PonderSeekDiscover

PonderSeekDiscover

Thursday, July 30, 2015

Liberating the Wanderers



Liberating the Wanders (Prismacolor on Watercolor paper)
by Wes Hansen Copyright all rights reserved

"Kyema! Out of unconscious karmic instinct I now wander in samsara. May the radiance of the primordial innate free me from all fear. O peaceful and wrathful deities, lead the way. O supreme consorts and great dakinis, hold me from behind. Show me how to cross the terrible path of the bardo. Point the way to the state of buddhahood itself.

When countless empty images appear as peaceful and wrathful forms, may the buddhas hold me with the hooks of their compassion. When the five great radiances arise, may I recognize them as my own mental projections. When the peaceful and wrathful deities appear, may I remain strong and without fear.

When the force of my own negative karma brings pain, may my mandala practice hold me free. When great thunderous sounds arise in the bardo, may I hear only om mani padme hum, the mantra of compassion.

May I rely upon Avalokiteshvara, Buddha of Compassion. May I rely upon samadhi, the meditation on inseparable bliss and void. May I see the five elements as friends, and not as enemies. May I see right now the realms of the five buddhas."

- from the Bardo Todol

The Wish-fulfilling Jewel


The Blessed Mother or Where the Honey Is (Prismacolor on Watercolor Paper)
by Wes Hansen Full Copyrights Reserved

"This Day-maker of the oral tradition, which dispels the darkness of the three worlds, rising out of the god's path, our investigations, is what makes the lotus of the correct view blossom. Hence, it is a treasure banquet for the hordes of bees, the great meditators."

- Khonton Rinpoche, “The Wish-fulfilling Jewel of the Oral Tradition,” as translated by Jose Ignacio Cabezon, the Chair of the Religious Studies Department at UC Santa Barbara, in the book, “Meditation on the Nature of Mind” (http://www.wisdompubs.org/book/meditation-nature-mind)





The Shadow of my Ancestors




The Shadow

I walk in the Shadow of Ancestors
thinking back on Misspent Youth,
Crystalized thought stabbing Life,
Killing the Psyche.

Deserted stillness comes like Death
While trying to find the last Kernel;
Happiness, Slippery as the worm
Avoiding the Hook

Evades the search, killing memories
Whose only purpose tricking one more Day,
Like a Whore to some unseen Pimp.
Monumental Madness disguised in

The Full, Blood-red Moon rocks the Tide
In a violent Trance; a beaten Roar.
Woman, Dance on my Corpse,
Help me ride the Threshold of

Time in need of a Partner.
Lapsed moments condensed into
Future Promises, maintaining the Shadow;
The Sickness cured by Dreams of

Self-mutilation. I hang from Two Hooks,
Pierced flesh a prelude to my Own
Private Peace; an Offering
To the Woman, Dancing on My Corpse.

I walk in the Shadow of Ancestors,
An outlaw aberration dedicated to
A Creative Mythology; My own Jihad.
A restless Native looking for Art in the

Land of Change. A neophyte tamed by
the course of War. Contempt for Life,
Cradled in Reverence for Death, redemption
Refused and Discarded, a spoiled, petulant

Philosophy rendered mute by carved Flesh
Hanging. Living sculpture in the form of Man
Suspended, Blood red dripping into the Mouth
Of the Woman riding the Moon.

I Love the Woman as She dances,
A celebration in honor of a Warrior’s
Death, and still I walk,
A meandering Journey in the timeless Shadow.

I escape the Tempest, turning Within,
Sitting in Silence, Doing nothing,
Seasons pass anew. The Shadow?
The Shadow takes care of Itself.

It’s something to ponder.


I set my own hooks and I enjoy it immensely . . . 

Ode to the Keeper of the White Lotus

An Ode to the Keeper of the White Lotus
by Wes Hansen Copyright all rights reserved




“I wish I could describe the feeling of being at sea; the anguish, frustration, and fear, the beauty that accompanies threatening spectacles, the spiritual communion with creatures in whose domain I sail. There is a magnificent intensity in life that comes when we are not in control but are only reacting, living, surviving. I am not a religious man per se. My own cosmology is convoluted and not in line with any particular church or philosophy. But for me, to go to sea is to glimpse the face of God. At sea I am reminded of my insignificance – of all men's insignificance. It is a wonderful feeling to be so humbled.”


We were separated at birth,
the torrent, Alluvion, came
sudden like and the
massacred ego, awash
in the tempest hue,
had no harbor against
time.

The images after, constructed
from Native spirits
untethered in the cold
inferno of an endless winter,
emerged from the
Wheel of Medicine.
They were
Spiritual, ephemeral, requisite . . .

I had . . .
I had so much to say,
but the separation was overwhelming;
I could only scream and yelp in
Beard pulling gibberish born
of the anguish . . .
the anguish of separation prior to
New Dawn.

“Come back to me . . .
come back to me,” I cried,
“and I will beat music
inspired by the love and
the Fury into your wintery
pelt.
And we will make the love sounds,
forlorn but elemental.
And we will cherish the blue depth
and ride the current together
until death,
dying,
dead.
And we will persevere into the
New Dawn.”

But the torrential wind
beat down and caste
my plea into the
deafening abyss of
icebound passage
and I was
stark,
naked,
alone . . .

Love was ripped from me
and I died an infinite
Death, transpired in bleak
ugliness, arisen in
Spiritual famine,
the youth sacrificed
to scarred flesh
Warriors . . .

And I became a man accustomed,
the ten-thousand horrors, the ten-thousand ecstasies,
the ten-thousand, ten-thousand,
meaningless fodder but for
the ancient hymns,
Dauphin elegies. [1]
And the truth became realized,
Eternal Reward,
A mantra of praise, beseeching:

“Have mercy on me,
a castaway drifting;
have mercy on me,
an initiate to the Wandering;
have mercy on me,
an intrepid traveler;
have mercy on me . . . “

And mercy was granted
in a blissful suffering
of color, sound, and fury;
a suffering reminiscent of
life before but fully engaged;
rapture without capture, free, but
suffering still . . .

And the cold Destroyer
beat down upon me,
fleeting moments substantial
in sheer volume.
I laughed, I cried, and
I screamed, “Come on . . .
come on with your furious
display.” The violent lust
of rapture flowed in
rivers of blood,
dark,
gaseous,
full . . .

But for a moment suspended,
my flesh torn and bleeding,
did I remember the riot of
Passion.
And the Passion was Love . . .


1.           Pelt, Dauphin Elegies, music for the Journey . . . 

Ponder, Seek, Discover

This little poem and the accompanying oil painting were both inspired by the excellent books of Dynamical Chaos theorist and computer scientist, Ben Goertzel: The Structure of Intelligence; The Evolving Mind; Chaotic Logic; From Complexity to Creativity; The Hidden Pattern. I highly recommend all of them!

The Contemplative
by Wes Hansen Copyright full rights reserved




Ponder this …
Life is just a Stream,
Thought, in Mind Divine,
It moves beyond Perception
Where time, collapsed, remains unknown.

Newly born Stars, Planets spinning,
Naught but New Ideas unfolding;
Plants and Animals, All of human consciousness,
Just a thread within the stream.

Inspiration comes, a Shock stirring the Nebulae,
Awakening potential, the birth of emergent form.

Cause becoming Effect, Effect becoming Cause,
a convoluted return to what has always been –
Infinity, boundless and eternal.

Scientific or Mystic, the approach matters not,
the conclusion, Universal, transcends Duality,
a Singularity giving birth to Thought Divine.

Relative stillness, demarcation unknown,
the result an Experience where
Time, collapsed, becomes Time Present.

Death and Re-birth, the slightest shift,
Awakens a New Paradigm. And yet,
Ancient and Perennial, It’s spoken of
often and Available to All.

Pure Heart, Pure Love …


Ponder …

A Simple Proof to Close the Binary Goldbach Conjecture

So that people won’t mistakenly deduce that I’m some cool kat magician, err . . . mathematician, I proclaim that we humans do not invent or create anything, rather, we discover. These discoveries are most aptly described by the mythic literature as boons from the gods and goddesses. This proof is just such a boon.

My own discoveries are granted by the infinitely brilliant and feminine presence I identify as the Muse. Awhile back I went to the art supply store to purchase some paint and in the same shopping complex was a Half Price Bookstore. I went into the bookstore hoping to find a copy of Ben Goertzel’s, The Hidden Pattern. I wasn’t lucky enough to find a copy of The Hidden Pattern but, while browsing the math and science section, I came across a textbook, Mathematical Ideas. When I saw the book Mathematical Ideas I received the “ping” in my mind which lets me know the Muse is permeating state space. I pulled the book and scanned through it.

Mathematical Ideas is a textbook, 11th edition, for liberal arts students. It gives a concise sketch of all the major mathematical concepts, such as number theory, set theory, group theory, functions, logic, etc.. I found the book interesting but when I encountered the Goldbach Conjecture the “pinging” just went crazy. I took due notice and put the book back on the shelf, knowing I could find a more in depth textbook online.

I purchased two books from the store: The Equation That Couldn’t Be Solved, by Mario Livio; The Quark and the Jaguar, by Murray Gell-Mann. The Goldbach Conjecture was mentioned three different times in The Quark and the Jaguar.

I conducted an online search and found the website of George Cain, Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, which is another synchronicity in and of itself. On Dr. Cain’s website I found Proofs and Concepts: The Fundamentals of Abstract Mathematics, a most extraordinary book.

Proofs and Concepts: The Fundamentals of Abstract Mathematics, by Dave and Joy Morris, Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, in Canada, is extremely user friendly; one concept flows into the next with unfettered ease. Furthermore, the exercises are designed such that one not only learns the rote of the language, but they force one to take the same journey the mathematical forefathers took; hence, one learns the why and whereof of the language as well. I finally have the core perspective of mathematics and I’m left with nothing but admiration, respect, and enthusiasm for the discipline. Proofs and Concepts: The Fundamentals of Abstract Mathematics is a masterwork, a boon from the gods and goddesses. And this little textbook led me straightaway to the following Proof which I hereby submit, with modifications, as a poem:

A Simple Proof To Close The Binary Goldbach Conjecture

By Wesley W. Hansen Copyright Creative Commons Attribution/NoDeriv

"Then there's the poetic truth, and it's different than the factual truth but has a better and more meaningful place, maybe, than any facts you think you could know. Knowing something on an intuitive or imaginative level is maybe a more true kind of knowledge than thinking you have some sort of fact."

-  Elisa Ambrogio in an interview by Rin Kelley, L. A. Record, Issue #117, with permission.

Abstract. We define the set of positive even integers, the set of prime numbers, and the Cartesian product on the set of prime numbers. We then define a set composed of the sums of all ordered pairs in the Cartesian product on the set of primes. Finally, we demonstrate the existence of a bijection between this set of sums and the set of positive even integers and conclude by demonstrating the existence of an identity between the domain and codomain of this bijection, thus closing the binary Goldbach Conjecture.

                1. Introduction. We wish to prove the “strong” or “binary” Goldbach Conjecture as reformulated by Leonhard Euler:

“All positive even integers greater than or equal to four can be expressed as the sum of two primes.”

By implication, this proof generalizes to the weak conjectures.

From elementary Number Theory:

The set of all positive integers greater than zero is equal to the set of all natural numbers. For any natural number n, n is even iff n = 2m, where m is any positive integer greater than zero, and n is odd iff n = 2m + 1, where m is any positive integer greater than or equal to zero. Since, in the expression, 2m + 1, m is any positive integer greater than or equal to zero, we can safely conclude that 2m + 1 defines a positive integer greater than zero. From this it follows that the sum of any two odd natural numbers yields an even natural number since, (2m + 1) + (2m + 1) = 4m + 2 = 2(2m + 1).

The only even number in the set of prime numbers is two. If we eliminate two from the set of prime numbers, we guarantee that the sum of any two of the remaining prime numbers will yield an even natural number; however, the positive even integer, four, can only be expressed as the sum, 2 + 2 = 4. We express this identity here allowing us to eliminate two from the set of prime numbers utilized in the main body of our work.

We conclude our introduction by stating an obvious fact of specific interest to the problem at hand: the set of all even integers greater than or equal to four is equal to the set of all even natural numbers greater than or equal to four.

                Notation. We use the following notation:

N+           │             the set of all natural numbers

                Acknowlegments. I wish to express sincere gratitude to the Drs. Dave and Joy Morris, Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada, for their unsurpassed free-access textbook, Proofs and Concepts: The Fundamentals of Abstract Mathematics; to Dr. Paul Dawkins, Department of Mathematics, Lamar University, Beaumont, Texas, for his equally unsurpassed Calculus Notes; to Dr. Duane Kouba, Department of Mathematics, University of California, Davis, California, and, by extension, the entire Mathematics Department at UC-Davis for exquisite problem sets and their internet Calculus Page; to Dr. George Cain, Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, for his website, Online Mathematical Textbooks; to Dr. William Tiller, Professor Emeritus, Stanford University, Stanford, California, and Dr. Ben Goertzel, Research Professor, Xiamen University, Xiamen, China, for invaluable inspiration.

                2. Definitions. We define our mathematical entities using standard terminology:

                Definition 2.1. We define the set of even natural numbers greater than or equal to six:

E = {2n │ n є N+, n ≥ 3}

                Definition 2.2. We define the set of prime numbers greater than two:

P = {p │ p is prime, p > 2}

                Definition 2.3. We define a set on the Cartesian product of Definition 2.2:

C = {(a,b) │ (a,b) є P X P}

                Definition 2.4. We define a set whose elements are defined on the elements of Definition 2.3:

D = {d │ d = a + b for every element, (a,b), of C}


                Definition 2.5. We define an ordered sequence on the elements of Definition 2.1 (reference Calculus II, Sequence and Series, in [DAW] and Chapter 16 in [MM]), where en < en+1:

{e1, e2, e3, … , en, … }

                Definition 2.6. We define a predicate representing the partial sums on Definition 2.5 (reference Calculus II, Sequence and Series, in [DAW] and Chapter 16 in [MM]):

P(n): e1 + e2 + e3 + … + en + … + 2(n + 2) = n2 + 5n

                Definition 2.7. We define an ordered sequence on the elements of Definition 2.4 (reference Calculus II, Sequence and Series, in [DAW] and Chapter 16 in [MM]), where dn < dn+1 and, by Definitions 2.2, 2.3, and 2.4, d1 = 6, d2 = 8, d3 = 10, d4 = 12, d5 = 14,    d6 = 16, d7 = 18, … , d20 = 44, …:

{d1, d2, d3, … , dn, … }

                Definition 2.8. We define a predicate on Definition 2.7:

P(n): dn = dn – 1 + 2 = 2(n + 2)

                3. Arguments. We demonstrate our arguments using standard terminology and theorems which form the foundation of abstract mathematics. Specifically, from the foundation of abstract mathematics, we’re given that the set of natural numbers, N+, is countably infinite and that any subset of N+ is either finite or countably infinite (reference Chapter 15 in [MM]).

                Lemma 3.1. The set E from Definition 2.1 is countably infinite.

                Proof. Given that N+ is countably infinite, we can list the elements of N+ in an infinite sequence, a1, a2, a3, …, where, for any an there exists an an+1. If we shift the index of this infinite sequence by any finite amount we still have an infinite sequence in that for any an there exists an an+1, hence, the set of all natural numbers greater than or equal to three is countably infinite. It then follows, by Definition 2.1, that there exists a bijection between the set of all natural numbers greater than or equal to three and the set E defined by, f(n) = 2n, hence, E is countably infinite, as desired.         □

                Lemma 3.2. The set P from Definition 2.2 is countably infinite.

                Proof. Given that the set of prime numbers is a subset of N+, we know that P is either finite or countably infinite. Since we can list the prime numbers and, hence, the elements of P, in an infinite sequence such that for any an there exists an an+1, the set P is countably infinite, as desired.              □

                Lemma 3.3. The set C from Definition 2.3 is countably infinite.

                Proof. Given that we can list the elements of set P, from Definition 2.2, in an infinite sequence, a1, a2, a3, …, then, by the definition of Cartesian product (reference Chapter 6, Section C, in [MM]), we can also list the elements of set C in an infinite sequence, (a,b)1, (a,b)2, (a,b)3, …, where for any (a,b)n there exists an (a,b)n+1 (reference Theorem 15.43(2) in [MM]), hence, C is countably infinite, as desired.              □

                Lemma 3.4 The set D from Definition 2.4 is countably infinite.

                Proof. As defined, for any element, d, of D, there exists an element, (a,b), of C, such that f(a,b) = a + b = d, hence, there exists a surjection, f : C → D, and, since              f : C → D is surjective, f(C) = D. Therefore, given that the image of a countably infinite set is countably infinite (reference Theorem 15.43(3) in [MM]), D is countably infinite, as desired.        □

                Lemma 3.5. The elements of set E, defined by Definition 2.1 and Definition 2.5, generate a series, ∑en, which diverges to infinity with an equation for partial sums,          n2 + 5n.

                Proof. We induct on n using Definition 2.6, where, by Definition 2.1, e1 = 6.

                Base case. For n = 1, we have:

6 + 8 + 10 + 12 + … + 2(n + 2) = 12 + 5(1) = 6

Since an identity exists, P(1) is true.

Assume P(k – 1) is true, then:

6 + 8 + 10 + 12 + … + 2((k – 1) + 2) = k2 + 3k – 4

Hence:


(6 + 8 + 10 + 12 + … + 2((k – 1) + 2)) + 2(k + 2) = k2 + 5k

k2 + 3k – 4 + 2k + 4 = k2 + 5k

k2 + 5k = k2 + 5k

Since an identity exists, P(k) is true; therefore, by the Principle of Mathematical Induction, Definition 2.5 generates a series, ∑en, with an equation for partial sums (reference Calculus II, Sequence and Series, in [DAW]), n2 + 5n, as desired.

Let {Sn} be the sequence of partial sums on the series ∑en, then, from calculus (reference Calculus II, Sequence and Series, in [DAW]), it’s given that ∑en diverges to infinity iff {Sn} diverges to infinity. It is plain to see that n2 + 5n goes to infinity as n goes to infinity, therefore, ∑en ­diverges to infinity, as desired.            □

                Lemma 3.6. For any dn in the sequence {dn}, defined by Definition 2.7,
dn = 2(n + 2).

                Proof. We induct on n using Definition 2.8 with strong induction and multiple base cases:

                Base case. For n = 1, we have:

6 = 2(1 + 2) = 2(3) = 6

Since an identity exists, P(1) is true.

                Base case. For n = 2, we have:

8 = 6 + 2 = 2(2 + 2) = 2(4) = 8

Since an identity exists, P(2) is true.

                Base case. For n = 3, we have:

10 = 8 + 2 = 2(3 + 2) = 2(5) = 10

Since an identity exists, P(3) is true.

                Base case. For n = 4, we have:

12 = 10 + 2 = 2(4 + 2) = 2(6) = 12

Since an identity exists, P(4) is true.

                Base case. For n = 5, we have:

14 = 12 + 2 = 2(5 + 2) = 2(7) = 14

Since an identity exists, P(5) is true.

Assume P(k – 5), P(k – 4), P(k – 3), P(k – 2), and P(k – 1) are all true, then:

P(k – 5) = 2(k – 5 + 2) = 2k – 6

P(k – 4) = 2k – 6 + 2 = 2k – 4 = 2(k – 4 + 2) = 2(k – 2)

P(k – 3) = 2k – 4 + 2 = 2k – 2 = 2(k – 3 + 2) = 2(k – 1)

P(k – 2) = 2k – 2 + 2 = 2k = 2(k – 2 + 2) = 2k

P(k – 1) = 2k + 2 = 2(k – 1 + 2) = 2(k + 1)

P(k) = 2k + 2 + 2 = 2k + 4 = 2(k + 2)

Since an identity exists, P(k) is true; therefore, by the Principle of Mathematical Induction using strong induction with multiple base cases, for any dn in the sequence {dn}, defined by Definition 2.7, dn = 2(n + 2), as desired.                 □
               
4. Proof of the conjecture. Our proof rests on demonstrating that set E, from Definition 2.1, is equal to set D, from Definition 2.4. Since, by Lemma 3.1 and Lemma 3.4, set E and set D are countably infinite, we know there exists a bijection between them (reference Chapter 12 and Chapter 15, Section E, in [MM]).

Assume the bijection, f: D → E, is defined by f(d) = d, then the elements of D, defined by Definitions 2.2, 2.3, 2.4, and 2.7, generate a series which diverges to infinity with an equation for partial sums, n2 + 5n. We define a predicate on Definition 2.7 where, by Lemma 3.6, dn = 2(n + 2):

P(n): 6 + 8 + 10 + 12 + … + 2(n + 2) = n2 + 5n

We now induct on n, making a note that, due to the distributed search for a counterexample, there existed, as of July 14, 2008, 6 x 1017 base cases (reference [WMW]).

Base case. For n = 1, we have:

6 + 8 + 10 + 12 + … + 2(n + 2) = 12 + 5(1) = 6

Since an identity exists, P(1) is true.

Assume P(k – 1) is true, then:

6 + 8 + 10 + 12 + … + 2((k – 1) + 2) = k2 + 3k – 4

Hence:


(6 + 8 + 10 + 12 + … + 2((k – 1) + 2)) + 2(k + 2) = k2 + 5k

k2 + 3k – 4 + 2k + 4 = k2 + 5k

k2 + 5k = k2 + 5k

Since an identity exists, P(k) is true; therefore, Definition 2.7 generates a series, ∑dn, with an equation for partial sums (reference Calculus II, Sequence and Series, in [DAW]), n2 + 5n, as desired.

Let {Tn} be the sequence of partial sums on the series ∑dn, then, from calculus (reference Calculus II, Sequence and Series, in [DAW]), it’s given that ∑dn diverges to infinity iff {Tn} diverges to infinity. It is plain to see that n2 + 5n goes to infinity as n goes to infinity, therefore, ∑dn ­diverges to infinity, as desired.            □

As the forgoing well demonstrates, the bijection, f: D → E, is defined by f(d) = d, hence, the domain D equals the codomain E proving the binary Goldbach Conjecture, as desired. □         


__________________
REFERENCES
___________________
[DAW]  P. Dawkins, Calculus Notes, http://tutorial.math.lamar.edu
[MM]    D. Morris and J. Morris, Proofs and Concepts: The Fundamentals of Abstract Mathematics, http://people.uleth.ca/~dave.morris/books/proofs+concepts.pdf

[WMW]                E. Weisstein, Goldbach Conjecture, http://mathworld.wolfram.com/GoldbachConjecture.html