PCT equations

CSGnet Archive CSG2011
1

( Gavin
Ritz 2011.07.26.9.43NZT)

I was looking at some
mathematical equations and I suddenly realised that PCT equations are probably Universal
Diophantine Equations. (or Diophantine equation computer, see Gregory Chaitin),
Chaitin created a new mathematical constant (omega) from his conclusions of
this problem. Not a new problem, but a very old one.

It startled me a bit, but
I think they are. That means they can never be solved.

It also made me think
that while I was trying to set up categories between the internal part of the
control systems and the external part there was a switch between function and variable.
(Which is impossible of course). I can’t be regarded as a
categorist by any means, so this doesn’t mean that much.

This sort of lines it
self with the subobject classifier of category theory, which says all possible objects
(sub sets) must be tested for the truth (truth value object), which is impossible
so it make this an open paradigm one can only test for a truth not the truth.

Feeling like a possum in
the headlights of an on coming car at the moment.

Regards

Gavin

···
2

[From Bill Powers (2011.07.25.1630 MDT)]

Gavin Ritz
2011.07.26.9.43NZT –

I was looking at
some mathematical equations and I suddenly realised that PCT equations
are probably Universal Diophantine Equations. (or Diophantine equation
computer, see
Gregory Chaitin),
Chaitin created a new mathematical constant (omega) from his conclusions
of this problem. Not a new problem, but a very old one.

It startled me a bit, but I think they are. That means they can never be
solved.

You can relax. They are not the Diophantine equations. A Diophantine
equation involves a polynomial expression in which the variables are
strictly exact integers. The one Fermat worked with is x^2 + y^n = z^n,
which he said he had proved to have no solutions except for n = 2, but
there wasn’t room for the proof in the margin where he wrote his famous
note. It was actually finally proven a few years ago.

The variables in the PCT equations are real numbers, infinitely variable,
and the equations definitely have solutions.

Best,

Bill P.

3

(Gavin Ritz 2011.07.26.11.02NZT)

[From Bill Powers
(2011.07.25.1630 MDT)]

Gavin
Ritz 2011.07.26.9.43NZT –

I was
looking at some mathematical equations and I suddenly realised that PCT
equations are probably Universal Diophantine Equations. (or Diophantine
equation computer, see Gregory Chaitin), Chaitin
created a new mathematical constant (omega) from his conclusions of this
problem. Not a new problem, but a very old one.

It startled me a bit, but I think they are. That means they can never be
solved.

You can relax. They are not the Diophantine equations. A Diophantine equation
involves a polynomial expression in which the variables are strictly exact
integers.

That’s a normal Diophantine equation;
I’m not talking about those. I’m talking about the Diophantine
computer. See Gregory Chaitin.

The one Fermat
worked with is x^2 + y^n = z^n, which he said he had proved to have no solutions
except for n = 2, but there wasn’t room for the proof in the margin where he
wrote his famous note. It was actually finally proven a few years ago.

I’m talking about the halting
problem and Hilbert’s 10th conjecture.

The variables in the PCT equations are real numbers, infinitely variable, and
the equations definitely have solutions.

I hope so, but looking at some of the equations
I’m not so sure.

Kind regards

Gavin

4

[From Rick Marken (2011.07.25.1630)]

Gavin Ritz 2011.07.26.11.02NZT)

BP: The variables in the PCT equations are real numbers, infinitely variable, and
the equations definitely have solutions.

GR: I hope so, but looking at some of the equations
I’m not so sure.

I’m willing to bet my “surplus” that those equations do have solutions and that you cannot solve them.

RSM

···

–
Richard S. Marken PhD
rsmarken@gmail.com

www.mindreadings.com

5

[From Bill Powers (2011.07.25.1725 MDT)]

Gavin Ritz 2011.07.26.11.02NZT –

BP earlier: You can relax. They are not the Diophantine equations.
A Diophantine equation involves a polynomial expression in which the
variables are strictly exact integers.

That’s a normal
Diophantine equation; I’m not talking about those. I’m talking about the
Diophantine computer. See Gregory Chaitin.

The halting problem and the Diophantine computer and Hilbert’s 10th
conjecture and all the rest of that genre are problems that arise from
working with discrete logic and integer arithmetic. The control system
equations refer to analog processes using continuous variables with
continuous derivatives. Some precautions are necessary to allow
simulating continuous systems on a digital computer, but they are easily
handled. The Diophantine equations look like ordinary algebra, but
they’re not, because only exact integer values of the variables are
allowed, nothing between 1 and 2, for example. This is the world of
counting numbers, not measuring numbers. In the world of counting numbers
there is no such thing as the square root of 2, though the square root of
4 does exist. That’s why those equations don’t necessarily have
solutions.

I had to look up the subject you mention since I’m not much of a
mathematician, but it wasn’t hard to see the above.

BP earlier: The variables in the PCT equations are real numbers,
infinitely variable, and the equations definitely have
solutions.

GR: I hope so, but
looking at some of the equations I’m not so
sure.

BP: You can’t tell just from looking at the equations. You have to know
what they represent. If they represent only discrete values, whole
numbers, then the whole problem of Diophantine equations has to be dealt
with. But if the phenomena are basically continuous, there’s no problem.
Neural firing frequencies are continuous variables; they’re not quantized
even though the individual impulses are. So the Diophantine equations are
irrelevant for dealing with neural signals.

Best,

Bill P.

6

(Gavin Ritz 2011.07.26.12.01NZT)

[From Bill Powers
(2011.07.25.1725 MDT)]

Gavin Ritz 2011.07.26.11.02NZT –

BP: You can’t tell just from looking at the equations. You have to know what
they represent.

GR: I don’t
think that’s what Chaitin saying. But then I’m not a very good mathematician
so I can’t be sure. There’s really not much on the internet about
this stuff. So not so sure what you looked at. I was reading his book Meta mathematics.

If
they represent only discrete values, whole numbers, then the whole problem of
Diophantine equations has to be dealt with. But if the phenomena are basically
continuous, there’s no problem. Neural firing frequencies are continuous
variables; they’re not quantized even though the individual impulses are. So
the Diophantine equations are irrelevant for dealing with neural signals.

GR: Okay, I don’t understand why you say neural
firing is continuous. The output variables in PCT look discrete to me, are they
not?

I’m
going to delve into this a bit deeper to see if I understand it a bit better.
The variables from input to outputs and the functions of the Diophantine
Computer Equation look very similar to the output variables and input variables
of PCT equations to me.

Kind regards

Gavin

7

[From Bill Powers (2011.07.25.1940 MDT)]

Gavin Ritz 2011.07.26.12.01NZT –

BP earlier: If they represent only discrete values, whole numbers,
then the whole problem of Diophantine equations has to be dealt with. But
if the phenomena are basically continuous, there’s no problem. Neural
firing frequencies are continuous variables; they’re not quantized even
though the individual impulses are. So the Diophantine equations are
irrelevant for dealing with neural signals.

GR: Okay, I don’t understand why you say neural firing is continuous. The
output variables in PCT look discrete to me, are they
not?

The frequency of neural firings, the main dimension in which a
neural signal can change, is continuous – that is, it doesn’t
necessarily jump suddenly from one frequency to a different frequency,
but can change slowly and smoothly, gradually increasing from one moment
to the next.

The output variables in PCT are also continuous for the most part, though
some are not. A muscle tension, for example, changes from one amount of
force to a different amount, passing through all the intermediate amounts
of force rather than jumping from one to another. An arm or leg changes
from one position to another, going through all intermediate positions –
the limb doesn’t disappear from one position and reappear instantly in a
different position, which is what a discrete variable would do.

GR: I’m going to
delve into this a bit deeper to see if I understand it a bit better. The
variables from input to outputs and the functions of the Diophantine
Computer Equation look very similar to the output variables and input
variables of PCT equations to me.

Yes, they do, but they’re continuous variables. Discrete variables can
change instantly from one value to a different value without going
through all the intermediate values, which is what is meant by
“discrete.” When you look at the equations, however, there is
nothing to indicate what kind of variables make it up. That information
is given separately as a condition on the solutions that are acceptable.
The condition that makes the equation Diophantine is that all variables
must be expressible as whole numbers, without any fractional numbers
between them at all.

In Diophantine equations, we have things like

x^2 + y^2 = z^2, which reduces to

z = square root of (x^2 + y^2).

You might think that
if x is 2 and y is 3, z would be equal to the square root of 13,
but the square root of 13 does not come out to an exact integer (as the
square root of 16 would) so it’s not a Diophantine solution, which allows
only whole numbers. However, if the same variables are stated to be
continuous, then the value of z is the square root of 13.000… or
3.60555. Those numbers to the right of the decimal place are
available only if this equation is in the domain of continuous variables,
where you can have all values between 3 and 4.

Best,

Bill P.