Author Archives: vinaire

I am originally from India. I am settled in United States since 1969. I love mathematics, philosophy and clarity in thinking.

KG MATH 3: Units & Fractions

A unit is what we count one at a time. When we count one cookie at a time then each cookie is a unit. When we count a ‘$10 note’ at a time then each ‘$10 note’ is a unit. When we count a ‘50¢ coin’ at a time then each ‘50¢ coin’ is a unit.  The word UNIT comes from a Latin word “unitus” which means “one.”

A fraction is a quantity that is part of a unit. Thus, a proper fraction is always less than a unit. If we are counting in units of a cookie then, a broken piece of a cookie is fraction of a cookie. If we are counting in units of ‘$10’ then, $5 is a fraction of ‘$10’. If we are counting in units of ‘50¢’ then, ‘10¢’ is fraction of ‘50¢’. The word FRACTION comes from a Latin word “fractere” which means, “a broken piece.”

Common fractions are ‘half’, ‘quarter’, etc. It must be stated that a fraction appears relative to the unit. Thus, if we are counting in units of ‘$10’ then, $5 is HALF of ‘$10’. Thus, we see large numbers being fractions of still larger numbers. Here are some exercises in this subject for the kindergarten level.

LEVEL K3: UNITS & FRACTIONS

Once again, a unit is what we count one at a time. Thus, if we are counting ‘half a cookie’ at a time then ‘half a cookie’ is the unit. This can appear very confusing unless we go strictly by the definitions of UNIT and FRACTION.

  • UNIT: what we count one at a time.
  • FRACTION: a part of a unit.

A unit can be arbitrary. A fraction is always relative to the unit. Fractions address ways to represent quantities, which cannot be represented by whole numbers.

We are always looking at a fraction of the universe.

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The Paradox of Unknowable

October 2, 2014
This issue is now obsolete. This was an early issue in this research and it is difficult to comprehend. For latest reference please see: Universe and Awareness and The Eighth Dynamic.
At the fundamental level, any curiosity shall represent a beginning of awareness. Any response to this curiosity would be more awareness as visualization.
Essentially, there is emergence of awareness and visualization. Prior to this there was simply non-awareness.  It is not known what mechanism brings this transition from non-awareness into awareness, but its later harmonic seems to be the mind.
Mind is a multi-dimensional matrix made up of definitions and logics. It outputs considerations based on input perception. ”Cause” and “effect” are considerations.
Prior to transition to awareness was non-awareness. Non-awareness can only be speculated upon. The barrier of non-awareness can be pushed back only by resolving inconsistencies one by one as they come up.

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One may say, “Cause considers (intends). The result is considerations.” That explains Cause only as an observation. It does not tell you how that observation came about. Thus, the idea of Cause itself is nothing more than a consideration. One may also say, “The resulting considerations are effect, and therefore not causative of anything.” This is an observation with some thought added to it. Thus, the idea of Effect also is just a consideration. What I have said here is consideration too by the same token.

We observe and decide what is there. Thus, what we see is our own assessment of what is there. This is one level of consideration.

One may say, “What about THAT which makes that observation? THAT, which cannot be named, is beyond that observation.” Well, is it? Now we are making an observation of ourselves. Believe me there is a considerable amount of observation here, such as, unmoved Mover, uncaused Cause, élan vital, God, soul, etc. They all fall under the same logic as above.

The “looking” beyond what we observe is just our thinking. This is another level of consideration. Whether we like it not, we are stuck with what we consider.

Is there experiencing of THAT, which is making the observations in the first place? Well, that is inherent in looking. When you are looking at any level, you are experiencing THAT. There is no mystery to that. So what is the issue?

The issue seems to be the desire to know. To know something, one needs to put it out there.

We do that very well by considering. Now we have completed the circle and back to where we started from. Can we step off this circle? Is there a dimension beyond this circle of considering and knowing? We may only hope and speculate.

I am sorry that the above “observation” seems to lead to an annoying sort of desperation. This is what seems to underlie the concept of UNKNOWABLE. I have been roundly criticized for it. Is there a way out? What follows as an answer to this question is just my speculation.

The UNKNOWABLE, which I have referred to so often, is so only from the viewpoint of the knowable universe. As long as one is attached to that viewpoint, the unknowable will remain outside one’s grasp.

Here is a possible prescription for overcoming this attachment.

  1. What makes the attachment persist is acceptance of inconsistencies.
  2. Removal of inconsistency paves the path for the removal of attachment.
  3. The removal of inconsistencies leads to the comprehension of the universe as a consistent whole.
  4. A complete review of that consistent whole may satisfy the desire to know.
  5. This may end this attachment to the viewpoint of the knowable universe.
  6. One may then enter the dimension of what appears to be “unknowable” at the moment.

I am sorry if this appears too abstract or mathematical. My current focus is on steps (1) and (2) above, with no further expectations about whether the rest would follow or not.

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KG MATH 2: Numbers & Place Values

A number answers the question, “How many?” Numbers are used in counting to find out how many things there are. One counts by sequentially calling out for each item, one, two, three, four, five, and so on.

Counting starts from ONE and not from zero. Zero is not used in counting because zero stands for “nothing.” Zero is useful in marking the absence of a count. Therefore, zero is used in writing numbers where an absence of count is implied.

Numbers are written by combining symbols called digits, much like words are written by combining symbols called letters. We have twenty-six letters that combine to make thousands of different words. We have only ten digits that combine to make infinity of different numbers.

There are ten different digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The number FIVE is written with only one digit 5. The number FIFTEEN is written with two digits 1 and 5 as 15. All possible numbers can be written just with these ten digits.

The numbers are written in a compact form by using the trick of regrouping. We regroup ten pennies as one dime. Similarly, we regroup ten ONES as one TEN; ten TENS as one HUNDRED; ten HUNDREDS as one THOUSAND; and so on.

The number 15 represents “1 TEN and 5 UNITS.” The number 264 represents “2 HUNDREDS, 6 TENS, and 4 UNITS.” The values of UNIT, TEN, HUNDRED, THOUSAND, etc., are called place values, because they are applied to a digit depending on where it appears in a number. The best way to understand place values is by means of a counting board called Abacus, which is described in the document referenced below.

Ten is very important in our numbering system as can be seen in this discussion. It was chosen because we have ten fingers that were used for counting in the beginning. Here are some exercises in this subject for the kindergarten level.

LEVEL K2: NUMBERS & PLACE VALUES

“Numbers and Place Values” forms the foundation of the subject of ARITHMETIC. Arithmetic literally means, “Skill with numbers.” Arithmetic helps one find the answers to problems involving numbers

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KG MATH 1: Orientation & Spatial Sense

Spatial sense is having the sense of direction, distance and location with respect to one’s environment. Orientation is getting adjusted to that environment.

To get oriented one needs to spot the relative locations of various things in one’s environment. To spot a location one only needs to know the direction it is in and its distance from one.

A direction is the line along which attention may be directed. There is infinity of directions radiating out from one’s location. The main directions are FRONT, BACK, ABOVE, BELOW, LEFT and RIGHT. The directions of LEFT and RIGHT are difficult for a child to recognize until he or she reaches Kindergarten.

A distance is the separation between two locations. There is infinity of different distances in any one direction. The distances may be identified roughly as NEAR or FAR.

A position of an object tells us how it is located in relation to other objects, such as, IN, OUT, ON, UNDER, MIDDLE, and NEXT TO. A child may learn these positions as part of learning the language, but may also be taught as part of mathematics.

Spatial locations combine into shape. A very common shape is rectangle that most doors and windows have. Other shapes are triangles, circles, etc. These shapes may be drawn on a plane surface. They are two dimensional because they have length and breadth, or width and height.

Most objects are three dimensional because they have a third dimension of thickness or depth. Examples of simple objects are cubes, spheres, cylinders, cones, etc.

Shapes and objects are symmetrical when one half is the mirror image of the other half. Most bodies are symmetrical. Objects may slide, flip or spin in space.

Our body is an object that exists in space. Therefore, orientation and spatial sense is an important subject for a child to become familiar with. Here are some exercises in this subject for the kindergarten level.

LEVEL K1: ORIENTATION & SPATIAL SENSE

“Orientation & Spatial Sense” forms the foundation of the subject of GEOMETRY. It introduces the elements of space and how these elements may relate to the student.

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Is there Divinity?

Ganesha

Reference: Religion

Socrates almost had his finger on it when he posed the question, “Can man be made self-determined and responsible for his own actions?”

Plato lost it when he recommended the use of religion (supernatural authority and fear) to control the wild beast nature latent in every person.

Aristotle came close to defining it, but the logic that brought him so close to an understanding of divinity, also prevented him from defining it precisely. Let us take a look at that one final step that he could not take.

Aristotle follows Socrates’ lead to examine such common terms as, justice, morality, virtue, etc., to uncover the unknowing assumptions made by people. He applies Plato’s Doctrine of Ideas to voluminous observations to define the concepts, laws, and principles that underlie all that we sense. He coins many new terms, such as, faculty, motive, energy, actuality, maxim, principle, etc., to communicate those concepts precisely. He formulates a scientific method so others may continue with this process.

Aristotle digs deep into observations, especially in the field of biology and natural sciences, and comes up with general frame of references (universals) from which to evaluate further observations. Thus he simplifies the management of voluminous observations by uncovering categories with logical connections.

He, then, digs deep into these categories to come up with a more fundamental frame of reference. He reduces all observations to (a) FORM (the shaping force), and (b) MATTER (the raw material being shaped).

To Aristotle, FORM is the inner necessity or impulse which exists in MATTER. MATTER is continually being formed into new, complex shapes by FORM that is inherent to it.

Aristotle considers MATTER to be without beginning. MATTER is worked into more complex and varied shapes by FORM. To him, God is “Prime Mover Unmoved.” God is the source of all motion. But, God has no motion within itself.

Aristotle never answers the question how MATTER arose in the first place. To him, this is like asking the question, “How God came to be in the first place?” And, that is as far as Aristotle goes. The inherent consideration here seems to be that ability, or potential, needs a “vessel” through which to express itself.

We find most viewpoints in the “Western thought” to be based on this frame of reference. It leads to the viewpoint that God must have a beingness in which to exist.

Can there be God without beingness? Can there be FORM without MATTER? Can there be Motion with no motion at its core? Can there be a Cause that is not itself caused?

 

DIVINITY

When we observe this universe, we cannot separate GOD from BEINGNESS, FORM from MATTER, MOTION from NO MOTION, and CAUSE from EFFECT.

These pairs, or dichotomies, appear simultaneously when a manifestation is perceived. Even the most fundamental ideas of MANIFESTATION and PERCEPTION seem to form a dichotomy. We all have struggled with the questions, “How does a manifestation appear?” “How is it perceived?” “Who or what creates?” “Who or what perceives?” The ultimate focus has been on “how,” “who” or “what.” It all boils down to the speculation that somebody or something must exist beyond all existence.

Essentially, the mind and its logic has hit a ceiling. Any attempt to pierce this ceiling runs into a fundamental  inconsistency, such as, “unmoved Mover” or ”uncaused Cause.”  This inconsistency seems to point to something that cannot even be conceived.

It would be beyond any mental conception. It would be beyond logic. It would be beyond any description. It would not be a form or cause. It would not even exist or be.

It would seem that

  1. A manifestation may occur without any prior consideration.
  2. A perception may occur without any prior consideration.

And in there, somewhere, may be Divinity, or may be not…

The Creation Hymn of Rig Veda

Neti, neti,”

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