Tractate 8 : The Error of Einstein (continued)

Real Numbers:
Let’s begin our examination of abstraction by examining the concept of Real numbers. The examination of Real numbers is best begun through the initiation of a discussion of Counting numbers.

The Counting numbers begin with what is called a ‘unit’ number, a number of which all the rest are composed. For example the number two is simply one more than the number one. The number three is simply one more two ones

This may not be of interest to most people, but to a theoretical metaphysician it is a truly exciting concept for it, in its simplicity, implies individuality, the individual is no ‘little’, useless, concept to be discarded for the ‘greater’ good, the ‘greater’ idea. In fact the concept of individuality is the bases of all counting numbers, including infinity itself.

Counting numbers lead to the concept of Whole numbers:

{Whole Numbers} = {0, 1, 2, 3, ….} The set of Whole Numbersis the set of numbers zero, one, two, three, etc. into infinity.

A new set evolves, or so it appears.

Now why would the phrase be added: ‘Or so it appears.’

For one thing, nothing was added to the set of counting numbers. Let me say that again, nothing, literally nothing, was added to the set of Counting Numbers

However, this appears to be the same set as the set of Counting Numbers. That’s true. So somehow, we have to distinguish the difference between the two sets. Somehow, we have to ‘represent’ nothingness being added to the set so that we can understand that ‘nothingness’ was not a part of the first set, the set of Counting Numbers but is part of the second set, the set of Whole Numbers.

So to distinguish that nothing is the difference between the two sets let’s use the symbol: &Mac198;. This will keep our drawings relatively simple.

Now we have:

Actually you could have:

Since ‘nothing has been added.

On the number line however, zero, nothing, shows up once and only once.

What are the marks on the left of zero, on the left of nothing? We will come back to that very soon. In the mean time let’s reconfigure the two dimensional diagram to better represent the one dimensional number line.

To reconfigure the two dimensional diagram, we need to develop a three dimensional tunnel within which we in essence could walk and observe the numbers which we pass as we walk through this tunnel.

To avoid getting too complicated too soon let’s go back to the set of counting numbers and construct the tunnel and then turn around and walk in the opposite direction through the tunnel.

The Tunnel of abstraction:
Counting Numbers = {1, 2, 3, …}

Tunnel of Counting Numbers:

Hmmmm, no the tunnel is going the wrong direction. We must revise this perception for we are individuals, we are the ‘one’ concept. We, each of us, is ‘an’ individual. Reversing the tunnel is not a difficult task.

By reversing the concept of the tunnel of numbers we obtain:

Counting Numbers = {1, 2, 3, …}

Tunnel of Counting Numbers:

Now we can proceed to the set of Whole Numbers. As we proceed to the tunnel of Whole Numbers, something very interesting develops.

Whole Numbers = {0, 1, 2, 3, …}

Tunnel of Whole Numbers:

Now ‘one’ becomes the entity of individuality, as such we must make a further revision to our tunnel of numbers. As we do so, we obtain:

Whole Numbers = {0, 1, 2, 3, …}

Tunnel of Whole Numbers:

Where is zero? Where is nothingness? Nothingness does not exist in the realm of the physical. We, however, are no longer in the realm of the concrete. We are in the realm of pure abstraction. In the abstract realm, zero, nothingness does exist. As such, because we are in the realm of the abstract, we have the ability to understand that the idea of nothingness exists. We are able to understand for we ourselves are now located within the purity of abstraction. As such, we can now place zero within our graphic.

In actuality, we have already placed zero, nothingness within our drawing, we just haven’t labeled nothingness. To correct this oversight, let’s redraw the graphic and include the label ‘nothingness’.

Whole Numbers = {0, 1, 2, 3 …}

Tunnel of Whole Numbers:

Within this drawing zero appears to take up no ‘space’ for zero is simply a circle represented by the drawing of a circle and everyone knows lines have only one dimension. Lines do not have depth or width. This perception is correct. Even in abstraction zero, nothing, is nothing and this drawing implies zero is nothing, only a boundary where one begins, where individuality begins, and moves forward with the concept of multiple individuality.

Expanding upon the concept of numbers and adding the concept of parts, pieces of the whole or what we call positive rational numbers such as 1/2 , 1 _, 5 _,…, we obtain:

Positive Rational Numbers = {#’s > 0 which can be expressed as a/b where a and b are Whole numbers and where b &Mac185; 0}

Tunnel of Positive Rational Numbers:

At this point we are going to ignore the concepts of infinite ‘largeness’ and infinite ‘smallness’. Instead we are simply going to consider the concept of size of infiniteness as being simply forms of infinity whose concept of infiniteness alone is what it is we wish to consider. By doing so we eliminate the complexity of +1/• as compared to +•/1.

This graphic leads to the termination regarding ever-expanding concepts of numbers unless we do something regarding nothing. Is doing something about nothing a paradox? Doing something about nothing is only a paradox if we continue our past actions of thinking that nothing has no functionality.

To eliminate the paradox of nothingness having no functionality while remaining what it is, nothing, we must revisit the concept of nothingness and continually remind ourselves that, within the realm of abstraction, all abstractions including nothingness have a function. The continual need to revisit the functionality of nothingness, leads one towards gaining an intuitive sense that nothingness is an important concept of abstraction.

Reintroduction of nothingness into the graphic gives us:

Positive Rational Numbers = {# > 0 which can be expressed as a/b where a and b are Whole numbers and where b &Mac185; 0}

Tunnel of Positive Rational numbers now becomes:

Now what?
Now we move into the abstract realm of the negative. The question becomes: Is this process of diagramming leading us anywhere? The diagrams are leading us to individuality, ‘being’. The diagrams are leading us to an understanding regarding the concept of nothingness itself. Lastly, the graphics may lead us to the very concept of the concrete, being – action, process/reality.