Mathematics, Pseudomathematics, Artificial Intelligence

From: david petry (david_lawrence_petry_at_yahoo.com)
Date: 02/11/05 

Date: 11 Feb 2005 13:20:09 -0800

Part I Mathematics

The essence of science is the ability to make predictions
about the results of experiments. Mathematical statements
can be interpreted as making predictions about the
results of computational experiments, and hence,
mathematics is a science. One example should suffice.

The Riemann Hypothesis is an example of a statement which
makes predictions about the results of computational
experiments. The Riemann Hypothesis asserts that if we
were to compute the value of one of the complex zeros of
the zeta function, then to within the accuracy of the
computation, the real part of that value will be 1/2. So
once we have ascertained that the zeros of the zeta
function can actually be computed to arbitrary precision
with a sensible amount of computation, then we clearly
see that the Riemann Hypothesis is making predictions
about the results of computing those zeros, and actually
doing the computations can be thought of as performing an
experiment. It makes sense to say that we can "observe"
the results of the experiment. And we can think of the
experiments taking place in a world of computation.

The power and utility and meaning of mathematics comes
from its ability to predict the results of computational
experiments. All of the mathematics which has the
potential to be applied to the task of modeling phenomena
in the real world necessarily has the ability to make
such predictions. The notion of "truth" in mathematics
has no concrete meaning without a connection to the
predictions of experimental results.

The astute reader will point out that some "mathematical"
statements cannot be interpreted as making predictions
about the results of computational experiments. For
example, the Continuum Hypothesis makes no such
predictions. And that's a good point; mathematics has
been infected by a virus, namely, Cantor's Theory.

Part II Pseudomathematics

Cantor's Theory (classical set theory) is a conceptual
framework which encompasses both mathematics (the science
of phenomena observable in the world of computation) and
a fabricated world of the super-infinite. It merges the
two worlds so cleverly and so seamlessly that those who
accept that framework of thought often lose the ability
to distinguish between the two worlds. (Note: henceforth,
I'll refer only to the fabricated world of the
super-infinite as Cantor's Theory)

Cantor's Theory is a pseudomathematics. It has the form
but not the function of mathematics. It follows the
definition-theorem-proof format of mathematics, but in
the end, the theorems have no interpretation as
predictions about the results of computational
experiments.

Cantor's Theory has been aptly described as "formal
operations on meaningless symbols". Formally, Cantor's
Theory is impeccable, but ultimately, it is absurd. It
builds upon notions of infinite sets and power sets of
those infinite sets, which it does not define in terms of
the reality that we observe (i.e. the world of
computatioin), but rather, it merely asserts their
existence by decree. The result is that the existential
quantifier becomes a meaningless symbol. To assert that
the infinite objects in Cantor's Theory "exist" is a lie.

Formal logic has the "garbage in, garbage out" property;
it does not have the power to transform meaningless
assumptions and symbols into meaningful assertions. The
"meaning" of mathematics comes from its ability to make
predictions about the results of computational
experiments. The theorems in Cantor's Theory are not
truths in any meaningful sense.

The problem with Cantor's Theory is not the notion of
infinity, nor the notion of power sets. A theory of the
infinite can be designed such that every concept and
every object in the theory has corresponding
approximations in the world of computation. In such a
theory, the notions of infinite sets and power sets will
be defined as abstractions derived from the finite world
we actually observe. Such a theory can serve as a
conceptual aid in mathematics, and is part of the science
of mathematics.

Mathematics has tremendous power to enlighten us about
the nature of reality. Cantor's Theory has no such power.
The only "power" Cantor's Theory has is the power to
bamboozle. The Cantorians have parlayed this power into
enormous political power within the mathematics
community. To maintain their power, the Cantorians must
impose their "theory" on everyone in the community, and
weed out from the community anyone who questions the
reality of the world implied to exist by their theory. In
general, a community which has a power structure built
around knowledge of a mythology about a world beyond what
can be observed is thought of as either a religious
community or a mystical cult, and either could describe
the Cantorian community.

The world of the super-infinite - a world lying beyond
the world that is accessible in the scientific paradigm -
is essentially theological, and it has theological
origins. Cantor himself was hoping to unify mathematics
and theology with his theory.

I'm hardly the first person to notice that there is no
mathematical content to Cantor's Theory. Consider the
following quote from a contemporary of Cantor:

"I don't know what predominates in Cantor's theory -
philosophy or theology, but I am sure that there is no
mathematics there" (Kronecker)

It would have been quite difficult in Cantor's time to
make an airtight case that Cantor's Theory is not
mathematics. At that time, one might have argued that
mathematics is a science which makes predictions about
the results of computational experiments, but if the
question were asked, "where do these experiments take
place?", the likely answer would have been, "in the
mind". But then it could have been argued that objects in
the theory of the infinite live in the imagination, and
hence in the mind, and hence computational experiments
and objects in the theory of the infinite have the same
ontological status.

Furthermore, in Cantor's time, the ability to compute was
so limited that the mathematics of the time was making
predictions about computational experiments which were
far beyond the ability of then current technology to
carry out, and so it might have seemed to the
mathematicians that the mathematics they were doing was
no more connected to reality than Cantor's Theory.

The computer revolution really has changed the way we
look at the world. The computer revolution has improved
our ability to compute by some 15 orders of magnitude,
which is a truly stunning number to any practical minded
person. It outdoes any other revolution in history. It
has opened up the world of computation to us. It has
given us a new paradigm for mathematics. It's changed our
thinking about the ontological status of mathematical
objects.

Mathematics can now be clearly seen as a science. The
computer serves as the mathematicians' microscope. It
serves as the mathematician's test tube in which
computational experiments are performed. The world of
computation - the world viewed through the
mathematicians' microscope - has an objective existence
independent of the mind, and mathematics is the science
which studies that world.

It should be emphasized that formalisms themselves are
objects that can be viewed in the mathematicians'
microscope. They are objects that are studied in the
science of mathematics. Hence, Cantor's Theory, as a
formalism, is something to be studied as part of
mathematics. But the formal theorems in Cantor's Theory
have no interpretation as theorems in the science of
mathematics. They do not make predictions about the
results of computational experiments. The objects in
Cantor's world of the infinite are merely figments of the
imagination. Cantor's Theory most certainly is not part
of the foundation of mathematics.

Mathematics belongs in the public arena. As long as we
agree that religion belongs in the private arena and not
in the public arena, as we do in the United States, then
mathematics must not include the Cantorian religion.

Part III Artificial Intelligence

When I was in graduate school, I thought it would be a
really cool idea to build a foundation for mathematics
incorporating as one of the cornerstones, the idea that
mathematical statements must make predictions about the
results of computational experiments. I believe it would
be a straightforward task to teach computers to
understand mathematics when it is built on such a
foundation, and then it would be seen that the foundation
of mathematics is also the foundation of artificial
intelligence. After all, mathematics is the tool we use,
and the tool computers could be using, to understand the
world in a precise and quantitative way. I believe that
making available a solid theoretical foundation for
artificial intelligence could have profound implications
for the future of mankind.

Even without the connection to artificial intelligence, I
believe I had a good idea. All of the mathematics which
has the potential to be applied must have the capacity
for making predictions, so I would have been building a
theoretical foundation for applied mathematics. If
nothing else, it could be a good pedagogical tool. And
failing that, intellectual curiosity alone should justify
pursuing such a goal.

Alas, the Cantorians told me essentially that my ideas
are worthless crackpot ideas with no connection to real
mathematics, and that I really have no right to pursue
such ideas until I first prove that I am fully competent
in Cantorian mathematics. It was discouraging. I ended up
believing that as long as Cantor's Theory is part of the
canon of the mathematics community, I don't fit in.

Maybe it's too much to ask that the current generation of
mathematicians should abandon Cantor's Theory. But it
can't possibly be too much to ask that the mathematicians
be tolerant of those students who have been deeply
affected by the radical technological changes taking
place in the world today.

As I've already pointed out, the computer revolution has
opened up the world of computation to us. Many of today's
students want to explore that world, and they see
mathematics as the science which studies that world. They
want to be mathematicians. So I'm asking that
mathematicians tolerate those students.

But even more significant is the dream of artificial
intelligence. Right now, it's only a dream, but it's a
powerful and compelling dream, and it has captured the
imaginations of some of the best and brightest students.
What the dream of artificial intelligence leads us to
believe is that there is no magic ingredient in the mind
which in not available to a computer. That is, there's an
equivalence between artificial intelligence and the mind,
and they both live in the world of computation; all of
the tools we need to understand the world around us, live
in the world of computation; all of the tools we need to
understand the world of computation itself, live in the
world of computation; mathematics lives in the world of
computation; the idea that there exists a world of the
super-infinite which has no connection to the world of
computation, is a fantasy; the Cantorian idea that we can
"prove" that such a world exists is preposterous. Again,
what I'm asking is that mathematicians tolerate the
students who hold that world view; encourage them, allow
them to be part of the mathematics community, don't think
of them as second rate minds, and don't call them
crackpots.

Part IV Babble

It's not the kind of thing I can prove beyond doubt, but
I believe that the mathematicians' adherence to Cantor's
Theory has held back progress in artificial intelligence
by as much as thirty years and counting. If mathematics
were built on a foundation which emphasized the
connection between mathematics and computation, then all
of the mathematical knowledge accumulated over the past
few thousand years would be immediately transferable to
the computer, and the computer would be then intelligent;
it would have the tools it needs to understand the
physical world, and it would have the tools it needs to
understand its own mind; it would be self-aware.
Computers simply need not understand the complexities and
nuances of human social behavior in order to be deemed
intelligent; the Turing Test is bunk (another of my pet
peeves).

There are lots of reasons for debunking pseudoscience. I
shouldn't have to elaborate on that for the relatively
sophisticated audience likely to be reading this article.
Sure, it's likely that those who have devoted themselves
to the pseudoscience will experience a stressfull period
of readjustment when that pseudoscience is debunked, and
we should do what we can to minimize that stress, but
that is no reason to refrain from the debunking process.

Cantor's Theory is pseudomathematics. It is
pseudoscience. It has theological origins. It's not part
of the foundation of mathematics. It doesn't belong in
the universities. It needs to be debunked.

Cantor's Theory has been debunked.

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