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Topic: Skepticism, mysticism, Jewish mathematics
Replies: 115   Last Post: Aug 7, 2006 1:30 AM

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zr

Posts: 6
Registered: 7/28/06
Re: Skepticism, mysticism, Jewish mathematics
Posted: Jul 28, 2006 1:04 AM
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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?"


"david petry" <david_lawrence_petry@yahoo.com> wrote in message
news:1153871230.238662.118160@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/ )
>
> 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.
>





Date Subject Author
7/25/06
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David Petry
7/25/06
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fishfry
7/25/06
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Dr. David Kirkby
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Dr. David Kirkby
7/25/06
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lloyd
7/25/06
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7/26/06
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Mike Kelly
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David Petry
7/26/06
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Gene Ward Smith
7/26/06
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Brian Quincy Hutchings
7/26/06
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dkfjdklj@yahoo.com
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herbzet
7/26/06
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David Petry
7/28/06
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herbzet
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herbzet
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herbzet
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dkfjdklj@yahoo.com
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herbzet
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zr
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8/4/06
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8/6/06
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8/7/06
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8/5/06
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7/30/06
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7/28/06
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7/28/06
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7/28/06
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7/28/06
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7/27/06
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T.H. Ray
7/28/06
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herbzet
7/29/06
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David Petry
7/30/06
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8/4/06
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8/4/06
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T.H. Ray
7/26/06
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David R Tribble
7/26/06
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Gene Ward Smith
7/26/06
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T.H. Ray
7/26/06
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toni.lassila@gmail.com
7/26/06
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Bennett Standeven
7/26/06
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Brian Quincy Hutchings
7/27/06
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Rotwang
7/27/06
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Craig Feinstein
7/27/06
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Toni Lassila
7/27/06
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Craig Feinstein
7/27/06
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7/27/06
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Rupert
7/28/06
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zr
7/28/06
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7/28/06
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7/29/06
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7/30/06
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LauLuna
7/30/06
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LauLuna

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