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Topic: Skepticism, mysticism, Jewish mathematics
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David Petry

Posts: 1,104
Registered: 12/8/04
Skepticism, mysticism, Jewish mathematics
Posted: Jul 25, 2006 7:47 PM
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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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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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Brian Quincy Hutchings
7/27/06
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Rupert
7/28/06
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zr
7/28/06
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herbzet
7/28/06
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T.H. Ray
7/29/06
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zr
7/29/06
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Virgil
7/29/06
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zr
7/29/06
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Virgil
7/30/06
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herbzet@cox.net
7/30/06
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T.H. Ray
7/30/06
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LauLuna
7/30/06
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LauLuna

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