Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

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Date: Fri, 19 Nov 2004 19:10:56 GMT

"Eray Ozkural exa" <examachine@gmail.com> wrote in message
news:320e992a.0411190852.7ce02e6c@posting.google.com...
> I have had such a discussion with an extremely intelligent and
> experienced mathematician. He told me that PCs are not Turing
> Machines, because they have "an infinite tape". I think he did not
> know anything about descriptive complexity. This infinite portion of
> the tape consists entirely of blank symbols, and therefore has
> descriptive complexity O(1), which is easily realized by a physical
> system. When I told him about Ullman's indefinite growing argument, he
> objected "But when the universe is filled up, it cannot grow any more!
> Then, it is not infinite", to which I responded "Yes, but there is
> *nothing* that is larger than the universe." To assume the contrary
> would be theology, which I despise.
>

Turing Machines are idealized which means they are not physically realized.
TMs are not meant to have physical constraints applied to them. That mixes
categories. Sometimes the question is asked, how many sentences can be
generated in some natural (say English) language? There are finitely many
words in the language, but the standard answer is that there are countably
infinite number of sentences, due to appending etc.

These kind of questions ask what is the potential in theory, not what is
practically possible. Observations like: a person can only articulated
finitely
many sentences in a lifetime, or any sentence has to be uttered before
somebody dies or a machine wears out, or that how many sentences
potentially exist is related to how many people generate sentences over
the lifetime of humanity in the universe are not relevant, because that is
not the question being asked. Every process within the universe is finite
due to heat death of the universe, so that makes all such questions trivial,
if one interprets them to mean or apply to a physical reality. A Turing
Machine or potential sentence of a language (there is no pre-existing
specification that the sentence has to be of finite length) is not of this
world.

The set of natural numbers is countably infinite and is has some use
theoretically. Would you claim infinite sets have no use because they
have more members that there are particles existing in the finite universe?

And the original description of a Turing Machine. It is common to call
this tape 'infinite' though some prefer finitely unbounded. There is no
physical time constraint applied to when the calculation has to be
completed. So there are calculations that a physical PC the size of
a galaxy could not complete before the universe ran out of power to
energize the computer. A Turing Machine can of course complete
such a calculation (because the calculation does not need to be infinite,
just finitely larger/longer in time that can be accomplished by any
physical device during the existence of the physical universe) because
the constraint of physical time is not applied to idealized situations.
Keeping those categories seperate, the idealized and the physical,
is definitional. The answer to theoretical questions is trivial and obvious
if you mix these categories. Mathematicians invented infinity without
the requirement that it be physically realized because it was useful.
Pure mathematics invents formal mathematical systems with no
requirement that this formal system represent any physical event or
process. Eray wrote:

> "Yes, but there is
> *nothing* that is larger than the universe." To assume the contrary
> would be theology, which I despise.

When you say *nothing* you mean no physical something. Mathematical
objects need not be physical. Ideas may be generated physically, but the
idea of a unicorn can exist without the idea being physically manifested.
The mathematical idea of a circle exists. We do find physical objects
which remind of this mathematical idea. Pi is the ratio of a circumference
of a circle to the diameter. Even if you think of Pi as finitely unbounded,
there is still no last digit of Pi, there is still no last digit of Pi. So
in theory
you can talk about the digit expansion of Pi after the decimal to a value
say, 10^10^10^10^ and so on to say millions of exponents of exponents
and this finite number will exceed the particles in the universe and no
computer could calculate within the lifetime of the universe.

That does not make Pi theology. You will find Pi used in Physics.
You will find infinity used in quantum theory which makes theoretical
predictions which match experiments to 10^11 of real world accuracy.

Mathematics has nothing to do with theology. It certainly does not require
one to adopt mathematical platonism, a metaphysical realm outside the
universe.
They do say that mathematics is 'unreasonably effective'. Mathematics starts
with observations of physical reality and then regularities are then
*represented*.
Mathematics is a logical relationship to reality, it is idealistic/symbolic,
especially when formalized, and is a useful tool to predict the behavior of
physical reality.
It is _not_ the same as physical reality. And that is why concepts of
mathematics can have theoretical objects; mathematics as abstract thinking
is not required to map one-to-one to existing physical events or objects.

I can imagine the successor function which adds one to the previous number
and which can generate the naturals 1,2,3,... and so on and so on into
infiinity,
even though I cannot mentally grasp infinity. But you are trying to make
this
analagous to grasping God or theology. I cannot visualize God as having a
1,2,3... foundation successor function so therefore having an abstract
existence.

When you made this comparison, infinity and theology/God, to the size
of the physical universe, you crossed over from debating potential vs.
actual infinities to declaring abstract thinking is just theology in another
guise.

> "Yes, but there is
> *nothing* that is larger than the universe." To assume the contrary
> would be theology, which I despise.

Statements like this are going to appear to others as displaying gaps
in your background education. Abstract thinking in mathematics does
not assume that there is a physical object under discussion so that
_size_ ("larger than the universe") is a pertinent factor. And it is
muddled to conflate the inability to grasp how a potential infinity
transforms into an actual infinity as a theological issue. Your statement
attacks mathematics using even the abstract concept of a *potential*
infinity as religious mumbo jumbo. A potential infinity is larger than
any aspect contained within the universe also. And actualized infinity
certainly has nothing to do with that infinity being manifested within
the physical universe.

Actualized infinity is another abstract mathematical construct
conceptualizing completing a potential infinity, neither of which
abstractions have physical size.
Your idea reminds me of, There can't be an acutalized infinity(number)
because
it would be too long to fit in the universe and nobody would live long
enough
to write it down anyway. Two factors having nothing to do with the
discussion.
After writing this, I think you may not have realized this.

-- 
"Mathematics - this may surprise or shock
 some - is never deductive in its creation.
The mathematician at work makes vague
guesses, visualizes broad generalizations,
and jumps to unwarranted conclusions.
He arranges and rearranges his ideas,
and he becomes convinced of their truth
long before he can write down a logical
proof....The deductive stage, writing the
results down, and writing its rigorous proof
are relatively trivial once the real insight
arrives: it is more the draftsman's work not
the architect's." - Paul Halmos
SH: Achieving or having a mathematical insight is not the
same thing as having a religious/theological experience.
The idea that it is the same thing, is itself, a mystical claim.
Language is abstract and symbolic,
Stephen

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