A lot of people are questioning the validity of Tao Mathematics. I do not blame them, it is mostly my fault as I jumped in with a lot of equations that do not seem to makes sense in contemporary mathematics. In fact, I would not be surprised if most people felt that I had lost the plot at this stage. Do not worry, it will make sense by the end of this article. This should really have been the first article, but like all discoveries it did not pan out that way.
For those who are not strong in mathematics, this article will not be too complex to follow. If you can read a bar chart and count to two, you will be fine.
To understand Tao mathematics I must bring everyone back to the foundations of mathematics itself. Modern mathematics is based upon Euclidean geometry and equivalent transforms within that geometry. You do not need to fully understand this, but you will have a solid grasp of it by the end of this article. I will run through some quick examples and you will clearly see how equations work and the theory it is based upon.
First, let's consider this image:
This is just a standard bar chart (Cartesian space), we have a value of one on the X-axis and a value of two on the Z-axis. We are going to use this to show everyone that what we define as an equation is the same thing as a transform, or rotation, in Cartesian space based upon equivalence. Before we do this, let us remind ourselves of where this came from and which disciplines make used of it.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing.
Let's now zoom in for a clearer view:
We will start with a very simple view on this. If we look down the Z-axis (vertical axis), we can see that we have two units of one. We also know that 1+1=2. We will now rotate the picture to look down the X-axis:
We can now see that we have one unit of two and we know from our math that this is just '2'. From this we can observe that 1+1=2 is telling us equality sign is a rotation (transform), in a particular coordinate system. In short, it is the same thing from a different perspective.
In this type of mathematics, a valid equation (rotation, transform) is always one that is equivalent. That is, the equation must balance, just as in the pictures above. In terms of physics, this is an unproven conjecture. In fact, the theory of Relativity demonstrates that we must always take into consideration the amount of time it takes to communicate changes. Nothing happens simultaneously, particles must move from one location to another to convey that changes have occurred. In practice, this means that any equation based upon equivalence is describing a state in which two conditions are true:
1. Information has propagated.
2. Nothing has happened to block that communication.
This means that for a certain amount of time, there is a level of inconsistency that violates the rules of mathematics and this permeates all of reality. Thus, for accurate description of reality current mathematics is unsuitable, it is too rigid.
Now that we understand this, we can begin to describe what Tao mathematics appears to be. If we take a look at the following picture, we can see the effect of a mathematical system based upon rotations (transforms) of equivalences.
Such a system provides an ever-reducing or ever-expanding series of ranges, just like the grid structure in the above diagram. It does not matter how this is expressed, it could be circles, triangles, polygons, etc., the effect is always the same.
Tao mathematics functions by eliminating the requirement of rotations to be based on equivalence. In effect, this melts, or liquefies, Cartesian space into an uncertain continuum by having no defined elements whatsoever. Tao mathematics is therefore the foundation of all mathematics and yet nothing at the same time. We can represent it by this picture:
Yes, that is intended to be a black square, or nothing. As we derive systems from this, which are obviously something, the Tao is therefore something but we will always be uncertain as to what.
This is why I chose the unit ? to represent Tao. I think it is very appropriate.
For programmers, you will most easily interpret this as an abstract base class of geometry and mathematics.
Now that we all understand the underlying theory, we can now address what exactly I have been doing in the last two articles. Hopefully, this will stop people banging their heads off tables and exploding in rage to my complete disregard for millennia of research.
Firstly, I chose a random aspect of reality that was well defined. In this case, I chose Elementary Charge. The choice was arbitrary and based upon the fact that I could locate sufficiently well defined values and relationships. This allowed me to represent an aspect of both the Electron and Proton and derive these from Tao. We can express this in this picture, the red dot is the Charge of an Electron and the green one the Charge of the Proton:
In the previous two articles, this is the definition 'e'. Now Charge is something that must happen within the context of time. That is, Charge itself can be described as an event, it is something that has happened. At some point in history there was no Charge of an Electron/Proton, then Charge appeared with the Electron/Proton. Thus, at all times it must be partnered with a particle of Time. We can represent this in the following diagram:
In the last two articles, this is the definition of 'e/s'. As you can see, rather starting with a coordinates system (such as Cartesian Space), I decided to start with the particles, understand the relationship of the forces between them and allow this to reveal the structure of space. In that respect, the model is very close to a simulation of particle formation just after the big bang. Next we had to express the fact that particles of Time exists within Time, a form of self-interaction. We can represent this in the following diagram as the two yellow dots in the center:
Rather than defining new units, to represent the concept of Tao I just expressed particles of Time in terms of 'e/s'. At first, given the level of ambiguity I considered that Elementary Charge was Tao, however, after further examination it was revealed that this was only a manifestation of Tao.
I then proceeded to use the mathematical relationships defined in physics to define everything in terms of 'e/s'. In terms of Tao mathematics what I was doing was essentially "pinning" transforms in units of Elementary Charge. Another way to express this, is that I created a form of space and coordinate system that used the unit value of Elementary Charge. In essence, it was just a form of Cartesian space and the units were of Elementary Charge.
This was done to be able to use the formulas defined by current physics, not that it would not work in any other derived interpretation of Tao such as non-Euclidean geometry like hyperbolic and elliptic.
I then used standard formulas that defined relationships to the Ohm to define Mass(Kg), Second(s) Distance(m) in terms of 'e/s'. The hope was that now the values were in the same units that the formulas would not break down when terms were rearranged. This hope proved short lived.
It then occurred to me that perhaps, due to the different units involved, that perhaps there was a ratio. That is, so many particles of Time equated to some many particles of Mass. Performing a standard ratio equation (i.e. 2:1) proved fruitless, but I could see that I could use an 'offset' in conjunction with the ratio that would arrive at the correct value.
The first value I targeted was the relationship between Joules(e/s) and Mass(e/s). The results of which are in the comments section of the previous article which can be found here:
The math is correct, but I have been informed that the notes on the equations are wrong, but I will clear this up later in another article. This equation (rotation, transform) revealed that in the process of converting Mass to Energy some space-time was also used. At first, I thought this was a mathematical phantom until I considered Einstein's formula of E=MC^2. It was also stating the same thing.
I still need to resolve what this means in practice and understand why an error in converting the result of the following formula to Joules provided a correct result:
Distance (e/s) ^ 2 = Mass (e/s) x (Current (e/s) ^2)
Based on my current understanding, it is the result of contemporary mathematics being somewhat arbitrary. This is why I can also create new mathematical operators, they are just transforms and my use of existing mathematical operators will diminish with time as I replace the rigidity and static nature that they imply.
Finally, a small note on mathematical rules. When deriving a new system from Tao, there are none. You establish rules based upon the properties of a system and allow the system to define the rules. For example, try dimensional analysis with the version of Tao math I used in the last two articles, it will make no sense. That is because the properties of the system under investigation do not allow for it.
Well, I hope this article goes someway to enlightening people as to what I have been doing and why these crazy equations seem to be working on one hand, but so confusing in another way.