[ExI] Simulating the brain

Stuart LaForge avant at sollegro.com
Wed Jul 12 08:15:24 UTC 2017

```John Clark wrote:

>​We already know the amount of electrical charge a object has is ​discrete,
>and yet calculus does a excellent job approximating what the electrical
>field produced by that discrete charge is like. If space and time are
>discrete the chunks are probably at the Planck level, and that is very very
>small making for very very good approximations.

But if space-time is discrete, it opens up a whole can of worms even at
the planck level once you enable more than a single dimension. For
example, lets say that for the sake of argument that the Planck length
(Lp) is the fundamental and thus indivisible unit of length. A pixle of
the universe if you will.

Then what is minimal unit of 2-dimensional area? If you say that it is
Lp^2, then that is wrong because the Pythagorean theorem says that the
length of the diagonal would be sqrt(2)*Lp but you can't have fractional
Lp (let alone an irrational number of Lp) so the smallest possible area
has to be larger than that. It has to be 12 Lp^2, because (3Lp)^2 +
(4Lp)^2 = (5Lp)^2  thus a 3x4 rectangle has a diagonal of 5 and 3Lp * 4Lp
= 12Lp.

And if you add a third dimension, it gets crazier. A three dimensional box
with edges, face diagonals, and space diagonals that are all integers is
called a perfect cuboid and nobody has ever found one.

http://mathworld.wolfram.com/PerfectCuboid.html

Furthermore, while I haven't checked his proof, Walter Wyss claims to have
proven perfect cuboids don't exist.

https://arxiv.org/pdf/1506.02215.pdf

Now I don't know if any work has been done on 4-dimensional perfect
hypercuboids, but if they exist, they are liable to be significantly
larger than the planck scale.

>
​>
> If infinities exist ontologically, then space-time is a continuum. In
> ​ ​
> which case classical computers would have difficulties with irrational
> ​ ​
> numbers.

​>I know this is a bit heretical but perhaps irrational numbers really do
>have a last digit. ​If the computational resources of the entire universe
>is insufficient to calculate the 10^100^100^100 digit of PI, and given that
>there are only about 10^81 atoms in the observable universe that seems
like >a reasonable assumption,
>could the ​
>10^100^100
​>^100​
>digit of PI
​> even be said to exist?​

There are two problems with this argument. First, the observable universe
is just a cosmological horizon, and we don't have any reason to believe
the universe ends at the horizon. It could go on forever with observers on
the edges of *our* visible universe seeing billions of galaxies that we
cannot and so on. Our universe could be an expanding bubble in an infinite
sea of expanding bubbles that are universes more or less like our own.

Secondly, given the set of all 10^81 atoms that you mention, there are
2^10^81 possible subsets of those atoms. Note that that this is just with
regards to the possible numbers of atoms in various subsets and ignores
any spatial or chemical relationship between those atoms. After factoring
in all the possible arrangements, relationships, and degrees of freedom
those 10^81 atoms can have with respect to each other, I would not be
surprised if the total number of possible states available to the atoms in
the visible universe becomes significantly larger than 10^100^100^100.

>If ​perfect circles don't exist is there anything about them to understand?

Yes. How did something that does not exist become so fundamental in
describing so much of what we can see and observe? Without perfect
circles, you can't have complex numbers. And without complex numbers you
can't have probability amplitudes and by extension, quantum mechanics.
Math is like the soul of the universe and an infinite number of angles can
dance on the head of a pin. Sorry, I couldn't resist the pun. :-)

Stuart LaForge

```