Wrapping
up the Chapter
If you ask a man or a woman in
English what “Julia” means, they will think of it as a first
name, for girls. But if you ask a mathematician they will think of it
as a last name, and they will reply “Gaston Julia discovered one of
the first sets...” such as the difference between asking anyone
what Julia means, and asking someone specifically. Right now,
hurricane Gaston is leaping out over the Atlantic with the intent of
slamming into the Azores – and Julia is on the way. Someone at NOAA
has a wicked streak; and every four years this will recur in the
Atlantic as his or her joke. The connections that we make when we are
not making connections, because it was this moment where the whole
concept of infinity's did not have a tome, fictional or not, for
people to see the scope of.
When one looks at the Gödel
incompleteness theorem – and remember he did great deal after
starting his career with a bang – you see it is of the nature of
aleph null. But aleph null is only the fundamental generator of
infinities. There must be a problem which has as its solution the
number of the continuum. And this is what we are going to talk about.
The problem exists in the area of game theory. We have talked about
Gödel,
Turning, Nash, and Cohen – and these all sit very well in game
theory – though Cohen is still a particularly hard subject. These
all set with the idea that each individual proof boils down to a
number, and since every symbol can be converted to a number, this
means that a great deal of human knowledge – not just mathematics,
but economics - and things that are based on economics – fit within
it quite nicely. But now we are going to show that points are not the
only thing which has truth in it, there are lines as well. And the
nature of proofs is different for lines.
The
story begins with a young man called Gaston Julia, who discovered
that there was a line which denoted a series of complex points on a
2-D number line, so that points outside of this line would go to
infinity, and points inside the number line would go to 0 – but
there was indeed a line which would neither go to infinity or go to
0. Eventually this would be discovered by Mandelbrot, and be called
the Mandelbrot set. With Julia discovering the set, and Mandelbrot
showing how it was quite different from the ordinary sets, it came in
to the public consciousness. But that consciousness did not
understand the deep significance.
If
you take a point, and either it goes to infinity, or goes to 0 –
then it will eventually behave like points in game theory. But it may
take a very very very very long time to do that. But if it never goes
to infinity or 0, it will never behave like them. For example, take
1. it will change between -1 and 0. this remember is part of the
proof of Gödel, but it branches off from this point. So if you look
at it in one way, it starts out as a proof of Gödel, but it does not
continue down with a proof of prime numbers – instead it looks at
everything as a line, which is to say, it delves into the region of
Cohen's proof of the continuum hypothesis. But we have not squared
this circle yet.
Before
going on to this problem, let us first go over the odd behavior that
the Julia and Mandelbrot set have.
The first thing that we notice, is
that all of the Julia sets that we encounter are not concave, which
in many parts of mathematics is an important start. Nor do the sets
have a finite number of non-concave points - for example, 1/x has
only one place where it is not concave: 0, where the negative side of
the equation heads towards negative infinity, and the positive side
heads towards infinity. (Remember on a machine that will mean that,
stealthily, they will head to the same place – because on a machine
it is not a plane but a circle. ) instead be Julia sets are
infinitely convex, and there is as many non-concave fracture points
ahead of us As there were before, in that one to one infinity
matchup. So, which infinity is it?
The first guess is aleph-0. But clearly some of the points are not on
the rational part of the number line. For example take √3/√5
Clearly it is not one of the rationals. But because by making it
accessible through a number plus a symbol, it can be Gödeliazable,
as we have just done. What the really mean by unfixed irrational, is
a number which cannot be easily accessed by this means. But any place
that we can look for a number, we can create a symbol and a simple
rule which makes it accessible by this means.
The only problem is that a physical mind has limited space – and
even if we took all of the power of many brains – there would still
be numbers for which Cantor's definition would still apply. That is,
it could not be related on the diagonal. So the problem is that we
know what will happen, but any time we look, we will find out that it
matches up 1 to 1. we blink, and it is gone – we know it would
happen, but when we look it does not. In other words the number of
the continuum is neither true nor false. At this is why the number of
the continuum is neither true nor false, no matter what Woodin will
argue. It is like the blind spot in our vision, when we look directly
on it, it vanishes.
Irrational numbers which not only do not repeat, but they cannot be
fixed – we put “...” in the definition. So there are at least
two kinds of irrational numbers, those which can be fixed and those
that cannot be fixed. And we know that if we limit our attention to
the fixed numbers, that they are part of aleph zero.That means that
√3/√5 is part of aleph 0, but that a continued fraction, that
cannot be expression in a finite form cannot. The problem with this
is we do not have a way of showing that a continued fraction can be
re-expressed in this way – for example pi can be re-expressed as a
host of continued fractions – it is only that we know that it can
also be re-expressed as pi. This also means that all physical numbers
are in the first category, where as mathematical numbers are not –
because the Plank number has a physical limit, and physically all
numbers can be reduced to the Plank number.
There are an unlimited number of fractions do not have such
proclivities as pi – indeed just change one number out of pi, and
it loses any meaning – just a string of infinite digits, which
means very little except as pi in some Gaussian form. In other words,
there are physical infinities, probable physical infinities, and
“god's infinities” - is just that we cannot tell the second class
from the third class.
But there are “god's infinities”? Yes, because we have Cantor's
proof to this already.What this does tell us is that there is an
infinity that is smaller than the number of the continuum, which was
what Cohen hoped to prove, and what Woodin may have done. This is why
Julia and Mandelbrot are important, because they attack the line
completely differently. Or rather, they are interested in the line,
and go about proving its fractal nature at a diagonal. This means
that they use Cantor's ideas in a different way – they use
recursion on the number line itself – the generator of the
Mandelbrot set is recursive. So instead of using a proof that is
recursive, they use a set which is recursive.
That is the beauty of Julia and Mandelbrot sets – because remember,
a proof in Gödel
form is also a “construction”. Thus the proof of Gödel is also
a construction of primes.
It
is here that we notice different styles of proof among the great
mathematicians of infinity. Cantor tries to establish the simple
proof – partially because he knows that no one else believes him.
He first publishes a difficult versions of his proofs, but then found easier
ones. Which is why he est. the counting by diagonals proof, which
showed that things that seemed different were the same; and he est.
the number of the continuum, which showed that it was different from
the lower order of infinities. In fact I have used these proofs to
show that some irrational numbers are the same as aleph-0. Gödel
was different, he est. a single proof that was flawless, and showed
that he was thinking about proofs that he could not describe. Cohen
however, these many proofs as an exercise for the student, if student
could describe the workings of a mind that can chew through his set
theory. Turing uses a different sort of proof, and it will often take
several times through – or even with help from a trusted source.
But he comes up with some astonishing proofs. Nash tends to write
simple proofs, that astonish with their complexity, which often he
did not think of.
We
should touch on the subject of madness. Nash was from time to time
imagining characters that he thought were real. Thus he spent a good
deal of his life not knowing if someone were real, and he talked with
them even when he knew they were not real. Sometimes I wonder would I
take the trade of having imaginary friends, and lots of stupendous
genius. Gödel was thought mad, but there is extremely little proof
of that. What there is is a sign that he did not want to be bothered
by any but few strange cases of brilliance; for example he
corresponded with Cohen, and there was no sign that he was anything
but polite.
