gsmith@brahms.BERKELEY.EDU (Gene Ward Smith) (02/20/86)
A while back I posted a comment that I found the math in Robert L.
Forward's _The Flight of the Dragonfly_ to be hilarious. Since there seems
to be a least a little interest out in net.land as to what might be wrong
with it, I post this to clear the problem up. The article is long because
of extensive quotes from Forward, which should provide plenty of yucks for
any mathematicians out there with nasty senses of humor.
The book introduces "flouwen", creatures with IQ's many times that of
humans (whatever that is supposed to mean). We come across a very old and
wise one, who has been thinking for years about a difficult research
problem:
Sour#Sapphire#Coo had taken for his research project the derivation
of an example of the fifth cardinal infinity. It had been twelve
seasons of the visitation of Warm since the massive blue elder had
left the pod and traveled to the Islands of Thought. [156]
[All page references are to the Baen Books paperback edition. The non-ASCII
orthography (stars, diamonds, etc.) is rendered with #,$,@ etc.]
Now what are we to make of this? It certainly seems that this brilliant
mathematician is trying to find an example of aleph_four. But this problem
was solved by Cantor: aleph_four is itself an example of something with
aleph_four elements in it (aleph_four is an ordinal number, as well as
a cardinal). What *can* he be thinking about? It certainly appears he is
an idiot, working on a non-problem. But more later.
"One of the unsolved problems of human mathematics was conjectured by
the human Fermat. There are many solutions to X^2 + Y^2 = Z^2. But
there is no solution to X^3 + Y^3 = Z^3, even if 3 is any number."
$That not problem! Deep Purple graveled.$
@That's a DUMB problem@ the red cloud exploded. @That problem not said
right. I say right way. X^2 + Y^2 = Z^2 has many solutions. Is there a
solution for U^3 + V^3 + W^3 = Z^3? That makes more sense. You have two
things X and Y. You multiply two times. You add two times. You get same
as Z multiplied two times. Two things three times is DUMB!! If you
multiply three times, then you should add three times!@ [236-237]
Now what? The human asks a problem about the existence of rational
points on a class of curves, and Red Cloud thinks the problem should be
about hypersurfaces. But why? If Red Cloud was 1/10th as smart as he is
supposed to be, he would know that U^3 + V^3 + W^3 = Z^3 is a rational
surface with an infinite number of points on it, with an easy parametrization
analogous to that of X^2 + Y^2 = Z^2. Euler proved this, whereas his proof
that X^3 + Y^3 = Z^3 has no non-trivial solutions had to be patched up.
Why? Because it's a harder problem. But the generalization is worse. It is
easy to find counterexamples with n > 3; and since the arithmetic genus is 0
it would seem to make more sense to conjecture there is always an infinity
of solutions for each n, not that there are none. But this is not only a
completely different problem, it looks like a much easier one. I might try
asking people about it; it wouldn't suprise me if someone has solved this
Fermat hypersurface problem. One the other hand, the Fermat curve, which
is supposed to be so dumb, is a natural one to look at as soon as you start
thinking about rational points on curves. It is connected to cyclotomicity
and lots of neat things, and the flouwen are only showing how ignorant they
are by not having thought of it and then dismissing it as dumb. DUMB flouwen!
Besides, the current status of the Fermat problem is extremely bright.
It has almost been reduced to the Weil-Taniyama conjecture. In light of
Faltings' recent brilliant and unexpected solution to the Mordell conjecture,
it seems highly plausible that by the time us DUMB humans reach these aliens,
the problem will no longer be a problem.
^What were you thinking about?^
%The fourth infinity.%
^Tell me about it!^
%Well ... I will someday. But first you have to learn about the second
infinity.%
^Tell me! Tell me!^
A yellow tendril poked a hole in the muddy bottom.
%Feel, youngling. There is a point.%
A delicate blue tendril felt into the murky bottom.
^That is a hole in the mud, older Warm*Amber*Resonance.^
There was a long pause as the yellow cloud rippled in annoyance.
However, the tone that resumed after the pause had all the warm
patience that it had contained previously.
%Imagine it is a point, with no dimensions.%
^Yes, older.^
The yellow tendril touched the surface of the soft mud again, leaving
another tiny spot in the smooth surface close to the first one.
%Here is another point.%
%Here is another.%
%Here is another.%
The line of close-spaced points grew.
%Imagine.%
%Imagine points so close they make a line. Infinitely long.%
There was a pause as the young one absorbed the sounds. Its blue
cloud enveloped the motions of the yellow wisp making a long string
of tiny dots in the ocean bottom.
^Infinite in both directions, older Warm*Amber*Resonance?^
=Yes. Very good, youngling.=
=Now ... Imagine a point not on the line.=
=Here is one.=
=Here is another.=
Soon a number of isolated spots were scattered above and below the
dotted line on the muddy sea floor.
=Imagine an infinite number of them.=
There was a slight pause.
=Are there more points *off* the line than *on* the line?=
The youngling thought carefully before answering, its wisps of
azure clumping and dissolving randomly. The older waited patiently.
Finally the youngling answered.
^No! They are the same.^
=Right!=
^That was too easy. Give me a harder one.^
=All right. Draw a line through any of those points I made.=
The blue cloud formed a tendril of its own and made a streak through one
of the isolated spots in the mud.
=Draw another through the same point. Make it wriggly if you want to.=
A wriggly line joined the streak.
=Draw more.=
Dainty^Blue^Warble concentrated, and soon dozens of distinctly different
lines were drawn through the same point. Then came the question.
=Imagine you did that to each point. Are there more wriggly lines than
points?=
The blue cloud stopped moving as it started to think. [246-248]
Now the awful truth emerges! The flouwen appear to assume, with
absolutely no justification, that the continuum hypothesis (which
Paul Cohen showed in 1962 is independent of ZFC) is true. This is clearly
the case, as he is giving an example of the second infinity, ie, aleph_one,
but as was discovered by Solomon Feferman and Robert Solovay, the only
generally definable subsets of the real line are denumerable and continuum.
(While it is consistent, for example, that V=HOD yet CH is false, giving
explicitly definable uncountable subcontinuum sets, this depends on the
model.) If the generalized continuum hypothesis is assumed by the flouwen,
which apparently is the case, then the nit-witted Sour#Sapphire#Coo was
trying to find an example of aleph_two, *assuming the GCH*!! Certainly the
collection of all order types of sets of cardinality at most aleph_one is
the more natural collection to consider. Indeed, the collection of "curves"
through a point is aleph_two sized if one considers in addition to the GCH
the notion of arbitrary function, not just continuous functions. Any being
who could spend years doing this is not cut out to be a mathematician. Also,
this "teacher" never even makes clear that the points on the line are anything
more than dense, so they could be countable for all Dainty^Blue^Warble could
tell. I was mystified as to where Forward was getting all this silly junk
about set theory from, until my colleague Matthew Wiener pointed out to me
that George Gamow's _One, Two, Three ... Infinity_ appears to have been the
high-powered reference work Forward relied on. For example, GG has a chapter
on set theory with the exact same misconceptions that the flouwen have.
The one science fictional advance Forward postulates is that the fourth
infinity (ie aleph_three) has been found (GG says it was an open problem.)
This is a wonderful book for bright ten year olds, and everyone should read
it at the right time in life ... but it isn't a substitute even for the
naive set theory all mathematicians (which should include flouwen) have to
know. To confuse this with *real* set theory is like mixing up first grade
arithmetic with number theory. (And to see what a mathematician means by
the word "arithmetic" look up J P Serre _A Course in Arithmetic_.)
#I have solved the motion of the lights in the sky!#
*Even the big circle?*
#All the lights except big circle. It is a swimmer of the
light. It is like us. Its motions are not that of logic.#
$But you can know the motions of all the rest? You can
know the risings of Hot and the fadings of Warm and the
tenacity of Sky@Rock?$
#All,# said White Whistler with confidence.
~How can you be sure?~
#The humans gave me the rule for simple spherical masses.
The rule was very simple. Yet it seemed complex when the
rule was used on more than two spheres. After some thinking,
I found the simple rule for many spheres.#
~Was it difficult?~
#No. A simple variable substitution combined with an
interesting coordinate transformation.#
$Let me taste.$
*Me too!*
^Me too!!!^
...
Warm%Amber%Resonance reveled in the cleanness of it. ~One
complex variable transformation, and then that simple, yet
unobvious, coordinate transformation! An nth-root dimension,
indeed! [249-250]
But there are no first integrals to the n-body problem! They
are quite off the mark here. Intelligent aliens can't find solutions;
they should obviously have piped up with something plausible. For
example, in the restricted 3-body problem, with parameter ratio of
heavy body to heavier body, the nondegeneracy condition is not
fulfilled when the parameter is zero (hopefully this is clear as it
reduces to a two body problem) but isoenergetic nondegeneracy IS
fulfilled. So a simple application of Kolmogorov's theorem gives that
almost all invariant tori with irrational frequency ratios are preserved
for small values of the parameter. Now that is interesting. And true.
As opposed to DUMB, DUMB!!!
To give them credit, the reference to nth-root dimensions might
be the rudiments of K-theory, or some other generalized cohomology
theory.
Then on pages 251-252 the Fermat conjecture is solved. (#Easy# the
lavender cloud responded. ... #DUMB problem#. That was the cue for Loud Red.
#I told you! DUMB problem!! DUMB!!!#). We then learn that the flouwen can't
even understand a point as elementary as why prove Fermat up to a certain
exponent. Apparently it doesn't occur to them that until it is proven, it
might be false; and if it is false, it might have a counter-example. DUMB
aliens, DUMB!!!
There are other things funny about this book like the use of tensor
product and diamond (math symbols both) inside of flouwen names. Another
boffo point is that the humans never even have any evidence other than the
word of a "semi-intelligent" computer and the unsupported assertions of
aliens who talk like brain-damaged teen surfers ("I could surf if I had a
surfboard," said Karin, her thoughts going back over six lightyears and
forty time years. [237]) that the flouwen are any good at mathematics at all.
I think if the human team had had a mathematician on it, he would have
concluded that the flouwen are retards. Maybe Forward should write a sequel
where this emerges?
From the back cover: "The man damn well knows what he is talking about."
-- Larry Niven. Dumb quote, DUMB!!! Dumb book, Dumb aliens, Dumb everything.
DUMB DUMB DUMB DUMB ... but you get the idea, I'm sure.
(written with)
ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
ucbvax!brahms!gsmith Gene Ward Smith/UCB Math Dept/Berkeley CA 94720
ucbvax!weyl!gsmith "Dumb problem. DUMB!!!"