Author Archives: vinaire

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

The Basics of Math

November 19, 2011 – 10:33 AM
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Mathematics presents “logical tools” for learning.

Arithmetic forms the first part of mathematics that presents the “number skill”. Arithmetic starts with counting.

Counting is a tool for learning how many things are there. Counting starts with one. The next count is one more.

A unit is the thing being counted one at a time. If one is counting houses, then each house is a unit. If one is counting inches of a length, then each inch is a unit.

The digits are the ten symbols – 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 – that are used to write numbers.

Numbers provide a system to represent the counts. A number is made up of one or more digits, just like words are made up of one or more letters.

Addition is counting together of numbers. Subtraction is opposite of addition.

Multiplication is repeated addition of a number. Division is opposite of multiplication.

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THE FIELD AXIOMS

Real numbers are undefined objects that satisfy certain properties.

AXIOM #1: CLOSURE PROPERTY OF ADDITION

If x and y are real numbers, then x+y is a unique real number.

Addition is an operation such that for every pair of real numbers x and y we can form the sum of x and y, which is another real number denoted by x+y. The sum x+y is uniquely determined by x and y.

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AXIOM #2: CLOSURE PROPERTY OF MULTIPLICATION

If x and y are real numbers, then xy is a unique real number.

Multiplication is an operation such that for every pair of real numbers x and y we can form the product of x and y, which is another real number denoted by xy or by x.y. The product xy is uniquely determined by x and y.

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AXIOM # 3: COMMUTATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x and y are real numbers, then x+y = y+x.

If x and y are real numbers, then xy = yx.

ADDITION: Two numbers may be added in any order. For example,

5 + 8       =       8 + 5

One may visualize the numbers as items of a one-dimensional array. For example,

$ $ $ $ $ $ $ $ $ $ $ $ $

One may count the items in this array as “5 first and 8 next”; or “8 first and 5 next”. The result is the same.

$ $ $ $ $      $ $ $ $ $ $ $ $

$ $ $ $ $ $ $ $      $ $ $ $ $

“Subtraction” is accounted by this law by treating the number being added as a negative integer. The sign moves with the following number. The first unsigned number is treated as having a positive sign. For example,

[8 – 5]     is           +8 –5    =     –5 +8

MULTIPLICATION: Two numbers may be multiplied in any order. For example,

5 x 8       =       8 x 5

One may visualize the numbers as items of a two-dimensional array. For example,

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

$  $  $  $  $  $  $  $

One may count the items in this array, as “5 rows of 8 each”, or “8 columns of 5 each”. The result is the same.

8+8+8+8+8     =     5+5+5+5+5+5+5+5

“Division” is accounted by this law by using the reciprocal (multiplicative inverse) of the divisor as the multiplicand.

[8 ÷ 2]     is           8 x ½    =     ½ x 8

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AXIOM # 4: ASSOCIATIVE PROPERTIES OF ADDITION & MULTIPLICATION

If x, y and z are real numbers, then (x+y)+z = x+(y+z).

If x, y and z are real numbers, then (xy)z = x(yz).

In addition, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   +  (5 + 8)      =        (3 + 5)   +  8

Similarly, in multiplication, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

3   x   (5 x 8)      =        (3 x 5)   x   8

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AXIOM 5: DISTRIBUTIVE PROPERTY

If x, y and z are real numbers, then x(y+z) = xy + xz.

Multiplication distributes over addition. For example, a factor may be multiplied by the other factor as a sum of two parts with the same outcome as follows.

5 x 17   =   5 x (10 + 7)   =   5 x 10 + 5 x 7   =   50 + 35   =   85

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AXIOM 6: IDENTITY ELEMENTS

The identity element for addition is 0, i.e., for any real number x, 

x+0 = x.

The identity element for multiplication is 1, i.e., for any real number x, 

x.1 or 1x = x.

There exist two real numbers, which we denote by 0 and 1, such that for every real x we have

0 + x  =  x + 0  =  x                    (the idea of adding nothing)

1 . x  =  x . 1  =  x                      (the idea of a single occurrence)

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AXIOM 7: INVERSES

A unique ADDITIVE INVERSE exists for every real number, i.e., for every x the additive inverse is -x such that

x     +     (-x)         =           0.

A unique MULTIPLICATIVE INVERSE exists for every real number, i.e., for every non-zero x the multiplicative inverse is 1/x such that

x      .       (1/x)      =           1.

For every real number x there is a real number y such that

x + y   =   y + x   =   0       (the idea of negating something into nothing)

For every real number x (except 0) there is a real number y such that

x . y   =   y . x   =   1         (the idea of reducing something to its unit)

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Note 1: 0 is an exception because it denotes nothing, whereas all other numbers denote something.

Note 2: 1 is unique because it denotes a unit, whereas all other non-zero numbers denote multiple units.

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Posted in Mathematics | Comments (0)

The Factors of Unknowable

November 13, 2011 – 11:29 AM

This is an effort to go as deep as possible into the Unknowable using the Vedic process “neti, neti.”

The most fundamental starting factor:

(1) There is observation (because this is an observation being registered).

Phenomena:

(2) There are things to be observed.

(3) These things are being manifested and unmanifested.

(4) A manifestation begins, continues, and then ends.

Manifestations:

(5) The universe is a manifestation that is continuing.

(6) There are sub-manifestations to this universe that begin, continue, end.

(7)  Manifestations may come and go, but the manifesting continues.

Existence and Time:

(8) It is the property of manifesting, which is known as existence.

(9) It means that something or other continues to be manifested.

(10) Thus, existence provides the idea of Time.

Form:

(11) A manifestation can be observed only because of its form.

(12) The form of a manifestation can be physical, such as, rocks, trees, chair, molecules, atoms, light and electromagnetic waves.

(13) The form of a manifestation can also be mental, such as, visualizations, thoughts, evaluations, conclusions, considerations and speculations.

Dimensions:

(14) The form of a manifestation has finite extents.

(15) It is the property of extents, which is perceived as dimensions.

(16) It means that something or other continues to be extended.

Space:

(17) These dimensions provide the idea of space.

(18) Physical dimensions provide the idea of physical space.

(19) Mental dimensions provide the idea of mental space.

Motion:

(20) As these extents reach out and withdraw there is change.

(21) It is the property of change, which is perceived as motion.

(22) It means that something or other continues to move.

Energy:

(23) This motion provides the idea of energy.

(24) Physical motion provides the idea of physical energy.

(25) Mental motion provides the idea of mental energy.

Patterns:

(26) Motion of extents may follow certain fixed patterns.

(27) It is the property of fixidity, which is perceived as patterns.

(28) It means that some pattern or other continues to be exhibited.

Matter:

(29) These patterns provides the idea of matter.

(30) Physical patterns provides the idea of physical matter.

(31) Mental patterns provides the idea of mental matter.

The Universe of Manifestation:

(32) The Universe of manifestation is made up of time, space, energy, and matter.

(33) Research into manifestations is continuing through physical and mental sciences.

Awareness:

(34) There is observation, so there is also the ability to observe.

(35) It is the ability to observe, which is recognized as awareness.

(36) Awareness continues as long as there is something to be aware of.

Perception:

(37) Thus, there is the ability to assess what is there.

(38) It is the ability to assess, which is recognized as perception.

(39) Perception continues as long as there is something to perceive.

Consideration:

(40) Thus, there is the ability to visualize what is assessed to be there.

(41) It is the ability to visualize, which is recognized as consideration.

(42) Consideration continues as long as there is something to consider.

The Universe of Consideration:

(43) The Universe of Consideration is made up of the awareness, perception and consideration of existence, form, motion and pattern.

(44) Research into manifestations and underlying considerations is continuing, primarily, by examining their consistency and inconsistency through mathematics.

Life and Beingness:

(45) It is the presence of abilities, which is perceived as life.

(46) These abilities are manifested  just like other things are manifested.

(47) It is the manifesting of abilities that we may call Beingness.

The universe:

(48) Thus, there are manifestations in terms of existence and beingness.

(49) Thus, there is a universe, which is aware of itself on the whole.

(50) There are considerations of existing and not existing.

(51) There are considerations of static and kinetic.

(52) There are considerations of cause and effect.

(53) What is beyond these considerations may only be speculated upon.

(54) The research into Unknowable thus continues.

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NOTES: Alfred North Whitehead said: “All systematic thought must start from  presuppositions.” The factors above start from what is observable. These may be considered to be presuppositions.

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Posted in Postulates | Comments (18)

Quantum Mechanics Beginnings

November 11, 2011 – 7:46 AM

The Nobel Prize in Physics 1932

Werner Heisenberg

Award Ceremony Speech

Your Majesty, Your Royal Highnesses, Ladies  and Gentlemen.

This year’s Nobel Prizes for Physics are dedicated to the new  atomic physics. The prizes, which the Academy of Sciences has at  its disposal, have namely been awarded to those men, Heisenberg,  Schrödinger, and Dirac, who have created and developed the  basic ideas of modern atomic physics.

It was Planck who, in 1900,  first expressed the thought that light had atomic properties, and  the theory put forward by Planck was later more exhaustively  developed by Einstein. The  conviction, arrived at by different paths, was that matter could  not create or absorb light, other than in quantities of energy  which represented the multiple of a specific unit of energy. This  unit of energy received the name of light quantum or photon. The  magnitude of the photon is different for different colours of  light, but if the quantity of energy of a photon is divided by  the frequency of oscillation of the ray of light, the same number  is always obtained, the so-called Planck’s constant h.  This constant is thus of a universal nature and forms one of the  foundation stones for modern atomic physics.

Since light too was thus divided into atoms it appeared that all  phenomena could be explained as interactions between atoms of  various kinds. Mass was also attributed to the atom of light, and  the effects which were observed when light rays were incident  upon matter could be explained with the help of the law for the  impact of bodies.

Not many years passed before the found connection between the  photon and the light ray led to an analogous connection between  the motion of matter and the propagation of waves being sought  for.

For a long time it had been known that the customary description  of the propagation of light in the form of rays of light, which  are diffracted and reflected on transmission from one medium to  another, was only an approximation to the true circumstances,  which only held good so long as the wavelength of the light was  infinitesimally small compared with the dimensions of the body  through which the light passed, and of the instruments with which  it was observed. In reality light is propagated in the form of  waves which spread out in all directions according to the laws  for the propagation of waves.

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[Click on the link above to read the whole speech]

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Posted in Science | Comments (9)

What is Unknowable?

November 10, 2011 – 7:54 AM

With the use of the Vedic process “neti, neti” I have finally arrived at the following understanding. This is also referred to as BRAHMAN in Vedic terminology.

(1) I shall use the word UNKNOWABLE for the moment in an attempt to describe my understanding at this level. It doesn’t really matter what word I use because word is not the “thing.”

(2) The problem here is that there is NOTHING to describe. As far as our knowingness is concerned we can only know the considerations that appear spontaneously and randomly… from where… that is impossible to know.

(3) We can never know what lies beyond these considerations. We may certainly CONSIDER what lies beyond; but then, we would only know the consideration we make.

(4) Thus, we can only know the considerations and not what is beyond them. As we try to dig past considerations, we simply find more considerations.

(5) Actually, as we dig deeper in our attempt to describe what is beyond, we may simply uncover and describe our own hidden considerations.

(6) Thus, this unknowable beyond is nothing more than a carrot, that may help us uncover the considerations that, maybe, lie hidden deep in our consciousness. This is more than wonderful.

Furthermore:

(7) When we think of beingness we think of the being (soul). That being is not the core of beingness. The core of beingness is unknowable.   

(8 ) Whether the being dissolves into the unknowable is neither here nor there. Beyond considerations there is no time. So, whether one is “Unknowable” for a billion years or for just a blink of an eye. It is the same thing.

(9) We will never know if we had ever dissolved into the unknowable or not. All we would ever know at the highest level is being a “soul” in its simplest form.

(10) As far as we are concerned, the idea of the unknowable is simply there to help us uncover the deepest of our considerations.

And beyond this:

(11) A being (soul) can be constrained only by its own considerations. Total freedom would mean not be constrained by one’s own considerations.

(12) Only those considerations would constrain a being that are being generated and held fixedly in place because of ignorance.

(13) A being may enjoy life fully; participate in any and all adventures, while also being aware of all considerations one is generating even at the deepest level.

(14) That would be Nirvana… that would be the unknowable beyond… that would be attaining Brahmanhood; for this doesn’t mean forsaking of life and adventure.

(15) Nirvana, or total freedom, simply means absence of hidden considerations even at the deepest level. It means total command over oneself. The concept of Brahman, as built into the process “neti, neti,” helps one move in that direction.

(16) Nirvana would simply mean the ability to generate a consideration, hold on to that consideration, and then destroy that consideration.

(17) Nirvana does not mean forsaking this universe. Nirvana simply means total control over one’s considerations. One can attain Nirvana while being in this universe. In fact one’s enjoyment of this universe would be infinitely greater after attaining nirvana.

(18) Buddha lived to a grand age of 80, quite rare for his time, after attaining nirvana.

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Research into Unknowable

November 3, 2011 – 6:36 AM
Little question
October 17, 2014
The concept of Unknowable is better expressed as the non-awareness part of the “awareness – non-awareness” dichotomy. Please see Universe and Awareness.

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This space shall contain comments that pertain to research into unknowable:

From Maria (Ref. Knowable and Unknowable):

I understand what you are describing here.  It aligned extremely well with my past experience but it does not align with my own experience very accurately these days.  Probably the language, which becomes very clumsy.

There is a level beyond knowing that we can and do “experience.”  I cannot say “I”  experience it to even halfway accurately describe this.  It does not have any manifestation associated with it and “I” becomes a meaningless word.  What follows is the best description I can presently offer:

Knowing, perceiving, experiencing in this reality (in which I am typing) as me is never anything more than a perception of the past and with a successive series of present moments. They are successive ONLY in interchange and they are not particularly precise.  Even the future in this reality is already past.  It is already past because it was already created as what will be.  So there is really only now and then in this reality.

Simultaneously and always there is a true now.  It is NOT in the time stream.  It is the source of the time stream.  It is not past, present or “future.”  And this is where me goes beyond the limited being me.

While it is true that this state is not knowable in any “concrete” way, including concept, thought, mental images, physical forms, emotion, perception (bodily or otherwise) it is also true that it is a some kind of wellspring or source of these things. I don’t “know about” it.  “I” don’t “know” it.  I am and I am more than I am no fixed state and the words fail miserably.

I am sorry I cannot be more clear about this.  I assure you this is real, more real than any manifestation of anything and I love it beyond all things.  In my “self” it is a state of complete ease that often reflects into emotion as unbelievable joy, and from there into the body as lightness and brightness and sheer exuberance.

It is unknowable in the terms of this world, in terms of things, yes.  But it is real beyond all limited reality.

Vinaire’s comment:

“Unknowable” is just an enticing placeholder that dares one to challenge it. It postulates that there always will be something that is not known no matter how deep you may dive into it…. somewhat like an infinte series. It is quite an exciting concept for me.

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The “unknowable” is postulated simply to warn, “DO NOT REST AFTER DISCOVERING A STABLE DATUM. THERE IS NO LAST WORD.”

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