Who said it?

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

"No Royal Road to Geometry?"

Click on the Picture

Are you an observant person? Look at the above picture. Why ask such a question as to, "No Royal Road to Geometry?" This presupposes that a logic is formulated that leads not only one by the "phenomenological values" but by the very principal of logic itself.

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

I don't think I could of made it any easier for one, but to reveal the answer in the quote. Now you must remember how the logic is introduced here, and what came before Euclid. The postulates are self evident in his analysis but, little did he know that there would be a "Royal Road indeed" to geometry that was much more complex and beautiful then the dry implication logic would reveal of itself.

It's done for a reason and all the geometries had to be leading in this progressive view to demonstrate that a "projective geometry" is the final destination, although, still evolving?

Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false. This means that it is possible to assign meanings to the terms "point" and "line" in such a way that they satisfy the first four postulates but not the parallel postulate. These are called non-Euclidean geometries. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.

In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead:

The projective axiom: Any two lines intersect (in exactly one point).

If you are "ever the artist" it is good to know in which direction you will use the sun, in order to demonstrate the shadowing that will go on into your picture. While you might of thought there was everything to know about Plato's cave and it's implication I am telling you indeed that the logic is a formative apparatus concealed in the geometries that are used to explain such questions about, "the shape of space."

The Material World

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.

Polytopes and allotrope are examples to me of "shapes in their formative compulsions" that while very very small in their continuing expression, "below planck length" in our analysis of the world, has an "formative structure" in the case of the allotrope in the material world. The polytopes, as an abstract structure of math thinking about the world. As if in nature's other ways.



This illustration depicts eight of the allotropes (different molecular configurations) that pure carbon can take:

a) Diamond
b) Graphite
c) Lonsdaleite
d) Buckminsterfullerene (C60)
e) C540
f) C70
g) Amorphous carbon
h) single-walled carbon nanotube

Review of experiments

Graphite exhibits elastic behaviour and even improves its mechanical strength up to the temperature of about 2500 K. Measured changes in ultrasonic velocity in graphite after high temperature creep shows marked plasticity at temperatures above 2200 K [16]. From the standpoint of thermodynamics, melting is a phase transition of the first kind, with an abrupt enthalpy change constituting the heat of melting. Therefore, any experimental proof of melting is associated with direct recording of the temperature dependence of enthalpy in the neighbourhood of a melting point. Pulsed heating of carbon materials was studied experimentally by transient electrical resistance and arc discharge techniques, in millisecond and microsecond time regime (see, e.g., [17, 18]), and by pulsed laser heating, in microsecond, nanosecond and picosecond time regime (see, e.g., [11, 19, 20]). Both kind of experiments recorded significant changes in the material properties (density, electrical and thermal conductivity, reflectivity, etc. ) within the range 4000-5000 K, interpreted as a phase change to a liquid state. The results of graphite irradiation by lasers suggest [11] that there is at least a small range of temperatures for which liquid carbon can exist at pressure as low as 0.01 GPa. The phase boundaries between graphite and liquid were investigated experimentally and defined fairly well.

Sean Carroll:But if you peer closely, you will see that the bottom one is the lopsided one — the overall contrast (representing temperature fluctuations) is a bit higher on the left than on the right, while in the untilted image at the top they are (statistically) equal. (The lower image exaggerates the claimed effect in the real universe by a factor of two, just to make it easier to see by eye.)

See The Lopsided Universe-.

#36.Plato on Jun 12th, 2008 at 10:17 am

Lawrence,

Thanks again.

“I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered.”Harold Scott Macdonald (H. S. M.) Coxeter

Moving to polytopes or allotrope seem to have values in science? Buckminister Fuller and Richard Smalley in terms of allotrope.

I was looking at Sylvestor surfaces and the Clebsch diagram. Cayley too. These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope[E8]“(where Coxeter[I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)

As the furthest extent of the Conjecture , how shall we place the dynamics of Sylvestor surfaces and B Fields in relation to the timeline of these geometries? Historically this would seem in order, but under the advancement of thinking in theoretics does it serve a purpose? Going beyond “planck length” what is a person to do?

Thanks for the clarifications on Lagrange points. This is how I see the WMAP.

Diagram of the Lagrange Point gravitational forces associated with the Sun-Earth system. WMAP orbits around L2, which is about 1.5 million km from the Earth. Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. The forces at L2 tend to keep WMAP aligned on the Sun-Earth axis, but requires course correction to keep the spacecraft from moving toward or away from the Earth.

Such concentration in the view of Sean’s group of the total WMAP while finding such a concentration would be revealing would it not of this geometrical instance in relation to gravitational gathering or views of the bulk tendency? Another example to show this fascinating elevation to non-euclidean, gravitational lensing, could be seen in this same light.

Such mapping would be important to the context of “seeing in the whole universe.”

See:No Royal Road to Geometry
Allotropes and the Ray of Creation
Pasquale Del Pezzo and E8 Origination?
Projective Geometries

Comparing Amplitutdes of the String

Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state.

So one tends to get in this mode of thinking, that the way in which one measures "aspects of reality" is somehow couched in the mathematics of a kind, and unbeknownst to us, a hoped a pattern is revealed?

Artist's impression of the setup.

The disks represent the bosonic condensate density and the blue balls in the vortex core represent the fermionic density. The black line is a guide to the eye to see the wiggling of the vortex line that corresponds to a so-called Kelvin mode, which provides the bosonic part of the superstring

To have gone to such extremes of thinking of the reality as one would of the elemental design, one hopes to see the "new perspective" will encourage new pathways to develping new phenomenological directions.

As absurd as this sounds, the current framework did not support the existing structure any longer in regards to sciences development, so theoretical insight needed to be pushed. New ways in which we could look at reality could provide for new experimental considerations?

It would not be unlike Bohm to introduce his system, and then ask us to think i the possible new way in which language has somehow been transformed. Does it not seem by our very nature, language accustoms perspective to have it's natural physical gestures as expression? Not just physical, but mental as well?

The History of Magnetic Vision

Grossmann is getting his doctorate on a topic that is connected with non-Euclidean geometry. I don’t know what it is.
Einstein to Mileva Maric,1902

Animal Navigation

The long-distance navigational abilities of animals have fascinated humans for centuries and challenged scientists for decades. How is a butterfly with a brain weighing less than 0.02 grams able to find its way to a very specific wintering site thousands of kilometers away, even though it has never been there before? And, how does a migratory bird circumnavigate the globe with a precision unobtainable by human navigators before the emergence of GPS satellites? To answer these questions, multi-disciplinary approaches are needed. A very good example of such an approach on shorter distance navigation is the classical ongoing studies on foraging trips of Cataglyphis desert ants. My Nachwuchsgruppe intends to use mathematical modelling, physics, quantum chemistry, molecular biology, neurobiology, computer simulations and newly developed laboratory equipment in combination with behavioral experiments and analyses of field data to achieve a better understanding of the behavioral and physiological mechanisms of long distance navigation in insects and birds.

Tony Smith has some interesting information in response to a post by Clifford of Asymptotia.

Clifford writes:

This is simply fascinating. I heard about it on NPR. While it is well known that birds are sensitive to the earth’s magnetic field, and use it to navigate, apparently it’s only been recently shown that this sensitivity is connected directly to the visual system (at least in some birds). The idea seems to be that the bird has evolved a mechanism for essentially seeing the magnetic field, presumably in the sense that magnetic information is encoded in the visual field and mapped to the brain along with the usual visual data

While my post has been insulted by cutting it short(and stamping it and proclaiming irrelevance,) I'd like to think otherwise, even in face of his streamlining that Clifford likes to do. His blog, he can do what he wants of course.

In any case, it seems reasonable to agree with Buhler, who concludes in his biography of Gauss that "the oft-told story according to which Gauss wanted to decide the question [of whether space is perfectly Euclidean] by measuring a particularly large triangle is, as far as we know, a myth."

So I'll repeat the post of mine here and the part, that he has deleted. You had to know how to see the relevance of the proposition of birds in relation to the magnetic field of the earth, to know why the bird relation is so important.

On Magnetic vision
Rupert Sheldrake has had similar thoughts on this topic.

"Numerous experiments on homing have already been carried out with pigeons. Nevertheless, after nearly a century of dedicated but frustrating research, no one knows how pigeons home, and all attempts to explain their navigational ability in terms of known senses and physical forces have so far proved unsuccessful. Researchers in this field readily admit the problem. 'The amazing flexibility of homing and migrating birds has been a puzzle for years. Remove cue after cue, and yet animals still retain some backup strategy for establishing flight direction.' 'The problem of navigation remains essentially unsolved.'

Many of academics might have steered clear because of the the thoughts and subject he has about this? It seems to me that if this information is credible, then some of Rupert's work has some substance to it and hence, brings some credibility to the academic outlook?

Update: Here I am adding some thoughts in regards to Rupert Sheldrake that I was having while reading his work. He had basically himself denounced the process of birds having an physiological connection to magnetic fields because of not having any information to support the magnetic vision Clifford is talking about. So Rupert moves beyond this speculation, to create an idea about what he calls Morphic resonance with regards to animals.

So Rupert presents future data and theoretics in face of what we now know in terms of the neurological basis is experimentally being talked about in the article in question Clifford is writing about.

On How to see in the Non Euclidean Geometrical World

8.6 On Gauss's Mountains

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface.

See:Reflections on Relativity

As a layperson, Riemann and Gauss were instrumental for helping me see beyond what we were accustom to in Euclidean, so I find Clifford's blog post extremely interesting as well. Maybe even a biological/physiological impute into our senses as well? Who knows?:)

Einstein's youth and the compass, becomes the motivation that drives the vision of what exists beyond what was acceptable in that youth. The mystery. Creates a new method on how we view the world beyond the magnetic, to help us include the view in the gravitational one as well.

From a early age, young Albert showed great interest in the world around him. When he was five years old, his father gave him a compass, and the child was enchanted by the device and intrigued by the fact the needle followed a invisible field to point always in the direction of the north pole.Reminiscing in old age, Einstein mentioned this incident as one of the factors that perhaps motivated him years later to study the gravitational field. God's Equation, by Amir D. Aczel, Pg 14

While something could exist that is abstract, like for instance the Gaussian arc, this inclusion in the value of general relativity is well known. Mileva's response in quote above was the key for Einstein's views on developing General Relativity, and without it "electromagnetism would not, and could not" have been included geometrically in the theory of GR.

It was a succession to "Gravitational wave production" that was understood in regards to Taylor and Hulse.

The theory of relativity predicts that, as it orbits the Sun, Mercury does not exactly retrace the same path each time, but rather swings around over time. We say therefore that the perihelion -- the point on its orbit when Mercury is closest to the Sun -- advances.

I would think this penduum exercise would make a deeper impression if held in concert with the way one might have look at Mercuries orbit.

Or, binary pulsar PSR 1913+16 of Taylor and Hulse. These are macroscopic valutions in what the pendulum means. Would this not be true? See:Harmonic Oscillation

I guess not every string theorist would know this? Maybe even Bee would understand that "German" is replace by another form of seeing using abstract language, for how everything can be seen in relation to the ground state? Where there are no gravitational waves, spacetime is flat.

You had to know how such views on the navigation of the birds could have a direct link to the evolutionary output of the biology and physiology of the species. What Toposense?

Yes it's a process where the mathematical minds look at knitting and such, in such modularc forms, to have said, "hey there is a space of thinking" that we can do really fancy twists and such.

One thing us humans can certainly do is construct the monumental world reality with straight lines and such in the Euclidean view. But nature was there before we thought to change all it's curves.

But the truth is, the Earth's topography is highly variable with mountains, valleys, plains, and deep ocean trenches. As a consequence of this variable topography, the density of Earth's surface varies. These fluctuations in density cause slight variations in the gravity field, which, remarkably, GRACE can detect from space. See: The Mind Field

See here for more info on Grace.

Look out into the wild world that nature itself presents and tell me what the ancient mind did not see. Native Americans lived closer to nature. Hopefully you'll understand why it is we must engage ourselves to experiencing the views of nature?:)

Mandalic Construction

See: The Last Mimzy

The "Ancient Medicine wheels" might have been place accordingly? Do you imagine seeing in the abstract world, the magnetic view we see of earth in it's different disguise?

So that last line about the "medicine wheels" probably caused Clifford to do what he did in regards to the post I wrote.

Yes I am creating a direct link between the Medicine Wheels and the Medicine Wheel as a Mandala constructed by early Native Americans. Where they were shamanically placed on the earth.



What is a Medicine Wheel?

The term "medicine wheel" was first applied to the Big Horn Medicine Wheel in Wyoming, the most southern one known. That site consists of a central cairn or rock pile surrounded by a circle of stone; lines of cobbles link the central cairn and the surrounding circle. The whole structure looks rather like a wagon wheel lain-out on the ground with the central cairn forming the hub, the radiating cobble lines the spokes, and the surrounding circle the rim. The "medicine" part of the name implies that it was of religious significance to Native peoples.

Figure 4 - Distribution of medicine wheel sites east of the Rockies

What was of importance is the underlying psychological patterns that exist in the forms of Mandalas. That such a thing like the Medicine wheel, would retain a impact from one's life, to another life.

There are various forms of mandalas with distinct concepts and different purposes. The individual representations range from the so-called Cosmic Mandalas, which transmit the ancient knowledge of the development of the universe and the world-systems which represents a high point among Mandalas dedicated to meditation; to the Mandalas of the Medicine Buddha which demonstrates how the Buddha-power radiates in all directions, portraying the healing power of the Buddha.

It would not be easy to understand this "seed mandala" as it makes it way into conscious recognition. It arises to awareness through the subconscious pathway during our susceptibility in dream time. This open accessibility is the understanding that there is a closer connection to the universality of being, and the realization that the degrees beyond the "emotive body" is developing the understanding of the "mental one" as well as, leading to "the spiritual one."

This comparative view is analogousness to development beyond the abstract view we see of earth in it's gravitational form.

However, the signals that scientists hope to measure with LISA and other gravitational wave detectors are best described as "sounds." If we could hear them, here are some of the possible sounds of a gravitational wave generated by the movement of a small body inspiralling into a black hole.

It would be much like a "energy packet" that would contain all that is demonstrated in "extravagant patterns." Look like a "flower in real life," or a "intricate pattern," while encouraging the person to explore these doorways and move on from.

That seed contains all of the history we have supplanted to it by how we built previously and embedded all the philosophy we had learnt from it.

The Emotional Body of the Earth

Would to me seem very emotive in terms of it's weather. How such weather patterns spread across the earth. Also, it would not seem so strange then that while we would have seen polarization aspects in the cosmos, in terms of magnetic field variances in relation to north and south, we would see "this of value" in the earth as well?

So would the earth have it's positive and negative developments in relation to aspect of it's weather? Most certainly psychological when the snows have lasted so long, one could indeed wish for warmer weather, but that's not what I mean. I mean on a physiological level, such ionic generations would indeed cause the state of the human body to react.

IN Search of Mandelstam's Holy Grail

There are two posts that reflect the purpose of this post today. One is Clifford's linked through Lee Smolin's comment and the other, at Backreaction. Good Physics is Conflict

A lot of you may never understand the significance of the mystery that follows the thinking of the Holy Grail. Yet is it more the knowledge that can be gained from all soul's day, that on this occasion we may have called it Halloween.

We celebrated the past, in the living of today? You philosophize, while you become the thoughts of models created by science leaders shared? I do not think any have a "personality disorder" like I do:)

Lee Smolin:

Here is an example of the kind of question I found I needed a book to explore: what to think of the problems that arise from the need for higher dimensions in string theory, such as the problem of moduli stabilization and the vast number of static solutions. To approach this I read books on the early history of GR and unified field theories and learned that higher dimensional compactifications were explored many times between 1914 and 1984 and that close to the beginning these problems were appreciated and discussed by Einstein and others. I weave this story into my book because I find it useful when trying to judge how serious the present issues in string theory are to know how Einstein and many others struggled with the same issues over decades.

So of course when we think of the persons of science who walked before us (shoulders of giants), what are their whole stories, but what is evidenced to us as we read those words? So you compile your data accordingly, and from it, we say at certain spots, how are we to react to the challenge now facing us?



Stanley Mandelstam, Professor Emeritus, Particle Theory

My present research concerns the problem of topology changing in string theory. It is currently believed that one has to sum over all string backgrounds and all topologies in doing the functional integral. I suspect that certain singular string backgrounds may be equivalent to topology changes, and that it is consequently only necessary to sum over string backgrounds. As a start I am investigating topology changes in two-dimensional target spaces. I am also interested in Seiberg-Witten invariants. Although much has been learned, some basic questions remain, and I hope to be able at least to understand the simpler of these questionsStanley Mandelstam-Professor Emeritus Particle Theory

As a lay person watching the debate it is difficult for me to discern the basis of these arguments. But I strive to go past what you think is surface in conduct in science's response, as some may show of themself in a reactionary pose. Should we all be so perfect, that the human condition is not also the example by which we shall progress in science?

Dealing in the Abstract



A sphere with three handles (and three holes), i.e., a genus-3 torus.

Of course the thinking may seem so detached from reality that one asks for some reason with which to believe anything. It required, that the history of this approached dust off models in glass cabinets, that were our early descendants of the museum today.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

How many of you know how to work in such abstract spaces, and know that what you are talking about has it's relationships in the physics of today? Or that, what satellites we use in measure of, have some correlation to how one may have seen "UV coordinates supplied by Gauss?"

Where are my keys?

"Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimensionality, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality." from Flatland, by E. A. Abbott

The Extra-Dimensions?

So you intuitively believe higher dimensions really exist?

Lisa Randall:I don't see why they shouldn't. In the history of physics, every time we've looked beyond the scales and energies we were familiar with, we've found things that we wouldn't have thought were there. You look inside the atom and eventually you discover quarks. Who would have thought that? It's hubris to think that the way we see things is everything there is.

And what is it that we don't see? I thought of a comment somewhere that spoke about what first started to make it's appearance in how we communicate?

Time is the Unseen fourth Dimension

They were able to create what we recognize today as the "elliptical" and "hyperbolic" non-Euclidean geometries. Most of Saccheri's first 32 theorems can be found in today's non-Euclidean textbooks. Saccheri's theorems are prefaced by "Sac."

One of my greatest "aha moments" came when I realized Non-euclidean geometries. I had to travel the history first with Giovanni Girolamo Saccheri, Bolya and Lobachevsky, for this to make an impression, and I can safely say, that learning of Gauss and Riemann, I was truly impressed.

Einstein had to include that "extra dimension of time." Greater then, or less then, 180 degrees and we know "this triangle" can take on some funny shapes when you apply them "to surfaces" that are doing funny things.?:)

Second, we must be wary of the "God of the Gaps" phenomena, where miracles are attributed to whatever we don't understand. Contrary to the famous drunk looking for his keys under the lamppost, here we are tempted to conclude that the keys must lie in whatever dark corners we have not searched, rather than face the unpleasant conclusion that the keys may be forever lost.

Let me just say that "it is not the fact that any drinking could have held the mind" of the person, but when they absentmindedly threw their car keys. The "point is" that if the light shines only so far, what conclusion should we live with?

Moving to the Fifth

So of course whatever real estate you are buying, make sure the light is shining on what your willing to purchase? Is this not a good lesson to learn?

Moving any idea to a fifth dimension I thought was important in relation to seeing what Einstein had done. See further: Concepts of the Fifth Dimension. I illustrate more ways in which we may see that has not been seen for most could have helped the mind see how this is accomplished in current day geometric methods.

Why was this thought "wrong" when one may of thought to include "gravity and light" together, after the conclusion of spacetime's 3+1? Gravity. What Had Maxwell done? What Had Riemann done?

You knew "the perfect symmetry" had to be reduced to General Relativity?

Greg Landsberg:

Two types of the extra-dimensional effects observable at collides.

A graviton leaves our world for a short moment of time, just to come back and decay into a pair of photons (the DØ physicists looked for that particular effect).

A graviton escapes from our 3-dimensional world in extra dimensions (Megaverse), resulting in an apparent energy non-conservation in our three-dimensional world.

So why would it matter to us if the universe has more than 3 spatial dimensions, if we can not feel them? Well, in fact we could “feel” these extra dimensions through their effect on gravity. While the forces that hold our world together (electromagnetic, weak, and strong interactions) are constrained to the 3+1-“flat” dimensions, the gravitational interaction always occupies the entire universe, thus allowing it to feel the effects of extra dimensions. Unfortunately, since gravity is a very weak force and since the radius of extra dimensions is tiny, it could be very hard to see any effects, unless there is some kind of mechanism that amplifies the gravitational interaction. Such a mechanism was recently proposed by Arkani-Hamed, Dimopoulos, and Dvali, who realized that the extra dimensions can be as large as one millimeter, and still we could have missed them in our quest for the understanding of how the universe works!

Of course these ideas are experimentally being challenged, like any good scientist would want of his theory. See EOT-WASH GROUP(4)

The Mathematikoi had Synesthesia?

Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism.

Later resurgence of ideas similar to those held by the early Pythagoreans are collected under the term Neopythagoreanism.

The Pythagoreans were called mathematikoi, which means "those that study all^1"

To say it is easy in knowing where to begin, is a understatement of what has been an enormous struggle to define the world around me. Indicative of the complications of how one may have seen this world in regards to the "views of a Synesthesist," would have taxed most "science minds" if they had "this inkling" of the complexity this brings to science. Think about what is implied here when one refers to "studying it all?"

So as I lay in the twilight hours of the mind's rest period, there are these things that I am asking of myself, as to how I may point to what is comparative in the "geometric views of science" and what is comparative to the views of that science in relation to examples given of the Synesthesist who sees from a certain position.

Again, my mind falls back in the history of humanities evolution and while the distinctiveness of sectors of that past history, it would not be unkind to draw from that history and present the question of what a Synesthesist might have seen in relation to the numbers?


Create and play with the most beautiful, hypnotic light illusions you have ever seen.

I seen the above in relation to Lubos's post. It would be nice to offer the "equation correlations" to these "colour displays" in string theory?:)

Numbers

Are you quicker then I then to see that numbers may have had the colour attached to their very nature, that "all things" then my have had this basis of "music" and "colour association" thrown "into the mix/cross over points"" to call it the Pythagorean?

So imagine being strapped with the job to start from some place, and move any mind to consider the complexity of "departing euclidean views" to meld with the "non-euclidean reality" assigned our everyday species to "what is natural" from straight lines and such. Has now moved to a dynamical world of "Faraday lines" Gauss's role as "teacher of Gaussian Co-ordinates" to views of his student, "Riemann?"


This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found.

Should one be so crude as to see that straight lines can have a "greater implication of design" that one would not have seen, had they not understood Gauss's work? That if you moved yourself to natures's domain, how many lines are really that straight?

Ask your self then what is natural and what was man-made? That these straight lines are indeed an order to mankind's "ode to building and living," while there are these "other worldly visions" supplied in the "non euclidean realm" existed free from man's definition of nature.

8.6 On Gauss's Mountains

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface. ²

The Interior Probabilities Manifests as Colour

How foolish would I be then to tell you that "Heaven' Ephemeral Qualities," are coloured to the degrees that "gravity defines itself in time?" That "model building" had to take place, so that the understanding of where this gravity explains itself, could find correlations to humans experiencing "durations of time" within in the living of day to day.

Again I move one back to what this "egg of fluttering does" as of physiological consequent, as the correlations of those same colours manifest in the qualities of those same thought patterns. Those experiences mapped to MRI imaging are condensible features "in the physical" do not explain the "Ephemeral Quality" assigned to each of these regions. Had one knew how to switch around the "value of consciousness" to the condensible feature as brain matter, one would have known about the happenings taking place "outside" of our bodies.

It is here to then that I take from the "metaphysical realm" and bring it into the relations of what is happening in the physical brain. While history has shown groups who gathered to see what was happening, saw "human experiencing" as they went through these colour modes.

¹ Hemmenway, Pryia – Divine Proportion pp66, Sterling Publishing, ISBN 1-4027-3522-7
² Reflections on Relativity8.6 On Guass's Mountain

Visual Abstraction to Equations

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

Some of you might have noticed the reference to the Ashmolean Museum?


Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.

It seems only the good scientist John Baez had epitomes the construction of the Platonic solids? A revision then, of the "time line of history" and the correction he himself had to make? Let's not be to arrogant to know that once we understand more and look at "the anomalies" it forces us to revise our assessments.

The Art form

I relayed this image and quote below on Clifford's site to encourage the thinking of young people into an art form that is truly amazing to me. Yes I get excited about it after having learnt of Gauss and Reimann's exceptional abilities to move into the non euclidean world.

Some think me a crackpot here? If you did not follow the history then how would you know to also include the "physics of approach," as well? Also, some might ask what use "this ability to see the visual abstraction" and I think this art form is in a way destined, to what was kept in glass cabinets and such, even while the glass cabinet in analogy is held in the brain/space of them) who have developed such artistic abilities.

It's as if you move past the layers of the evolution of the human being(brain casings) and it evolution and the field that surrounds them. Having accomplished the intellect( your equations and such), has now moved into the world of imagery. Closet to this is the emotive field which circumvents our perspective on the greater potential of the world in the amazing thought forms of imagery. This move outward, varies for each of us from time to time. Some who are focused in which ever area can move beyond them. This paragraph just written is what would be considered crackpot(I dislike that word)because of the long years of research I had gone through to arrive at this point.

Of course, those views above are different.

Mapping



Is it illusionary or delusional, and having looked at the Clebsch's Diagonal Surface below, how is it that "abstraction" written?

The enthusiasm that characterized such collections was captured by Francis Bacon [1, p. 247], who ironically advised "learned gentlemen" of the era to assemble within "a small compass a model of the universal made private", building

... a goodly, huge cabinet, wherein whatsoever the hand of man by exquisite art or engine has made rare in stuff, form or motion; whatsoever singularity, chance, and the shuffle of things hath produced; whatsoever Nature has wrought in things that want life and may be kept; shall be sorted and included.

There is no doubt that the long road to understanding science is the prerequisite to mapping the images from an equation's signs and symbols. While not sitting in the classroom of the teachers it was necessary to try and move into the fifth dimensional referencing of our computer screen to see what is being extolled here not just in image development, but of what the physics is doing in relation.

In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27

So of course to be the historical journey was established like most things, Mandelstam current and what is happening there as an interlude, as well as helping to establish some understanding of the abstractions that had been developed.



But yes, before moving to current day imagery and abstraction, I had to understand how these developments were being tackled in today's theoretical sciences.

Reaching for the Stars

Mars in 6 weeks? And back in a total of four months? That's the prediction of a design team working on antimatter rocket concepts at Pennsylvania State University. But first, you have to get the stuff - and store it. (PSU)

The popular belief is that an antimatter particle coming in contact with its matter counterpart yields energy. That's true for electrons and positrons (anti-electrons). They'll produce gamma rays at 511,000 electron volts.

But heavier particles like protons and anti-protons are somewhat messier, making gamma rays and leaving a spray of secondary particles that eventually decay into neutrinos and low-energy gamma rays.

And that is partly what Schmidt and others want in an antimatter engine. The gamma rays from a perfect reaction would escape immediately, unless the ship had thick shielding, and serve no purpose. But the charged debris from a proton/anti-proton annihilation can push a ship.

"We want to get as close as possible to the initial annihilation event," Schmidt explained. What's important is intercepting some of the pions and other charged particles that are produced and using the energy to produce thrust."

So our history here in this blog has detailed how we see the issues of "collision processes developed(Cern), that we may now see the cosmological playground teaming with the opportunities to produce this "stuff" that would send our spaceships to Mars?

The extension of the thinking of experimental development, has allowed us to think of "what is possible" and what this propulsion system can do, as we make our way into the new territories? As we set sail our ships, searching for those new lands.


A Penn State artist's concept of n antimatter-powered Mars ship with equipment and crew landers at the right, and the engine, with magnetic nozzles, at left.

Of course "storage" is always a troubling issue here so they developed what is call the Penning Trap. But it is not without some insight that our geometrical understanding developed in the events in the cosmos, could not be transformed in that same geometrical sense to propel those ships?


This "Penning trap" developed at Penn State University stores antiprotons.

It sounds like science fiction, but researchers are learning to create and store small amounts of antimatter in real-life labs. A portable electromagnetic antimatter trap at Penn State University, for example, can hold 10 billion antiprotons. If we could learn how to use such antimatter safely, we could impinge some on a thin stream of hydrogen gas to create thrust. Alternatively, a little antimatter could be injected into a fusion reactor to lower the temperatures needed to trigger a fusion reaction.

So you ask how is that possible?

The gravitational collapse sets up the very ideas for us as we make use of that "propulsion system" to move that space ship. So in a sense, "the collider process" at Cern is a gigantic model of what we want in the developmental process as the new engine of our spaceship.


A schematic of the heart of a Penning trap where a cloud of antiprotons (the fuzzy bluish spot) is kept cold and quiet by liquid nitrogen and helium and a stable magnetic field. (PSU)

Anti-protons, explained Dr. Gerald Smith of Pennsylvania State University, can be obtained in modest quantities from high-energy accelerators slamming particles into solid targets. The anti-protons are then collected and held in a magnetic bottle

.

While previously here I have spoken about how we may use Susskind's thought experiment as a monitoring system of gravitational considerations, it is also this thought process that helps us adjust the ship according to how much thrust is needed in face of the lagrangian views we encounter in star systems?

However, by using "matter/antimatter annihilation", velocities just below the speed of light could be reached, making it possible to reach the next star in about six years.

I think Stephen Hawking is going to have to work faster, in order to elucidate his thoughts on this travel. That while I may have started this lesson from the idea of 1999, it is much more advanced then many had understood. The "experimental process" of Cern is much greater then most of us had realized.

Also there is a developmental "thought pattern" that needs to be understood as we speak about how such a geometrics could have been seen underneath the very structures of our realities. Not only within the cosmos at large but in the dynamical processes of the quantum world.

Angels and Demons



Cern IMagery takes a "dramatic position" on what it is saying about itself? :) I would like to think that the fun is in how "mirror world" has somehow been transposed into what we know of the develpmental processes we are given as we now lok at what may help us move into the cosmos.

If as a society we were "uncultured" we might have thought the tribal influence of the "bad side" of all things? But in that exploratory sense al the tidbits had to add up to something, yet without our understanding of what lies beneath, one might have never gone "past" Robert Mclaughlin, to realize, the geometrical nature that imbues the process we are developing.



This was Riemann lesson to Gauss in his thesis, who like his student had thought for sure "vision capable now," would also have been transferred into a "whole new world" of understanding of the non euclidean geometries.

What do they say about the devil being in the details?

Plato:

This image had horns drawn on it, with a tail attached. Something about “angels and demons?” I don’t think we should take the “anti” too literal in face of an outcome, or should we?

It's about how we can take a legitimate process and build ideas on it, according to the very nature of the "negative and positive expressions" of what Riemann set out to do.

ON a large scale, we see the dynamics of this process, yet failed to see it work at a microcosmic sense as we deal with the colliders? As we move forward in the propulsion systems, it is importance how we see this developmental process take on dynamic views.

Types of Blogging Software

Ask yourself this? What is the new kernel to be, if we had for one moment presented the opportunities for the using Riemann hypothesis, and contained the very idea as a philosophy presented within this blog?

A VIEW OF MATHEMATICS by Alain CONNES

Each generation builds a mental picture" of their own understanding of this world and constructs more and more penetrating mental tools to explore previously hidden aspects of that reality.

Would such a "paradigmal change" allow for insightual software development to take a turn for the better if the understanding existed, that one had already left the cave, and saw the aspects of probable outcomes, as more then the primes and it's integrations with physics mentality, along with theoretical development?

Micro-quantum structures that are exemplfiled, in Monte Carlo methods?

Are we "FREE" to Express?

While I have enjoyed the blogging experience of Blogger.com, and the integration of development that had been going on, the questions remain, as to where this information is deposited and how the moderation of "such a tool" is enforced?

Like many important concepts, Web 2.0 doesn't have a hard boundary, but rather, a gravitational core. You can visualize Web 2.0 as a set of principles and practices that tie together a veritable solar system of sites that demonstrate some or all of those principles, at a varying distance from that core.

I have a certain ideology about trying to bring together as much information as possible, by asking, if image linking, and phrase connections, do not involve copyright infringements, and allow the versatility of blogging experience, while respecting the owners of images and wording, while connected directly to their source.

Linux is subversive. Who would have thought even five years ago (1991) that a world-class operating system could coalesce as if by magic out of part-time hacking by several thousand developers scattered all over the planet, connected only by the tenuous strands of the Internet?

Certainly not I. By the time Linux swam onto my radar screen in early 1993, I had already been involved in Unix and open-source development for ten years. I was one of the first GNU contributors in the mid-1980s. I had released a good deal of open-source software onto the net, developing or co-developing several programs (nethack, Emacs's VC and GUD modes, xlife, and others) that are still in wide use today. I thought I knew how it was done.

Linux overturned much of what I thought I knew. I had been preaching the Unix gospel of small tools, rapid prototyping and evolutionary programming for years. But I also believed there was a certain critical complexity above which a more centralized, a priori approach was required. I believed that the most important software (operating systems and really large tools like the Emacs programming editor) needed to be built like cathedrals, carefully crafted by individual wizards or small bands of mages working in splendid isolation, with no beta to be released before its time.

This has been on my mind as I brought together many aspects of the information that is out there. From the respectable information posted by scientists and their personal experiences, to those shared by all, through such blogging experiences. So what was the battle brewing about from those early days and the struggle to develope communities, sharing information, and who are these people today?

AOL=Netscape? Microsoft? Google? Yahoo?

How would such blogging experiences allow the movement forward of society, and the thinking brain, this internet has become?

Are there concerns, that the human being once exposed to the vastness of this information, could bring it together in such a way, as to insight the "new idea" that would forward research and developement? Encourage our minds to percieve in other ways that we are not accustom? I gave an example at the very beginning of this post in regards to the Riemann Hypothesis.


Witten:

One thing I can tell you, though, is that most string theorist's suspect that spacetime is a emergent Phenomena in the language of condensed matter physics.

This is important to ask, because if such an ability is focused through the individuals efforts using such a medium, how could/would it be exploited, that it could be brought to the forefront of the "thinking brain/internet" and find indeed, that such information is useful?

Meddle then in the internal structure and enforce the rights of deposition as to the respository, and deal with it as you like?

The information depository costs money, I know? Image transference costs money. Then how shall "the dream of the thinking mind" ask, that if the repositories are the resources held in abeyance, until used as seen fit, then why not/should disrupt the information gathering and make it disjointed, while we/you look at it? Before it reveals it's state secret? An open society, right? People who are free?

Robert Laughlin:

Likewise, if the very fabric of the Universe is in a quantum-critical state, then the "stuff" that underlies reality is totally irrelevant-it could be anything, says Laughlin. Even if the string theorists show that strings can give rise to the matter and natural laws we know, they won't have proved that strings are the answer-merely one of the infinite number of possible answers. It could as well be pool balls or Lego bricks or drunk sergeant majors.

This would mean that the very ideas of the internet explosion and control of it becomes in question, as well as, the provders we use to express ourselves on the internet?

How will these repositiories change then in technologies that you and I are very quickly connected in ways that the human mind/internet becomes quite capable of seeing, in ways it is not accustom?

What revolution/paradigmalchange will then happen, that the very experiences we now enjoy, will be defuncedt with all the software solutons to metigate the ability for the individual to do what any of us can do freely, without any ofthe blogging software now demonstrated below?

Shall we choose carefully, read the requirements of, and what conclusion have you reached?

b2evolution

bBlog

Blogger

Bloxsom

Blojsom

Drupal

ExpressionEngine

Geeklog

Greymatter

iUpload

LifeType

LiveJournal

Movable Type

MvBlog

Nucleus CMS

PostNuke

Roller Weblogger

Serendipity

Slash

TypePad

Typo

TYPO3

WordPress

Xanga

If we are looking for the new "idea" where shall it arise from then? It is apparent that the early thinking in cosmology has been changed(to include strings ina time sequence of events evn thoguh they be micro seconds) and so too, the values of measure in "time," recognized as problematic, in terms of it's discrete value, when it is very well understood that continuity of expression can be very smooth(yet is it?)?

The count of Primes begins in Chaos. If we were to think of the Riemann Hypothesis assigned to a scale as an approximation to the prime distribution function, then how woud any pattern suffice to be an "emergent property" of that chaos?

What a Good String Theorist Should Know?

Arthur Miller
Einstein and Schrödinger never fully accepted the highly abstract nature of Heisenberg's quantum mechanics, says Miller. They agreed with Galileo's assertion that "the book of nature is written in mathematics", but they also realized the power of using visual imagery to represent mathematical symbols.

I am a bit of a fanatic when it comes to the visualizations. What benefit might these have for any good theorist? What creative ability is developed, when one sees this way?

To me, as it has been described with Dirac wording that I have spell out many a time, there is also all this "other information" that has to be followed up. I know it. Many science people know it. Maybe sometimes, caught up in all the aspirations for truth, I might not remember it. So this post is here for this purpose.

You have to trust me that I will not be knocking on any good scientists door, being the crackpot that I am, with some amazing discovery.I just don't have time to bother you good science people.:)

Anyway, I thought I should clear up some ideas people have about learning. Getting some insight into what is being talked about in regards to theoretical ideas being borne, what learning the older folk like me can look forward too. The last part of this post is in regards to Think Quest comments on string theory.

Personally, I think a good theoretician needs to know a lot.

I found information provided by Gerard t’ Hooft which gives one a a good base to what he thought we should be doing. So I wanted to include some of that here as well. Also by including each of the links, typing into the "search fucntion," this post, should come up, and the related subjects, as to what should be known.

I created one on the requirements of mathematics sometime ago as well so this would be a good source link as well to the requirements needed to work within the string theory realm. I am still looking for it. You cna see now why this post is good for memory retention being somewhat lost as to where it is put under.

Is your motivation and pursuance of knowledge up to it?

HOW to BECOME a GOOD THEORETICAL PHYSICISTby Gerard 't Hooft

Theoretical Physics is like a sky scraper. It has solid foundations in elementary mathematics and notions of classical (pre-20th century) physics. Don't think that pre-20th century physics is "irrelevant" since now we have so much more. In those days, the solid foundations were laid of the knowledge that we enjoy now. Don't try to construct your sky scraper without first reconstructing these foundations yourself. The first few floors of our skyscraper consist of advanced mathematical formalisms that turn the Classical Physics theories into beauties of their own. They are needed if you want to go higher than that. So, next come many of the other subjects listed below. Finally, if you are mad enough that you want to solve those tremendously perplexing problems of reconciling gravitational physics with the quantum world, you end up studying general relativity, superstring theory, M-theory, Calabi-Yau compactification and so on. That's presently the top of the sky scraper. There are other peaks such as Bose-Einstein condensation, fractional Hall effect, and more. Also good for Nobel Prizes, as the past years have shown. A warning is called for: even if you are extremely smart, you are still likely to get stuck somewhere. Surf the net yourself. Find more. Tell me about what you found. If this site has been of any help to someone while preparing for a University study, if this has motivated someone, helped someone along the way, and smoothened his or her path towards science, then I call this site successful. Please let me know. Here is the list.

Languages

Primary Mathematics

Classical Mechanics

Optics

Statistical Mechanics and Thermodynamics

Electronics

Electromagnetism

Quantum Mechanics

Atoms and Molecules

Solid State Physics

Nuclear Physics

Plasma Physics

Advanced Mathematics

Special Relativity

Advanced Quantum Mechanics

Phenomenology

General Relativity

Quantum Field Theory

Superstring Theory

Think Math

While I quickly jumped to the end of the third page of reference below, it summarizes a bit as to what culminations might be found with the math in all it's aspects describe as the language. The language(herein described as the math), brings it together nicely. Whole.

Guide to Math, by Superstringtheory.com
Noncommutative geometry (NCG for short)

Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.

Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.

In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.

The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.

While the truer quest of seeing is in the world of mathematics used besides english, is the real language of commonality among scientists. It serves them well to understand how all these maths could add up too, what is required of those students of youth, and youth of mind of those advacing in age, that we see this described someplace.


Nature's patterns

So who is right? Well, there is much that is attractive in the Platonist point of view. It's tempting to see our everyday world as a pale shadow of a more perfect, ordered, mathematically exact one. For one thing, mathematical patterns permeate all areas of science. Moreover, they have a universal feel to them, rather as though God thumbed His way through some kind of mathematical wallpaper catalogue when He was trying to work out how to decorate His Universe. Not only that: the deity's pattern catalogue is remarkably versatile, with the same patterns being used in many different guises. For example, the ripples on the surface of sand dunes are pretty much identical to the wave patterns in liquid crystals. Raindrops and planets are both spherical. Rainbows and ripples on a pond are circular. Honeycomb patterns are used by bees to store honey (and to pigeonhole grubs for safekeeping), and they can also be found in the geographical distribution of territorial fish, the frozen magma of the Giant's Causeway, and rock piles created by convection currents in shallow lakes. Spirals can be seen in water running out of a bath and in the Andromeda Galaxy. Frothy bubbles occur in a washing-up bowl and the arrangement of galaxies.

Imagine calling someone with this background "flaky" because of a "strange idea" that might be borne in mind, while it is encompassed by all this knowledge of science, respectively? People who had been well intentioned, hiding all the information because they might have been taunted by those who were not respectful of the age of reason, with which they had applied them self.

I think every teacher, Mother, Father understands the best they have for their student, child respectively, and what they strive to encourage in regards to the independence and strength, to move forward with the motivation that is borne in every good seeker of truth?


ThinkQuest
Think Quest is all about students thinking and learning together. Students work in teams to create the best educational websites and compete for exciting prizes, including a trip to Think Quest Live, an educational extravaganza celebrating their achievements.

Sponsored by the Oracle Education Foundation, the competition offers a unique project-based learning experience to students and teachers around the world. Globally relevant subjects and diverse teams are encouraged.
The teams' websites are published for the world to see in the Think Quest Library. This rich online resource contains over 5,500 educational websites, created by students for students. Search the library and you'll be sure to find a site that intrigues you.

Information Links Below Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.

The History of String Theory:

Introduction to String Theory:

Gravity and String Theory:

Supersymmetry:

The Dimensions of String Theory:

Dimensions, Wound Strings, Branes, and Calabi-Yau Spaces:

The Many Types of String Theory:

New Developments in String Theory:

Well so easily explained in the english language, Gerard's comments about explaining what we are doing now bears fruit? My inept capilities with this of courses draws recognition, let alone, the need to write those visionary qualities to algebraic equations. So Penrose has more words for us, besides his change of heart?:)

You think it easy to change the ingraininess of our methods that we should let them drop away easily? Find a new path/math with a heart? It is not without thinking that such decisions are made.

[ROGER PENROSE]

"One particular thing that struck me... [LAUGHTER]...is the fact that he found it necessary to translate all the results that he had achieved with such methods into algebraic notation. It struck me particularly, because remember I am told of Newton, when he wrote up his work, it was always exactly the opposite, in that he obtained so much of his results, so many of his results using analytical techniques and because of the general way in which things at that time had to be explained to people, he found it necessary to translate his results into the language of geometry, so his contemporaries could understand him. Well, I guess geometry… [INAUDIBLE] not quite the same topic as to whether one thinks theoretically or analytically, algebraically perhaps. This rule is perhaps touched upon at the beginning of Professor Dirac's talk, and I think it is a very interesting topic."

A more direct link to quote above on page 12.