable consequences if it makes predictions about the results of
computational experiments.
fishfry
"david petry" <david_lawr...@yahoo.com> wrote:
> Is there any reason to be skeptical about modern mathematics?
It's good to be skeptical about everything. Blind faith in any human
endeavor or way of thought is never advised.
That said, it's a long way from healthy skepticism to outright crankery.
And generally when people talk about being skeptical of math, crankery
is exactly what they have in mind.
Robert J. Kolker
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
Mathematics done abstractly in a purely formal manner has zero empirical
content. The only way a mathematical structure acquires empirical
content is to be mapped onto the real world in some fashion. This is not
rocket science.
Bob Kolker
Robert J. Kolker
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
P.S. There is no such thing as Jewish mathematics, Jewish physics,
Jewish geology etc.. Mathematics is objective to the extent once can
determine if what purports to be a proof really is. The physical
sciences are corroberated or falsified by objective examination of the
real world. There is no Jewish aspect to this.
This notion of Jewish X is the stigma of an anti-semite. Are you an
anti-semite?
Bob Kolker
Jim F.
"Robert J. Kolker" <now...@nowhere.com> wrote in message
news:4inr8bF...@individual.net...
Well, this David Petry certainly seems to be a crank.
Unfortunately, certain trends in postmodern thought
have given the kinds of arguments that this David Petry
presented a certain surface plausibility that they would
not otherwise have. Alan Sokal very effectively satirized
this form of pomo, a decade ago, with his witty hoaxing
of the pomo journal, Social Text. Alas, the lessons of
that have apparently not quite sunk in with everyone yet.
>
> Bob Kolker
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
Wrong!
It is the assumption that the abstract mathematics actually models to
something not abstract that is being tested. and that is not a
mathematical assumption but an engineering assumption.
Mathematics can be perfectly valid as mathematics and still be wrongly
applied to non-mathematical situations.
>For
> one thing, bridges would fall down, airplanes wouldn't fly, computers
> wouldn't work, and weather reports would be wrong half the time if
> mathematics were flawed,
Or correct mathematics being misapplied.
There is the tale of the supposed mathematical proof that a bumble bee
cannot fly. The mathematics is correct but the physical assumptions on
which the "proof" was based were wrong.
>
> We can think of mathematics as a science which studies the phenomena
> observed in the world of computation.
One may simplify arithmetic with computers, but that is more accounting
than mathematics. And anyone who thinks of mathematics as only what can
be done with computers has no idea of what mathematics rally is.
Victor Eijkhout
> There is no such thing as Jewish mathematics,
"I told Banach about an expression Johnny had once used in conversation
with me in Princeton before stating some non-Jewish mathematician's
result, "Die Goim haben den folgenden Satz bewiesen" (The goys have
proved the following theorem). Banach, who was pure goy, thought it was
one of the funniest sayings he had ever heard."
Stanislaw Ulam, Adventures of a Mathematician.
But yeah, you're right.
Victor.
--
Victor Eijkhout -- eijkhout at tacc utexas edu
ph: 512 471 5809
Mike Kelly
david petry wrote:
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
>
> I'm like the reader to consider the possibility that the answer to all
> those questions is yes.
>
> So now, let's pretend that our skeptic ventures into sci.math and
> sci.logic to explain her reasons for being skeptical of modern
> mathematics. What will happen? It's not a pretty picture; the
> mathematicians will go on the offensive. They'll call her a crackpot.
> They'll claim that she is unqualified to even have skeptical thoughts
> about mathematics, and that she is unreasonably demanding that the
> experts come down to her level of understanding. They'll claim that she
> is trying to impose her religion on others, and that she is trying to
> take away the mathematicians' freedom. They'll borrow vocabulary from
> the liberals, and accuse her of being a closed minded, ignorant nut
> case. They'll try to dismiss all of the skeptical objections as a
> result of an inability to deal with abstract thought.
Ah, so the intelligent skeptic in your story is supposed to be you. You
could have been honest and told us this was a sob-story from the start.
So little bigoted nazi-fuckwit davey get his feeling hurt by the big
bad mathematicians who rightly rubbish his crankish ideas as being born
of ignorance? Aww, diddums.
--
mike.
abo
david petry wrote:
>
> 4) Fermat's Last Theorem.
I don't know whether your numbers add up, but it's a cute argument...
> They'll borrow vocabulary from
> the liberals, and accuse her of being a closed minded, ignorant nut.
Well there's a self-verifying assertion (i.e. an assertion which, by
being uttered, makes itself true) if there ever was one! You obviously
are a close minded, ignorant nut.
Your critique of mathematics touches some very common bases, but it's
muddled. Here, however, you blaze new ground... unfortunately I think
it's the ground under you which you're burning.
> Both Cantorian mathematics and Godel's results, if taken
> seriously, would appear to validate the liberal view that truth and
> reality and logic are merely social constructs, and I suggest that that
> is a big part of the reason why those "theories" are accepted as part
> of mathematics. As I see it, there is a really big problem here which
> needs to be addressed; those who question the liberal dogma are
> severely discriminated against in our universities.
Do you have any examples in mind? You've cited Fefermann as being
anti-Cantorian, but he had a pretty nice post at Stanford...
> I believe that the
> preferred solution to the humanism (the religion behind liberalism)
> problem is to recognize humanism as a religion and apply the laws that
> keep religion separate from government.
> Likewise, the Cantorian
> religion (i.e. the belief in the existence of a world of the infinite
> lying beyond what we can observe) doesn't belong in the publicly funded
> universities.
>
You're the Timothy McVeigh of sci.logic. I sincerely hope you keep the
expression of your views limited to ranting mindlessly on sci.logic.
herbzet
david petry wrote:
>
[...]
>
> 2) Godel's Theorem (loosely, no consistent formalism can prove its own
> consistency) Informally, our skeptic claims, a proof is a compelling
> argument. It seems clear to our skeptic that if we are to believe that
> the formal theorems in our formalism should be accepted as compelling
> arguments, then at the very least it must be the case that we already
> believe that our formalism is consistent, and hence, no possible formal
> proof within that formalism could be considered to be the evidence that
> compels us to believe that the formalism is consistent.
This last sentence is well-written. I strongly agree with it,
though it's possible that I might consider such a proof to be
in the nature of corroborating evidence.
> And our skeptic
> asks, is that not already the essential content of Godel's theorem?
No.
> Even if you argue that Godel's proof is superior because it is actually
> formal, you still have to deal with the informal notion of proof: does
> Godel's proof compel us to believe that Godel's theorem is actually
> true?
Yes. To a mathematical certainty.
> So, our skeptic asks, what is the concrete content to Godel's
> theorem?
Godel's First Incompleteness theorem states that if arithmetic
(number theory, Peano arithmetic, the theory P which Godel
actually was using) is (omega) consistent, then it is incomplete:
there will be formulae phi such that neither phi nor not-phi are
provable in arithmetic.
The Second Incompleteness theorem states that one of those
undecideable-in-P formulae is the formula that formalizes
the proposition "P is consistent".
> What does it tell us that is not implicit in the definition of
> "proof"?
The question is mis-asked, and has no answer, since it is based
on a false assumption.
> How can it be tested?
It doesn't need to be tested, IMO, any more than any other theorem
needs to be tested. In any case, it could be falsified by
giving a proof in P (or PA, or any equivalent theory) of the
formula Con-P (or, respectively, Con-PA, or Con-(eqivalent T)).
Is it anything more than clever
> argumentation?
No more or less than any other mathematical theorem.
> How can such a theorem be regarded as one of the most
> important theorems in all of mathematics?
That's a good question. The significance of the theorems is
still being investigated, 75 years later. I personally think
it may not be as significant as its reception has suggested.
( see http://tinyurl.com/zyzzu )
> Why don't mathematicians
> raise these kind of questions?
They do.
> Why aren't they at least a little bit
> skeptical?
They are.
--
hz
'Even the crows on the roofs caw about the nature of conditionals.'
-- Callimachus --
herbzet
david petry wrote:
>
[...]
>
I ran a search on your post. I couldn't find the string "Jew" or
"Jewish" anywhere in the article except for the title, and the
cross-posting to soc.culture.jewish.
1) What does your article have to do with "Jewish mathematics"?
2) What is Jewish mathematics?
3) While we're on the subject, what is a Jew?
BTW, neither Cantor nor Godel were Jewish.
T.H. Ray
Scientists don't "test the mathematics," whether one
speaks of engineering models or compututational
experiments. They test theories written in mathematical
language. The theory is independent of the result.
Why is it so common that people who know the least
about mathematics and mathematicians are compelled to
hold forth on it the most?
And what could you possibly mean by "Jewish mathematics,"
except to parrot some stupidly racist screed that we have
rarely heard since Nazi propagandists condemned relativity
as "Jewish science?"
As a troll, this article is superior. As criticism, it is
stupid, obnoxious and anti-intellectual.
Tom
toni.l...@gmail.com
[blather snipped]
> We can think of mathematics as a science which studies the phenomena
> observed in the world of computation. All of the mathematics that has
> the potential to be applied can be thought of that way. As a conceptual
> aid, we can think of the (abstract) computer as both a microscope and a
> test tube: it helps us peer deeply into the world of computation, and
> it gives us a way to perform experiments within the world of
> computation. Mathematics studies what we observe when we look through
> that microscope. Then, roughly speaking, a statement may be said to
> have testable consequences if it makes predictions about the results of
> computational experiments.
Mathematical concepts involving uncountable infinities such as
infinite-dimensional vector spaces can be used to model real-life
phenomena. That such concepts do not directly adhere to the physical
reality we experience is irrelevant. Things like optimization
algorithms rely on fundamental properties that exist outside the narrow
confines of R^n (or worse, pseudo-physical R^n as perpetrated by cranks
who think mathematics is a subset of physics).
That David Petry has a philosophical problem against uncountable
infinities does not remove the utility of such models, nor will it
persuade mathematicians to recast all their work in terms of simplistic
finite computations which amount to nothing more than worshipping a
particular tool.
Down with constrictivism!
Jim F.
"herbzet" <her...@yahoo.com> wrote in message
news:1153907813....@i42g2000cwa.googlegroups.com...
>
>
> david petry wrote:
>>
> [...]
>>
>
> I ran a search on your post. I couldn't find the string "Jew" or
> "Jewish" anywhere in the article except for the title, and the
> cross-posting to soc.culture.jewish.
>
> 1) What does your article have to do with "Jewish mathematics"?
>
> 2) What is Jewish mathematics?
>
> 3) While we're on the subject, what is a Jew?
>
> BTW, neither Cantor nor Godel were Jewish.
Goedel was certainly not Jewish. Cantor may well have
been of Jewish descent, although he was certainly not
brought up as a Jew but rather was raised as a Lutheran,
a faith that he adhered to all of his life.
Well, according to footnotes in http://www.jinfo.org/Philosophers.html,
"4. In Men of Mathematics, Eric Temple Bell described Cantor as being "of
pure Jewish descent on both sides," although both parents were baptized. In
a 1971 article entitled "Towards a Biography of George Cantor," the British
historian of mathematics Ivor Grattan-Guinness claimed (Annals of Science
27, pp. 345-391, 1971) to be unable to find any evidence of Jewish ancestry
(although he conceded that Cantor's wife, Vally Guttmann, was Jewish).
However, a letter written by Georg Cantor to Paul Tannery in 1896 (Paul
Tannery, Memoires Scientifique 13, Correspondance, Gauthier-Villars, Paris,
1934, p. 306) explicitly acknowledges that Cantor's paternal grandparents
were members of the Sephardic community of Copenhagen. In a recent book, The
Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
(Four Walls Eight Windows, New York, 2000. pp. 94, 144), Amir Aczel provides
new evidence in the form of a letter, recently uncovered by Nathalie
Charraud, that was written by Georg Cantor's brother Louis to their mother.
This letter seems to indicate that she was also of Jewish descent, as Bell
had claimed originally."
In any case, Cantor's Jewishness ,or lack thereof, has nothing to do with
the validity of his work in set theory.
herbzet
Taking all this at face value, we have that Cantor's grandparents
were Jewish, his parents were baptized, that he was raised Lutheran
and adhered to that all his life, and that he had a Jewish wife.
May I ask, as neutrally as possible, what does any of
this have to do with my assertion that Cantor was not Jewish?
I mean, other than the part about him being Lutheran.
abo
herbzet wrote:
>
> May I ask, as neutrally as possible, what does any of
> this have to do with my assertion that Cantor was not Jewish?
>
> I mean, other than the part about him being Lutheran.
>
> > In any case, Cantor's Jewishness ,or lack thereof, has nothing to do with
> > the validity of his work in set theory.
>
>
I am by no means an expert on this, and I am relying on a memory of a
conversation I had when I was 13, but according to Jewish law (?)
Cantor would be considered Jewish if his mother had only been baptized
after his birth.
Not that it matters (this is beginning to sound like a Seinfeld
episode...).
herbzet
abo wrote:
>
[...]
> I am by no means an expert on this, and I am relying on a memory of a
> conversation I had when I was 13, but according to Jewish law (?)
> Cantor would be considered Jewish if his mother had only been baptized
> after his birth.
>
> Not that it matters (this is beginning to sound like a Seinfeld
> episode...).
LOL -- Not that there's anything wrong with that!
--
hz
david petry
Mike Kelly wrote:
> > So now, let's pretend that our skeptic ventures into sci.math and
> > sci.logic to explain her reasons for being skeptical of modern
> > mathematics. What will happen? It's not a pretty picture; [...]
> Ah, so the intelligent skeptic in your story is supposed to be you.
Not really. There are lots of people with essentially the same
skeptical arguments who come into these newsgroups. The skeptical
views I present in this article do roughly correspond to the views I
had as an undergraduate.
david petry
herbzet wrote:
> I ran a search on your post. I couldn't find the string "Jew" or
> "Jewish" anywhere in the article except for the title, and the
> cross-posting to soc.culture.jewish.
>
> 1) What does your article have to do with "Jewish mathematics"?
As I pointed out in the article, the best explanation I can find for
why we have mysticism in mathematics is that it seems to validate the
beliefs of the Kabbalists.
david petry
herbzet wrote:
> david petry wrote:
> >
>
> [...]
>
> >
> > 2) Godel's Theorem (loosely, no consistent formalism can prove its own
> > consistency) Informally, our skeptic claims, a proof is a compelling
> > argument. It seems clear to our skeptic that if we are to believe that
> > the formal theorems in our formalism should be accepted as compelling
> > arguments, then at the very least it must be the case that we already
> > believe that our formalism is consistent, and hence, no possible formal
> > proof within that formalism could be considered to be the evidence that
> > compels us to believe that the formalism is consistent.
>
> This last sentence is well-written. I strongly agree with it,
> though it's possible that I might consider such a proof to be
> in the nature of corroborating evidence.
>
> > And our skeptic
> > asks, is that not already the essential content of Godel's theorem?
>
> No.
The pretense of this article is that a skeptic who has read popular
books on mathematics is asking the question. She believes that
mathematics must be justified by helping us understand the observable
world in which we live. Your answer is rather unsatisfying to her.
Randy Poe
david petry wrote:
> herbzet wrote:
>
> > I ran a search on your post. I couldn't find the string "Jew" or
> > "Jewish" anywhere in the article except for the title, and the
> > cross-posting to soc.culture.jewish.
> >
> > 1) What does your article have to do with "Jewish mathematics"?
>
> As I pointed out in the article, the best explanation I can find for
> why we have mysticism in mathematics
Where do we have "mysticism in mathematics"?
By "mysticism" do you mean "concepts that are beyond
David Petry"? That's not the conventional meaning
of "mysticism".
- Randy
mensa...@aol.com
Randy Poe wrote:
> david petry wrote:
> > herbzet wrote:
> >
> > > I ran a search on your post. I couldn't find the string "Jew" or
> > > "Jewish" anywhere in the article except for the title, and the
> > > cross-posting to soc.culture.jewish.
> > >
> > > 1) What does your article have to do with "Jewish mathematics"?
> >
> > As I pointed out in the article, the best explanation I can find for
> > why we have mysticism in mathematics
>
> Where do we have "mysticism in mathematics"?
Don't mathematicians believe in imaginary numbers?
Gene Ward Smith
david petry wrote:
> As I pointed out in the article, the best explanation I can find for
> why we have mysticism in mathematics is that it seems to validate the
> beliefs of the Kabbalists.
Kabbalah (as opposed to the Cabal) has had zero effect on mathematics.
Gene Ward Smith
david petry wrote:
> Not really. There are lots of people with essentially the same
> skeptical arguments who come into these newsgroups.
The word for someone who claims modern mathematics derives from Jewish
mysticism (or Neoplatonism, or Gnosticism, or Advaita Vedanta, or
Freemasonry) is not "skeptic".
Gene Ward Smith
Jim F. wrote:
> Goedel was certainly not Jewish. Cantor may well have
> been of Jewish descent, although he was certainly not
> brought up as a Jew but rather was raised as a Lutheran,
> a faith that he adhered to all of his life.
Just to be complete here, let me point out that neither Fermat nor
Wiles were/are Jewish either. It seems "Jewish mathematics" has been
invented entirely by gentiles, and since there were and are many Jewish
mathematicans, whom one might have supposed could have been safely left
to invent Jewish mathematics if it needed inventing, it makes me wonder
why they bothered.
be...@pop.networkusa.net
david petry wrote:
[...]
>
> The Marxists produce beautiful theories. They produce complex, clever,
> precise, and apparently logically consistent arguments, which must pass
> a rigorous peer review process. The Marxists believe themselves to be
> open minded, unbiased, compassionate, independent thinkers. But, of
> course, the skeptics don't see it that way at all. According to the
> skeptics, market forces are simply part of reality, and responding to
> those forces is a natural and compassionate thing for humans to do, and
> hence the implementation of Marxism (which tries to replace market
> forces with governmental planning) requires a brutally oppressive and
> intrusive government willing to criminalize human nature, so that no
> matter how beautiful the Marxist theories may be, there is something
> fundamentally very very wrong with the Marxist world view. Many
> skeptics in academia believe that they are severely discriminated
> against by the Marxists.
>
This is not really true, of course. It's just market forces at work.
There is more demand for Marxists in humanist circles than for
anti-Marxists.
[...]
Sounds like our skeptic is a staunch positivist (and not even a logical
positivist, since she apparently doesn't accept the validity of logic),
if she thinks "The idea that there must exist more unidentifiable
objects than identifiable objects appears to be silly word play." If
she really had no "axe to grind", she would be perfectly willing to
accept this claim, given sufficient evidence. Now I will grant that she
might feel that Cantor's argument is insufficient evidence for the
claim, but that of course is a different matter than rejecting the
claim out of hand, as you describe her doing.
> Clearly, to our skeptic, set theory includes an element of make-believe. So
> she concludes that much of what she has been reading is nothing more than
> clever argumentation with no concrete content; what possible testable
> consequences are there to the assertion that unidentifiable objects
> exist? And why don't the books she has been reading address the obvious
> skeptical objections to such ideas?
It is usually estimated that there are roughly 10^80 atoms in the
universe (give or take a few orders of magnitude). Obviously we cannot
hope to "identify" all of them. So evidently, our "skeptic" must either
reject atomic theory, or accept that the existence of unidentifiable
objects has testable consequences.
> 2) Godel's Theorem (loosely, no consistent formalism can prove its own
> consistency) Informally, our skeptic claims, a proof is a compelling
> argument. It seems clear to our skeptic that if we are to believe that
> the formal theorems in our formalism should be accepted as compelling
> arguments, then at the very least it must be the case that we already
> believe that our formalism is consistent, and hence, no possible formal
> proof within that formalism could be considered to be the evidence that
> compels us to believe that the formalism is consistent. And our skeptic
> asks, is that not already the essential content of Godel's theorem?
Godel's theorem shows that there cannot be a formal proof that the
formalism is consistent. Our non-logical positivist's reasoning, while
valid, only shows that such a formal proof would be useless, not that
it cannot exist.
[...delete argument from false premises...]
>
> 4) Fermat's Last Theorem. Our skeptic notes that while FLT itself has
> clear meaning and a concrete content, there's nevertheless something
> fishy about the idea that it has been proved. There is something that
> is immediately clear to anyone who has dared to search for a
> counterexample to FLT: just due to chance alone, it seems unlikely that
> there is a counterexample. That is, for an exponent 'p' of modest size
> or larger, the set of integers which are p'th powers is a very very
> sparse set of integers, and for an arbitrary set of integers that is
> that sparse, straightforward probabilistic reasoning tells us that it
> is very unlikely that the sum of two of its elements will turn out to
> be another element of the set. In fact, for example, a
> back-of-the-envelope calculation suggests that for a set as sparse as
> the set of 50'th powers, the probability that two of its elements will
> sum to a third element of the set is about 1/10^200, and this can be
> loosely interpreted as giving a probabilistic proof that FLT is almost
> certainly true for exponent 50.
Well, this is certainly a consistent argument for our non-logical
positivist. Or maybe not. She does not care about "proof", but only
about what can be identified, so it is only natural that a proof of FLT
would hold no interest to her. On the other hand, in her eyes, FLT is
proven by the fact that noone has ever found a counterexample, quite
independently of any probability arguments. So even your probability
argument is of no value to her. In any event, she likely rejects the
existence of probabilty because it cannot be tested without
circularity. But perhaps she is a "heuristic positivist".
[...]
> Our skeptic notes that "real" mathematics (i.e. the mathematics which
> has the potential to help us understand the observable world in which
> we live; the mathematics used by physicists, computer scientists,
> statisticians, economists, and applied mathematicians) has testable
> consequences (see appendix).
i.e. mathematics with testable consequences has testable
consequences. I have to agree with this...
> But how can we test statements about a world of the infinite lying
> beyond what we can observe?
How can we test statements about atoms that are too small to see? I say
that the validity of calculus demonstrates the "existence" of the
infinite in the same sense that the validity of modern chemistry and
thermodynamics proves the existence of atoms.
> What test could we perform to compel us to believe that Godel's proof tells us more
> than what a simple and immediate informal argument compels us to
> believe?
A test can't "compel" us to beleive anything, it can only support or
refute a claim. But one is always free to accept or reject the claim
anyway, by adding auxiliary claims if necessary.
> How can we test the assertion that paradox is something other
> than pure nonsense?
If consideration of the paradox leads to useful or interesting results,
such as Godel's theorem, then it must not be _pure_ nonsense.
> How can we test the proof of a theorem which tells
> us nothing more than what we should expect from simple probabilistic
> reasoning?
Look at individual steps of the proof, to see if they are valid.
> Should we not be skeptical of modern mathematics?
You have not given any reason to be any more skeptical of "modern
mathematics" than of, say, the mathematics of Euclid. Euclid also
included lengthy and complicated proofs of apparently self-evident
results.
> Is it possible that modern mathematics is built on a flawed model of
> reality?
Conventional wisdom among mathematicians is that it is not based on a
model of reality at all. So I would have to say no. If it were based on
some
world model, they would surely know that, no?
[...]
> So now, let's pretend that our skeptic ventures into sci.math and
> sci.logic to explain her reasons for being skeptical of modern
> mathematics. What will happen? It's not a pretty picture; the
> mathematicians will go on the offensive. They'll call her a crackpot.
Quite rightly, too, seeing as she rejects the validity of logic, and
possibly atomic theory as well.
> They'll claim that she is unqualified to even have skeptical thoughts
> about mathematics, and that she is unreasonably demanding that the
> experts come down to her level of understanding. They'll claim that she
> is trying to impose her religion on others, and that she is trying to
> take away the mathematicians' freedom.
Quite rightly; she is indeed trying to impose her heuristic positivism
on other
people, and doing so under the false pretense that it is identical with
"skepticism".
[...]
> We can think of mathematics as a science which studies the phenomena
> observed in the world of computation. All of the mathematics that has
> the potential to be applied can be thought of that way.
True. In fact, all known mathematics can be thought of that way,
although in some cases, the world of computation must be generalized.
(For example, recursion theory routinely deals with the concept of
relative computability, where one considers computability given a
single non-computable primitive.)
> As a conceptual aid, we can think of the (abstract) computer as both a
> microscope and a test tube: it helps us peer deeply into the world of
> computation, and it gives us a way to perform experiments within the world of
> computation.
No, you need a real computer to do experiments. You can't do
experiments on something that doesn't acutally exist.
David R Tribble
>> [...]
>
herbzet wrote:
> I ran a search on your post. I couldn't find the string "Jew" or
> "Jewish" anywhere in the article except for the title, and the
> cross-posting to soc.culture.jewish.
>
> 2) What is Jewish mathematics?
Perhaps that's the branch of mathematics dealing with Jewish numbers.
"How many apples in that bag of yours? Ten, maybe?"
"Um, ten-ish".
david petry
I don't know why you think you can say that with such certainty. The
point of the article is that a belief in a world of the infinite is not
a culturally neutral idea, but it is part of the Kabbalistic culture.
Also, Cantor's mathematical ideas were very strongly influenced by his
religious beliefs, and those were mystical beliefs, and he grew up in a
Jewish environment. That amounts to suggestive evidence that Kabbalah
has influenced mathematics.
Gene Ward Smith
David R Tribble wrote:
> "How many apples in that bag of yours? Ten, maybe?"
> "Um, ten-ish".
Who wants to know?
Patricia Shanahan
What exactly are the mystical elements of the Lutheran religion, and how
do they relate to the mathematical concept of infinity?
Patricia
Gerry Myerson
Perhaps it's related to the American invention of French fries.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Robert J. Kolker
> I don't know why you think you can say that with such certainty. The
> point of the article is that a belief in a world of the infinite is not
> a culturally neutral idea, but it is part of the Kabbalistic culture.
> Also, Cantor's mathematical ideas were very strongly influenced by his
> religious beliefs, and those were mystical beliefs, and he grew up in a
> Jewish environment. That amounts to suggestive evidence that Kabbalah
> has influenced mathematics.
Georg Cantor was a Lutheran, born and baptized. His grandparents were
Jewish. His parents were converts to the Lutheran Church.
Bob Kolker
>
Gene Ward Smith
Patricia Shanahan wrote:
> What exactly are the mystical elements of the Lutheran religion, and how
> do they relate to the mathematical concept of infinity?
Certainly the most famous Lutheran mystic would be Jacob Boehme. Good
luck to anyone trying to link that to mathematics.
Randy Poe
So if you learned that some famous mathematician liked, say,
Agatha Christie, you would take that as conclusive evidence
that Agatha Christie has influenced mathematics?
- Randy
Aluminium Holocene Holodeck Zoroaster
-- dude *cited* Sokal, heh-heh --
not the Director's Cut of "Contact".
there is no sufficient reason for you or me to comprhened Wiles' proof,
which you can take to be alleged til such time as you might do that;
there are probably two-many (at least) proofs of Fermat's "last"
theorem,
wich appears to be a very early result of his & indicative
of his method, perhaps the first use of it, which might even
have subsumed "infinite descent" in some sense ... that is,
you may as well think about his theorem with other math, or
geometry, or what ever, with the possible exception
of "probability," since that can never prove any thing (after all,
per your example, their are an infinity of 50th powers).
in other words,
Well it looks like Fermat could have proved it, and
this British guy in a closet has been reported to have
in a Famous Math Journal out of New Jersey;
maybe there are elementary proofs in between or around them.
NB: Fermat made not even one mistake,
that I know of, including his conjecture about the Fermat numbers, but
it took a hundred years, aside from F"L"T, to show thatso,
what was his mehtod?
thus:
how simple of a reference do you want,
relating to "really, really tall office-buildings
that are hit by a plane filled with fuel for a transontinental flight,
bigger than a flock of V2s?..." just guessing, about the V2
equivalence....
do you want it to be in only Self-consistent Muslim Physics?...
the important thing, of course, is that they were offices
for the "earned income" folks, hte folks who are bringing you the
Protocol
of the Elders of Kyoto (sik), whether or not you agree
with the tactics, or even on who had used them on us.
> Yeah, and while you're at it why don't you gravitation experts explain
> how the WTC buildings can spontaneously gravitationally collapse into
thus:
note the key phrase, which I hasn't noticed
til a day after I posted it, that
they actually did have an amendment
to *extend* the God-am preclearance rules
-- how much further, I know not -- although
it may have been illegal to pass it.
>Of the four amendments defeated in the House, three had the support of a
>majority of the chamber's 231 Republicans. Only the suggestion that the
>Voting Rights Act be extended to cover other jurisdictions was defeated by
>majorities of both Republicans and Democrats.
thus:
compression is only & always deployed around tension.
--it takes some to jitterbug!
http://members.tripod.com/~american_almanac
http://www.21stcenturysciencetech.com/2006_articles/Amplitude.W05.pdf
http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html
http://larouchepub.com/other/2006/3322_ethanol_no_science.html
http://www.wlym.com/pdf/iclc/howthenation.pdf
Aluminium Holocene Holodeck Zoroaster
of "Uncle Bertie is lying, again," with the temporal reference left
out;
the Village Barber e.g. is easily resolved in plain language
(barbers don't usually cut their own hairs;
they go to another village for that). if you want to know
about his more fullsome and "productive" lies
as an alleged Peace Activist,
you can look it up on the LaRouche website.
the books you cite are to heterogenous;
Singh and Aczel are pretty awful, striving to be somewhat mathless;
Kline, I haven't read as much of as Klein, who's not For Dummies;
Rucker's 4D stuff is just a joke, continuing the British Psychical
Research crap,
Abbot Abbot Squeezed Flatlandmania superfluousness (modern,
even more deathless exercises of this "school" are
from Ian Stewart, Dewdney etc. -- don't have to read the whole thing);
Penrose and Hofstadter are good, as little as I've read of them (NB:
Penrose's "tribar" is just a God-am Mobius strip);
Smullyan's book on reverse-engineering chess is great,
if I ever read it, set as a Homes novel -- I doubt if
you could find any author that is more "expository."
there is nothing difficult about Godel's proof, and
there are many expositions of it. I mean,
how can you know any thing -- if you can,
you can know this. some tines, the solution
to a "math problem" is just "more mathematics"
(*mathemata*, the *study* of music, geometry, astronomy, arithemtic).
> mathematics. For example, books by Hofstadter, Penrose, Rucker,
> Smullyan, Kline, Singh, Aczel, and maybe others. She is troubled that a
> formal, you still have to deal with the informal notion of proof: does
> Godel's proof compel us to believe that Godel's theorem is actually
> of what we are saying. Hence, an utterance such as "I am lying" (i.e.
> the Liar paradox) must be analyzed as if it contained its implicit
> Our skeptic notes that consistency is the concern of the liar. Those
uh-oh, here cometh the Last Digit of Pi
-- dude *cited* Sokal, heh-heh --
not the digit in the Director's Cut of "Contact"....
there is no sufficient reason for you or me to comprhened Wiles' proof,
which you can take to be alleged til such time as you might do that;
there are probably two-many (at least) proofs of Fermat's "last"
theorem,
wich appears to be a very early result of his & indicative
of his method, perhaps the first use of it, which might even
have subsumed "infinite descent" in some sense ... that is,
you may as well think about his theorem with other math, or
geometry, or what ever, with the possible exception
of "probability," since that can never prove any thing (after all,
per your example, their are an infinity of 50th powers)....
in other words,
Well it looks like Fermat could have proved it, and
this British guy in a closet has been reported to have
in a Famous Math Journal out of New Jersey;
maybe there are elementary proofs in between or around them....
T.H. Ray
> herbzet wrote:
>
> > I ran a search on your post. I couldn't find the
> string "Jew" or
> > "Jewish" anywhere in the article except for the
> title, and the
> > cross-posting to soc.culture.jewish.
> >
> > 1) What does your article have to do with "Jewish
> mathematics"?
>
> As I pointed out in the article, the best explanation
> I can find for
> why we have mysticism in mathematics is that it seems
> to validate the
> beliefs of the Kabbalists.
>
You don't know enough mathematics to make even an
ignorant statement about it.
No theorem assigns value to personal belief.
Tom
Richard Herring
"mensa...@aol.compost" <mensa...@aol.com> writes
>
>Randy Poe wrote:
>> david petry wrote:
>> > herbzet wrote:
>> >
>> > > I ran a search on your post. I couldn't find the string "Jew" or
>> > > "Jewish" anywhere in the article except for the title, and the
>> > > cross-posting to soc.culture.jewish.
>> > >
>> > > 1) What does your article have to do with "Jewish mathematics"?
>> >
>> > As I pointed out in the article, the best explanation I can find for
>> > why we have mysticism in mathematics
>>
>> Where do we have "mysticism in mathematics"?
>
>Don't mathematicians believe in imaginary numbers?
No. They manipulate symbols which they call "imaginary numbers" when
they need a name for them. Can't you see the difference?
--
Richard Herring
Rotwang
"...So let's say that our skeptic has been reading popular books about mathematics. For example, books by Hofstadter, Penrose, Rucker, Smullyan, Kline, Singh, Aczel, and maybe others..."
This is what happens when you read popular maths books. If you want to learn mathematics in such a way that it makes sense, read a text book. They are considerable more rewarding.
Rotwang
"I don't know why you think you can say that with such certainty. The point of the article is that a belief in a world of the infinite is not a culturally neutral idea, but it is part of the kabbalistic culture."
In your original post you claim that for mathematics to have worth it should have real-world applications. The real numbers are an uncountable set. Tell me, how much physics or engineering do you think you could do without using the real numbers?
mensa...@aol.com
It was a joke. Just as "imaginary" was a joke on the fact that
the symbols in question were "not real".
>
> --
> Richard Herring
Craig Feinstein
david petry wrote:
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
>
> I'm like the reader to consider the possibility that the answer to all
> those questions is yes.
>
> First of all, it would be absurd to say that the modern academic system
> based on peer review would preclude fraud and cultural bias. The "Sokal
> Affair" seems to have proven that when academic writing becomes
> indistinguishable to the non-expert from buzzword salad, then it's
> likely that even experts can be fooled. (see
> http://en.wikipedia.org/wiki/Sokal_Affair) Even physicists can be
> fooled (see http://math.ucr.edu/home/baez/bogdanoff/ )
>
> But even ignoring the possibility of outright fraud, cultural bias can
> creep into academia slowly and progressively so that it's hardly
> recognized by the majority of people involved. The best example might
> be the takeover of the humanities by the Marxists (or extreme left
> wing).
>
> The Marxists produce beautiful theories. They produce complex, clever,
> precise, and apparently logically consistent arguments, which must pass
> a rigorous peer review process. The Marxists believe themselves to be
> open minded, unbiased, compassionate, independent thinkers. But, of
> course, the skeptics don't see it that way at all. According to the
> skeptics, market forces are simply part of reality, and responding to
> those forces is a natural and compassionate thing for humans to do, and
> hence the implementation of Marxism (which tries to replace market
> forces with governmental planning) requires a brutally oppressive and
> intrusive government willing to criminalize human nature, so that no
> matter how beautiful the Marxist theories may be, there is something
> fundamentally very very wrong with the Marxist world view. Many
> skeptics in academia believe that they are severely discriminated
> against by the Marxists.
>
> consistency of the Marxists' arguments, those arguments are built upon
> a defective (or at least, a culturally biased) model of reality. So the
> question becomes, is it possible that modern mathematics is built upon
> a flawed model of reality which gives it a cultural bias? Does this
> bias have any connection to the left wing bias in the humanities?
>
> For the sake of this argument, let's consider an idealized skeptic. Our
> skeptic will be intelligent and honest to a fault. She will have a
> technical background, and will be fully aware of the power of
> mathematics in technology. She will have no axe to grind, and she will
> have no philosophical, religious or political biases, other than a
> propensity for skepticism. That is, she looks for concrete evidence,
> observable implications, and testable consequences. She is skeptical of
> mere clever argumentation. And she refuses to be intimidated by appeals
> to authority or ad hominem attack.
>
> mathematics. For example, books by Hofstadter, Penrose, Rucker,
> nothing more than clever argumentation with no concrete content and no
> testable consequences. She wants to know how the ideas she has been
> reading about can help us to understand the world in which we live.
>
> Let's look at specific examples of the mathematical ideas she is
> skeptical about.
>
> 1) Set theory and Cantor's Theorem. It seems obvious to our skeptic
> that the mathematical constructs we actually deal with must be
> identifiable, and that we can only identify a countable number of such
> constructs (since our language is countable). So Cantor's Theorem
> asserting the existence of uncountable sets (and hence the existence of
> objects which cannot be identified) cannot have any concrete content.
> The idea that there must exist more unidentifiable objects than
> skeptic, set theory includes an element of make-believe. So she
> concludes that much of what she has been reading is nothing more than
> clever argumentation with no concrete content; what possible testable
> consequences are there to the assertion that unidentifiable objects
> exist? And why don't the books she has been reading address the obvious
> skeptical objections to such ideas?
>
> consistency) Informally, our skeptic claims, a proof is a compelling
> argument. It seems clear to our skeptic that if we are to believe that
> the formal theorems in our formalism should be accepted as compelling
> arguments, then at the very least it must be the case that we already
> believe that our formalism is consistent, and hence, no possible formal
> proof within that formalism could be considered to be the evidence that
> compels us to believe that the formalism is consistent. And our skeptic
> asks, is that not already the essential content of Godel's theorem?
> formal, you still have to deal with the informal notion of proof: does
> Godel's proof compel us to believe that Godel's theorem is actually
> theorem? What does it tell us that is not implicit in the definition of
> "proof"? How can it be tested? Is it anything more than clever
> argumentation? How can such a theorem be regarded as one of the most
> important theorems in all of mathematics? Why don't mathematicians
> raise these kind of questions? Why aren't they at least a little bit
> skeptical?
>
> 3) Self-reference and paradox. (note that some of the popular books our
> skeptic has been reading do suggest that this is of great importance in
> mathematics, and essential for understanding Godel's theorem). First of
> all, our skeptic notes, the assertion that paradox is in some sense
> "real" (i.e. something more than an illusion or a game or a joke),
> would appear to be almost equivalent to the assertion that logical
> reasoning can be used to prove that logic is flawed, which is
> immediately highly suspicious. But it can be analyzed further: one of
> the "ground rules" in communication is that we should always intend to
> tell the truth. That is, when we speak, we are implicitly claiming to
> be telling the truth, and we need to explicitly comment on the truth
> value of our assertions (e.g. with modifiers such as 'probably',
> 'possibly', or 'not') only when we do not feel certain about the truth
> of what we are saying. Hence, an utterance such as "I am lying" (i.e.
> the Liar paradox) must be analyzed as if it contained its implicit
> "(implicitly) I am telling the truth; (explicitly) I am lying", which
> is nothing more than a simple contradiction, with nothing paradoxical
> about it. So our skeptic wonders, how can the study of paradox can be
> anything more than a game; how can the contemplation of paradox
> possibly help us understand the world in which we live; how can it
> possibly have testable consequences? And yet, whole books have been
> devoted to its study -- why? Why don't mathematicians ask and address
> these questions? And as far as self-reference goes, clearly humans can
> talk about themselves, but to claim that sentences can talk about
> themselves would seem to be a bizarre anthropomorphization of abstract
> symbols; natural language gives us no way to create sentences which
> unambiguously refer to themselves.
>
> 4) Fermat's Last Theorem. Our skeptic notes that while FLT itself has
> clear meaning and a concrete content, there's nevertheless something
> fishy about the idea that it has been proved. There is something that
> is immediately clear to anyone who has dared to search for a
> counterexample to FLT: just due to chance alone, it seems unlikely that
> there is a counterexample. That is, for an exponent 'p' of modest size
> or larger, the set of integers which are p'th powers is a very very
> sparse set of integers, and for an arbitrary set of integers that is
> that sparse, straightforward probabilistic reasoning tells us that it
> is very unlikely that the sum of two of its elements will turn out to
> be another element of the set. In fact, for example, a
> back-of-the-envelope calculation suggests that for a set as sparse as
> the set of 50'th powers, the probability that two of its elements will
> sum to a third element of the set is about 1/10^200, and this can be
> loosely interpreted as giving a probabilistic proof that FLT is almost
> proven for all exponents up to 10^6 before Wiles came along, using the
> same heuristic argument, the probability (in the Bayesian sense where a
> probability is a degree of belief) that there could be a counterexample
> to FLT could be taken to be about 1/10^10^7. So, in other words, Wiles
> spent seven years locked in his attic (so the story goes) to do nothing
> more than remove that last little bit (1 part in 10^10^7) of
> uncertainty that FLT is true, assuming that we generously assign a
> value of less than 1/10^10^7 to the probability that his proof is
> flawed! Since the proof tells us nothing that we do not already believe
> to be true with very very high probability, searching for
> counterexamples to the theorem in no way can be deemed a test of the
> proof. So our skeptic has to wonder whether a man who is willing to
> devote so much energy to such an insignificant task, for no apparent
> reason other than to seek fame, would he not be willing to pull off a
> hoax? How could we know? And furthermore, the proof itself is
> presumably accessible to only the top one tenth of one percent of
> mathematicians, so our skeptic notes that she has no realistic hope of
> ever determining for herself whether the proof is consistent. But why
> should she trust the "experts"? Why should the proof of FLT qualify as
> headline news? Why don't the books our skeptic has been reading address
> these kinds of questions?
>
>
> So, does Cantor's proof compel us to believe that there exist
> mathematical objects that cannot be identified? Does it compel us to
> believe that there are more unidentifiable objects than identifiable
> objects? Of course not! For example, we could simply assert that as
> part of the definition, mathematics only studies identifiable objects,
> and then with less magic than was used to prove the existence of
> unidentifiable objects in the first place, all of the unidentifiable
> objects would vanish from the mathematical universe! And to be sure,
> the mathematics that does have testable consequences would hardly be
> affected at all by such a change of definition.
>
> So what's going on? Our skeptic will note that somehow mathematicians
> are cheating. When they use words like "proof", "truth", "exists",
> "logic", and even the word "mathematics" itself, they are not using
> them in the way the rest of the world uses them. The mathematicians
> have chosen convenient definitions and convenient axioms which let the
> mathematicians formally "prove" what they want to prove; they have
> completely abandoned the idea that mathematics should have testable
> consequences; they are playing word games; they have insulated
> themselves from reality.
>
> So now our skeptic asks, given that we see what games the
> mathematicians play, is it even remotely plausible that the
> mathematicians could be capable of coming up with important insights
> into the nature of proof, truth, existence, and logic? For one thing,
> the mathematicians seem to be totally clueless about what is
> "important" to anyone but themselves, given that they think testable
> consequences are not important.
>
> As far as the proof of Fermat's Last Theorem goes, our skeptic admits
> that she has no special insight. But she has to wonder why
> mathematicians apparently refuse to even think about such things from a
> probabilistic point of view. Probabilistic reasoning does produce
> results with testable consequences, and if we regard mathematics as a
> science with the purpose of explaining the phenomena we observe within
> the world of computation (a view completely compatible with the views
> of those who apply mathematics), then probabilistic reasoning should be
> accepted as part of mathematics. And furthermore, once we recognize
> that formal reasoning and probabilistic reasoning complement each other
> (i.e. probabilistic reasoning works especially well where formal
> reasoning fails, and vice versa), we have to reexamine the content of
> Godel's theorem (in this case, the assertion that there exist true
> statements not provable in a given formalism). Clearly we can come up
> with an unlimited number of statements which can be shown by
> probabilistic reasoning to be almost certainly true but for which we
> have no reason whatsoever to believe that they can be proven true in
> any particular formalism. So Godel's theorem is true with a vengeance,
> but it's not Godel's proof which compels us to believe that. And the
> important question -- are there true mathematical statements having
> testable consequences which cannot be understood and explained with
> some combination of formal and probabilistic reasoning -- is most
> certainly not answered by Godel's theorem.
>
> Our skeptic notes that consistency is the concern of the liar. Those
> believing that mathematics is consistent is compelling, but it
> necessarily comes from outside mathematics itself, and here's the
> argument: we simply have to believe that we are capable of consistent
> reasoning (clearly we could not "reason" about the possibility that we
> lack the ability to reason consistently), and the best model we have of
> our own minds is that our minds are equivalent to computers. And, the
> basic laws of mathematics are implicit in our best models of
> computation. Together, those three assertions compel us to believe that
> the basic laws of mathematics must be consistent.
>
> To our skeptic, the mathematicians are playing a very twisted game when
> they try to "prove" that mathematics (e.g. PA) is consistent. They
> start with the basic principles of mathematics, and then they add on a
> mythology about a world of the infinite, and then they claim that
> within this bigger theory they can construct a "model" for the more
> basic mathematics, and then they claim that that constitutes a proof
> that the basic mathematics is consistent. Our skeptic notes that the
> mathematicians' "proof" does not compel us to believe anything that we
> do not already believe.
>
> Our skeptic notes that "real" mathematics (i.e. the mathematics which
> has the potential to help us understand the observable world in which
> we live; the mathematics used by physicists, computer scientists,
> statisticians, economists, and applied mathematicians) has testable
> world of the infinite lying beyond what we can observe? What test could
> we perform to compel us to believe that Godel's proof tells us more
> than what a simple and immediate informal argument compels us to
> than pure nonsense? How can we test the proof of a theorem which tells
> us nothing more than what we should expect from simple probabilistic
> mathematics anything more than clever argumentation with no content? Is
> it possible that modern mathematics is built on a flawed model of
> little content, and that it is almost completely inaccessible to the
> average person, is it not plausible that the mathematicians have
> created an environment in which outright fraud is possible?
>
> So now, let's pretend that our skeptic ventures into sci.math and
> sci.logic to explain her reasons for being skeptical of modern
> mathematics. What will happen? It's not a pretty picture; the
> mathematicians will go on the offensive. They'll call her a crackpot.
> about mathematics, and that she is unreasonably demanding that the
> experts come down to her level of understanding. They'll claim that she
> is trying to impose her religion on others, and that she is trying to
> the liberals, and accuse her of being a closed minded, ignorant nut
> case. They'll try to dismiss all of the skeptical objections as a
> result of an inability to deal with abstract thought. Ultimately our
> skeptic will be told that mathematics is rightly defined by the
> experts, and by definition, mathematics is what expert mathematicians
> do, and that skepticism is simply not part of what mathematicians do,
> and hence, by definition, mathematicians cannot be doing anything
> wrong, and they have no obligation to respond to skepticism. And our
> skeptic notes that this last argument is an almost perfect example of
> the mathematicians' clever but vacuous, circular argumentation methods.
>
> >From our skeptic's point of view, those brilliant mathematicians simply
> cannot respond to skepticism honestly and intelligently, or even
> civilly and coherently. They just play games. The skeptic is claiming
> that all statements must have testable consequence, and hence, it's
> possible for both 'A' and 'not A' to be meaningless if neither has
> testable consequences. The mathematicians will insist that if 'A' is a
> grammatically correct sentence, then either 'A' or 'not A' must be
> true, and they will try to force the skeptic to commit to one or the
> other. They willfully refuse to even understand the skeptic's position.
> Our skeptic notes that the problem with modern mathematics lies in the
> language itself; in order to accommodate Cantor's world of the
> infinite, the mathematicians had to expunge the notion of testable
> consequences from their language, and now there is essentially no way
> to express that idea using their language, and communication between
> the skeptics and mathematicians has become impossible.
>
> So what's going on? It may not be entirely clear, but one of the books
> that our skeptic has been reading offers an interesting perspective on
> the situation. A book by Amir Aczel, "The Mystery of the Aleph :
> Mathematics, the Kabbalah, and the Search for Infinity" draws parallels
> between modern mathematics and the Kabbalah. The book itself speaks
> positively of both modern mathematics and the Kabbalah; it seems to
> suggest that we should marvel that the Medieval mystics (Kabbalists)
> were able to anticipate the important results of twentieth century
> mathematics. But there's another way of looking at what that book is
> saying: maybe mathematics has been corrupted by Medieval mysticism.
>
> Does the mathematicians' belief in a world of the infinite lying beyond
> what we can observe have its roots in Kabbalism? There is certainly
> evidence pointing in that direction. For one thing, Cantor was strongly
> influenced by his religious beliefs, and those were mystical beliefs.
> And other mystical influences can be found in the historical record.
> Some evidence suggests that the mathematicians' belief that important
> insights are to be gained from the contemplation of paradox has
> Kabbalistic origins. Where does the mathematicians apparent belief
> that the knowledge with the very least significant implications for the
> world we observe is somehow the most important knowledge, come from?
> Why do mathematicians place so much emphasis on clever argumentation
> and attach no importance to the testable consequences of their
> theories? Why are mathematicians unable to respond rationally to
> skepticism? Such questions suggest that mathematics has been corrupted
> by mysticism.
>
> If we accept the notion that truth necessarily has observable
> implications, and some cultures do emphasize that idea, then we must
> admit that modern mathematics is built on a lie. Modern mathematics is
> not culturally neutral. Although there is little evidence that anyone
> is discriminated against on the basis of race or gender or nationality
> or religious affiliation, there is an extreme bias against those who
> accept the idea that truth necessarily has observable implications;
> those people are regarded as untermenschen (crackpots) by the
> mathematics community.
>
> At the start of this article, I posed the question of whether
> mathematics has been influenced by the liberalism (humanism, Marxism)
> which has taken over the humanties and the social sciences. I believe
> it has. Both Cantorian mathematics and Godel's results, if taken
> seriously, would appear to validate the liberal view that truth and
> reality and logic are merely social constructs, and I suggest that that
> is a big part of the reason why those "theories" are accepted as part
> of mathematics. As I see it, there is a really big problem here which
> needs to be addressed; those who question the liberal dogma are
> severely discriminated against in our universities. I believe that the
> preferred solution to the humanism (the religion behind liberalism)
> problem is to recognize humanism as a religion and apply the laws that
> keep religion separate from government. Likewise, the Cantorian
> religion (i.e. the belief in the existence of a world of the infinite
> lying beyond what we can observe) doesn't belong in the publicly funded
> universities.
>
>
>
>
>
>
> Appendix Testable consequences of mathematics
>
>
> Mathematics that is applied necessarily has testable consequences. For
> one thing, bridges would fall down, airplanes wouldn't fly, computers
> wouldn't work, and weather reports would be wrong half the time if
> mathematics were flawed, and those things could be considered to be
> tests of mathematics. But even in a more abstract sense, mathematics
> has testable consequences.
>
> We can think of mathematics as a science which studies the phenomena
> observed in the world of computation. All of the mathematics that has
> aid, we can think of the (abstract) computer as both a microscope and a
> test tube: it helps us peer deeply into the world of computation, and
> it gives us a way to perform experiments within the world of
> that microscope. Then, roughly speaking, a statement may be said to
> have testable consequences if it makes predictions about the results of
> computational experiments.
This is a good post. The truth is that modern Cantorian mathematics is
a castle built on sand. But it gives jobs to mathematicians, keeps them
off the street and occupied, so we should really thankful for it.
Furthermore, if you look at the history of mathematics, you'll find
that it is only until recently that Jews have had an impact on
mathematics. And their impact has not even been as a group, but as
individuals. Nothing really distinguishes their work from the work of
Gentiles. The Greeks are really the ones who have had the greatest
impact on modern math. The big-name modern mathematicians who happen to
be Jewish are really just copy-cats, following in the footsteps of the
Ancient Greeks. You don't go to a Yeshiva (institution of Jewish
learning) to learn modern mathematics. You go to a university, which is
built on Ancient Greek ideals, that Truth=Beauty=Good. (The Jewish
ideal is "whatever G-d says goes".)
If you look at Jewish Talmudic sources, you'll see that rigorous and
sophisticated mathematics was never really so high on the list of
priorities for the rabbis. For instance, pi is considered to be 3,
according to the Talmud. The Jews weren't stupid and certainly knew
that pi was not a whole number, but for the purposes of Talmudic
discussion, it was considered to be three, because the circumference of
King Solomon's molten sea was 30 amos and its diameter was 10 amos,
according to the Book of Kings and also Chronicles. (It seems that the
Book of Kings and Chronicles go out of their way to be inaccurate - the
Book of Kings and Chronicles could have said 31 amos around and 10 amos
diameter.)
In short, the following is the closest that you'll come to "Jewish
mathematics":
Beryl: What's 6Q + 4Q?
Shmeryl: 10Q.
Beryl: You're velcome!
Craig
Richard Herring
"mensa...@aol.compost" <mensa...@aol.com> writes
--
Richard Herring
Toni Lassila
wrote:
>If you look at Jewish Talmudic sources, you'll see that rigorous and
>sophisticated mathematics was never really so high on the list of
>priorities for the rabbis.
This claim, like many by Craig Feinstein, is dubious:
Game theoretic analysis of a bankruptcy problem from the Talmud
Robert J. Aumann and Michael Maschler
The Hebrew University, 91904, Jerusalem, Israel
Abstract
For three different bankruptcy problems, the 2000-year old Babylonian
Talmud prescribes solutions that equal precisely the nucleoli of the
corresponding coalitional games. A rationale for these solutions that
is independent of game theory is given in terms of the Talmudic
principle of equal division of the contested amount; this rationale
leads to a unique solution for all bankruptcy problems, which always
coincides with the nucleolus. Two other rationales for the same rule
are suggested, in terms of other Talmudic principles. (Needless to
say, the rule in question is not proportional division).
Craig Feinstein
I repeat, "rigorous and sophisticated mathematics was never really so
high on the list of priorities for the rabbis". Basically, the math
they did was simply the application of grade-school math and some basic
geometry to Jewish Law. You have to go to a university, in this case
"The Hebrew University" (which is really built on Greek, not Jewish
ideals, despite its name) in order to get a game-theoretic analysis of
this particular Mishnah in the Talmud. The Talmud isn't concerned with
the game-theoretic properties of this solution to the bankruptcy
problem. (This is something that the Ancient Greeks would have been
interested in.) The Talmud just discusses the application of the
particular solution given in the Mishnah to Jewish Law. Average Jewish
children can and do learn this passage in the Talmud without any
background in higher level mathematics.
Craig
Aluminium Holocene Holodeck Zoroaster
of the awful Bible Codes, a while ago. monsieur Satnover's "experts"
were apparently associated with Rabin's assassin;
thus, the singular "prediction" in Drosnin's flimflam.
http://meru.org/Codes/biblcode.html
> this particular Mishnah in the Talmud. The Talmud isn't concerned with
> the game-theoretic properties of this solution to the bankruptcy
> problem. (This is something that the Ancient Greeks would have been
> interested in.) The Talmud just discusses the application of the
> particular solution given in the Mishnah to Jewish Law. Average Jewish
> children can and do learn this passage in the Talmud without any
> background in higher level mathematics.
thus:
how simple of a reference do you want,
relating to "really, really tall office-buildings
that are hit by a plane filled with fuel for a transcontinental flight,
bigger than a flock of V2s?..." just guessing, about the V2
equivalence....
do you want it to be in only Self-consistent Muslim Physics?...
the important thing, of course, is that they were offices
for the "earned income" folks, the folks who are bringing you the
Protocol
of the Elders of Kyoto (sik), whether or not you agree
with the tactics, or even on who had used them on us. of course,
there were more British Subjects than any other foreigners,
who died in the conflagration. (just remember:
many more excaped, than died .-)
Rupert
david petry wrote:
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
>
> I'm like the reader to consider the possibility that the answer to all
> those questions is yes.
>
> First of all, it would be absurd to say that the modern academic system
> based on peer review would preclude fraud and cultural bias. The "Sokal
> Affair" seems to have proven that when academic writing becomes
> indistinguishable to the non-expert from buzzword salad, then it's
> likely that even experts can be fooled. (see
> http://en.wikipedia.org/wiki/Sokal_Affair) Even physicists can be
> fooled (see http://math.ucr.edu/home/baez/bogdanoff/ )
>
The question of whether an argument goes through in a particular formal
system is an objective matter which qualified people will come to agree
on. There is no reason to think that any fraud is going on in the
process of peer review. If you don't like a particular formal system,
you are welcome to raise philosophical objections to it. If you think
that a result that has been accepted as provable in a particular formal
system is in fact not provable in this formal system, you are welcome
to point out the flaw in the argument. Unsubstantiated accusations of
fraud are ridiculous.
herbzet
david petry wrote:
>
> herbzet wrote:
>
> > I ran a search on your post. I couldn't find the string "Jew" or
> > "Jewish" anywhere in the article except for the title, and the
> > cross-posting to soc.culture.jewish.
> >
> > 1) What does your article have to do with "Jewish mathematics"?
>
> As I pointed out in the article, the best explanation I can find for
> beliefs of the Kabbalists.
There has been mysticism in mathematics since its beginnings. Read
up on the Pythagoreans, for a start.
--
hz
'Even the crows on the roofs caw about the nature of conditionals.'
-- Callimachus -
herbzet
david petry wrote:
>
> herbzet wrote:
> > >
> >
> > [...]
> >
> > >
> > > 2) Godel's Theorem (loosely, no consistent formalism can prove its own
> > > consistency) Informally, our skeptic claims, a proof is a compelling
> > > argument. It seems clear to our skeptic that if we are to believe that
> > > the formal theorems in our formalism should be accepted as compelling
> > > arguments, then at the very least it must be the case that we already
> > > believe that our formalism is consistent, and hence, no possible formal
> > > proof within that formalism could be considered to be the evidence that
> > > compels us to believe that the formalism is consistent.
> >
> > though it's possible that I might consider such a proof to be
> > in the nature of corroborating evidence.
> >
> > > And our skeptic
> > > asks, is that not already the essential content of Godel's theorem?
> >
>
> The pretense of this article is that a skeptic who has read popular
> books on mathematics is asking the question. She believes that
> mathematics must be justified by helping us understand the observable
> world in which we live. Your answer is rather unsatisfying to her.
1) From the tone of your original post, I would think think your
skeptic would be more unsatisfied with a "yes" answer. I
should think she would be relieved to hear that the answer
is "no".
2) I think a better word than "pretense" would be "conceit". But
that's a matter of taste.
herbzet
Gene Ward Smith wrote:
>
> david petry wrote:
>
> > As I pointed out in the article, the best explanation I can find for
> > why we have mysticism in mathematics is that it seems to validate the
> > beliefs of the Kabbalists.
>
> Kabbalah (as opposed to the Cabal) has had zero effect on mathematics.
What is the Cabal?
--
hz
'Even the crows on the roofs caw about the nature of conditionals.'
-- Callimachus --
herbzet
Easy.
Jacob Boehme is a Lutheran mystic.
Cantor is a Lutheran.
Therefore, Cantor's mathematics are mystical.
If you weren't so over-educated this would be obvious.
herbzet
david petry wrote:
>
> Gene Ward Smith wrote:
> > david petry wrote:
> >
> > > As I pointed out in the article, the best explanation I can find for
> > > why we have mysticism in mathematics is that it seems to validate the
> > > beliefs of the Kabbalists.
> >
> > Kabbalah (as opposed to the Cabal) has had zero effect on mathematics
>
> I don't know why you think you can say that with such certainty. The
> point of the article is that a belief in a world of the infinite is not
> a culturally neutral idea, but it is part of the Kabbalistic culture.
It is a part of many cultures.
> Also, Cantor's mathematical ideas were very strongly influenced by his
> religious beliefs, and those were mystical beliefs, and he grew up in a
> Jewish environment. That amounts to suggestive evidence that Kabbalah
> has influenced mathematics.
Have you heard that Madonna is studying Kabbalah? Soon pop music
will be infected with Kabbalistic mysticism.
Arguably, that may be an improvement.
--
hz
'Even the crows on the roofs caw about the nature of conditionals.'
-- Callimachus --
Gene Ward Smith
herbzet wrote:
> Gene Ward Smith wrote:
> > Kabbalah (as opposed to the Cabal) has had zero effect on mathematics.
>
>
> What is the Cabal?
zr
We need a suite for 4 hours to talk business.
Well says the concierge I have one smaller room available. It will cost you
$30.00 for 4 hours.
Fine said the three, and they each put in $10.00 and took the elevator up to
the 5th floor.
The concierge once they had left realized that he had overcharged them by
$5. He called the Bell Captain and explaining the overcharge handed the
Captain $5 to be returned to the three men.
In the elevator on his way to their room the Bell Captain decided he
couldn't return the $5 evenly to the three.
Upon arriving at the room, he explained that the three were overcharged and
he was returning $1 each, on leaving he pocketed the $2 for his time.
Each man got back 1$ from his $10 meaning they each paid $9 or $27 for the
room.
Now 3 times $9 is $27 and the Bell Captain kept $2 that adds up to $29.
"What happened to the other $1?"
"david petry" <david_lawr...@yahoo.com> wrote in message
news:1153871230.2...@m73g2000cwd.googlegroups.com...
>
> Is there any reason to be skeptical about modern mathematics? Is it
> possible that modern mathematics is culturally biased? Is it possible
> that there is an element of fraud in modern mathematics? Has
> mathematics become clever argumentation with no concrete content?
>
> I'm like the reader to consider the possibility that the answer to all
> those questions is yes.
>
> First of all, it would be absurd to say that the modern academic system
> based on peer review would preclude fraud and cultural bias. The "Sokal
> Affair" seems to have proven that when academic writing becomes
> indistinguishable to the non-expert from buzzword salad, then it's
> likely that even experts can be fooled. (see
> http://en.wikipedia.org/wiki/Sokal_Affair) Even physicists can be
> fooled (see http://math.ucr.edu/home/baez/bogdanoff/ )
>
> 2) Godel's Theorem (loosely, no consistent formalism can prove its own
> consistency) Informally, our skeptic claims, a proof is a compelling
> argument. It seems clear to our skeptic that if we are to believe that
> the formal theorems in our formalism should be accepted as compelling
> arguments, then at the very least it must be the case that we already
> believe that our formalism is consistent, and hence, no possible formal
> proof within that formalism could be considered to be the evidence that
> compels us to believe that the formalism is consistent. And our skeptic
> asks, is that not already the essential content of Godel's theorem?
mensa...@aol.com
herbzet wrote:
> Gene Ward Smith wrote:
> >
> > david petry wrote:
> >
> > > As I pointed out in the article, the best explanation I can find for
> > > why we have mysticism in mathematics is that it seems to validate the
> > > beliefs of the Kabbalists.
> >
> > Kabbalah (as opposed to the Cabal) has had zero effect on mathematics.
>
>
> What is the Cabal?
A form of television.
herbzet
zr wrote:
>
> Three men walk into a hotel in downtown Tel Aviv.
> We need a suite for 4 hours to talk business.
> Well says the concierge I have one smaller room available. It will cost you
> $30.00 for 4 hours.
> Fine said the three, and they each put in $10.00 and took the elevator up to
> the 5th floor.
> The concierge once they had left realized that he had overcharged them by
> $5. He called the Bell Captain and explaining the overcharge handed the
> Captain $5 to be returned to the three men.
> In the elevator on his way to their room the Bell Captain decided he
> couldn't return the $5 evenly to the three.
> Upon arriving at the room, he explained that the three were overcharged and
> he was returning $1 each, on leaving he pocketed the $2 for his time.
> Each man got back 1$ from his $10 meaning they each paid $9 or $27 for the
> room.
> Now 3 times $9 is $27 and the Bell Captain kept $2 that adds up to $29.
> "What happened to the other $1?"
There is no other $1. 3 times $9 is $27 and the Bell Captain kept $2.
$27 - $2 = $25 which is in the hotel cash drawer.
herbzet
Thank you.
--
hz
herbzet
mensa...@aol.compost wrote:
> herbzet wrote:
> > Gene Ward Smith wrote:
> > > david petry wrote:
> > >
> > > > As I pointed out in the article, the best explanation I can find for
> > > > why we have mysticism in mathematics is that it seems to validate the
> > > > beliefs of the Kabbalists.
> > >
> > > Kabbalah (as opposed to the Cabal) has had zero effect on mathematics.
> >
> >
> > What is the Cabal?
>
> A form of television.
LOL -- well, in that case, its effect could not have been good.
--
hz
T.H. Ray
>
> -- Callimachus --
>
Boy, that's an oldie. It occurs to me, though, that
the story is quite relevant to this thread -- because it
demonstrates the false reasoning that grows from asking
the wrong question so as to create a pseudo-problem,
exactly as the OP has done.
In the story, the question, "what happened to the missing
dollar?" is equivalent to the OP's implied question,
"why does modern mathematics resemble ancient Jewish
mysticism?"
In both cases, the question is merely a red herring trail
to mythical notions -- assigning value to what one
prefers to believe is true, as opposed to the truth that
lies before one, "like a vast ocean" as Newton put it.
One does not acquire knowledge -- truly objective
knowledge -- by practicing the intellectually lazy scheme
of assigning the value of causation to correlation.
This is observably the way most of the world thinks.
What does one gain, however, by abandoning truth, merely
to assuage lazy skepticism?
Tom
david petry
herbzet wrote:
> david petry wrote:
> > As I pointed out in the article, the best explanation I can find for
> > why we have mysticism in mathematics is that it seems to validate the
> > beliefs of the Kabbalists.
>
>
> There has been mysticism in mathematics since its beginnings. Read
> up on the Pythagoreans, for a start.
The current form of mysticism in mathematics is definitely not
Pythagorean in nature. The Pythagoreans strongly rejected a belief in
a world of the infinite lying beyond what we can observe, as far as I
know.
zr
The Till has $25
The Clients paid $9 each
Regardless there is still $1 missing
"herbzet" <her...@yahoo.com> wrote in message
news:1154064570....@s13g2000cwa.googlegroups.com...
Virgil
"zr" <ng...@cogeco.ca> wrote:
> No the Bellboy has $2
> The Till has $25
> The Clients paid $9 each
> Regardless there is still $1 missing
3 clients paid &9 each for a total of $27 paid out.
The till has $25 and the bellboy has $2 for $27 received.
What dollar is missing?
zr
"Virgil" <vir...@comcast.net> wrote in message
news:virgil-829AE2....@news.usenetmonster.com...
Virgil
"zr" <ng...@cogeco.ca> wrote:
> The 30th.
The 28th, 29th and 30th dollars were returned to the clients, $1 each.
>
> "Virgil" <vir...@comcast.net> wrote in message
> news:virgil-829AE2....@news.usenetmonster.com...
> > In article <2_Qyg.53033$Uy1....@read1.cgocable.net>,
> > "zr" <ng...@cogeco.ca> wrote:
> >
> >> No the Bellboy has $2
> >> The Till has $25
> >> The Clients paid $9 each
> >> Regardless there is still $1 missing
> >
> > 3 clients paid &9 each for a total of $27 paid out.
> > The till has $25 and the bellboy has $2 for $27 received.
> >
> > What dollar is missing?
Each client paid $10 and received $1 back. So they paid out only $27
total.
LauLuna
david petry wrote:
>It seems clear to our skeptic that if we are to believe that
>the formal theorems in our formalism should be accepted as compelling
>arguments, then at the very least it must be the case that we already
>believe that our formalism is consistent, and hence, no possible formal
>proof within that formalism could be considered to be the evidence that
>compels us to believe that the formalism is consistent. And our skeptic
>asks, is that not already the essential content of Godel's theorem?
You are conflating two meanings of "prove": proving as proving someting
true (let's say "PROVE") and proving as generating as string of symbols
the way a formal system could do (let's say "prove"). And,
consequently, you confuse two meanings of "sentence": sentence as
"formal sentence" and sentence as the proposition resulting from some
interpretation of some formal sentence.
Gödel PROVED that some formal systems don't prove some formal
sentences, while your skeptic PROVES that those systems don't PROVE
some interpretations of those sentences.
Regards
T.H. Ray
> No the Bellboy has $2
> The Till has $25
> The Clients paid $9 each
> Regardless there is still $1 missing
>
I am going to make the assumption that you are not just
trying to pull our legs.
After all, if we accepted your version of the problem,
there would be 2 + 25 + (3 X 9) = 54 dollars in
existence. Not missing just 1 dollar, but 24. :-)
Reminds me of an Abbot and Costello routine, where
the clever Bud Abbott cheats his friend Lou out of
money, by borrowing and repaying in quick succession.
The trick is to start off owing money, borrow to repay
it, pay back and then borrow more, so that it appears
the original amount is "missing" and no longer owed.
I expect that some similar scam long ago led to the
invention of double entry bookkeeping.
The innkeeper riddle is simple. I.e., 30-5+3+2=30.
One gets confused by the shell game operation of
dividing out among the customers and bellboy -- but
looking at the arithmetic in the equation above, one
can clearly see that no money has disappeared; 3 + 2
is still 5 and -5 + 5 is still 0. The original $30
is all accounted for. 30-5+3+2-30=0
A deeper point can be made of this simplicity. When
the arithmetic is written in a clear numerical
equivalence, the properties of identity, reflexivity
and transitivity that characterize an arithmetic -- or
algebraic -- equation, are manifestly obvious. One
shouldn't think of these terms as mere mathematical
jargon; they underpin the logic of problem solving in
both literary and mathematical language. They map
abstract symbols to facts. Learning how to do that in
ever more meaningful ways may prevent one from making
irrational conclusions about missing money that isn't
really missing. Or, more importantly, about "Jewish
mathematics."
Tom
her...@cox.net
Virgil, you're obviously failing to grasp that this is
an example of "Jewish" mathematics, as evidenced by your
perverse insistence on subtracting $2 from $27, instead
of adding.
--
hz
"Against stupidity, the gods themselves contend in vain."
-- Schiller --
her...@cox.net
Oh, I get it ... the mysticism of Cantor's mathematics is
the _wrong kind_ of mysticism.
--
hz
"If the doors of perception were cleansed
everything would appear to man as it is, infinite."
-- William Blake --
Han.de...@dto.tudelft.nl
> Also, Cantor's mathematical ideas were very strongly influenced by his
> religious beliefs, and those were mystical beliefs, and he grew up in a
> Jewish environment. That amounts to suggestive evidence that Kabbalah
> has influenced mathematics.
Affirmative:
http://pirate.shu.edu/~wachsmut/ira/history/cantor.html
Quote:
Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg,
Russia, on March 3, 1845. Georg's background was very diverse.
His father was a Danish _Jewish merchant_ that had converted to
Protestantism while his mother was a Danish Roman Catholic.
Underline by me: _Jewish merchant_. I think the "merchant" is equally
important as the "Jewish". Perhaps the "Jewish" has lead to Cantor's
preoccupation with the infinite, but the "merchant" has lead Cantor to
his believe that the whole world is a set and nothing but a set. Read
"The Political Economy of Sets":
http://groups.google.nl/group/sci.math/msg/19e5174536f49c32?hl=en&
Han de Bruijn
Barb Knox
Han.de...@DTO.TUDelft.NL wrote:
The opening paragraph of that shows considerable ignorance of the actual
history of mathematics (and misrepresenting History is a sin for a
Marxist, no?) --
> Virtually any kind of Modern Math is based upon Set Theory. Despite
> the fact that ST suffered from (Russell's) paradoxes from the very
> beginning. This would have assassinated any other kind of
> mathematical theory. It is remarkable that Set Theory survived its
> shortcomings in the first place. Big surprise; it even became the
> foundation "par exellance" whereupon Modern Mathematics is based.
Consider that calculus too started out as a half-baked theory laden with
paradoxes, and that one of the great mathematical achievements was to
put it on a rigourous footing. And indeed, set theory was one of the
important tools in that enterprise.
zr
Mysticism is even more of a conundrum.
"Everyone knows the speed of light"
Can anyone tell us the speed of dark?
"Barb Knox" <s...@sig.below> wrote in message
news:see-FF2685.0...@lust.ihug.co.nz...
Virgil
"zr" <ng...@cogeco.ca> wrote:
> Everyone knows that mathematics is no longer an exact science.
When has mathematics ever been a "science" at all?
T.H. Ray
> Everyone knows that mathematics is no longer an exact
> science.
> Mysticism is even more of a conundrum.
> "Everyone knows the speed of light"
> Can anyone tell us the speed of dark?
>
Maybe the missing light went to the same place as
the missing dollar.
(And please -- the convention here is to not top-post.)
Tom
Jack Campin - bogus address
>> religious beliefs, and those were mystical beliefs, and he grew up in a
>> Jewish environment. That amounts to suggestive evidence that Kabbalah
>> has influenced mathematics.
>
> Affirmative:
>
> http://pirate.shu.edu/~wachsmut/ira/history/cantor.html
>
> Quote:
>
> Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg,
> Russia, on March 3, 1845. Georg's background was very diverse.
> His father was a Danish _Jewish merchant_ that had converted to
> Protestantism while his mother was a Danish Roman Catholic.
That says nothing to indicate that Cantor had any knowledge of or
interest in any kind of Judaism, let alone the Kabbalah. My mum
was an ex-Catholic - I couldn't have told you who St Anne was until
ten years after I'd left home. Why would an ex-Jew educate his
child in Jewish arcana?
That page also credits Cantor with arguments that are due to Galileo.
> Underline by me: _Jewish merchant_. I think the "merchant" is equally
> important as the "Jewish". Perhaps the "Jewish" has lead to Cantor's
> preoccupation with the infinite, but the "merchant" has lead Cantor to
> his believe that the whole world is a set and nothing but a set. Read
> "The Political Economy of Sets":
>
> http://groups.google.nl/group/sci.math/msg/19e5174536f49c32?hl=en&
where you write:
> How can somebody for example conceive the thought that the whole of
> mathematics is made up from nothing else but Sets? This would be
> impossible if not society itself had'nt adopted the shape of an
> "ungeheure Warensammlung" (unprecedented collection of goods: Karl
> Marx in "Das Kapital").
The idea of constructing an entire ontology out of a single substratum
goes back at least as far as Thales, and is a commonplace of theogonies
all over Eurasia. Why on earth look for an economic parallel?
If Cantor's dad wanted his son to be an engineer he can't have been that
enthused about trade.
And in case you hadn't noticed, the Kabbalah makes a number of quite
specific, non-metaphorical, fudge-free assertions about the nature of
reality. Which of these do you think Cantorian set theory adopts?
============== j-c ====== @ ====== purr . demon . co . uk ==============
Jack Campin: 11 Third St, Newtongrange EH22 4PU, Scotland | tel 0131 660 4760
<http://www.purr.demon.co.uk/jack/> for CD-ROMs and free | fax 0870 0554 975
stuff: Scottish music, food intolerance, & Mac logic fonts | mob 07800 739 557
Gene Ward Smith
It's always been a science. As is often the case, you've confused
"science" with "natural science". Mathematics is what's sometimes
called a "formal science".
Here's a pretty good rundown, except that it neglects the historical
aspect:
Gene Ward Smith
zr wrote:
> Everyone knows that mathematics is no longer an exact science.
Thanks for playing, but no prize.
http://en.wikipedia.org/wiki/Exact_science
"Mathematics, the natural sciences, and the applied sciences are
considered exact."
"Exact sciences are distinguished from the social sciences on the one
hand, and from the humanities, theology, the arts on the other."
Those two sentences about cover it.
Ioannis
news:O%9zg.91236$hp.8...@read2.cgocable.net...
> "Everyone knows the speed of light"
> Can anyone tell us the speed of dark?
The speed of darkness, is exactly -c, where c is the speed of light. Btw,
the physicists got it backwards: Stars, and light sources like lamps, are
not photon-emitters. They are "darkness-absorbers". The sun is absorbing the
darkness around it at speed -c. Specifically, light sources absorb
"darkions" which are the dual particles of photons.
Wherever you detect a photon traveling somewhere using a photon-sensing
device, actually your device emits a darkion traveling in the opposite
direction.
Your eyes, for example, emit darkions, which is what our souls are naturally
made of: darkness.
--
Ioannis
Robert J. Kolker
> Your eyes, for example, emit darkions, which is what our souls are naturally
What soul?
> made of: darkness.
You must be aware that this is balderdash.
Bob Kolker
zr
"Virgil" <vir...@comcast.net> wrote in message
news:virgil-FAE03B....@comcast.dca.giganews.com...
zr
The speed of light absorbed?
"Ioannis" <morp...@olympus.mons> wrote in message
news:1154300056.296164@athnrd02...
Dave Rusin
>"zr" <ng...@cogeco.ca> wrote in message
>news:O%9zg.91236$hp.8...@read2.cgocable.net...
>
>> "Everyone knows the speed of light"
>> Can anyone tell us the speed of dark?
>
>The speed of darkness, is exactly -c, where c is the speed of light. Btw,
>the physicists got it backwards: Stars, and light sources like lamps, are
>not photon-emitters. They are "darkness-absorbers".
I believe they're called "dark-suckers". Reference:
http://www.siliconhell.com/humour/darksucker.htm
Han.de...@dto.tudelft.nl
> The opening paragraph of that shows considerable ignorance of the actual
> history of mathematics (and misrepresenting History is a sin for a
> Marxist, no?) --
Ignorance? Denied.
> Consider that calculus too started out as a half-baked theory laden with
> paradoxes, and that one of the great mathematical achievements was to
> put it on a rigourous footing. And indeed, set theory was one of the
> important tools in that enterprise.
Set Theory important tool? Denied as well.
Han de Bruijn
Aluminium Holocene Holodeck Zoroaster
that one should not stick one's reply at the beginning?
assuming adequate referential skills, on both sides of the screen,
doesn't it save one from some repeatative strange injury?
> (And please -- the convention here is to not top-post.)
thus:
Pierre Duh, that's what our Muslim Fisikist, Schonfeld, is trying to
say:
that the biggest bombs ever to have hit the biggest buildings,
could not have resulted in such a "free fall" collapse, although
that's just a fisikal hypothesis.
> makes this much more different then free fall. Why would you make such
> simple assumptions about such a complex process?
thus:
I may be extrememly socially retarded but I'm not stupid --
this is a *classic* analysis of Muslim Fisiks,
which also tend to be hardcore examples of it.
the real question is,
Why should Earth's tallest, rather highly tensile structure not
collapse
at the speed of freefall?
I said, Why not?
anyway, there's a good analysis of the comparison
between a surreptitious bombing, and
an inside controlled demo, in the current issue of that MIT mag
-- *Technology & Innovation*, or some thing --
using the Murrah Building for the example.
still, it is high time to impeach Trickier Dick Cheeny,
who did *what* in the Nixon Administration with Don Rumfseld?... oh,
you were convinced by that braindead guy, that he did it?
--it takes some to jitterbug!
http://members.tripod.com/~american_almanac
http://www.21stcenturysciencetech.com/2006_articles/Amplitude.W05.pdf
http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html
http://larouchepub.com/other/2006/3322_ethanol_no_science.html
http://www.wlym.com/pdf/iclc/howthenation.pdf
T.H. Ray
> what Miss Manners or Wikipedia Authority enshrines
> that,
> that one should not stick one's reply at the
> beginning?
>
Neither Miss Manners nor Wikipedia entered the
dialogue, except via you.
Good sense and mutually agreeable convention, however,
have always figured in honest and clear communication.
We can have rules, or we can have babble.
Tom
Virgil
"zr" <ng...@cogeco.ca> wrote:
> Since Euclid.
Scientists who are not mathematicians may claim it so, but
mathematicians know better.
David Bernier
Experimental physicists are still looking for a particle predicted
to exist by the Standard Model: the Higgs boson. (Named
after the physicist who first got the idea).
Photons are bosons that mediate the electromagnetic force. The
Higgs boson and associated Higgs field "explain mass".
I've never heard of an anti-photon, so perhaps in the Standard Model
a photon is its own anti-particle.
One of the goals of the ATLAS experiment is to find the predicted
particle, or evidence of its non-existence:
http://atlas.ch/etours_physics/etours_physics09.html
and
http://hepwww.ph.qmul.ac.uk/epp/higgs.html
David Bernier
David R Tribble
>> "Everyone knows the speed of light"
>> Can anyone tell us the speed of dark?
>
I know a kid so fast that he can turn out the light and jump into
bed before the room gets dark.
David R Tribble
>> Can anyone tell us the speed of dark?
>
Ioannis wrote:
> The speed of darkness, is exactly -c, where c is the speed of light. Btw,
> the physicists got it backwards: Stars, and light sources like lamps, are
> not photon-emitters. They are "darkness-absorbers". The sun is absorbing the
> darkness around it at speed -c. Specifically, light sources absorb
> "darkions" which are the dual particles of photons.
They could be called "scotons", a term a friend of mine invented from
the Greek word some twenty-odd years ago for this very thing.
Aluminium Holocene Holodeck Zoroaster
> Good sense and mutually agreeable convention, however,
> have always figured in honest and clear communication.
thus:
with the M-set, you've found a new field,
"squaring the cardioid," although it must be said that
the M-set is *entirely* an artifact of the floatingpoint spec, and
its many, many implimentations in hardware & software
(it's IEEE-755, or some thing, with a more recent update);
this was confirmed when monsieur M. begged my question
about this, at a rather dull "general audience" talk
that he gave at Royce Hall, UCLA, some time ago.
others have tried this, called Quadray for a spatial/tetrahedral one,
and
it doesn't seem to offer any utility, although it's possible that
some interesting numbertheory could lurk therein (whereas,
the quadray folks were rather more utilitatrian, seeking only
to establish "Synergetics" as better than "cartesianism,"
with no really interesting result -- and
when they both largely suck).
"spacetime" is a hopelessly useless abstraction,
since it was really already highly formalized
as phase-spatialization, using hamiltonians & lagrangians;
unfortunately, it tends to give the "time travel" crowd a funny
platform,
like time is "going to go" some where, some how. well,
"there's no where, therein," thanks to a momentary lapse
by AE's teacher, Minkowski -- good N-d numbertheorist,
as far as you could go with it!
> > http://bandtechnology.com/PolySigned/PolySigned.html
thus:
speaking of Kyoto,
there is an almost historical article on the emmissions trading schemes
of yore & today, in yesterday's NYTMagazine,
via its billionaire proponent from the junkbond biz. only thing
missing:
how many billions of dollars per year is CCX hedging, and
has it ever had any effect on, like, the price of oil?
> apt phraseology, "in sympathy," compared
> to the typical Muslim Fisikist hypothesis that
> there is no essential connection between the buildings
> on the site -- when there are very many.
>
> the simplest one is that there is a *huge* concourse
> running under the site, containing a subway, parking,
> malls, utilities etc., into which the towers collapsed.
>
> > WTC7 seems to be forgotten here.
> thus:
> what Miss Manners or Wikipedia Authority enshrines that,
> that one should not stick one's reply at the beginning?...
> assuming adequate referential skills, on both sides of the screen,
> doesn't it save one from some repeatative strange injury?
> thus:
> Pierre Duh, that's what our Muslim Fisikist, Schonfeld,
> is trying to say:
> that the biggest bombs ever to have hit the biggest buildings,
> could not have resulted in such a "free fall" collapse, although
> that's just a fisikal hypothesis.
> thus:
> I may be extrememly socially retarded but I'm not stupid --
> this is a *classic* analysis of Muslim Fisiks,
> which also tend to be hardcore examples of it....
> the real question is,
> Why should Earth's tallest, rather highly tensile structure not
> collapse
> at the speed of freefall?...
> I said, Why not?...
> anyway, there's a good analysis of the comparison
> between a surreptitious bombing, and
> an inside controlled demo, in the current issue of that MIT mag
> -- *Technology & Innovation*, or some thing --
> using the Murrah Building for the example....
R. Srinivasan
>
> Consider that calculus too started out as a half-baked theory laden with
> paradoxes, and that one of the great mathematical achievements was to
> put it on a rigourous footing. And indeed, set theory was one of the
> important tools in that enterprise.
The presently accepted foundations of the calculus still does not
resolve Zeno's paradoxes. These foundations have been challenged and
new foundations have been proposed. See Sec. 4 of
<http://arxiv.org/abs/math.LO/0506475>. The failure of the academic
community to respond to this work still seems completely inexplicable
to me.
The prroblem with academics today is not that it is dominated by any
particular race or clan. The problem is that institutionalization of
research has led to systematic suppression of dissent. And from my own
perspective, it looks to me like protecting turf has at some stage
become more important than honesty, ethics and the true spirit of
enquiry that research is all about. When professionals in a field of
research fail to acknowledge significant new work in their own field
just because it happens to represent dissent, then they are as much
politicians as they are researchers. And there is something seriously
wrong with that.
Regards, RS
Ioannis
news:1154477394....@75g2000cwc.googlegroups.com...
That's right. I was actually aware of the Greek word, but didn't want to
impose it on the english reading audience here. The dual of "phos" (=light)
is of course "scotos" (or "skotos") (=darkness), of the same root as, say,
"scotopic vision".
I guess the physicists now need to explain what happens when an electron
transitions from an orbit of higher energy to a lower energy one. Perhaps a
"scoton" collides with an electron and it reduces the electron's energy by
Delta(E)=h*nu.
So maybe atoms are the smallest darkness absorbers.
--
Ioannis
Rupert
R. Srinivasan wrote:
> Barb Knox wrote:
> >
> > Consider that calculus too started out as a half-baked theory laden with
> > paradoxes, and that one of the great mathematical achievements was to
> > put it on a rigourous footing. And indeed, set theory was one of the
> > important tools in that enterprise.
>
> The presently accepted foundations of the calculus still does not
> resolve Zeno's paradoxes.
Why not? What's the paradox?
R. Srinivasan
> R. Srinivasan wrote:
> > Barb Knox wrote:
> > >
> > > Consider that calculus too started out as a half-baked theory laden with
> > > paradoxes, and that one of the great mathematical achievements was to
> > > put it on a rigourous footing. And indeed, set theory was one of the
> > > important tools in that enterprise.
> >
> > The presently accepted foundations of the calculus still does not
> > resolve Zeno's paradoxes.
>
> Why not? What's the paradox?
The paradox can be stated in two ways. Consider the example of Achilles
chasing the tortoise along a straight line at a constant velocity
higher than that of the tortoise's (constant velocity). The paradox is
that Achilles has to "complete" infinitely many opertaions to catch up
with the torotoise -- from the starting point he sees the tortoise at a
particular location ahead, and he first has to reach that location.
When he reaches, he sees the toroise ahead at another location, and
Achilles has to reach there. And ad infinitum. The Greeks thought that
this was a paradox presumably because they viewed infinity as
"potential", i.e., the infinitely many operations required to reach the
torotoise cannot never be completed in finite time.
>From the modern point of view the paradox can be stated as follows.
How can infinitely many finite, non-zero, non-inifnitesimal intervals
of reals sum to a finite interval (why isn't the sum infinite)? I..e,
suppose starting from location zero, Achilles first reaches 1/2, then
3/4, then 7/8,...... Then Achilles covers the distance
1/2+1/4+1/8.....=1, where he catches up with the tortoise. The paradox
is -- why isn't the sum infinite, given that there are infinitely many
finite, non-zero and non-infinitesimal intervals being summed (assume
we are using some standard version of real analysis).
The NAFL resolution of these paradoxes is given in Sec. 4 of
<http://arxiv.org/abs/math.LO/0506475> (see Remarks 14-16). Basically
open/semi-open intervals of reals do not exist in the NAFL version of
real analysis -- so the proposition that Achilles is confined to the
interval [0,1) fails and cannot even be stated. Secondly it is not
legal in NAFL to ask *how many* intervals (or reals) are present in the
super-class of intervals ([0,1/2], [1/2,1/4] ....[1,1]), because direct
quantifiication over reals (or intervals of reals), which are infinite
classes/super-classes, is banned. But there is a way to quantify
indirectly, as explained in my paper -- and represent intervals, etc.
as "super-classes" mentioned above.
Regards, RS
R. Srinivasan
> The NAFL resolution of these paradoxes is given in Sec. 4 of
> <http://arxiv.org/abs/math.LO/0506475> (see Remarks 14-16). Basically
> open/semi-open intervals of reals do not exist in the NAFL version of
> real analysis -- so the proposition that Achilles is confined to the
> interval [0,1) fails and cannot even be stated. Secondly it is not
> legal in NAFL to ask *how many* intervals (or reals) are present in the
> super-class of intervals ([0,1/2], [1/2,1/4] ....[1,1]), because direct
> quantifiication over reals (or intervals of reals), which are infinite
> classes/super-classes, is banned. [...]
Correction -- read as "super-class of intervals ([0,1/2], [1/2,3/4],
...[1,1]),"
Regards, RS
Rupert
Well, it just isn't. I don't see any reason why it should be. You
haven't shown a contradiction in standard real analysis.
R. Srinivasan
That is true. The same can be said of the Banach-Tarski paradox or many
of the other paradoxes of classical measure theory. But these are
paradoxes nevertheless, and highly counter-intuitive.
Regards, RS
Rupert
Well, they may be counter-intuitive for some people. I agree the
Banach-Tarski paradox is a rather surprising result. But the fact that
the sum of an infinite series of positive numbers can be finite I don't
find counterintuitive at all, myself. Do you really claim to have a
formulation of analysis where this result is avoided? Can you tell me
what it is?
R. Srinivasan
Another way to state the paradox is as follows. Consider the infinite
series of nested real intervals [-1,1], [-1/2, 1/2], [-1/4,1/4],... The
intersection of these intervals contains the single point 0, but *each*
of these infinitely many intervals is of non-zero and non-infinitesimal
length. So why doesn't their intersection contain infinitely many
points? Again you may not find this counter-intuitive, but many do.
The NAFL version of real analysis is outlined in the reference [arxiv:
math.LO/0506475] given in my previous posts. The result is "avoided" in
the following senses -- the above infinite series of intervals, when
represented as a super-class by the method outlined in that ref.,
*must* also include [0,0], i.e. the interval of zero length. So it is
not possible to talk of infinitely many non-zero, non-infinitesimal
intervals of the above series in NAFL. Secondly it is *not* valid to
ask the question "How many intervals are present in the super-class
defined above" because quantification over reals (proper-classes) or
intervals of reals (super-classes) is not permitted in my logic NAFL. I
have not yet formulated in detail how I am going to do various aspects
of real analysis by this method. As of now I can no longer continue
this work because my bosses at IBM have given me other work and told me
that this logic stuff should be not be done on official time. Frankly,
I am now struggling for survival here.
Regards, RS
Mike Kelly
Because they keep getting smaller, but never "reach" 0????
>Again you may not find this counter-intuitive, but many do.
How many?
--
mike.