Re: randomness


With the disclaimer that what we are doing is just purely experiemntal -
just fiddling around -

We really have to explain some things about a + ~b before we go any further.

a + ~b must be either continuous or discrete.

Either
a + ~b = a + Rnd{b1, b2, ...bn}, or
a + ~b = a + Rnd[b_1, b_2]

It is very important at this point to note that it is tempting to think that
for example

a + ~b = a + Rnd{b1, b2, ...bn} = Rnd{a+b1, a+b2, ...a+bn}, but this is
absolutely a huge problem that must be avoided, and this is exactly where
you start having problems.

In words -
Alice tosses a pair of dice. One die is unfair and always shows a 6, the
other die is fair.
Bob also tosses a pair of dice, but they are both fair.

What Alice is doing is fundamentally different than what Bob is doing, in a
physical sense. But also, if Alice and Bob both roll their dice and both get
a total face value of say 8, what you must say is that Alice's 8 is not the
same as Bob's 8.

This is where it starts getting tricky. You must be able to believe that an
8 is different from an 8, because of some loaded dice issues. It will get
pretty wierd I think, it will still be sensible. But you can see that we
will need some notation to really carefully keep things straight. Would'nt
it be neat if we could apply some results from complex analysis to all of
this ? I sure think so. Would we learn some new things about randomness,
just maybe.

So, Alice did'nt really roll an 8. She rolled a 6 + ~2.
Bob rolled a ~6 + ~2, which is an ~8.
And of course, 6 + ~2 is not the same as ~8.

It all makes sense, one of the dice was loaded.

As we said above,
a + ~b =/= ~{a+b1, a+b2, ...a+bn}

Likewise,

a + ~b =/= ~ [ a + b_1, a + b_2 ]

The reason for these last two statements becomes clear from the mental
picture of the physical signifigance, and it's relation to the algebra.

Things are starting to resmble the complex numbers, we'll see how far we can
push it before it melts down.


Another thing which we need to understand is the physical signifigance of
the expression ~ ( ~a) .

If we simply define ~ ( ~ a ) = - a, then we can start doing calculus
right away. However, it seems that ~( ~a) does not make any sense, unless
you introduce the notion of dimension. To explain :

Let ~a = ~[a1, a2] a random on an interval
and let ~b = ~[b1, b2] a random on an interval

~[a1, a2] * ~[b1, b2] = ~ ( ~ ( [a1, a2] * [b1, b2] ) ) a random on
a 2-dimensional tile.

So, perhaps it can be said that ~ ( ~a) does make sense.

Even in the discrete case where
~a = ~{a1, a2, ...an}, it seems that ~ ( ~a) might actually make sense
somehow, but it seems that we are introducing dimensionality into this
formulation. We need to understand some key concepts such as conjugacy,
additive and multiplicative inverse, and some other things which will sort
of look similar to complex number calculations. But, as we proceed with
this, we must continually try to find philosophical justifications for the
procedures that we are doing. We are primarily concerned with randomness,
and not so much with Srt(-1). We already have one huuge difference between
the complex numbers and what we are playing with, namely the dimensional
considerations. If we do get to the point where we are doing complex-like
analysis, it might get kind of weird. We still need to tackle some
preliminaries.


Here is a problem that might eventually crop up. We know what

~b = ~{b1, b2, ...bn}, and that

~b * ~b = ~{b1, b2, ...bn} * ~{b1, b2, ...bn} = ~( ~( {b1, b2, ...bn}*
{b1, b2, ...bn} ) )


This last item, ~( ~( {b1, b2, ...bn}* {b1, b2, ...bn} ) ), can be
understood as an nxn matrix. The tildes corresponding to random column, and
random row.

What if you had an nxn matrix with only one tilde ? How would you interpret
that ?

Would you say that it is an entity with a random part and also a nonrandom
part, but you dont know which is which ? Well, this is starting to sound
very much like the Heisenberg Uncertainty Principle, and I think that this
may be the only intuitive way to model it that I have ever seen.

Do the same thing with real intervals

~[a1, a2] * ~[b1, b2] = ~ ( ~ ( [a1, a2] * [b1, b2] ) )

This is a 2-dimensional tile, and the two tildes represent choosing randomly
along either axis. So, what happens when you take away a tilde ?

~ ( [a1, a2] * [b1, b2] )

This is a 2-dimensional tile, and you are choosing at random along an axis,
but you dont know which axis !! This is really incredible stuff you have
ever heard of HUP - at least in my opinion it should be exciting ! To
implement something like that within the framework of a complex-like
calculus, that would be pretty cool.









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