Saturday, July 12, 2008

The Geologist and the Mathematician

In an ordinary 2-sphere, any loop can be continuously tightened to a point on the surface. Does this condition characterize the 2-sphere? The answer is yes, and it has been known for a long time. The Poincaré conjecture asks the same question for the 3-sphere, which is more difficult to visualize.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year," the first time this had been bestowed in the area of mathematics


I have been following the Poincaré work under the heading of the Poincaré Conjecture. It would serve to point out any relation that would be mathematically inclined to deserve a philosophically jaunt into the "derivation of a mind in comparative views" that one might come to some conclusion about the nature of the world, that we would see it differences, and know that is arose from such philosophical debate.

Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.


Previous links in label index on right and relative associative posts point out the basis of the Poincaré Conjecture and it's consequent in developmental attempts to deduction about the nature of the world in an mathematical abstract sense?

Jules Henri Poincare (1854-1912)

The scientist does not study nature because it is useful. He studies it because he delights in it, and he delights in it because it is beautiful.


HENRI POINCARE

Mathematics and Science:Last Essays

8 Last Essays

But it is exactly because all things tend toward death that life is
an exception which it is necessary to explain.

Let rolling pebbles be left subject to chance on the side of a
mountain, and they will all end by falling into the valley. If we
find one of them at the foot, it will be a commonplace effect which
will teach us nothing about the previous history of the pebble;
we will not be able to know its original position on the mountain.
But if, by accident, we find a stone near the summit, we can assert
that it has always been there, since, if it had been on the slope, it
would have rolled to the very bottom. And we will make this
assertion with the greater certainty, the more exceptional the event
is and the greater the chances were that the situation would not
have occurred.


How simple such a view that one would speak about the complexity of the world in it's relations. To know that any resting place on the mountain could have it's descendants resting in some place called such a valley?

Stratification and Mind Maps

Pascal's Triangle

By which path, and left to some "Pascalian idea" about comparing some such mountains in abstraction to such a view, we are left to "numbered pathways" by such a design that we can call it "a resting" by nature selection of all probable pathways?


Diagram 6. Khu Shijiei triangle, depth 8, 1303.
The so called 'Pascal' triangle was known in China as early as 1261. In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei (as in diagram 6) in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century' (Stillwell, 1989, p136). They used it as we do, as a means of generating the binomial coefficients.

It wasn't until the eleventh century that a method for solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian 'generalised the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one useable for solving polynomial equations of any degree' (Katz, 1993, p191.)



Even the wisest of us does not realize what Boltzmann in his expressions would leave for us that such expression would leave to chance such pebbles in that valley for such considerations, that we might call this pebble, "some topological form," left to the preponderance for us in our descriptions to what nature shall reveal in those same valleys?

The Topography of Energy Resting in the Valleys

The theory of strings predicts that the universe might occupy one random "valley" out of a virtually infinite selection of valleys in a vast landscape of possibilities

Most certainly it should be understood that the "valley and the pebble" are two separate things, and yet, can we not say that the pebble is an artifact of the energy in expression that eventually lies resting in one of the possible pathways to that energy at rest.

The mountain, "as a stratification" exists.



Here in mind then, such rooms are created.

The ancients would have us believe in mind, that such "high mountain views do exist." Your "Olympus," or the "Fields of Elysium." Today, are these not to be considered in such a way? Such a view is part and parcel of our aspirate. The decomposable limits will be self evident in what shall rest in the valleys of our views?

Such elevations are a closer to a decomposable limit of the energy in my views. The sun shall shine, and the matter will be describe in such a view. Here we have reverted to such a view that is closer to the understanding, that such particle disseminations are the pebbles, and that such expressions, have been pushed back our views on the nature of the cosmos. Regardless of what the LHC does not represent, or does, in minds with regards to the BIG Bang? The push back to micros perspective views, allow us to introduce examples of this analogy, as artifacts of our considerations, and these hold in my view, a description closer to the source of that energy in expression.

To be bold here means to push on, in face of what the limitations imposed by such statements of Lee Smolin as a statement a book represents, and subsequent desires now taken by Hooft, in PI's Status of research and development.

It means to continue in face of the Witten's tiring of abstraction of the landscape. It means to go past the "intellectual defeatism" expressed by a Woitian design held of that mathematical world.

12 comments:

Phil Warnell said...

Hi Plato,

A truly interesting post and yet as usual a little hard to follow. That is I’m not sure what Poincare’s conjecture has in common with the current directions in physics. My own definition of intellectual defeatism I've made pretty clear and that being when one believes that there are not answers to found based on reason yet only answers that suggest connections without there being any. That is for me to surrender reason to understand existence is in itself illogical.

The thing that does stand out for me is that the contention made by Poincare, although simply intuitive for many at the same time was so difficult to prove within the limits of mathematical logic. Things like this has always indicated for me there exists a logic or a system of reason if you will that lays beyond the mathematical and therefore mathematics be only the shadowy projection of what it truly be. So for me this is what I would hope for those who feel defeated will also realize and find hope and confidence that it will be discovered so that one day all may be revealed.

As a final word, although I have complained in the past that science relies to heavily on the inductive process as its primary method of discovery, there was one statement made by the modern founder of this view that I not simply believe yet rather hold as true which is:

“But if there be any man who, not content to rest in and use the knowledge which has already been discovered, aspires to penetrate further; to overcome, not an adversary in argument, but nature in action; to seek, not pretty and probable conjectures, but certain and demonstrable knowledge — I invite all such to join themselves, as true sons of knowledge, with me, that passing by the outer courts of nature, which numbers have trodden, we may find a way at length into her inner chambers.”

-Francis Bacon from his Novum Organum (New Instrument)

Best,

Phil

John G said...

The Pascal Triangle of course has the Clifford Algebra gradings and I really think as Tony Smith says, that Clifford Algebra is the protospace for an SO(10) Lie Algebra emergent spacetime. The 8-fold periodicity of Clifford Algebra makes it quite useful foamy-wise. Cl(8) apparently can describe the local neighborhood but things might get quite monstrous globally.

http://www.valdostamuseum.org/hamsmith/E8physicsbook.pdf

Plato said...

Phil:My own definition of intellectual defeatism I've made pretty clear and that being when one believes that there are not answers to found based on reason yet only answers that suggest connections without there being any

I do not believe I have deviated from this acknowledgement of your intent and reason supported. Only that you believe this to be so here without understanding the implications, of what moves were made whether in context of the mathematician, or those who had difficulties solving, as an indication of not understanding the logic of approach.

Poincaré was enamoured, as Einstein was with the the non-euclidean geometries that were developed in terms not only of "time clocks" but of the departure from "euclidean perspectives."

Both Klein and Poincaré worked together and further posts will be dedicated to this as I have stated previous. There is a point to what is instinctual to me working in areas that one might say I am insufficient with the maths, yet work hard to understand this history. This, being drawn to the flame:)

Plato said...

Phil,

If you get a chance read Poincaré's "last essays" that I link, and see if the reference to the geologist and mathematician helps?:)

The basis is a tribute to "philosophy of science?":)

Plato said...

Hi John G,

I understood the work of Tony in a subtle way, with my approach( although you may see Tony's "historical context here" in reference), although I differ much greatly on the basis of the I CHING, as to it's nature and it's existence as a explanation to Probabilities and outcome.

I'd like to thank you for the further elucidation's of Tony's work.

Phil Warnell said...

Hi Plato,


"I do not believe I have deviated from this acknowledgement of your intent and reason supported. Only that you believe this to be so here without understanding the implications"

Now you have taken me wrong for I did not intend it to be believed that I thought you had deviated, simply I thought I would let you know that the connections you were making lacked clarity for me. There are times when one must be subtle and times when one must be blunt; I am thus sorry you misunderstood my bluntness for contention. I then hope it becomes more clear to me as you expand the thought.

Best,

Phil

Phil Warnell said...

Hi John,

The pdf of Tony Smith's you cited appears to be very interesting and I thank you for offering it. I wouldn't claim that I have the mathematical skill to comprehend all of this E8 significance and yet will read it all in time for what I might gather.

One thing I'm fascinated with is this connection to the Bohmian perspective in particular the quantum potential aspect to it. It is should also be noted that most modern Bohmian find this to be more of a contrivance and such a embarrassment yet Bohm never considered it so nor I. It is then interesting to learn that this plays a central role in the new direction. It was also a revelation to me that Oppenheimer attempted to have Bohm censured for his views as ironically that in the end it was he that was ostracized and subsequently ignored.

Best.

Phil

Phil Warnell said...

Hi Plato,

Oh yes, I forgot to tell you how much I appreciate you pointing out the Poincare book. As like Bee when it comes to material of any length or serious content (which this is both)I find I can only read it properly when it's in what she refers to as "Dead tree format":-) It was then nice to discover that it is also downloadable in pdf format which I can print out and thus read it in the manner preferred and promise to render homage to the trees and show proper respect by having it recycled if it ever is discarded, which I doubt will happen for as long as I remain:-)

Best,

Phil

John G said...

Plato, I was in to Jungian psychology and the Enneagram work of John Fudjack (of liminocentric fame) before Tony so I kind of run into the I Ching a lot. The I Ching's structure and movements have nice math and apparently the nice math is helpful for Jungian synchronicity.

Phil, yes the thing Tony does is give a nice math for a Bohmian implicite order structure but Tony's also giving a nice math for a Deutsch many-worlds structure so Tony is kind of a blend of Bohm and Deutsch in this area.

Plato said...

John:The I Ching's structure and movements have nice math and apparently the nice math is helpful for Jungian synchronicity.

I remember you now John.

I noticed the, "was into" as in, past tense? Is this correct? Do you practise Jungian psychology?

Off and on, over the years I have had brief exchanges on this topic, with my part being less then knowledgeable.

I do not know if one can ever leave such a "liminocentric structure" once they discover it embedded within one's soul?:)

Best,

Plato said...

Phil:Now you have taken me wrong for I did not intend it to be believed that I thought you had deviated, simply I thought I would let you know that the connections you were making lacked clarity for me.

Sorry Phil, and thanks for clarification.

I am a bit slow on the materialization of this thought process, yet I know it's culmination in two gentleman's work.

I don't want to go there yet without finishing with Felix Klein and his relationship with Poincaré.

The possibilities in relation to General relativity, and who should got there first in it's description although a mute point since Einstein did, is still interesting to me. To think that these two men had the Gumption to actually succeed in it's creation. As well, given the environment they were working in, with such abstractions,
that we might see such tendencies in nature being described for us.

Poincaré speaks on this in another essay that he did, that I will have to find.

Intuition and Logic in Mathematics by Henri Poincaré

Among the German geometers of this century, two names above all are illustrious, those of the two scientists who have founded the general theory of functions, Weierstrass and Riemann. Weierstrass leads everything back to the consideration of series and their analytic transformations; to express it better, he reduces analysis to a sort of prolongation of arithmetic; you may turn through all his books without finding a figure. Riemann, on the contrary, at once calls geometry to his aid; each of his conceptions is an image that no one can forget, once he has caught its meaning.

I allude to this statement of Poincaré in the expression of various posts and statements of Dirac as well, in regards to the geometer.

John G said...

Plato, I'm certainly still into esoteric Jungian related things. I have three papers in Fudjack's journal and they could use an update. Fudjack kind of stopped his journal to do political activism/war protesting stuff though I probably could get him to add an update to my third paper if I really wanted to (it's an online journal).

I was an engineer/programmer at IBM at the time I wote the papers and there was an organizational behavior consultant that I got some information from so my career did help a little but it's mostly been just a hobby I've had since I was a teenager. Tony has a physicist friend (Ark Jadczyk) who along with his wife is into some very esoteric things and I'm active on their forum as well as on an Enneagram forum.