Lab Notes 5
Lab Notes
William Kuchta
January 1, 2016
I have intended to submit some type of comprehensive paper
discussing the kinds of models I have been working with for some time, but
for some reason it seems impossible to organize such a broad collection of
ideas into a paper which would serve the purpose of explaining
key concepts of the work I’ve been doing.
If all of this material were presented properly with greater depth and
rigor I think that there is easily enough material for at least one book and it is
unliklely that I will ever be able to compile a such a treatise for various reasons
and so I am releasing my personal Lab Notes with annotations, to explain
these modelling methods in a very informal style.
There is a very small amount of philosophy involved with these
modelling techniques and I will try to add annotations in appropriate places
so that the reader can understand the proper context of what is being done.
Everything in this paper is purely exploratory in nature, it is the result
of my own independent research and I had no assistance in this, nor are there
any references because as far as I know it has never been done before.
This is an ongoing work in progress and if you see a yellow box
like this one it is really just a note to myself to do more work in a particular
area which will appear in subsequent versions of Lab Notes.
[1] When we are doing mathematics it is perfectly reasonable to work
with magnitudes and other constructs which are said to ‘exist’.
[2] It would make no sense whatsoever to work with constructs which are
assumed to be ‘nonexistent’.
[3] However, there is a question about mixing the existent with the nonexistent.
Would it make any sense if we were to consider ‘partial existence’ or things
which have a ‘potential to exist’ ? If we devise a hybrid which is partly existent
and partly nonexistent ... would that make any sense and how would it behave ?
These initial questions were a strong influence early in this research, and I ask
the reader to keep an open mind because methods will be revealed shortly which
should inspire some interest in this area.
Here is a legend with some key concepts which will be needed to understand
the ‘algebra-like’ derivations which are given elsewhere in this paper. This legend
is a very handy summary which has a lot of philosophy ‘built-in’ to these devices.
In order to combine existent with nonexistent magnitude, I was forced to invent
a new representation of ‘number’, which has an inherent duality built into it which will
be discussed in greater detail later. I call this ‘number’ a Mixed Manitude.
Anatomy of a Mixed Magnitude
Existent Part
Nonexistent Part
( a, b )
[ c, d ]
Conjectured Magnitude
A single, given
Magnitude.
Legend for Graphic Representations
of Mixed Magnitudes
exists
does not exist
Existential Potential
This is a Mixed Magnitude.
( a, b )
[ c, d ]
It is a hybrid of existent and nonexistent length.
It is a single magnitude and it has an inherent duality.
The top part resembles Addition.
The bottom part resembles Multiplication.
Because both of these aspects are
inherent to the magnitude we
regard this as a Duality.
When we try to combine several mixed magnitudes, either by adding or
multiplying them together, some strange things can happen. I will attempt
to do this and understand how the duality manifests itself throughout
the resulting algebra, and hopefully draw some connections between
these results and some similar relationships from probability theory.
These are some basic properties of these Mixed Magnitudes. Please note that
a Mixed Magnitude as discussed here is definitely NOT the same thing as the
well known concept of either ‘number’ or ‘magnitude’ from standard mathematics.
Contemporary math does not deal with entities which are partially existent. We
present this new concept of magnitude, and a context which will be helpful to explain
it’s relationship to standard mathematics.
Anatomy of a
Mixed Magnitude
Existent Part
Nonexistent Part
(A, b)
[ C, d]
Existential Potential
Conjectured Magnitude
A ‘Mixed Magnitude’ is a kind of number. It represents a single value, or magnitude.
It may look like 4 distinct numerical values, but they are all taken together to have
one single collective meaning.
These mixed magnitudes can be used for any purpose in place of traditional
numbers, and in fact the numbers you are accustomed to using can easily be
invoked with this notation simply by allowing the nonexistent part to be everywhere
equal to zero.
If the existent part is everywhere equal to zero then your calculations may
still come out looking correct but from a technical standpoint it would be nonsense
by definition.
If both the existent and nonexistent parts of a mixed magnitude are nonzero,
then you are doing stuff which is very much like probability theory.
needs better development of the concept of mixed magnitude, duality and
equivalence of continuous / discrete aspects ... then talk about multiplication
of several mixed magnitudes & algebra etc etc
A worksheet with some examples of multiplication.
(1,0)
[1,1]
White pieces Exist
(1,0)
[1,1]
Red pieces are Non-Existent
(1,0)
[1,1]
(1,0) (1,0) (1,0)
=
[1,1] * [1,1] [1,1]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1) (0,1) = (0,1)
[1,0] * [1,0] [1,0]
(0,1)
[1,1]
(1,0)
[1,1]
(1,0)
[1,1]
(0,1)
[1,1]
(1,0) (0,1) (1/2,1/2)
=
[1,1] * [1,1]
[1,1]
(3,0)
[3,1]
(1,0)
[1,1]
(1,0) (3,0) (3,0)
=
[1,1] * [3,1] [3,1]
(3,0)
[3,1]
(1,1)
[2,1/2]
(1,1) (3,0) (4.5, 1.5)
=
[2,1/2] * [3,1]
[6, 3/4]
A worksheet with some examples of multiplication.
(2,0)
[2,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
(1,1)
[2,1/2]
(1,1)
[2,1/2]
(0,2) (2,0)
(2, 2)
=
[2,0] * [2,1] [4, 1/2]
(2,0)
[2,1]
(1,1)
(3,1) = (5, 3)
*
[2,1/2] [4,3/4] [8, 5/8]
(1,1)
[2,1/2]
(0,4) = (2, 6)
(1,1)
*
[8, 1/4]
[2,1/2] [4,0]
(0,3)
(3,0) (4.5, 4.5)
=
[3,0] * [3,1]
[9, 1/2]
(3,1)
[4,3/4]
(3,1)
[4,3/4]
(4,0) (2,2)
(12, 4)
=
[4,1] * [4,1/2] [16, 3/4]
(3,0)
[3,1]
(0,3)
[3,0]
(1,1) (4,0) = (6, 2)
*
[2,1/2] [4,1] [8, 3/4]
(2,2)
[4,1/2]
(4,0)
[4,1]
(4,0) (2,2)
(12, 4)
=
[4,1] * [4,1/2] [16, 3/4]
(4,0)
[4,1]
(1,1)
[2,1/2]
(1,1)
(3,1) = (5, 3)
*
[2,1/2] [4,3/4] [8, 5/8]
(4,0)
[4,1]
(2,2)
[4,1/2]
(2,0) (2,2) = (6, 2)
*
[8, 3/4]
[2,1] [4,1/2]
(3,1)
[4,3/4]
(0,4)
[4,0]
(1,1)
[2,1/2]
(2,2)
[4,1/2]
(3,1)
[4,3/4]
(1,1)
[2,1/2]
(0,2) (4,0)
(4, 4)
=
[2,0] * [4,1] [8, 1/2]
(1,1)
(2, 2)
(1,1)
=
*
[2,1/2] [2,1/2] [4, 1/2]
(1,1)
(1,1) = (2, 2)
*
[2,1/2] [2,1/2]
[4, 1/2]
(4,0)
[4,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
(3,1)
(3,1)
(12, 4)
=
[4,3/4] * [4,3/4] [16, 3/4]
(1,2)
[3,1/3]
(2,1)
[3,2/3]
(1,2)
[3,1/3]
(1,2)
(2,1)
(4.5, 4.5)
=
[3,1/3] * [3,2/3]
[9, 1/2]
(2,1)
[3,2/3]
(2,1)
(4.5, 4.5)
(1,2)
=
*
[9, 1/2]
[3,1/3] [3,2/3]
A worksheet with some examples of multiplication.
(1,3)
[4,1/4]
(0,1)
[1,0]
(1,3)
[4,1/4]
(0,1)
[1,0]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(1,3)
[4,1/4]
(1,1)
[2,1/2]
(1,1)
(1,3) = (3, 5)
*
[2,1/2] [4,1/4] [8, 3/8]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(1,1)
(2,2) = (4, 4)
*
[2,1/2] [4,1/2] [8, 1/2]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(4,0)
[4,1]
(3,1)
[4,3/4]
(2,2)
[4,1/2]
(1,1)
[2,1/2]
(1,3)
[4,1/4]
(1,3)
[4,1/4]
(1,1)
[2,1/2]
(5, 3)
(3,1)
(1,1)
=
*
[2,1/2] [4,3/4] [8, 5/8]
(1,1)
[2,1/2]
(6, 2)
(4,0)
(1,1)
* [4,1] = [8, 3/4]
[2,1/2]
A worksheet with some examples of addition.
(1,0)
(1,0)
+
=
[1,1]
[1,1]
(2,0)
[2,1]
(0,1)
(0,1)
+
=
[1,0]
[1,0]
(0,1)
(1,0)
+
=
[1,1]
[1,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
Important Note:
This concept of ‘number’ or ‘magnitude’ has an inherent duality which is fundamentally built in to it.
=
Discrete, looks like addition
(1,1)
[2,1/2]
=
(1,1)
[2,1/2]
Continuous, looks like multiplication
These are simply two different representations of the exact same thing.
The magnitudes and the left and right sides are identical.
DISCRETE-LIKE ADDITION
(3,1)
[4,3/4]
THESE BOTH SAY
THE SAME EXACT
THING, THEY ARE
EQUIVALENT FORMS
AND BOTH ARE
PERFECTLY VALID
SIMULTANEOUSLY.
+
(5,3)
[8,5/8]
=
(8,4)
[12,3/4]
(3,1)
(5,3)
(8,4)
=
[4,3/4] + [8,5/8]
[12,3/4]
CONTINUOUS-LIKE ADDITION, ACTUALLY RESEMBLES MULTIPLICATION
(3,1)
[4,3/4]
+
(5,3)
[8,5/8]
=
(3,1)
(5,3)
(8,4)
=
[4,3/4] + [8,5/8]
[12,3/4]
In case you haven’t noticed,
THIS IS A DUALITY
(8,4)
[12,3/4]
A worksheet with some examples of addition.
(8,8)
[16,1/2]
(8,1)
[9,8/9]
+
(16,9)
[25,16/25]
=
(8,1)
(8,8)
(16,9)
+
=
[16,1/2] [9,8/9] [25,16/25]
(8,8)
[16,1/2]
(3,1)
[4,3/4]
+
(11,9)
[20,11/20]
=
(8,1)
(8,8)
(16,9)
+
=
[16,1/2] [9,8/9] [25,16/25]
(16,0)
[16,1]
+
(0,1)
[1,0]
(16,0) + (0,1)
[1,0]
[16,1]
=
=
(16,1)
[17,16/17]
(16,1)
[17,16/17]
An extremely interesting looking result.
A mixed magnitude multiplied
with it’s own conjugate
( 3, 1)
( 1, 3)
*
=
[ 4, 3/4] [ 4, 1/4]
( 3, 2)
[ 5, 3/5]
*
Legend:
=
*
( 8, 8)
[ 16, 1/2]
*
=
*
( 2, 3)
=
[ 5, 2/5]
( 12.5, 12.5)
[ 25, 1/2]
=
( 2, 1)
( 1, 2)
=
*
[ 3, 2/3] [ 3, 1/3]
( 4.5, 4.5)
[ 9, 1/2]
( 8, 8)
[ 16, 1/2]
( 3, 0)
, 0 > + < 0, [ 3, 1] > = < ( 4, 0) , ( 3, 0) >
υ1 + υ2 = < ([ 4,4, 0)
1]
[ 4, 1] [ 3, 1]
||υ3|| =
<
( 4, 0) , 0
[ 4, 1]
2
>+<
( 16, 0)
[ 16, 1]
||υ3|| =
||υ3|| =
( 3, 0)
0, [ 3, 1]
2
>
+ ([ 9,9, 0)
1]
( 25, 0)
[ 25, 1]
=
( 5, 0)
[ 5, 1]
υ3
υ2
υ1
Example 1
Mixed Magnitude as Vectors
υ1 = <
( 1, 1) , 0
[ 2, 1/2]
>
( 1, 1)
[ 2, 1/2]
>
υ2 = < 0,
( 1, 1)
( 1, 1)
( 1, 1)
1)
υ3 = υ1 + υ2 = < [(2,1,1/2]
, 0 > + < 0, [ 2, 1/2] > = < [ 2, 1/2] , [ 2, 1/2] >
||υ3|| =
<
( 1, 1)
[ 2, 1/2] , 0
2
( 1, 1)
[ 2, 1/2]
=
2
+
>
<
( 1, 1)
0, [ 2, 1/2]
( 1, 1)
( 1, 1)
∗
[ 2, 1/2] [ 2, 1/2]
( 2, 2)
[ 4, 1/2]
||υ3|| =
||υ3|| =
checking
υ2
υ1
>
=
( 2, 2)
[ 4, 1/2]
( 2, 2)
+ [ 4, 1/2]
( 4, 4)
[ 8, 1/2]
( 2 , 2)
( 2 , 2)
∗
[ 2 2 , 1/2] [ 2 2 , 1/2]
||υ3|| =
υ3
2
( 2 , 2)
[ 2 2 , 1/2]
Example 2
4)
= [(8,4,1/2]
υ1 = <
υ2 = <
( 1, 2) , 0
[ 3, 1/3]
( 2, 1)
0, [ 3, 2/3]
>
>
( 2, 1)
( 1, 2) ( 2, 1)
+
>
<
υ1 + υ2 = < ([ 1,3, 2)
,
0
=
> 0, [ 3, 2/3]
< [ 3, 1/3], [ 3, 2/3] >
1/3]
||υ3|| =
<
( 1, 2)
[ 3, 1/3] , 0
||υ3|| =
2
>
2
+ < 0, ( 2, 1) >
[ 3, 2/3]
( 3, 6)
[ 9, 1/3]
+
||υ3|| =
( 6, 3)
[ 9, 2/3]
( 9, 9)
[ 18, 1/2]
( 3/ 2 , 3/ 2 )
υ
|| 3|| = [ 2 , 1/2]
/
6
/ , 3/ ) ( 3/ , 3/ )
/ , 1/2] [6/ , 1/2]
(3
[6
2
2
2
2
2
( 9, 9)
[ 18, 1/2]
υ2
υ3
υ1
Example 3
2
2, 8) , 0
υ1 = < [ (10,
>
1/5]
5, 5)
υ2 = < 0, [ (10,
1/2] >
( 5, 5)
2, 8)
( 2, 8) ( 5, 5)
+
,0
υ3 = υ1 + υ2 = < [ (10,
,
=
>
0,
>
<
<
>
[ 10, 1/2]
1/5]
[ 10, 1/5] [ 10, 1/2]
||υ3|| =
<
( 2, 8)
[ 10, 1/5], 0
||υ3|| =
2
> +<
( 5, 5)
0, [ 10, 1/2]
2
>
( 5, 5) 2
( 2, 8) 2
+
[ 10, 1/5] [ 10, 1/2]
||υ3|| =
( 50, 50)
(20, 80)
+
[ 100, 1/5] [ 100, 1/2]
||υ3|| =
( 70, 130)
[ 200, 7/20]
=
( 7 2/2 ,13 2/2)
[20 2/2, 7/20]
υ3
υ2
υ1
Example 4
2, 13)
,0>
υ1 = < [(15,
2/15]
2)
υ2 = < 0, [(4,2,1/2]
>
( 2, 13)
[ 15, 2/15] ,0
2, 13)
( 2, 2)
2)
,
>+ =
υ3 = υ1 + υ2 = <
( 2, 13)
,0
[ 15, 2/15]
||υ3|| =
<
||υ3|| =
2
> +<
2
( 2, 2)
0, [ 4, 1/2]
2
>
2
( 2, 13)
+ ( 2, 2)
[ 15, 2/15] [ 4, 1/2]
||υ3|| =
(30, 195)
( 8, 8)
+
[ 225, 30/225] [ 16, 1/2]
||υ3|| =
(38, 203)
[ 241, 38/241]
=
(
, 203/ 241)
[ 241,38/241]
38/ 241
υ3
υ2
υ1
Example 5
In standard mathematics, we define the dot product as follows,
Let
a = < a1 , a 2 , a 3 >
b = < b1 , b2 , b3 >
then
.
a b = a 1 b1 + a 2 b2 + a3 b3
We’ll do some similar things with Mixed Magnitudes and see what happens.
Let
a=<
(α, α) , (α, α) , (α, α)
[α, α]1 [α, α] 2 [α, α] 3
β)
b = < (β,
[β, β]
Then
.
ab=
,
1
(β, β) (β, β)
,
[β, β]2 [β, β] 3
(α, α) (β, β)
(α, α) (β, β)
[α, α]1 [β, β] 1 + [α, α] 2 [β, β]2
+
>
>
(α, α) (β, β)
[α, α] 3 [β, β] 3
Im not completely satisfied with the notation above, but to avoid excessive use
of superscripts and subscripts Im leaving the shorthand version as is. We’ll do a
few examples and see what it looks like in practice.
Example 6
a=<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
1)
b = < (0,
[1, 0]
,
(0, 1) (0, 1)
,
[1, 0] [1, 0]
>
>
. (1,[1, 0)1] (0,[1, 1)0] + (1,[1, 0)1] (0,[1, 1)0] + (1,[1, 0)1] (0,[1, 1)0]
1/2)
(1/2, 1/2)
(1/2, 1/2)
a . b = (1/2,
+
+
[1, 1/2]
[1, 1/2]
[1, 1/2]
3/2)
a . b = (3/2,
[3, 1/2]
a b=
Example 7
a=<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
>
b =<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
>
. (1,[1, 0)1] (1,[1, 0)1] + (1,[1, 0)1] (1,[1, 0)1] + (1,[1, 0)1] (1,[1, 0)1]
(1, 0)
(1, 0)
0)
a . b = (1,
+
+
[1, 1]
[1, 1]
[1, 1]
0)
a . b = (3,
[3, 1]
a b=
Example 8
(1, 1) , (1, 1) , (1, 1)
a = < [2,
1/2] [2, 1/2] [2, 1/2] >
b =<
.
a.b =
a.b =
a b=
(1, 1) , (1, 1) , (1, 1)
[2, 1/2] [2, 1/2] [2, 1/2]
>
(1, 1) (1, 1)
(1, 1) (1, 1)
(1, 1) (1, 1)
+
+
[2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2]
(2, 2)
[4, 1/2]
(6, 6)
[12, 1/2]
+
(2, 2)
[4, 1/2]
+
(2, 2)
[4, 1/2]
If we were doing standard mathematics we would have the following
relationships for vectors.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
a+b=b+a
a + (b + c) = (a + b) + c
a+0=a
a + (-a) = 0
c(a + b) = ca + cb
(c + d)a = ca + da
(cd)a = c(da) = d(ca)
1a = a
0a = 0 = a0
It’s pretty interesting that when using Mixed Magnitudes you have situations where
you do have associativity under multiplication, but in other cases you do not have
associativity under multiplication. That’s an important thing to take note of, but the
good news is that this behavior does not seem to conflict with any of the well known
relationships above. So, based on that, I’m pretty sure that Mixed Magnitudes will
behave pretty much like standard vector mathematics without any problem. The real
challenge (and strength) of doing vector math with Mixed Magnitudes will be in
cases where the existent and nonexistent parts are both nonzero, because such
calculations will have a totally different philosophical interpretation associated with
them. So it is my hope that you would have some very powerful new tools for doing
probability theory, and I think that’s probably the best use for these methods that I am
aware of.
I wont attempt to prove that all of these relatinships hold for Mixed Magnitudes,
mainly because on philosophical grounds I do not believe that a genuine standard
proof of anything is possible when you are working in a system based on a logic
where truth values 1 and 0 are not allowed, so I shall not write a proof. But I will
demonstrate consistency with standard math, and that it sufficient for a claim
of Equivalence.
Let
(α1 , α2)
a = [α , α ]
4
3
(β1 , β2 )
b = [β , β ]
4
3
c, d are constants which are written as
c=
(γ1 , γ2 )
[γ3 , γ4 ]
(c, 0) , (d, 0)
respectively
[c, 1] [d, 1]
(c, 0)
[c, 1]
The zero element is given here as (d, 0)
[d, 1]
The identity element is given here as
At some point I’ll go through properties [1] through [9] and show that they are all valid
relationships when using Mixed Magnitudes, I just have no time for it at the moment so
I’m leaving it as an exercise or something to be added in the future. It is my belief that
these things can be demonstrated easily.
Ok what follows is a series of graphics and after seeing these
graphics there are a few things that should be absolutely clear
in your mind. First, you will get an immediate and sensible
comprehension of why Planck Length does in fact make perfect
sense. You should also be able to see that we have a method for
bending space using the idea of stochastic existence applied to
geometry. Also, we can apply these ideas to time to get a whole
new understanding of time which I call ‘stochastic time’. Should
be pretty easy to visualize the motivation behind these concepts
just by looking at these ridiculously simple graphics.
Note - the WHITE parts are existent, the RED parts are nonexistent,
and if it is PINK then is has an existential potential somewhere
between zero and 1.
Enjoy Important note
To understand these graphics, imagine that the the Red parts
and the White parts true locations are indeterminate. Imagine
that there is a superpositioning, or that the configurations are
changing so rapidly cycling through all possible configurations
that the Red and White become a Pink BLUR. Once you see this,
then you will understand why Planck Length DOES make sense.
Ok so that is some motivation to continue looking further into
this nonsense.
The important thing about this graphic is that the arrangement or
permutations of the red and white chunks of length may be regarded as
being totally indeterminate. Why ? Because 3 of them exist, 1 of them does
not exist, and therefore due to triviality it is easy to say that the nonexistent
piece could be located anywhere at any time, alternatively you could
argue that it is in all possible positions simultaneously. A superpositioning
of permutations.
An alternative way to think of the “trivial superpositioning of permutations”,
this graphic shows that it behaves like a continuous segment of length.
The important thing about this graphic is that the arrangement or
permutations of the red and white tiles may be regarded as
being totally indeterminate. Why ? Because 3 of them exist, 1 of them does
not exist, and therefore due to triviality it is easy to say that the nonexistent
piece could be located anywhere at any time, alternatively you could
argue that it is in all possible positions simultaneously. A superpositioning
of permutations.
An alternative way to think of the “trivial superpositioning of permutations”,
this graphic shows that it behaves like a continuous manifold.
We could extend this basic concept to 3-dimensional volumes, or any
higher dimension you wish. I have not got time to make all the graphics
for that but the idea would be exactly the same when going to higher
dimensions, as long as you utilize the “trivial superpositioning of permutations”.
It should be possible to examine some constructs similar
to Cellular Automata which obey all of these probabilistic
considerations. I am only giving it a brief mention here, I actually
dont have anything of value at this point except the basic idea
of applying these ideas in this fashion and formalizing it at
some point in the future.
I think that it would make some amazing models and open the door
to lots of new kinds of dynamics but I have done almost nothing
in this area, except to say that I am aware that such a model seems
possible, it’s very exciting to think about it, and there’s huge
potential for making all kinds of strange new models that nobody
has ever imagined.
Gedanken Experiment on Faster than Light
Suppose that we have a stack of 10 coins which is located at
the furthest possible distance away from us in the universe.
And that we are going to create a 2nd stack of coins here on
Earth. We start adding coins one at a time. Eventually, we will
add the 10th coin to this new stack of coins.
At the precise moment when we add the 10th coin, the two
stacks will have the same number of coins.
The question is simply this. How long does it take for these two
stacks of coins to achieve equality ? What is the velocity of equality
in a vacuum such as space ? Does a condition of equality propagate
instantaneously over any distance, or is it limited to the speed
of light ?
It seems that equality should be regarded as propagating
instantaneously over any distance. I can see no reason why it
would be limited to the speed of light.
We’ll use this result to make the same exact argument regarding
other relational quantifiers for example “”, “=/=”, and most
importantly equivalence.
This paper makes the assumption that Equivalence propagates
Instantaneously over any distance, in nature. That assumption
will be assumed valid throughout the entirety of this paper and
will be especially helpful when looking at physics experiments.
Young’s Double Slit Revisited
an attempt to model the Double Slit based on these modeling methods
A gun which fires electrons, one at a time
When we ask which-way then the electron acts like a particle
When we do not ask which-way then the electron acts like a wave
A quantity of energy with a continuous distribution of some kind.
Same quantity of energy with a discrete distribution.
EQUIVALENCE
Anatomy of the Electron as
a defomormation of spacetime fabric
EQUIVALENCE
The underlying assumption being modeled here is that the
fabric of spacetime is such that: “Discrete Spacetime and
Continuous Spacetime are Equivalent”. That is the hypothesis
that we would like to test using empirical science.
The electron is hypothesized to be a ‘blob-like’ deformation in the
fabric of spacetime, very much like gravity. For the purposes of this
crude model we do not care (at the moment) about the more
complicated aspects of the electron, it’s shape, or anything else.
All we care about here is what it is composed of. We hypothesize that
it is composed of bent space, and that the spacetime fabric from which
it is composed has certain characteristics which are now discussed.
The only thing that we need to form the foundation of our
model of the electron is the idea that it is a deformation of
spacetime, and that spacetime itself has an inherent dualistic
nature which is therefore imparted to the electron in it’s entirety,
regardless of it’s shape or any other properties it may have.
EQUIVALENCE
An electron.
This will be our model of the electron. It is a generalized blob
with some peculiar properties.
So we now have a very different kind of model of what an electron
actually is, based on the hypothesis that it is composed of a
deformation in the fabric of spacetime, and that the fabric itself has
some peculiar properties. The property that we are concerned with
is the apparent dualistic nature of this spacetime fabric. That it seems
perfectly reasonable to model it as if it were discrete, and yet perfectly
reasonable to model it as if it were continuous. As part of this hypothesis
we assume further that both are simultaneously correct, due to
equivalence.
So we now have a very different kind of model of what an electron
actually is, based on the hypothesis that it is composed of a
deformation in the fabric of spacetime, and that the fabric itself has
some peculiar properties. The property that we are concerned with
is the apparent dualistic nature of this spacetime fabric. That it seems
perfectly reasonable to model it as if it were discrete, and yet perfectly
reasonable to model it as if it were continuous. As part of this hypothesis
we assume further that both are simultaneously correct, due to
equivalence.
EQUIVALENCE
We now consider what would happen if such an electron
were subjected to Young’s Double Slit experiment.
Electrons are fired one at a time toward a double slit. The Electron
has an inherent ability to exhibit both continuous and discrete
behaviour based on the model we created.
If we ask “which way” the particle went, then we have broken the equivalence
by requiring an answer which is formatted to match the question, and the
answer we get is “particle like”.
If we do not ask “which way” the particle went, then we have also broken
the equivalence by requiring an answer which is formatted to match the
question, in this case it defaults to the “continuous-like” mode, and the
answer we get is “wave like”.
Starting with the well known “Ship of Theseus” paradox
we reformulate just a little bit to restate it as follows:
“A thing may be regarded as a single totality, a whole,
or alternatively it may be regarded as a collection of
individual discrete components. There is _no_way_ to
say that one is more or less correct than the other.”
*** We could elaborate on this a little to note that the two
cases are qualitatively different, but quantitatively
identical. That would be helpful elsewhere, but for now
we’d like to attack the WP Duality and we dont need that
fact at the moment. ***
Some properties of things which are discrete
[1] You can ask “which way”
[2] You can ask “how many”
[3] You can ask “which is first and which is last”
[4] You can ask “when did each ...”
[5] You can ask “in what order is a,b,c ...”
[6] You can ask “How many combinations or permutations”
[7] You can ask “On or Off ?”
Some properties of things which are continuous
[1] Not asking “which way”
[2] You cannot ask “how many”
[3] You cannot ask “which is first and which is last”
[4] Not asking “when did each ...”
[5] You cannot ask “in what order is a,b,c ...”
[6] You cannot ask “How many combinations or permutations”
[7] You cannot ask “On or Off ?” ,
you must ask “What is the potential”
Stern-Gerlach type Experiment
Filter Blocks either
Up or Down
THIS AREA IS
INCOMPLETE.
Up
Down
1
1/2
0
Down
Up
Up
Up
1/2
1
Down
Down
Up
Up
Left
1
1/2
Down
1/8
1/4
Right
Down
ADD philosophical discussion of wave particle
duality in physics and why these tools offer a way
to explain the phenomena sensibly.
The Laws of Thought (implicitly deterministic)
[1] Law of Identity: A thing is itself.
[2] Law of Non-contradiction: Nothing can both be and also not be.
[3] Law of Excluded Middle: Everything must either be or not be.
Laws of Thought Expressed in the format of Uncertainty
[1] Law of Identity: A thing is probably itself.
[2] Law of Non-contradiction: Probably nothing can both probably be
and also probably not be.
[3] Law of Excluded Middle: Probably everything must either probably
be or probably not be.
Ok well there are no doubt better ways to write this out, this particular
wording is after Russel but other formats are possible. At some point
it will be neccessary to write it in first order logic but I just dont have time
at the moment.
The main point of this section is that what we have here is two different
ways to write the Laws of Thought. And we know that we can write them
this way because 0.999... = 1 and so we can easily regard a statement
which has truth value 1 as having truth value 0.999... and take this value
as a truth potential ans the statement becomes an uncertainty. Trivially.
This is pretty sloppy writing but here’s the bottom line ... at the very
most fundamental level we see that we can write the Laws of Thought
in two different formats. We regard these as being equivalent, and
create an Axiom of Equivalence which establishes the foundations of
further work, development of a kind of logic and math which is very
similar to Fuzzy Logic in many ways, but it is different because of this
crucial Axiom of Equivalence that we are putting in place.
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
Our Axiom of Equivalence states that
these two things are Equivalent.
Ok so if we assume that this Axiom of Equivalence is correct, then
we can do all kinds of things and experiment with it and see what
happens.
The Laws of Thought form the basis of logic and mathematics. If we set up
this equivalence, as suggested by our Axiom, then the result will be two
quantitative systems which are also equivalent. Math, and something that
looks like a ‘dual’ of mathematics itself.
‘Dual’ of
Mathematics
All of
Mathematics
The only
possible or
allowable
Truth values
are T and F.
These Are
Equivalent
looks like the
mirror image
of mathematics
but is based on
absolute
uncertainty
Truth values
T and F do
not exist
over here.
These Are
Equivalent
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
These Are
Equivalent
Systems of Reason
“X is certainly True
with truth value 1”
“X might be True
with truth value 0.999...”
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
“X certainly Exists, likelihood = 1”
“X might Exist, likelihood = 0.999...”
Standard Logic yields
traditional mathematics
Logic based on total uncertainty
yields a tool which is totally based
on uncertainties. So it is qualitatively
different than math, but it is
quantitatively identical and will
yield identical quantitative
results as uncertainties.
Equivalent
Foundations
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
Fuzzy Logic simply takes these two very different things and puts
them together into a single domain. A system of reason based on that
domain of truth values can produce an infinite number of contradictions.
Using Fuzzy Logic, the Laws of Thought can be written in both
Deterministic Format AND in the format of Uncertainty.
This can cause problems because as seen in the next graphic, ALL of
the statements are simultaneously valid, even though they may cause
some contradictions or other problems.
“X is certainly True
with truth value 1”
“X might be True
with truth value 0.999...”
“X might Exist, likelihood = 0.999...”
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
“X certainly Exists, likelihood = 1”
ALL of these statements can be written simultaneouslyusing Fuzzy Logic.
The reason why there is a problem is explained in the next graphic.
The Liar Paradox can be written in two formats according to
the axioms of Fuzzy Logic. the problem is that one of these versions
is paradoxical, and the other is NOT a paradox.
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
Under Fuzzy Logic, we can do the same thing with Russell’s Paradox
and lots of others. My feeling is that if a statement both is and also
is notparadoxical, then you must have some kind of contradiction.
Of course this is not a ‘direct’ contradiction, because you have
two very different forms of the same sentence. So there is a
good argument to make if you wanted to claim that it’s not
a contradiction, but it’s also very easy to argue that indeed it is.
The bottom line is that the Liar Paradox can be written in two distinct
formats. These formats are Qualitatively different, but they are
Quantitatively identical. So the most reasonable thing to say is that
they are Equivalent.
If you allow BOTH versions of the Liar Paradox to float freely in a logical
system, I have no idea what might result from doing that.
However, it is possible to keep them separated and STILL have
something which resembles Fuzzy Logic, and we can do this by saying
that you have this dual system that Im proposing.
In classical Logic, truth values can only have the values 0 or 1.
Partial truth values were not considered sensible to the Greeks.
0
1
In Fuzzy Logic, truth values can have the value 0, 1, or anything in between.
0
1
In the system I am proposing, you can have truth values of 0 or 1, and this
allows you to create an entire system of mathematics. You can also have
truth values on the open interval 0
William Kuchta
January 1, 2016
I have intended to submit some type of comprehensive paper
discussing the kinds of models I have been working with for some time, but
for some reason it seems impossible to organize such a broad collection of
ideas into a paper which would serve the purpose of explaining
key concepts of the work I’ve been doing.
If all of this material were presented properly with greater depth and
rigor I think that there is easily enough material for at least one book and it is
unliklely that I will ever be able to compile a such a treatise for various reasons
and so I am releasing my personal Lab Notes with annotations, to explain
these modelling methods in a very informal style.
There is a very small amount of philosophy involved with these
modelling techniques and I will try to add annotations in appropriate places
so that the reader can understand the proper context of what is being done.
Everything in this paper is purely exploratory in nature, it is the result
of my own independent research and I had no assistance in this, nor are there
any references because as far as I know it has never been done before.
This is an ongoing work in progress and if you see a yellow box
like this one it is really just a note to myself to do more work in a particular
area which will appear in subsequent versions of Lab Notes.
[1] When we are doing mathematics it is perfectly reasonable to work
with magnitudes and other constructs which are said to ‘exist’.
[2] It would make no sense whatsoever to work with constructs which are
assumed to be ‘nonexistent’.
[3] However, there is a question about mixing the existent with the nonexistent.
Would it make any sense if we were to consider ‘partial existence’ or things
which have a ‘potential to exist’ ? If we devise a hybrid which is partly existent
and partly nonexistent ... would that make any sense and how would it behave ?
These initial questions were a strong influence early in this research, and I ask
the reader to keep an open mind because methods will be revealed shortly which
should inspire some interest in this area.
Here is a legend with some key concepts which will be needed to understand
the ‘algebra-like’ derivations which are given elsewhere in this paper. This legend
is a very handy summary which has a lot of philosophy ‘built-in’ to these devices.
In order to combine existent with nonexistent magnitude, I was forced to invent
a new representation of ‘number’, which has an inherent duality built into it which will
be discussed in greater detail later. I call this ‘number’ a Mixed Manitude.
Anatomy of a Mixed Magnitude
Existent Part
Nonexistent Part
( a, b )
[ c, d ]
Conjectured Magnitude
A single, given
Magnitude.
Legend for Graphic Representations
of Mixed Magnitudes
exists
does not exist
Existential Potential
This is a Mixed Magnitude.
( a, b )
[ c, d ]
It is a hybrid of existent and nonexistent length.
It is a single magnitude and it has an inherent duality.
The top part resembles Addition.
The bottom part resembles Multiplication.
Because both of these aspects are
inherent to the magnitude we
regard this as a Duality.
When we try to combine several mixed magnitudes, either by adding or
multiplying them together, some strange things can happen. I will attempt
to do this and understand how the duality manifests itself throughout
the resulting algebra, and hopefully draw some connections between
these results and some similar relationships from probability theory.
These are some basic properties of these Mixed Magnitudes. Please note that
a Mixed Magnitude as discussed here is definitely NOT the same thing as the
well known concept of either ‘number’ or ‘magnitude’ from standard mathematics.
Contemporary math does not deal with entities which are partially existent. We
present this new concept of magnitude, and a context which will be helpful to explain
it’s relationship to standard mathematics.
Anatomy of a
Mixed Magnitude
Existent Part
Nonexistent Part
(A, b)
[ C, d]
Existential Potential
Conjectured Magnitude
A ‘Mixed Magnitude’ is a kind of number. It represents a single value, or magnitude.
It may look like 4 distinct numerical values, but they are all taken together to have
one single collective meaning.
These mixed magnitudes can be used for any purpose in place of traditional
numbers, and in fact the numbers you are accustomed to using can easily be
invoked with this notation simply by allowing the nonexistent part to be everywhere
equal to zero.
If the existent part is everywhere equal to zero then your calculations may
still come out looking correct but from a technical standpoint it would be nonsense
by definition.
If both the existent and nonexistent parts of a mixed magnitude are nonzero,
then you are doing stuff which is very much like probability theory.
needs better development of the concept of mixed magnitude, duality and
equivalence of continuous / discrete aspects ... then talk about multiplication
of several mixed magnitudes & algebra etc etc
A worksheet with some examples of multiplication.
(1,0)
[1,1]
White pieces Exist
(1,0)
[1,1]
Red pieces are Non-Existent
(1,0)
[1,1]
(1,0) (1,0) (1,0)
=
[1,1] * [1,1] [1,1]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1) (0,1) = (0,1)
[1,0] * [1,0] [1,0]
(0,1)
[1,1]
(1,0)
[1,1]
(1,0)
[1,1]
(0,1)
[1,1]
(1,0) (0,1) (1/2,1/2)
=
[1,1] * [1,1]
[1,1]
(3,0)
[3,1]
(1,0)
[1,1]
(1,0) (3,0) (3,0)
=
[1,1] * [3,1] [3,1]
(3,0)
[3,1]
(1,1)
[2,1/2]
(1,1) (3,0) (4.5, 1.5)
=
[2,1/2] * [3,1]
[6, 3/4]
A worksheet with some examples of multiplication.
(2,0)
[2,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
(1,1)
[2,1/2]
(1,1)
[2,1/2]
(0,2) (2,0)
(2, 2)
=
[2,0] * [2,1] [4, 1/2]
(2,0)
[2,1]
(1,1)
(3,1) = (5, 3)
*
[2,1/2] [4,3/4] [8, 5/8]
(1,1)
[2,1/2]
(0,4) = (2, 6)
(1,1)
*
[8, 1/4]
[2,1/2] [4,0]
(0,3)
(3,0) (4.5, 4.5)
=
[3,0] * [3,1]
[9, 1/2]
(3,1)
[4,3/4]
(3,1)
[4,3/4]
(4,0) (2,2)
(12, 4)
=
[4,1] * [4,1/2] [16, 3/4]
(3,0)
[3,1]
(0,3)
[3,0]
(1,1) (4,0) = (6, 2)
*
[2,1/2] [4,1] [8, 3/4]
(2,2)
[4,1/2]
(4,0)
[4,1]
(4,0) (2,2)
(12, 4)
=
[4,1] * [4,1/2] [16, 3/4]
(4,0)
[4,1]
(1,1)
[2,1/2]
(1,1)
(3,1) = (5, 3)
*
[2,1/2] [4,3/4] [8, 5/8]
(4,0)
[4,1]
(2,2)
[4,1/2]
(2,0) (2,2) = (6, 2)
*
[8, 3/4]
[2,1] [4,1/2]
(3,1)
[4,3/4]
(0,4)
[4,0]
(1,1)
[2,1/2]
(2,2)
[4,1/2]
(3,1)
[4,3/4]
(1,1)
[2,1/2]
(0,2) (4,0)
(4, 4)
=
[2,0] * [4,1] [8, 1/2]
(1,1)
(2, 2)
(1,1)
=
*
[2,1/2] [2,1/2] [4, 1/2]
(1,1)
(1,1) = (2, 2)
*
[2,1/2] [2,1/2]
[4, 1/2]
(4,0)
[4,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
(3,1)
(3,1)
(12, 4)
=
[4,3/4] * [4,3/4] [16, 3/4]
(1,2)
[3,1/3]
(2,1)
[3,2/3]
(1,2)
[3,1/3]
(1,2)
(2,1)
(4.5, 4.5)
=
[3,1/3] * [3,2/3]
[9, 1/2]
(2,1)
[3,2/3]
(2,1)
(4.5, 4.5)
(1,2)
=
*
[9, 1/2]
[3,1/3] [3,2/3]
A worksheet with some examples of multiplication.
(1,3)
[4,1/4]
(0,1)
[1,0]
(1,3)
[4,1/4]
(0,1)
[1,0]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(1,3)
[4,1/4]
(1,1)
[2,1/2]
(1,1)
(1,3) = (3, 5)
*
[2,1/2] [4,1/4] [8, 3/8]
(0,1)
[1,0]
(0,1)
[1,0]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(1,1)
(2,2) = (4, 4)
*
[2,1/2] [4,1/2] [8, 1/2]
(0,1) (1,3)
(1/2, 7/2)
=
[1,0] * [4,1/4] [8,1/16]
(4,0)
[4,1]
(3,1)
[4,3/4]
(2,2)
[4,1/2]
(1,1)
[2,1/2]
(1,3)
[4,1/4]
(1,3)
[4,1/4]
(1,1)
[2,1/2]
(5, 3)
(3,1)
(1,1)
=
*
[2,1/2] [4,3/4] [8, 5/8]
(1,1)
[2,1/2]
(6, 2)
(4,0)
(1,1)
* [4,1] = [8, 3/4]
[2,1/2]
A worksheet with some examples of addition.
(1,0)
(1,0)
+
=
[1,1]
[1,1]
(2,0)
[2,1]
(0,1)
(0,1)
+
=
[1,0]
[1,0]
(0,1)
(1,0)
+
=
[1,1]
[1,1]
(0,2)
[2,0]
(1,1)
[2,1/2]
Important Note:
This concept of ‘number’ or ‘magnitude’ has an inherent duality which is fundamentally built in to it.
=
Discrete, looks like addition
(1,1)
[2,1/2]
=
(1,1)
[2,1/2]
Continuous, looks like multiplication
These are simply two different representations of the exact same thing.
The magnitudes and the left and right sides are identical.
DISCRETE-LIKE ADDITION
(3,1)
[4,3/4]
THESE BOTH SAY
THE SAME EXACT
THING, THEY ARE
EQUIVALENT FORMS
AND BOTH ARE
PERFECTLY VALID
SIMULTANEOUSLY.
+
(5,3)
[8,5/8]
=
(8,4)
[12,3/4]
(3,1)
(5,3)
(8,4)
=
[4,3/4] + [8,5/8]
[12,3/4]
CONTINUOUS-LIKE ADDITION, ACTUALLY RESEMBLES MULTIPLICATION
(3,1)
[4,3/4]
+
(5,3)
[8,5/8]
=
(3,1)
(5,3)
(8,4)
=
[4,3/4] + [8,5/8]
[12,3/4]
In case you haven’t noticed,
THIS IS A DUALITY
(8,4)
[12,3/4]
A worksheet with some examples of addition.
(8,8)
[16,1/2]
(8,1)
[9,8/9]
+
(16,9)
[25,16/25]
=
(8,1)
(8,8)
(16,9)
+
=
[16,1/2] [9,8/9] [25,16/25]
(8,8)
[16,1/2]
(3,1)
[4,3/4]
+
(11,9)
[20,11/20]
=
(8,1)
(8,8)
(16,9)
+
=
[16,1/2] [9,8/9] [25,16/25]
(16,0)
[16,1]
+
(0,1)
[1,0]
(16,0) + (0,1)
[1,0]
[16,1]
=
=
(16,1)
[17,16/17]
(16,1)
[17,16/17]
An extremely interesting looking result.
A mixed magnitude multiplied
with it’s own conjugate
( 3, 1)
( 1, 3)
*
=
[ 4, 3/4] [ 4, 1/4]
( 3, 2)
[ 5, 3/5]
*
Legend:
=
*
( 8, 8)
[ 16, 1/2]
*
=
*
( 2, 3)
=
[ 5, 2/5]
( 12.5, 12.5)
[ 25, 1/2]
=
( 2, 1)
( 1, 2)
=
*
[ 3, 2/3] [ 3, 1/3]
( 4.5, 4.5)
[ 9, 1/2]
( 8, 8)
[ 16, 1/2]
( 3, 0)
, 0 > + < 0, [ 3, 1] > = < ( 4, 0) , ( 3, 0) >
υ1 + υ2 = < ([ 4,4, 0)
1]
[ 4, 1] [ 3, 1]
||υ3|| =
<
( 4, 0) , 0
[ 4, 1]
2
>+<
( 16, 0)
[ 16, 1]
||υ3|| =
||υ3|| =
( 3, 0)
0, [ 3, 1]
2
>
+ ([ 9,9, 0)
1]
( 25, 0)
[ 25, 1]
=
( 5, 0)
[ 5, 1]
υ3
υ2
υ1
Example 1
Mixed Magnitude as Vectors
υ1 = <
( 1, 1) , 0
[ 2, 1/2]
>
( 1, 1)
[ 2, 1/2]
>
υ2 = < 0,
( 1, 1)
( 1, 1)
( 1, 1)
1)
υ3 = υ1 + υ2 = < [(2,1,1/2]
, 0 > + < 0, [ 2, 1/2] > = < [ 2, 1/2] , [ 2, 1/2] >
||υ3|| =
<
( 1, 1)
[ 2, 1/2] , 0
2
( 1, 1)
[ 2, 1/2]
=
2
+
>
<
( 1, 1)
0, [ 2, 1/2]
( 1, 1)
( 1, 1)
∗
[ 2, 1/2] [ 2, 1/2]
( 2, 2)
[ 4, 1/2]
||υ3|| =
||υ3|| =
checking
υ2
υ1
>
=
( 2, 2)
[ 4, 1/2]
( 2, 2)
+ [ 4, 1/2]
( 4, 4)
[ 8, 1/2]
( 2 , 2)
( 2 , 2)
∗
[ 2 2 , 1/2] [ 2 2 , 1/2]
||υ3|| =
υ3
2
( 2 , 2)
[ 2 2 , 1/2]
Example 2
4)
= [(8,4,1/2]
υ1 = <
υ2 = <
( 1, 2) , 0
[ 3, 1/3]
( 2, 1)
0, [ 3, 2/3]
>
>
( 2, 1)
( 1, 2) ( 2, 1)
+
>
<
υ1 + υ2 = < ([ 1,3, 2)
,
0
=
> 0, [ 3, 2/3]
< [ 3, 1/3], [ 3, 2/3] >
1/3]
||υ3|| =
<
( 1, 2)
[ 3, 1/3] , 0
||υ3|| =
2
>
2
+ < 0, ( 2, 1) >
[ 3, 2/3]
( 3, 6)
[ 9, 1/3]
+
||υ3|| =
( 6, 3)
[ 9, 2/3]
( 9, 9)
[ 18, 1/2]
( 3/ 2 , 3/ 2 )
υ
|| 3|| = [ 2 , 1/2]
/
6
/ , 3/ ) ( 3/ , 3/ )
/ , 1/2] [6/ , 1/2]
(3
[6
2
2
2
2
2
( 9, 9)
[ 18, 1/2]
υ2
υ3
υ1
Example 3
2
2, 8) , 0
υ1 = < [ (10,
>
1/5]
5, 5)
υ2 = < 0, [ (10,
1/2] >
( 5, 5)
2, 8)
( 2, 8) ( 5, 5)
+
,0
υ3 = υ1 + υ2 = < [ (10,
,
=
>
0,
>
<
<
>
[ 10, 1/2]
1/5]
[ 10, 1/5] [ 10, 1/2]
||υ3|| =
<
( 2, 8)
[ 10, 1/5], 0
||υ3|| =
2
> +<
( 5, 5)
0, [ 10, 1/2]
2
>
( 5, 5) 2
( 2, 8) 2
+
[ 10, 1/5] [ 10, 1/2]
||υ3|| =
( 50, 50)
(20, 80)
+
[ 100, 1/5] [ 100, 1/2]
||υ3|| =
( 70, 130)
[ 200, 7/20]
=
( 7 2/2 ,13 2/2)
[20 2/2, 7/20]
υ3
υ2
υ1
Example 4
2, 13)
,0>
υ1 = < [(15,
2/15]
2)
υ2 = < 0, [(4,2,1/2]
>
( 2, 13)
[ 15, 2/15] ,0
2, 13)
( 2, 2)
2)
,
>+ =
υ3 = υ1 + υ2 = <
( 2, 13)
,0
[ 15, 2/15]
||υ3|| =
<
||υ3|| =
2
> +<
2
( 2, 2)
0, [ 4, 1/2]
2
>
2
( 2, 13)
+ ( 2, 2)
[ 15, 2/15] [ 4, 1/2]
||υ3|| =
(30, 195)
( 8, 8)
+
[ 225, 30/225] [ 16, 1/2]
||υ3|| =
(38, 203)
[ 241, 38/241]
=
(
, 203/ 241)
[ 241,38/241]
38/ 241
υ3
υ2
υ1
Example 5
In standard mathematics, we define the dot product as follows,
Let
a = < a1 , a 2 , a 3 >
b = < b1 , b2 , b3 >
then
.
a b = a 1 b1 + a 2 b2 + a3 b3
We’ll do some similar things with Mixed Magnitudes and see what happens.
Let
a=<
(α, α) , (α, α) , (α, α)
[α, α]1 [α, α] 2 [α, α] 3
β)
b = < (β,
[β, β]
Then
.
ab=
,
1
(β, β) (β, β)
,
[β, β]2 [β, β] 3
(α, α) (β, β)
(α, α) (β, β)
[α, α]1 [β, β] 1 + [α, α] 2 [β, β]2
+
>
>
(α, α) (β, β)
[α, α] 3 [β, β] 3
Im not completely satisfied with the notation above, but to avoid excessive use
of superscripts and subscripts Im leaving the shorthand version as is. We’ll do a
few examples and see what it looks like in practice.
Example 6
a=<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
1)
b = < (0,
[1, 0]
,
(0, 1) (0, 1)
,
[1, 0] [1, 0]
>
>
. (1,[1, 0)1] (0,[1, 1)0] + (1,[1, 0)1] (0,[1, 1)0] + (1,[1, 0)1] (0,[1, 1)0]
1/2)
(1/2, 1/2)
(1/2, 1/2)
a . b = (1/2,
+
+
[1, 1/2]
[1, 1/2]
[1, 1/2]
3/2)
a . b = (3/2,
[3, 1/2]
a b=
Example 7
a=<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
>
b =<
(1, 0) , (1, 0) , (1, 0)
[1, 1] [1, 1] [1, 1]
>
. (1,[1, 0)1] (1,[1, 0)1] + (1,[1, 0)1] (1,[1, 0)1] + (1,[1, 0)1] (1,[1, 0)1]
(1, 0)
(1, 0)
0)
a . b = (1,
+
+
[1, 1]
[1, 1]
[1, 1]
0)
a . b = (3,
[3, 1]
a b=
Example 8
(1, 1) , (1, 1) , (1, 1)
a = < [2,
1/2] [2, 1/2] [2, 1/2] >
b =<
.
a.b =
a.b =
a b=
(1, 1) , (1, 1) , (1, 1)
[2, 1/2] [2, 1/2] [2, 1/2]
>
(1, 1) (1, 1)
(1, 1) (1, 1)
(1, 1) (1, 1)
+
+
[2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2] [2, 1/2]
(2, 2)
[4, 1/2]
(6, 6)
[12, 1/2]
+
(2, 2)
[4, 1/2]
+
(2, 2)
[4, 1/2]
If we were doing standard mathematics we would have the following
relationships for vectors.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
a+b=b+a
a + (b + c) = (a + b) + c
a+0=a
a + (-a) = 0
c(a + b) = ca + cb
(c + d)a = ca + da
(cd)a = c(da) = d(ca)
1a = a
0a = 0 = a0
It’s pretty interesting that when using Mixed Magnitudes you have situations where
you do have associativity under multiplication, but in other cases you do not have
associativity under multiplication. That’s an important thing to take note of, but the
good news is that this behavior does not seem to conflict with any of the well known
relationships above. So, based on that, I’m pretty sure that Mixed Magnitudes will
behave pretty much like standard vector mathematics without any problem. The real
challenge (and strength) of doing vector math with Mixed Magnitudes will be in
cases where the existent and nonexistent parts are both nonzero, because such
calculations will have a totally different philosophical interpretation associated with
them. So it is my hope that you would have some very powerful new tools for doing
probability theory, and I think that’s probably the best use for these methods that I am
aware of.
I wont attempt to prove that all of these relatinships hold for Mixed Magnitudes,
mainly because on philosophical grounds I do not believe that a genuine standard
proof of anything is possible when you are working in a system based on a logic
where truth values 1 and 0 are not allowed, so I shall not write a proof. But I will
demonstrate consistency with standard math, and that it sufficient for a claim
of Equivalence.
Let
(α1 , α2)
a = [α , α ]
4
3
(β1 , β2 )
b = [β , β ]
4
3
c, d are constants which are written as
c=
(γ1 , γ2 )
[γ3 , γ4 ]
(c, 0) , (d, 0)
respectively
[c, 1] [d, 1]
(c, 0)
[c, 1]
The zero element is given here as (d, 0)
[d, 1]
The identity element is given here as
At some point I’ll go through properties [1] through [9] and show that they are all valid
relationships when using Mixed Magnitudes, I just have no time for it at the moment so
I’m leaving it as an exercise or something to be added in the future. It is my belief that
these things can be demonstrated easily.
Ok what follows is a series of graphics and after seeing these
graphics there are a few things that should be absolutely clear
in your mind. First, you will get an immediate and sensible
comprehension of why Planck Length does in fact make perfect
sense. You should also be able to see that we have a method for
bending space using the idea of stochastic existence applied to
geometry. Also, we can apply these ideas to time to get a whole
new understanding of time which I call ‘stochastic time’. Should
be pretty easy to visualize the motivation behind these concepts
just by looking at these ridiculously simple graphics.
Note - the WHITE parts are existent, the RED parts are nonexistent,
and if it is PINK then is has an existential potential somewhere
between zero and 1.
Enjoy Important note
To understand these graphics, imagine that the the Red parts
and the White parts true locations are indeterminate. Imagine
that there is a superpositioning, or that the configurations are
changing so rapidly cycling through all possible configurations
that the Red and White become a Pink BLUR. Once you see this,
then you will understand why Planck Length DOES make sense.
Ok so that is some motivation to continue looking further into
this nonsense.
The important thing about this graphic is that the arrangement or
permutations of the red and white chunks of length may be regarded as
being totally indeterminate. Why ? Because 3 of them exist, 1 of them does
not exist, and therefore due to triviality it is easy to say that the nonexistent
piece could be located anywhere at any time, alternatively you could
argue that it is in all possible positions simultaneously. A superpositioning
of permutations.
An alternative way to think of the “trivial superpositioning of permutations”,
this graphic shows that it behaves like a continuous segment of length.
The important thing about this graphic is that the arrangement or
permutations of the red and white tiles may be regarded as
being totally indeterminate. Why ? Because 3 of them exist, 1 of them does
not exist, and therefore due to triviality it is easy to say that the nonexistent
piece could be located anywhere at any time, alternatively you could
argue that it is in all possible positions simultaneously. A superpositioning
of permutations.
An alternative way to think of the “trivial superpositioning of permutations”,
this graphic shows that it behaves like a continuous manifold.
We could extend this basic concept to 3-dimensional volumes, or any
higher dimension you wish. I have not got time to make all the graphics
for that but the idea would be exactly the same when going to higher
dimensions, as long as you utilize the “trivial superpositioning of permutations”.
It should be possible to examine some constructs similar
to Cellular Automata which obey all of these probabilistic
considerations. I am only giving it a brief mention here, I actually
dont have anything of value at this point except the basic idea
of applying these ideas in this fashion and formalizing it at
some point in the future.
I think that it would make some amazing models and open the door
to lots of new kinds of dynamics but I have done almost nothing
in this area, except to say that I am aware that such a model seems
possible, it’s very exciting to think about it, and there’s huge
potential for making all kinds of strange new models that nobody
has ever imagined.
Gedanken Experiment on Faster than Light
Suppose that we have a stack of 10 coins which is located at
the furthest possible distance away from us in the universe.
And that we are going to create a 2nd stack of coins here on
Earth. We start adding coins one at a time. Eventually, we will
add the 10th coin to this new stack of coins.
At the precise moment when we add the 10th coin, the two
stacks will have the same number of coins.
The question is simply this. How long does it take for these two
stacks of coins to achieve equality ? What is the velocity of equality
in a vacuum such as space ? Does a condition of equality propagate
instantaneously over any distance, or is it limited to the speed
of light ?
It seems that equality should be regarded as propagating
instantaneously over any distance. I can see no reason why it
would be limited to the speed of light.
We’ll use this result to make the same exact argument regarding
other relational quantifiers for example “”, “=/=”, and most
importantly equivalence.
This paper makes the assumption that Equivalence propagates
Instantaneously over any distance, in nature. That assumption
will be assumed valid throughout the entirety of this paper and
will be especially helpful when looking at physics experiments.
Young’s Double Slit Revisited
an attempt to model the Double Slit based on these modeling methods
A gun which fires electrons, one at a time
When we ask which-way then the electron acts like a particle
When we do not ask which-way then the electron acts like a wave
A quantity of energy with a continuous distribution of some kind.
Same quantity of energy with a discrete distribution.
EQUIVALENCE
Anatomy of the Electron as
a defomormation of spacetime fabric
EQUIVALENCE
The underlying assumption being modeled here is that the
fabric of spacetime is such that: “Discrete Spacetime and
Continuous Spacetime are Equivalent”. That is the hypothesis
that we would like to test using empirical science.
The electron is hypothesized to be a ‘blob-like’ deformation in the
fabric of spacetime, very much like gravity. For the purposes of this
crude model we do not care (at the moment) about the more
complicated aspects of the electron, it’s shape, or anything else.
All we care about here is what it is composed of. We hypothesize that
it is composed of bent space, and that the spacetime fabric from which
it is composed has certain characteristics which are now discussed.
The only thing that we need to form the foundation of our
model of the electron is the idea that it is a deformation of
spacetime, and that spacetime itself has an inherent dualistic
nature which is therefore imparted to the electron in it’s entirety,
regardless of it’s shape or any other properties it may have.
EQUIVALENCE
An electron.
This will be our model of the electron. It is a generalized blob
with some peculiar properties.
So we now have a very different kind of model of what an electron
actually is, based on the hypothesis that it is composed of a
deformation in the fabric of spacetime, and that the fabric itself has
some peculiar properties. The property that we are concerned with
is the apparent dualistic nature of this spacetime fabric. That it seems
perfectly reasonable to model it as if it were discrete, and yet perfectly
reasonable to model it as if it were continuous. As part of this hypothesis
we assume further that both are simultaneously correct, due to
equivalence.
So we now have a very different kind of model of what an electron
actually is, based on the hypothesis that it is composed of a
deformation in the fabric of spacetime, and that the fabric itself has
some peculiar properties. The property that we are concerned with
is the apparent dualistic nature of this spacetime fabric. That it seems
perfectly reasonable to model it as if it were discrete, and yet perfectly
reasonable to model it as if it were continuous. As part of this hypothesis
we assume further that both are simultaneously correct, due to
equivalence.
EQUIVALENCE
We now consider what would happen if such an electron
were subjected to Young’s Double Slit experiment.
Electrons are fired one at a time toward a double slit. The Electron
has an inherent ability to exhibit both continuous and discrete
behaviour based on the model we created.
If we ask “which way” the particle went, then we have broken the equivalence
by requiring an answer which is formatted to match the question, and the
answer we get is “particle like”.
If we do not ask “which way” the particle went, then we have also broken
the equivalence by requiring an answer which is formatted to match the
question, in this case it defaults to the “continuous-like” mode, and the
answer we get is “wave like”.
Starting with the well known “Ship of Theseus” paradox
we reformulate just a little bit to restate it as follows:
“A thing may be regarded as a single totality, a whole,
or alternatively it may be regarded as a collection of
individual discrete components. There is _no_way_ to
say that one is more or less correct than the other.”
*** We could elaborate on this a little to note that the two
cases are qualitatively different, but quantitatively
identical. That would be helpful elsewhere, but for now
we’d like to attack the WP Duality and we dont need that
fact at the moment. ***
Some properties of things which are discrete
[1] You can ask “which way”
[2] You can ask “how many”
[3] You can ask “which is first and which is last”
[4] You can ask “when did each ...”
[5] You can ask “in what order is a,b,c ...”
[6] You can ask “How many combinations or permutations”
[7] You can ask “On or Off ?”
Some properties of things which are continuous
[1] Not asking “which way”
[2] You cannot ask “how many”
[3] You cannot ask “which is first and which is last”
[4] Not asking “when did each ...”
[5] You cannot ask “in what order is a,b,c ...”
[6] You cannot ask “How many combinations or permutations”
[7] You cannot ask “On or Off ?” ,
you must ask “What is the potential”
Stern-Gerlach type Experiment
Filter Blocks either
Up or Down
THIS AREA IS
INCOMPLETE.
Up
Down
1
1/2
0
Down
Up
Up
Up
1/2
1
Down
Down
Up
Up
Left
1
1/2
Down
1/8
1/4
Right
Down
ADD philosophical discussion of wave particle
duality in physics and why these tools offer a way
to explain the phenomena sensibly.
The Laws of Thought (implicitly deterministic)
[1] Law of Identity: A thing is itself.
[2] Law of Non-contradiction: Nothing can both be and also not be.
[3] Law of Excluded Middle: Everything must either be or not be.
Laws of Thought Expressed in the format of Uncertainty
[1] Law of Identity: A thing is probably itself.
[2] Law of Non-contradiction: Probably nothing can both probably be
and also probably not be.
[3] Law of Excluded Middle: Probably everything must either probably
be or probably not be.
Ok well there are no doubt better ways to write this out, this particular
wording is after Russel but other formats are possible. At some point
it will be neccessary to write it in first order logic but I just dont have time
at the moment.
The main point of this section is that what we have here is two different
ways to write the Laws of Thought. And we know that we can write them
this way because 0.999... = 1 and so we can easily regard a statement
which has truth value 1 as having truth value 0.999... and take this value
as a truth potential ans the statement becomes an uncertainty. Trivially.
This is pretty sloppy writing but here’s the bottom line ... at the very
most fundamental level we see that we can write the Laws of Thought
in two different formats. We regard these as being equivalent, and
create an Axiom of Equivalence which establishes the foundations of
further work, development of a kind of logic and math which is very
similar to Fuzzy Logic in many ways, but it is different because of this
crucial Axiom of Equivalence that we are putting in place.
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
Our Axiom of Equivalence states that
these two things are Equivalent.
Ok so if we assume that this Axiom of Equivalence is correct, then
we can do all kinds of things and experiment with it and see what
happens.
The Laws of Thought form the basis of logic and mathematics. If we set up
this equivalence, as suggested by our Axiom, then the result will be two
quantitative systems which are also equivalent. Math, and something that
looks like a ‘dual’ of mathematics itself.
‘Dual’ of
Mathematics
All of
Mathematics
The only
possible or
allowable
Truth values
are T and F.
These Are
Equivalent
looks like the
mirror image
of mathematics
but is based on
absolute
uncertainty
Truth values
T and F do
not exist
over here.
These Are
Equivalent
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
These Are
Equivalent
Systems of Reason
“X is certainly True
with truth value 1”
“X might be True
with truth value 0.999...”
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
“X certainly Exists, likelihood = 1”
“X might Exist, likelihood = 0.999...”
Standard Logic yields
traditional mathematics
Logic based on total uncertainty
yields a tool which is totally based
on uncertainties. So it is qualitatively
different than math, but it is
quantitatively identical and will
yield identical quantitative
results as uncertainties.
Equivalent
Foundations
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
The Laws of Thought
Deterministic Format
The Laws of Thought
Formatted as Uncertainties
Fuzzy Logic simply takes these two very different things and puts
them together into a single domain. A system of reason based on that
domain of truth values can produce an infinite number of contradictions.
Using Fuzzy Logic, the Laws of Thought can be written in both
Deterministic Format AND in the format of Uncertainty.
This can cause problems because as seen in the next graphic, ALL of
the statements are simultaneously valid, even though they may cause
some contradictions or other problems.
“X is certainly True
with truth value 1”
“X might be True
with truth value 0.999...”
“X might Exist, likelihood = 0.999...”
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
“X certainly Exists, likelihood = 1”
ALL of these statements can be written simultaneouslyusing Fuzzy Logic.
The reason why there is a problem is explained in the next graphic.
The Liar Paradox can be written in two formats according to
the axioms of Fuzzy Logic. the problem is that one of these versions
is paradoxical, and the other is NOT a paradox.
“This statement is certainly False”
“This statement might be False
with likelihood = 0.999...”
Under Fuzzy Logic, we can do the same thing with Russell’s Paradox
and lots of others. My feeling is that if a statement both is and also
is notparadoxical, then you must have some kind of contradiction.
Of course this is not a ‘direct’ contradiction, because you have
two very different forms of the same sentence. So there is a
good argument to make if you wanted to claim that it’s not
a contradiction, but it’s also very easy to argue that indeed it is.
The bottom line is that the Liar Paradox can be written in two distinct
formats. These formats are Qualitatively different, but they are
Quantitatively identical. So the most reasonable thing to say is that
they are Equivalent.
If you allow BOTH versions of the Liar Paradox to float freely in a logical
system, I have no idea what might result from doing that.
However, it is possible to keep them separated and STILL have
something which resembles Fuzzy Logic, and we can do this by saying
that you have this dual system that Im proposing.
In classical Logic, truth values can only have the values 0 or 1.
Partial truth values were not considered sensible to the Greeks.
0
1
In Fuzzy Logic, truth values can have the value 0, 1, or anything in between.
0
1
In the system I am proposing, you can have truth values of 0 or 1, and this
allows you to create an entire system of mathematics. You can also have
truth values on the open interval 0