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

exists

does not exist ( a, b )

[ c, d ]

Existent Part Nonexistent Part

Existential Potential Conjectured Magnitude

Anatomy of a Mixed Magnitude

( a, b ) [ c, d ]

Legend for Graphic Representations of Mixed Magnitudes

This is a Mixed Magnitude. The top part resembles Addition.

The bottom part resembles Multiplication.

It is a hybrid of existent and nonexistent length. It is a single magnitude and it has an inherent duality.

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.

A single, given Magnitude.

[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.

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.

(A, b)

[ C, d]

Existent Part

Nonexistent Part

Existential Potential

Conjectured Magnitude

Anatomy of a

Mixed 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.

A worksheet with some examples of multiplication. (1,0) [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) [1,1] (1,0) [1,1] (1,0) [1,1] (1/2,1/2) [1,1] *(0,1)_[1,1]=

(0,1) [1,1] * = (0,1) [1,0] (0,1) [1,0] (0,1) [1,0] (0,1) [1,0] (0,1) [1,0] (0,1) [1,0] White pieces Exist

Red pieces are Non-Existent

(1,0) [1,1] (3,0) [3,1] * = (1,0) [1,1] (3,0) [3,1] (3,0) [3,1] (1,1) [2,1/2] (3,0) [3,1]

*(3,0)_[3,1]=(4.5, 1.5) _{[6, 3/4]} (1,1)

A worksheet with some examples of multiplication.

(0,2) [2,0]

(2,0) [2,1]

* = (2, 2)_{[4, 1/2]} (0,2) [2,0] (2,0) [2,1] (0,2) [2,0] (4,0) [4,1]

* = (4, 4)_{[8, 1/2]} (0,2) [2,0] (4,0) [4,1] (2,2) [4,1/2] (4,0) [4,1]

* = (12, 4)_{[16, 3/4]} (4,0) [4,1] (2,2) [4,1/2] (1,1) [2,1/2]

* = (2, 2)

[4, 1/2] (1,1) [2,1/2] (1,1) [2,1/2] (1,1) [2,1/2] (3,1) [4,3/4]

* = (5, 3)

[8, 5/8] (1,1) [2,1/2] (1,1) [2,1/2] (3,1) [4,3/4] (2,2) [4,1/2]

* = (6, 2)

[8, 3/4] (2,0) [2,1] (2,2) [4,1/2] (2,0) [2,1] (3,1) [4,3/4] (3,1) [4,3/4]

* = (12, 4)_{[16, 3/4]} (2,2)

[4,1/2]

(4,0) [4,1]

* = (12, 4)_{[16, 3/4]} (4,0) [4,1] (2,2) [4,1/2] (3,1) [4,3/4] (3,1) [4,3/4] (0,3) [3,0] (3,0) [3,1]

* = (4.5, 4.5) _{[9, 1/2]} (0,3) [3,0] (3,0) [3,1] (1,2) [3,1/3] (2,1) [3,2/3]

* = (4.5, 4.5) _{[9, 1/2]} (1,2) [3,1/3] (2,1) [3,2/3] (1,2) [3,1/3] (2,1) [3,2/3]

* = (4.5, 4.5) _{[9, 1/2]} (1,2) [3,1/3] (2,1) [3,2/3] (1,1) [2,1/2] (4,0) [4,1]

*(4,0)_[4,1]= (6, 2)_{[8, 3/4]} (1,1) [2,1/2] (1,1) [2,1/2] (3,1) [4,3/4]

* = (5, 3)_{[8, 5/8]} (1,1) [2,1/2] (3,1) [4,3/4] (1,1) [2,1/2] (0,4) [4,0]

* = (2, 6)_{[8, 1/4]} (1,1) [2,1/2] (0,4) [4,0] (1,1) [2,1/2]

= (2, 2) [4, 1/2] (1,1)

[2,1/2]

(1,1) [2,1/2]

(1,1) [2,1/2]

(2,2) [4,1/2]

* = (4, 4)_{[8, 1/2]} (1,1) [2,1/2] (2,2) [4,1/2] (1,1) [2,1/2] (1,3) [4,1/4]

* = (3, 5)_{[8, 3/8]} (1,1)

[2,1/2]

(1,3) [4,1/4]

A worksheet with some examples of multiplication.

(0,1) [1,0] (1,3) [4,1/4] * = (0,1) [1,0] (1,3) [4,1/4] (1/2, 7/2) [8,1/16] (0,1) [1,0] (1,3) [4,1/4] * = (0,1) [1,0] (1,3) [4,1/4] (1/2, 7/2) [8,1/16] (0,1) [1,0] (1,3) [4,1/4] * = (0,1) [1,0] (1,3) [4,1/4] (1/2, 7/2) [8,1/16] (0,1) [1,0] (1,3) [4,1/4] * = (0,1) [1,0] (1,3) [4,1/4] (1/2, 7/2) [8,1/16] (1,1) [2,1/2] (3,1) [4,3/4]

* = (5, 3)_{[8, 5/8]} (1,1) [2,1/2] (1,1) [2,1/2] (4,0) [4,1]

A worksheet with some examples of addition.

(1,1) [2,1/2] (1,0)

[1,1] + =

(2,0) [2,1] (1,0)

[1,1]

(0,1)

[1,0] + =

(0,2) [2,0] (0,1)

[1,0]

(3,1)

[4,3/4] + =

(8,4) [12,3/4] (5,3)

[8,5/8]

(3,1)

[4,3/4] + =

(8,4) [12,3/4] (5,3)

[8,5/8] = (1,1)

[2,1/2] Important Note:

This concept of ‘number’ or ‘magnitude’ has an inherent duality which is fundamentally built in to it. (1,0)

[1,1]

(0,1) [1,1] =

+ (1,1)_[2,1/2]

Discrete, looks like addition 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.

=

(3,1)

[4,3/4] + =

(8,4) [12,3/4] (5,3)

[8,5/8]

(3,1)

[4,3/4] + =

(8,4) [12,3/4] (5,3)

[8,5/8]

DISCRETE-LIKE ADDITION

CONTINUOUS-LIKE ADDITION, ACTUALLY RESEMBLES MULTIPLICATION THESE BOTH SAY

THE SAME EXACT THING, THEY ARE EQUIVALENT FORMS AND BOTH ARE PERFECTLY VALID SIMULTANEOUSLY.

A worksheet with some examples of addition.

(8,8) [16,1/2]

+ =

(8,1) [9,8/9]

+ =

(16,9) [25,16/25]

(8,8) [16,1/2]

(8,1) [9,8/9]

(16,9) [25,16/25]

(8,8) [16,1/2]

+ =

(3,1) [4,3/4]

+ =

(11,9) [20,11/20]

(8,8) [16,1/2]

(8,1) [9,8/9]

(16,9) [25,16/25]

(16,0) [16,1]

+ =

(0,1) [1,0]

+ =

(16,1) [17,16/17]

(16,0) [16,1]

(0,1) [1,0]

A mixed magnitude multiplied

with it’s own conjugate

*

( 3, 1)

[ 4, 3/4] * ( 1, 3)_{[ 4, 1/4]} =

= ( 8, 8) [ 16, 1/2]

*

( 3, 2)

[ 5, 3/5] *

( 2, 3) [ 5, 2/5]

=

= ( 12.5, 12.5) [ 25, 1/2]

*

( 2, 1)

[ 3, 2/3] * ( 1, 2)_{[ 3, 1/3]} =

= ( 4.5, 4.5) [ 9, 1/2]

exists

does not exist Legend:

( 8, 8) [ 16, 1/2]

<- Addition like aspect <- Multiplication like aspect The expression above is a “Mixed Magnitude”. It has an existent part, and a nonexistent part. We mix these parts into a single magnitude and impose CONSERVATION so that the nonexistent part does nto simply collapse or vanish trivially. The only reason any of these operations make sense or hold together is due to conservation. This is the only example I am aware of where conservation is used directly on a logic structure, instead of using the abstract model to describe a conservation which is assumed to exist elsewhere in physics for example.

( 1, 5) [ 6, 1/6]

( 1, 5) [ 6, 1/6]

( 1, 5) [ 6, 1/6]

=

+

+

( 0, 18)

[ 18, 0] ( 0, 6)

[ 6, 0]

( 0, 6) [ 6, 0]

( 0, 6) [ 6, 0]

( 1, 17) [ 18, 1/18]

State Change Obeys CONSERVATION

Here is a useful diagram which shows the interesting similarity between a mixed magnitude and some standard problems from elementary probability theory. These are the actual lab notes and graphics that I have devised which help me to keep track of things when considering these methods.

There seems to be an inescapable resemblance between these magnitudes and a random variable. So naturally I have been digging around looking for relationships and trying to formalize some things.

The following graphics seem to be a good guide for devising models. At some point I realized that the ONLY WAY that such a magnitude even made any sense is if you impose conservation in order to keep the nonexistent part from simply collapsing. Amazingly, conservation also serves an additional purpose when a possible

outcome (potentially existent) is transformed into an actualized (existent) outcome, and I regard this as an Existential State Change. The possible relationship to

( 0, 6) [ 6, 0]

( 0, 6) [ 6, 0]

( 0, 6) [ 6, 0]

( 0, 6)

[ 6, 0]

*

*

=

( 0, 36)

[ 36, 0]

*

=

=

( 0, 216) [ 216, 0] ( 0, 6)

[ 6, 0]

( 0, 6) [ 6, 0] ( 0, 6) [ 6, 0]

( 0, 216) [ 216, 0]

( 1, 215) [ 216, 1/216]

Here is an example of how equivalence can be applied to the coin toss problem. We solve the problem using two completely different philosophies, and show that they both give identical answers. Therefore, you have quantitative equivalence between these two entirely different systems of reason.

P( X ) = { x | H, T }

Problem 1A (standard math)

Consider a fair, two sided coin. If we want to model a single coin toss trial using standard mathematics we can use the well known methods from Probability Theory and it would look something like this:

You have a random variable, an outcome space, both possible outcomes H and T are equally likely to occur and so they each have a likelihood of 1/2.

P( H ) = 1/2 P( T ) = 1/2

You can use this model to accurately predict the behavior of a physical coin toss experiment and the important thing to emphasize here is that all of the math involved is based on the Law of Excluded Middle. Very standard math where something either exists or it does not, there is no in-between.

If you use this modelling approach you do not even need to consdier the existence of H and T because the model seems to assume that neither exists until the outcome event has occured, at which time either H will exist or T will exist, but you cannot have both. This fact is not reflected in the mechanism of the algebra, but it is definitely a prominent aspect of the underlying philosophy of the model itself and is generally revealed through philosophical discourse on the topic.

Problem 1B (using partial existence)

Consider a fair, two sided coin. We are going to model what happens when we toss the coin using a complete different set of tools, other than standard mathematics. We consider a time line with several events. You have the initial toss, an pre-event period, the outcome event, and the post-event period.

There is another route to the same solution, as follows:

initial toss

pre-event period

outcome event

( 0, 1) [ 1, 0]

initial toss

pre-event period

outcome event

post-event period

( 0, 2) [ 2, 0]

( 0, 1) [ 1, 0]

( 1, 0) [ 1, 1]

( 0, 1) [ 1, 0]

H

T

H

T

( 1, 0) [ 1, 1]

H

State Change Obeys CONSERVATION

Initially, there are 2 possible outcomes and they are both nonexistent, because the outcome event has not yet occured. Therefore, by virtue of being nonexistent, they share some unique attributes which are peculiar to nonexistent things. They behave trivially.

We can decompose into two possible outcomes. At this stage each outcome is still nonexistent, for the same reason as before. And therefore these outcomes can behave in ways which would seem trivial, because they are nonexistent. This process of triviality gives a much better foundation for the claim that these two outcomes are “entangled” in much the same way that particles are considered entangled in physics.

At the precise moment of the outcome event, the entangled pair of outcomes will experience a “state change”. In our

example above the outcome H is transformed from being nonexistent to existent. At that point the outcome H will transform from a potential to a new state of “tangible & existent”, and the outcome T will also transform from being a potential to a new state “tangible nonexistent”. The outcome event destroys the state of

( 0, 1) [ 1, 0]

initial toss

pre-event period

outcome event

post-event period

( 0, 2) [ 2, 0]

( 0, 1) [ 1, 0]

( 1, 0) [ 1, 1]

( 0, 1) [ 1, 0]

H

T

H

T

( 1, 0) [ 1, 1]

H

State Change Obeys CONSERVATION

( 0, 1) [ 1, 0]

T

( 1, 1) [ 2, 1/2]

We now have an outcome which exists, and another outcome which had the potential to exist but remains nonexistent. These outcomes cannot be “entangled”

at this time because while they are both tangible, one of them exists and the other does not. Also note that while these

outcomes are in an entangled state prior to the outcome event that we can also say that it is ‘equivalent’ whether we predict am outcome of H or T. There are many other ways to consider equivalence with respect to this entangled state, Im leaving it as an exercise to contemplate it.

Ok so we have one very simple problem (the coin toss), and we have solved it in two different ways. And it’s reasonable to ask just why in the world someone would solve a problem twice instead of just once, and why the hell are we doing this anyway. The reason why we are doing this is to demonstrate that there is more than one way to skin a cat. We can solve the problem doing standard math and prbability theory, or we can use tools which utilize the idea of ‘partial existence’, arguably these methods arent even mathematics at all, but we still get an identical quantitative result.

Personally I think that any reasonable person might be a little intrigued by that. So what does it mean.

It should be clear that since both solutions yield the same numerical answers, and the only difference between them is the vastly different philosophical considerations which are the basis of the respective models, then it is clear that either of these is just as good as the other and that the only reasn to choose one over the other is human bias. They are qualitatively different because they have totally different philosophical qualities, but

quantitatively they are identical in terms of teh answers they yield and so the only conclusion one can draw from the situation is that the two models are EQUIVALENT. In other words, we may accurately and correctly construct models based on the Law of the Excluded Middle, standard math and probability theory and those models will be correct. AND that if we instead creat models which are based on the Excluded Middle

instead of Exclusive Middle then we will get models which are equally correct, they just use different philosophical apparatus to explain them in common language.

T H HH TH HT TT

( 1, 1) [ 2, 1/2]

( 1, 3) [ 4, 1/4] = *

( 0, 2) [ 2, 0]

= *

( 0, 2) [ 2, 0]

( 1, 1) [ 2, 1/2]

( 1, 6) [ 6, 1/6]

( 0, 3) [ 3, 0]

( 1, 12) [ 12, 1/12]

( 0, 2) [ 2, 0]

( 1, 3) [ 4, 1/4]

* = ( 1, 7)_{[ 8, 1/8]}

* =

TH HH

( 0, 2) [ 2, 0]

( 1, 23) [ 24, 1/24]

( 0, 2) [ 2, 0]

( 1, 6) [ 6, 1/6]

( 1, 12) [ 12, 1/12]

* =

* =

( 0, 2) [ 2, 0]

( 1, 12) [ 12, 1/12]

( 1, 23) [ 24, 1/24]

* =

* =

=

PHASE DIAGRAM / STATE CHANGE HHH THH HTH HHT TTH THT HTT TTT HHH HHT THH THT

( 0, 2) [ 2, 0]

( 0, 2) [ 2, 0]

( 0, 3) [ 3, 0]

( 0, 2) [ 2, 0]

( 0, 2) [ 2, 0]

* * * = ( 0, 24)

[ 24, 0]

STATE CHANGE

( 0, 24) [ 24, 0]

( 1, 23) [ 24, 1/24]

Some further efforts, attempting to make a Boolean type tree diagram and

explain the numbers using State Changes. The precise meaning of a “State Change” in this regard is as follows. For example, when a coin is tossed, the outcome does

T H

HH

TH HT

TT

( 1, 1) [ 2, 1/2]

( 1, 3) [ 4, 1/4] =

* ( 0, 2) [ 2, 0]

= *

( 0, 2) [ 2, 0] ( 1, 1)

[ 2, 1/2]

( 1, 3) [ 4, 1/4]

( 0, 2) [ 2, 0]

( 1, 7) [ 8, 1/8]

( 0, 2) [ 2, 0]

( 1, 3) [ 4, 1/4]

* = ( 1, 7)_{[ 8, 1/8]}

* =

Some further efforts, attempting to make a Boolean type tree diagram and explain the numbers.

THIS AREA IS INCOMPLETE.

Basic trigonometry based on Mixed Magnitudes

θ x

y r

θ x

y r

θ

=

( 22.5, 22.5)

[ 45, 1/2]

θ

=

( 22.5, 22.5)

[ 45, 1/2]

This angle is represented as

22.5 existent PLUS 22.5 nonexistent, added together to yield 45 degress. We call this a “Conjectured Angle of 45 degrees” because it is partly nonexistent.

This angle is represented as 45 degrees with an Existential Potential of 1/2. That is why it is pink instead of red. We simply take the value 45 and MULTIPLY by 1/2 to get smoething which is everywhere 1/2 existent.

We also call this a “Conjectured Angle of 45 degrees” because it is partly nonexistent.

These are two EQUIVALENT ways to represent the same exact thing. Both representations are simultaneously valid. You have a duality, and it should be clear that both ways of thinking about this angle are simultaneously correct. It can be written in DISCRETE format as seen on the left, or in CONTINUOUS format as shown on the right. Both are simultaneously correct, and we represent this DUALITY in a compact form by simple stating the angle with our notation as follows:

θ

=

( 22.5, 22.5)

θ x

y r

sin(θ) = y/r cos(θ) = x/r tan(θ) = y/x

sin(45) = 1/√2

cos(45) = 1/√2

tan(45) = (1/√2)/(1/√2) = 1

sin( cos( tan( ) = ) = ) =

( 45, 0) [ 45, 1] ( 45, 0) [ 45, 1] ( 45, 0) [ 45, 1]

Example:

Consider the unit circle where r=1. Then we have that x = 1/√2, y=1/√2

( 1/√2, 0) [ 1/√2, 1]

( 1, 0) [ 1, 1] ( 1, 0) [ 1, 1]

/

/

/

( 1/√2, 0) [ 1/√2, 1] ( 1/√2, 0) [ 1/√2, 1]

( 1/√2, 0) [ 1/√2, 1]

= = = ( 1, 0)

[ 1, 1] ( 1/√2, 0) [ 1/√2, 1] ( 1/√2, 0) [ 1/√2, 1]

θ =

( 1/√2, 0) [ 1/√2, 1]

( 1, 0) [ 1, 1] x = y = r =

( 45, 0) [ 45, 1]

θ x

y r

sin(θ) = y/r cos(θ) = x/r tan(θ) = y/x sin( cos( tan( ) = ) = ) =

Example 2, again on the unit circle:

/

/

/

= =

= ( 1/2, 1/2)_{[ 1, 1/2]}

θ =

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

( 1/2, 1/2) [ 1, 1/2] x = y = r =

( 22.5, 22.5) [ 45, 1/2]

( 22.5, 22.5) [ 45, 1/2]

( 22.5, 22.5) [ 45, 1/2] ( 22.5, 22.5) [ 45, 1/2]

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

( 1/2, 1/2) [ 1, 1/2]

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2] ( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

( 1/2, 1/2) [ 1, 1/2]

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

= =

= ( 1/2, 1/2) _{[ 1, 1/2]}

( 1/(2√2), 1/(2√2)) [ 1/√2, 1/2]

Basic trigonometry based on Mixed Magnitudes

( 22.5, 22.5) [ 45, 1/2]

θ

θ = ( 67.5, 22.5) _{[ 90, 3/4]} θ

θ = ( 45, 45) _{[ 90, 1/2]} θ

θ =

( 180, 90) [ 270, 2/3]

θ

θ = ( 135, 45) _{[ 180, 3/4]} θ

θ = ( 180, 180) _{[ 360, 1/2]} θ

θ =

( 360, 0) [ 360, 1]

θ

θ = ( 0, 360) _{[ 360, 0]} θ

θ =

( 180, 0) [ 180, 1]

θ

θ = ( 0, 180) _{[ 180, 0]} θ θ = θ x y r

sin(θ) = y/r cos(θ) = x/r tan(θ) = y/x

sin(45) = 1/√2

cos(45) = 1/√2

tan(45) = (1/√2)/(1/√2) = 1

sin( cos( tan( ) = ) = ) =

( 0, 45) [ 45, 0] ( 0, 45) [ 45, 0] ( 0, 45) [ 45, 0]

Example:

Consider the unit circle where x, y, r and θ are all purely nonexistent, this system allows the algebra to remain consistent even in this extreme case which is for illustrative purposes only.

( 0, 1/√2) [ 1/√2, 0]

( 0, 1) [ 1, 0] ( 0, 1) [ 1, 0]

/

/

/

( 0, 1/√2) [ 1/√2, 0] ( 0, 1/√2) [ 1/√2, 0]

( 0, 1/√2) [ 1/√2, 0]

= = = ( 0, 1)

[ 1, 0]

θ =

( 0, 1/√2) [ 1/√2, 0]

( 0, 1) [ 1, 0] x = y = r =

( 0, 45) [ 45, 0]

ADD WORKED EXAMPLES HERE

( 67.5, 22.5) [ 90, 3/4]

( 22.5, 22.5) [ 45, 1/2]

( 90, 45) [ 135, 2/3]

+ =

+ =

( 67.5, 22.5) [ 90, 3/4]

( 22.5, 22.5) [ 45, 1/2]

( 90, 45) [ 135, 2/3]

( 1, 0) [ 1, 1]

( 1, 0) [ 1, 1]

+ + ( 1, 0)_{[ 1, 1]}

[

]

( 2, 2) [ 4, 1/2] *

= ( 1, 0)_{[ 1, 1]} + ( 1, 0)_{[ 1, 1]} ( 1, 0) [ 1, 1] +

( 2, 2) [ 4, 1/2] *

( 2, 2) [ 4, 1/2] *

( 2, 2) [ 4, 1/2] *

= ( 2, 2) + + [ 4, 1/2]

( 2, 2) [ 4, 1/2]

( 2, 2) [ 4, 1/2] = ( 6, 6)_{[ 12, 1/2]}

Examples of basic algebra

( 1, 0) [ 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] *[ 1, 1]}( 1, 0) + ( 1, 0)_{[ 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]

( 1, 0) [ 1, 1]

+ +

= ( 4, 0)_{[ 4, 1]}

( 1, 0) [ 1, 1]

( 1, 0) [ 1, 1] +

[

]

* ( 1, 0)[ 1, 1]

( 1, 0) [ 1, 1] +

[

]

= ( 2, 0)_{[ 2, 1] *} ( 2, 0)_{[ 2, 1]} = ( 4, 0)_{[ 4, 1]}

( 2, 0) [ 2, 1] ( 2, 0) [ 2, 1]

= ( 1, 0)_{[ 1, 1]}

( 2, 2) [ 4, 1/2]

( 2, 2) [ 4, 1/2]

= ( 1, 0)_{[ 1, 1]}

=

(1,1) [2,1/2]

(3,0) [3,1]

*(3,0)_[3,1]=(4.5, 1.5) _{[6, 3/4]} (1,1) [2,1/2] (4.5, 1.5) [6, 3/4] (3,0) [3,1] (1,1) [2,1/2] = (4.5, 1.5)

[6, 3/4] _(3,0) [3,1] (1,1)

[2,1/2]

Mixed Magnitude as Vectors

υ + υ

1 2

=

( 3, 0) [ 3, 1] ( 4, 0) [ 4, 1]

υ =

υ =

1

2

<

>

<

>

, 0

0,

υ

₁

υ

₂

υ

₃

( 4, 0) [ 4, 1]

<

, 0

>

+

<

0, ( 3, 0)_{[ 3, 1]}

>

υ

₃

||

=

2 2

||

=

<

( 4, 0)_{[ 4, 1]},( 3, 0)

[ 3, 1]

>

( 4, 0) [ 4, 1]

<

, 0

>

+

<

0, ( 3, 0)_{[ 3, 1]}

>

υ

₃

||

=

||

( 16, 0)_{[ 16, 1]}

+

( 9, 0)_{[ 9, 1]}

υ

₃

||

=

||

( 25, 0)_{[ 25, 1]}

=

( 5, 0)_{[ 5, 1]}

Example 1

Mixed Magnitude as Vectors

υ + υ

_{1 2}

=

( 1, 1) [ 2, 1/2] ( 1, 1)

[ 2, 1/2]

υ =

υ =

1

2

<

>

<

, 0

>

0,

<

, 0

>

+

<

0,

>

υ

₁

υ

₂

υ

3

υ

₃

<

_{, 0}

>

2

+

<

0,

>

2

( 1, 1) [ 2, 1/2] ( 1, 1)

[ 2, 1/2]

( 1, 1) [ 2, 1/2] ( 1, 1)

[ 2, 1/2]

= <

( 1, 1)_{[ 2, 1/2]}, ( 1, 1)_{[ 2, 1/2]}

>

( 1, 1)

[ 2, 1/2]

∗

( 1, 1)[ 2, 1/2]

=

( 2, 2)[ 4, 1/2]

2

=

( 1, 1) [ 2, 1/2]

( 2, 2)

[ 4, 1/2]

+

( 2, 2)[ 4, 1/2]

( 4, 4) [ 8, 1/2]

υ =

₃

=

||

||

=

=

=

( , )_{[ 2 , 1/2]} 2 2 ₂

( , )

[ 2 , 1/2] 2 2 2

( , )

[ 2 , 1/2] 2 2 2

∗

=

( 4, 4)_{[ 8, 1/2]}

checking

υ

₃

||

||

υ

₃

||

||

υ

₃

||

||

Example 2

υ + υ

_{1 2}

=

( 2, 1) [ 3, 2/3] ( 1, 2)

[ 3, 1/3]

υ =

υ =

1

2

<

>

<

, 0

>

0,

( 1, 2) [ 3, 1/3]

<

, 0

>

+

<

0,( 2, 1)_{[ 3, 2/3]}

>

υ

₁

υ

₂

υ

₃

( 1, 2) [ 3, 1/3]

<

, 0

>

2

+

<

0,( 2, 1)_{[ 3, 2/3]}

>

2

= <

( 1, 2)_{[ 3, 1/3]}, ( 2, 1)_{[ 3, 2/3]}

>

=

=

( 3, 6)

[ 9, 1/3] ( 6, 3)[ 9, 2/3]

=

( 9, 9)[ 18, 1/2]

+

( , ) [ , 1/2]

( 9, 9) [ 18, 1/2]

6

υ

₃

||

||

υ

₃

||

||

υ

₃

||

||

3

/

2 3

/

2

/

2

( , )

[ , 1/2]6

3

/

2 3

/

2

/

2

=

υ

₃

||

||

( , ) _{[ , 1/2]6}3

/

2 3

/

2

/

2

Example 3

υ

₁

υ

₂

υ

₃

υ + υ

_{1 2}

=

( 5, 5) [ 10, 1/2] ( 2, 8)

[ 10, 1/5]

υ =

υ =

1

2

<

>

<

, 0

>

0,

<

,0

>

+

<

0,

>

= <

,

>

υ =

₃ ( 5, 5)_{[ 10, 1/2]} ( 5, 5)_{[ 10, 1/2]}

<

, 0

>

2

+

<

0,

>

2

=

υ

₃

||

||

=

=

+

υ

₃

||

||

υ

₃

||

||

=

υ

₃

||

||

( 2, 8) [ 10, 1/5]

( 2, 8)

[ 10, 1/5] ( 2, 8)_{[ 10, 1/5]}

( 5, 5) [ 10, 1/2]

( 5, 5) [ 10, 1/2] ( 2, 8)

[ 10, 1/5]

2 2

( 50, 50) [ 100, 1/2]

+

(20, 80) [ 100, 1/5]

( 70, 130) [ 200, 7/20]

( , ) [ , 7/20]

=

20 7 2 2/2/213 2/2

Example 4

υ

₁

υ

₂

υ

₃

υ + υ

1 2

=

( 2, 2) [ 4, 1/2] ( 2, 13) [ 15, 2/15]

υ =

υ =

1

2

<

>

<

, 0

>

0,

<

,0

>

+

<

0,

>

=<

<

,

>

>

υ =

₃

<

, 0

>

2

+

<

0,

>

2

=

υ

₃

||

||

=

=

+

υ

₃

||

||

υ

₃

||

||

=

υ

₃

||

||

2 2

+

(30, 195) [ 225, 30/225]

=

Example 5

( 2, 13)

[ 15, 2/15] ( 2, 2)[ 4, 1/2] ( 2, 13)[ 15, 2/15]

( 2, 2) [ 4, 1/2]

( 2, 13) [ 15, 2/15]

( 2, 13) [ 15, 2/15]

( 2, 2) [ 4, 1/2]

( 2, 2) [ 4, 1/2]

( 8, 8) [ 16, 1/2]

(38, 203) [ 241, 38/241]

( , )

[ , ]241

Example 6

In standard mathematics, we define the dot product as follows,

Let a = < a , a , a > b = < b , b , b >

then a b = a b + a b + a b

.

_{1 1 2 2 3 3}

1 2 3

1 2 3

We’ll do some similar things with Mixed Magnitudes and see what happens.

<

(α, α)_{[α, α]}

>

Let

a

=

b

=

(α, α)

[α, α] (α, α)[α, α]

(β, β) [β, β]

<

(β, β)_{[β, β]} (β, β)_{[β, β]}

>

1 2 3

1 2 3

, ,

, ,

Then

a b

.

=

(α, α)_{[α, α]} (α, α)_{[α, α]} (α, α)_{[α, α]}

1 2 3

(β, β) [β, β]

(β, β)

[β, β] (β, β)[β, β]

1 2 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.

<

(1, 0)_{[1, 1]}

>

a

=

b

=

<

(0, 1)_{[1, 0]} , ,

>

, (1, 0) ,

[1, 1] (0, 1) [1, 0]

(1, 0) [1, 1] (0, 1) [1, 0]

a b

.

=

(1, 0)_{[1, 1]}(0, 1)_{[1, 0]}

+

(1, 0)_{[1, 1]}(0, 1)_{[1, 0]}

+

(1, 0)_{[1, 1]}(0, 1)_{[1, 0]}

a b

.

=

(1/2, 1/2) _{[1, 1/2]}

+

(1/2, 1/2) _{[1, 1/2]}

+

(1/2, 1/2) _{[1, 1/2]}

Example 7

a

=

<

(1, 0)_{[1, 1]}

>

b

=

<

>

, (1, 0) ,

[1, 1] (1, 0)[1, 1]

a b

.

=

(1, 0)_{[1, 1]}(1, 0)_{[1, 1]}

+

(1, 0)_{[1, 1]}(1, 0)_{[1, 1]}

+

(1, 0)_{[1, 1]}(1, 0)_{[1, 1]}

a b

.

=

(1, 0)_{[1, 1]}

+

+

a b

.

=

(3, 0)_{[3, 1]}

(1, 0)

[1, 1] , (1, 0)[1, 1] , (1, 0)[1, 1]

(1, 0)

[1, 1] (1, 0)[1, 1]

Example 8

a

=

<

(1, 1)_{[2, 1/2]}

>

b

=

<

>

, (1, 1) ,

[2, 1/2] (1, 1)[2, 1/2]

a b

.

=

+

+

a b

.

=

(2, 2)_{[4, 1/2]}

+

a b

.

=

(6, 6)_{[12, 1/2]}

(1, 1)

[2, 1/2], (1, 1)[2, 1/2], (1, 1)[2, 1/2]

(1, 1)

[2, 1/2] (1, 1)[2, 1/2] (1, 1)[2, 1/2] (1, 1)[2, 1/2] (1, 1)[2, 1/2] (1, 1)[2, 1/2]

(2, 2)

[4, 1/2]

+

If we were doing standard mathematics we would have the following relationships for vectors.

[1]

a

+

b

=

b

+

a

[2]

a

+ (

b

+

c

) = (

a

+

b

) +

c

[3]

a

+ 0 =

a

[4]

a

+ (-

a

) = 0

[5]

c

(

a

+

b

) =

c

a

+

c

b

[6] (

c

+

d

)

a

=

c

a

+

d

a

[7] (

c

d)

a

=

c

(

d

a

) =

d

(

c

a

)

[8] 1

a

=

a

[9] 0

a

= 0 =

a

0

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.

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.

Let a = b = c =

c, d are constants which are written as

(α , α ) [α , α ]

(β , β ) [β , β ]

(γ , γ ) [γ , γ ]

(c, 0) [c, 1]

(d, 0) [d, 1]

The identity element is given here as

The zero element is given here as

(c, 0) [c, 1] (d, 0) [d, 1]

respectively ,

1 2 3 4

1 2 3 4

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.

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

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

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

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

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.

Young’s Double Slit Revisited

an attempt to model the Double Slit based on these modeling methods

When we ask which-way then the electron acts like a particle

EQUIVALENCE

EQUIVALENCE

Anatomy of the Electron as

a defomormation of spacetime fabric

A quantity of energy with a continuous distribution of some kind.

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.

EQUIVALENCE

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.

An electron.

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.

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

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”.

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”

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

Stern-Gerlach type Experiment

Filter Blocks either Up or Down

Up

Down

Left

Right

Up

Down Up

Down

Up

Down Up

Down Up

Down

1

1/2 0

1

1 1/2 1/4 1/8

THIS AREA IS INCOMPLETE.

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.

The Laws of Thought

Deterministic Format

The Laws of Thought

Formatted as Uncertainties

Our Axiom of Equivalence states that

these two things are Equivalent.

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought

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.

These Are Equivalent These Are Equivalent All of

Mathematics

The only possible or

allowable Truth values

are T and F.

‘Dual’ of Mathematics

looks like the mirror image of mathematics

but is based on absolute uncertainty Truth values

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought Equivalent

Foundations 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

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought

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” _{with truth value 0.999...”}“X might be True

“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...”

“This statement is certainly False” _{“This statement might be False} with likelihood = 0.999...” 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.

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

0

1

In Fuzzy Logic, truth values can have the value 0, 1, or anything in between. 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

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<p<1 (non-inclusive), and this will also yield a perfectly valid system which is much like a mirror image of math, and loosely speaking it may appear to be a kind of ‘dual’ of math.

I’ll try to explain why it must be a dual, and also cannot be a dual, and why this seems to imply that paradox is tautological under the assumption that Excluded and Exclusive Middles are Equivalent.

This

One of the best arguments I have found to justify this use of Equivalence is the fact that 0.999... = 1 identically. We may interpret a truth value T=1 as a kind of “deterministic” truth, a construct of absolute certainty. And if we create a truth value of 0.999..., we may interpret this truth value to be probabilistic. A construct based on uncertainty.

The interesting thing is that while they are radically different in terms of their qualitative aspects, one being deterministic and the other being uncertain, it is clear that they both have the same ‘magnitude’ because 0.999... = 1. And so the best way to reconcile this is by saying that these two very different kinds of truth are in fact Equivalent.

This application of Equivalence is also interesting for a very wide number of reasons. I’ll just name a few. Equivalence is extremely important in the Theory of Relativity. If Equivalence makes sense at this fundamental level, we have a very strong connection to

Relativity and it is my belief that subsequent modelling along these lines would allow us to derive the theory of General Relativity from these ‘first principles’.

Another hamdy thing about this approach is that it allows us to have some very new kinds of ways to think about dimension, namely time and length, and it is very easy to devise a kind of modified Minkowski Spacetime which has this inherent duality built into it. It is a very powerful axiom. Should be able to explain wave-particle duality pretty easily by modelling with these methods, and that is also an objective of this line of inquiry.

Also note that the algebraic manipulations shown elsewhere absolutely require that the ratio of existence to nonexistence be strictly conserved otherwise this would all be pure nonsense. This conservation which is

inherent to the algebra, is in fact commanded by the Axiom of Equivalence (Exclusive and Exclusive Middles), and gives a VERY strong argument for physics ... that conseravtion in physics is absolutely mandated, and

The Laws of Thought Deterministic Format

One very nice thing about this application of Equivalence is that standard mathematics is ‘formally recoverable’ from this construct. If we simply delete or disregard all of the ‘Quasi-Logic’ based

on the Exclusive Middle, (and we can), then what we are left with

is standard mathematics and it is not altered in any way by this action.

Exclusive Middle

Excluded Middle

A potential difficulty of Fuzzy Logic is that these things are not so easily separable. Because the foundations of Fuzzy Logic basically takes the union of the Excluded and Exclusive Middles, it seems that there is no easy way to separate them if you wanted to do that. In other words, it is a system where partial truth values are mixed with structures based on truth values { T, F }, and there is no way to separate these

The Laws of Thought Deterministic Format

However in the system seen below it is clear that mathematics is

completely recoverable simply by deleting all of the Exclusive Middle logic, and so you have this powerful ability to isolate them because the Axiom of Equivalence only says that these things are Equivalent. It does not say that they both need to be present at any given time.

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought

There is no function or operation available in standard mathematics which would allow you to transform from say %100 existence to %50 existence. There is no way to do it. It may be possible if you are using Fuzzy Logic somehow, but in standard mathematics it is not possible and I think you could prove that it is beyond the capability of math perhaps by using Peano Arithmetic or some First Order Logic.

These Are Equivalent These Are Equivalent All of

Mathematics

The only possible or

allowable Truth values

are T and F.

‘Dual’ of Mathematics

looks like the mirror image of mathematics

but is based on absolute uncertainty Truth values

T and F do not exist over here.

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought These Are

Equivalent These Are Equivalent

At this point it is extremely useful to point out that the empirical sciences use math as a tool for quantitative modelling and the only thing that empiricism cares about is that it can quantify things. Equivalence is very

powerful in this regard because our usage says that we can completely delete either one of these two branches of this duality and you will be left with a tool which would work for physics in either case. Here, we delete the

The Laws of Thought

Deterministic Format _{Formatted as Uncertainties}The Laws of Thought These Are

Equivalent These Are Equivalent

Here we shall delete mathematics, and what we are left with is a structure which is Equivalent to mathematics, but it is based on the Exclusive Middle. It generates the exact same out put that mathematics would generate, except for the fact that it can only output uncertainties. However, since the quantitative outputs are identical in either case, you have a completely alternate tool which would obviously work perfectly for all empirical

Exclusive Middle

Excluded Middle

It is possible to make a similar argument about the physical sciences and their use of mathematics with respect to Fuzzy Logic, but things are rather different because the two different kinds of logic are mixed together. They do not seem separable. This creates some philosophical difficulties, but the worst aspect is that you are forfeiting the use of Equivalence at the

foundational level which is potentially a very nice smooth link that will get you directly to General Relativity.

I have no idea how I would go about reconciling Equivalence in GR as an emergent property in physics. It seems that this would be very

difficult for the philosopher to grapple with. They cant even seem to figure out emergence of order/disorder, so it seems that asking them to devise the emergence of Equivalence as it is used in GR would be too great a challenge.

This is perhaps the greatest strength of my argument in favor of an Axiom which states that Exclusive and Excluded Middles are Equivalent. We already have laboratory experiments which confirm Relativity, and so this math

(A, b)

[ C, d]

Existent Part

Nonexistent Part

Existential Potential

Conjectured Magnitude

Anatomy of a

Mixed Magnitude

Using my system, standard mathematics is ‘fully recoverable’ within the system. What that means is that if we write numbers in the format that I have given above, all you need to do is let the nonexistent part be zero, let the existential potential be always equal to 1 .... and you have mathematics in an unaltered form.

In other words, if you write numbers using this format, it is possible to do

purely standard mathematics, it is possible to do exclusively ‘Intuitionist Logic’ type mathematics, and it is possible to do Fuzzy Logic and all kinds of Fuzzy Math. And, these things are separable. They can be isolated from one another.

Using my system, standard mathematics is ‘fully recoverable’ within the system. What that means is that if we write numbers in the format that I have given above, all you need to do is let the nonexistent part be zero, let the existential potential be always equal to 1, and the result will be standard mathematics in an unaltered form.

In other words, if you write numbers using this format, it is possible to do

purely standard mathematics, it is possible to do exclusively ‘Intuitionist Logic’ type mathematics, and it is possible to do Fuzzy Logic and all kinds of Fuzzy Math. And, these things are separable. They can be isolated from one another.

This representation of magnitude, this representation of ‘number’ allows you to do all of that. Together, with the Axiom of Equivalence you have a greatly expanded ability to model all kinds of things, of particular interest we have quantum mechanics, wave particle duality, and many other things to numerous to mention here.

(1,0) [1,1]

As an example and to demonstrate what is writtem in the preceeding paragraph, consider how we add 1 + 1 = 2 in standard math. It is actually pretty easy and well known we simply write:

1 + 1 = 2

We can write the exact same thing using this alternate notation, but the meaning will be exactly the same. We write it as follows:

(1,0) [1,1]

(2,0) [2,1]

If we want to write this same equation in the format of uncertainty based on the Exclusive Middle we have the flexibility to do so. First, recall that 0.999... = 1 identically and then all you need to do is make some simple substitutions.

(1.999... , 0 ) [1.999... , 0.999...] + (0.999... , 0 ) =

[0.999... , 0.999...] (0.999... , 0 )

[0.999... , 0.999...]

(A, b)

[ C, d]

Existent Part

Nonexistent Part

Existential Potential

Conjectured Magnitude

Anatomy of a

Mixed Magnitude

You could actually argue (correctly) that there are 3 different ways to do empirical science, based on 3 totally different kinds of

quantification and they are all perfectly valid, they all yield the exact same quantitative answers !!!

This is standard math,

a valid tool for quantification in the empirical sciences

This also yields a perfectly valid tool for quantification in the empirical sciences

We say that they are Equivalent

and put them together into a unified tool, a tool which has an inherent duality, and is unified by Equivalence.

This Works

This Also Works

This Works BEST of all because it can easily explain wave-particle duality and other effects

In the development of empirical science, people could have chosen ANY of these 3 different approaches and they all would have worked. The only reason that mankind has chosen the path of mathematics seems to be that man has a cultural, human bias which favors

deterministic structures like standard math, the answers seem more credible and the models are more appealing to common sense.

However there is absolutely no quantitative difference between reasoning with certainty and reasoning with uncertainty. The difference is NOT quantitative, it is qualitative.

If I say “It shall rain with absolute certainty” it sounds pretty credible.

But if I say “It might rain, with likelihood 0.999...” it sounds as if I am unsure of my conclusion.

BUT, the truth values in both statements are identical

because 0.999... = 1 identically. So both of those statements are equally true, equally valid, and they are just two different

One difficulty of this approach is that standard mathematics is comprised of a vast amount of structure which could be called deterministic, but we also have probability theory which is regarded as a kind of subset of that larger deterministic construct. If this approach is ever going to make any sense then this needs to be addressed and the next graphic shows one way to represent the situation.

Probability Theory

Inverse of Probability

Theory is “Determinacy

Theory”

lots of deterministic mathematical

structure

quasi-mathematical

structure based on

total uncertainty Equivalence

The preceeding graphic is attempting to demonstrate what must be done in order for this Equivalence to work properly and make sense. What we are saying basically is that ‘deterministic and probablistic are Equivalent. And the reason we say this is because that is what is implied by our formalism when we say that Excluded and Exclusive Middles