What do all things have in common?

Posts tagged “Metamathematics

Lateral Numbers – How ‘Imaginary Numbers’ May Be Understood


First, allow me to rename theses numbers during the remainder of this post to lateral numbers, in accordance to the naming convention as was recommended by Gauss. I have a special reason for using this naming convention. It will later become apparent why I’ve done this.

If we examine lateral numbers algebraically, a pattern emerges:

i^0 = 1

i^1 = i

i^2 = -1

i^3 = -i

i^4 = (i^2)^2 = (-1)^2 = 1

i^5 = i \cdot i^4 = i

i^6 = i^2 \cdot i^4 = (-1)(1) = -1

i^7 = i^2 \cdot i^5 = (-1)i = -i

i^8 = i^4 \cdot i^4 = (1)(1) = 1

When we raise lateral numbers to higher powers, the answers do not get higher and higher in value like other numbers do. Instead, a pattern emerges after every 4th multiplication. This pattern never ceases.

All other numbers, besides laterals, have a place on what currently is called the ‘Real number line’.

I qualify the naming of the Real Numbers, because even their conceptualisation has come into question by some very incisive modern mathematicians. That is a very ‘volatile’ subject for conventional mathematicians and would take us off on a different tangent, so I’ll leave that idea for a different post.

If we look for laterals on any conventional Real number line, we will never ‘locate’ them. They are found there, but we need to look at numbers differently in order to ‘see’ them.

Lateral numbers solve one problem in particular: to find a number, which when multiplied by itself, yields another negative number.
Lateral numbers unify the number line with the algebraic pattern shown above.

ComplexNumbers Example 001

2 is positive and, when multiplied by itself, yields a positive number. It maintains direction on the number line.

ComplexNumbers Example 002 - Negative

When one of the numbers (leaving squaring briefly) being multiplied is negative, the multiplication yields a negative number. The direction ‘flips’ 180° into the opposite direction.

ComplexNumbers Example 003 - Negative Squaring

Multiplying -2 by -2 brings us back to the positive direction, because of the change resulting in multiplying by a negative number, which always flips our direction on the number line.

So, it appears as if there’s no way of landing on a negative number, right? We need a number that only rotates 90°, instead of the 180° when using negative numbers. This is where lateral numbers come into play.

ComplexNumbers Example 004 - Negative Squaring using lateral numbers01

If we place another lateral axis perpendicular to our ‘Real’ number line, we obtain the desired fit of geometry with our algebra.

When we multiply our ‘Real’ number 1 by i, we get i algebraically, which geometrically corresponds to a 90° rotation from 1 to i.

Now, multiplying by i again results in i squared, which is -1. This additional 90° rotation equals the customary 180° rotation when multiplying by -1 (above).

ComplexNumbers Example 004 - Negative Squaring using lateral numbers

We may even look at this point as if we were viewing it down a perpendicular axis of the origin itself (moving in towards the origin from our vantage point, through the origin, and then out the back of our screen).

[If we allow this interpretation, we can identify the ‘spin’ of a point around the axis of its own origin! The amount of spin is determined by how much the point moves laterally in terms of i.
We may even determine in which direction the rotation is made. I’ll add how this is done to this post soon.]

Each time we increase our rotation by multiplying by a factor of i, we increase our rotation another 90°, as seen here:

ComplexNumbers Example 004 - Negative Squaring using lateral numbers03


ComplexNumbers Example 004 - Negative Squaring using lateral numbers04

The cycle repeats itself on every 4th power of i.

We could even add additional lateral numbers to any arbitrary point. This is what I do in my knowledge representations of holons. For example a point at say 5 may be expressed as any number of laterals i, j, k,… simply by adding or subtracting some amount of i, j, k,…:

5 + i + j +k +…

Or better as:

[5, i, j, k,…]

Seeing numbers in this fashion makes a point n-dimensional.

HUD Fly-by Test


Link to video.

Don’t take this as an actual knowledge representation; rather, simply a simulation of one. I’m working out the colour, transparent/translucent, camera movements, and other technical issues.
In any case you may find it interesting.
The real representations are coming soon.

Universal Constants, Variations and Identities – #16 (Creation/Discovery)

Constants and Variances 1MathCreatedOrDiscovered

Universal Constants, Variations and Identities
#16 Creation and discovery compliment each other and are the means in which the Universe fundamentally unfolds and enfolds itself (Creation/Discovery)

We tend not to identify them, because there are so many variations in their harmony. Please do overestimate your thoughts… as you will see they are the beginning of your expression to and of the world.

Both Creation and Discovery will work in unison, if we allow them.
Discovery is to recognize/relate what is in your world.
Creation is to transform/synthesize it too.
Each is alone without the other.

Creation=Right ‘brain’ (right+mind)
Discovery=Left ‘brain’ (left+mind)

Their ‘magick’ (sic.) manifests not when you synchronize them; rather, when you harmonize them.

(Please take the time to watch the 4 minute video.)

Universal Constants, Variations and Identities – #15 (Change/Time)

Constants and Variances 1clock clock

#15 Time is a temporally ‘linear’ (directed) form of change that is not limited by dimension. (Change/Time)

Time has been arbitrarily and wrongly assigned to dimension. Change is not restricted to any dimension: therefore time is also not limited to it.

I know it’s trendy to see time as a dimension, but dimension is something completely different. Stay tuned to find out what and why.

Update: There are many reasons why time needs a proper definition. Here are a few of them:

The chemical reactions in the vessel are not really effected by some mysterious thing called time, but by the number of contacts or collisions that take place in the soup of atoms or molecules. That is what the factor ‘T’ really stands for.MathematicalTime

1) Eternity may be a somewhat mystical overarching reality outside of the physical universe, but time is not. Nor is time a thing that anybody can do anything to. In other words: it cannot be reified.

2) The universe doesn’t exist in time, but time exists in the universe.

3) The proper definition of time is exactly:  the sequence of events in the material universe.

Universal Constants, Variations and Identities #13 (Knowledge)

Knowlege Is What Awareness Does

Universal Constants, Variations and Identities
#13 Knowledge is what awareness does. (Knowledge)

I’ve published this before elsewhere, but it must be restated now for what is to follow (I’m starting a new octave).

#Knowledge #Wisdom #Understanding #Learning #Insight #Constants #Variances #Metaphysics #Philosophy #MathesisUniversalis #ScientiaUniversalis #PhilosophiaUniversalis #LogicaUniversalis #MetaMathematics #MetaLogic #MetaScience #MetaPhysics #MetaPhilosophy #Awareness

Universal Constants, Variations and Identities

 Constants and Variances 1  Constants and Variances 2
 Constants and Variances 5 - We are the ones we have been waiting for!

Universal Constants, Variations and Identities
#12 The ends are determined by their means. (Integrity of purpose)

(Dispelling the lie of: “The ends justify their means.” The truth is that ends are inextricably tied to the means used to achieve them. If we employ destructive means, then only destructive ends can result. Intention means nothing with regard to means… those who believe that learn: “The road to Hell is paved with good intentions.”

See http://mathesis-universalis.com for more information.

Universal Constants, Variations and Identities

Constants and Variances 1Constants and Variances 2

Universal Constants and Variances
#11 Observation is communication between its participants (observer and observed). (Reference)

See http://mathesis-universalis.com for more information.