Intuition or Gibberish?

The physical universe

And the spiritual universe

Are subsets of the Universe of Consideration.

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The Universe of Consideration seems to come about

Because of the effort of the Unknowable

To know itself.

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It is Looking that seems to be creating space.

It is Looking that seems to be accelerating the expansion of the universe.

It is Looking that is collapsing the wave function.

Could this looking be that dreaded Unknowable?

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Space fills itself with space.

Space is its own fabric.

Space undulates…

And that is light.

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Space undulating is “space + time”

And that is energy (light is one expression of that energy)

Presence of time indicates presence of additives.

An additive is the basic inconsistency.

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The true nature of a number is “irrational.”

Could the rational number be a “collapsed irrational number.”

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The true nature of matter and energy also seems to be “irrational.”

Could the discreteness of matter and energy be “collapsed matter and energy.”

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Could the discreteness of consideration be a “collapsed consideration.”

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Are there some connections…

Or, is it all just insane gibberish?

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MILESTONE 4: Multiplication

Multiplication is “repeated addition.”

To explore further logic that follows from the idea of multiplication as “repeated addition,” go to the link

MATH MILESTONES #A4, MULTIPLICATION

Many a time the problem with multiplication gets resolved by getting the student to count on the ten fingers “by two,” “by three,” “by four,” etc. When the student is counting on his fingers, say, “by three,” each finger has a value of three. As he counts “by three,” he adds three at each count. He may then write down the count in a column. This will then be the multiplication table for three.

By repeated addition, one may easily produce multiplication tables, and then use them to solve problems.

The proper approach is to create the multiplication table, as above, many times using “repeated addition.” Knowing the techniques of addition from MS 02: ADDITION can be of great help. You are building the skill in multiplication as an extension of the skill in addition. This approach is better than simple memorization.

The real shortcuts in math come from thinking with the basics, such as,

  1. A number multiplied  by zero  = the number “added repeatedly” zero times = zero

  2. A number multiplied by one =  the number “added repeatedly” once = the number

  3. A number multiplied by 10 = the digits shift one place value to the left = the number with a zero attached (e.g., 3 x 10 = 30;  12 x 10 = 120).  

Check out the above with repeated addition on abacus per MILESTONE 1: Numbers & Place Values]

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The following is the most amazing property of repeated addition.

4 x 17     = 4 added repeatedly 17 times

= 4 added repeatedly 10 times + 4 added repeatedly 7 times

This may be written as,

4 x 17  =  4 x (10 + 7)  = (4 x 10) + (4 x 7)  

One may now compute this mentally as 40 +28 = 68. Here, the key property is

4 x (10 + 7)  = (4 x 10) + (4 x 7)

This ia also written in its general form as

a x (b + c)  = (a x b) + (a x c)

And we recognize it as the Distributive Property.

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Since one operation of multiplication contains many opearation of addition, multiplication takes precedence over addition in mixed operations.  For example,

3  +  2  x  4  =  11  (not 20)

Here are some videos from the Khan Academy that explain multiplication:

Basic Multiplication

Multiplication 2: The Multiplication Tables

Multiplication 3: 10,11,12 times tables

Multiplication 4: 2-digit times 1-digit number

Multiplication 5: 2-digit times a 2-digit number

Multiplication 6: Multiple Digit Numbers

Multiplication 7: Old video giving more examples

Mulitplication 8: Multiplying decimals (Old video)

Lattice Multiplication

Why Lattice Multiplication Works

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Going Beyond Self

The well-known sequence of “Body – Mind – Spirit” may be expanded as follows:

  • Body (or Sensory input)
  • Perception
  • Experience
  • Information
  • Hypothesis
  • Theory
  • Principles
  • Axioms
  • Spirit (or Self)

All these stages between Body and Spirit comprise the mind. The function of the mind is to assess and determine what is there. The sensory input is assessed as perception. Perception is then assessed as experience. Experience is then assessed as Information. And so on down the line. We may look at these stages as the “Scale of Mind.”

At the top of this scale we have the Sensory Input, which may be characterized as material from some unknown primordial state, which is basic to all existence. At the bottom we have identifiable existence, whether animate or inanimate, the basic unit of which may be regarded as “self.”

One may say that sensory input from the primordial state precipitates through these stages recursively, thus bringing about self, and an evolution of self. “Self” may be looked upon as a UNIT of existence, whether it is a thought or an atom. To understand the make-up of “self”, we may borrow some ideas from mathematics.

  1. A UNIT is what we count one at a time. “Self” has the characteristic of individuality that renders it to be counted one at a time.

  2. A rational number posits a unit that divides into both numerator and the denominator. Such a unit has a finite magnitude, no matter how large or small. Similarly, we may consider “self” to be a finite individuality, regardless of its characteristics.

  3. An irrational number reveals that a “unit” is essentially an arbitrarily constructed concept. This discovery created quite an upset in the mathematical world. Please see Going Beyond Counting. If we look at the parallel here, maybe “self,” though useful, is essentially an arbitrarily constructed concept.

Prior to the emergence of self (whether as thought, or as atom, or whatever lies at the core), there is simply a primordial state that cannot be described. It cannot be called chaotic for there is no reference point to assess chaos. That state is its own reference point.

The primordial state creates its own perception, experience, information, hypothesis, thesis, principles, axioms etc. to finally come up with a fabric of self on a totally arbitrary basis. That fabric then seems to create its own perception, experience, information, etc. to evolve into self. This recursive process seems to evolve a self out of “no self,” much like an irrational number evolves out of “no unit.”

Another parallel may be drawn from the formation of a particle (photon) out of a “no particle” (electromagnetic wave). The electromagnetic wave seems to be a phenomenon of space. Space may be looked upon as empty, or as nothing; but space is an observable phenomenon on its own right.

We all perceive and experience the same physical universe. Is it because we agree with each other consciously or unconsciously? Or, is it because we share the same sensory input as the fabric of our very makeup? Does thought and atom have the same basis?

From sensory input to self there seems to be a progression from impressions to visualization.  This may be looked upon as the gradual introduction of self-determination, or free will. Again it is a recursive phenomenon and it cannot be said if thought came first, or the atom, in the equation of “self.”

Cognitive functions may be an outcome of this developing ability to visualize after much such iteration. Memory may be a combination of impressions and visualizations. The consistency between impressions and visualizations may manifest as honesty and integrity.

Are we really separate from the physical universe? Or, is the physical universe simply an extension of us? Let’s leave that to future contemplation (looking) for now.

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MILESTONE 3: Subtraction

The idea of subtraction as “reverse addition” is explored in greater detail in this document.

MATH MILESTONE #A3: SUBTRACTION

Subtraction is taking a quantity away from another quantity. For example, suppose you have 15 marbles; and you give 6 marbles to your friend. You can find the remaining marbles in two different ways:

  1. Take 6 away from 15:  start from 15 and count back 6 to see the number you end up with.
  2. Find 6 and “what” is 15: start from 6 and see how many counts it takes to arrive at 15.

The method in (a) requires a new skill of counting backwards. The method in (b) utilizes the already learned skill of addition. Both these methods work because

Subtraction is the opposite of addition.

The better one can do addition, the more skillful one gets at subtraction. In subtraction, one also learns the concept of regrouping the place values. The following videos from Khan Academy demonstrate subtraction.

Basic Subtraction

Subtraction 2

Subtraction 3: Introduction to Borrowing or Regrouping

Alternate mental subtraction method

Level 4 Subtraction

Why borrowing works

When we take away all the units, which are there, we are left with nothing. Thus, subtraction provides us with the idea of “nothing” as a “number.” We call it zero.

When we need to take away more than what is there, we are faced with a definite shortage. Thus, subtraction also provides us with a measure of “shortage.” We call it a negative number.

Calculators may be used to save time with subtraction; but still mental awareness is necessary to know if the calculated answer is correct. This awareness comes from the practice with mental math as outlined in the document above.

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MILESTONE 2: Addition

One may easily learn to add numbers mentally up to one hundred using the procedures given here.

MATH MILESTONE #A2: ADDITION

Addition is an extension of counting. For example, by counting you know that you have 3 marbles. By counting again you know that you got 4 more marbles from your friend. You put all these marbles in a box and count them together. You find that you now have 7 marbles.

Addition is counting together.

In the initial stages one learns to add small numbers by counting on one’s fingers. For larger numbers, one may use abacus. The first concept that one learns here is “carry-over” across the place values. After gaining sufficient familiarity with objects, one may add on paper using digits.

It is easy to visualize addition with objects. We may also visualize addition using a number line. Basic addition is demonstrated in the following video from Khan Academy.

Basic Addition

When we have larger numbers we may add them more easily by column. See the demonstration in the following videos from Khan Academy

Addition 2

Level 2 Addition

Addition 3

Addition 4

Here are some comments on addition.

(1) Addition depends on the concept of the same units. For example, we may add 2 cats and 3 cats to come up with the sum of 5 cats. However, we cannot express the sum of 2 cats and 3 dogs in terms of a single unit, unless we change the unit to “animal”, which is inclusive of both cats and dogs.

(2) Mathematical units may be the same, but units in the real world are never exactly the same. For example, any two oranges would never be exactly be the same in all aspects including the number of atoms they contain.

Thus, mathematics may come very close to describing the physical universe, but it is never exact.

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