Inside Out

3.1 As Cytowic notes, Plato and Socrates viewed emotion and reason as in a kind of struggle, one in which it was vitally important for reason to win out. Aristotle took a more moderate view, that both emotion and reason are integral parts of a complex human soul--a theory proposed by Aristotle in explicit opposition to Platonism (De Anima 414a 19ff). Cytowic appears to endorse the Platonic line, with the notable difference that he would apparently rather have emotion win out.

I am trying to "create a image" that will use the one above. It is important that the select quoted comment below is understood. This can't be done without some reference.

So while the exercise may be going on "inside" things are happening on the outside. Scientists have never been completely honest with themselves, while some may concern themselves with whose name said what?

I use Plato as a namesake obviously, because of what I saw of some of our influential minds speaking, all the while making inferences to Plato. When ever you read something that resonates with you, it is of value because it correlates to something that you already know. This is what I tried to get across in the previous post, about what is "self evident." Little do some people recognize that while I may have inferred the point of some philosophical foundations, it is not without recognizing that the "qualitative phrases" have to be reduced as well to a logic. To reason.

How do you do that? Well I'll tell you what I found and then you can think whether I understood reason in it's proper format. Whether I understood the "shadows of Plato" to mean something other then what could have been interpreted as being wrong. What is that analogy of the Cave really mean?

Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.

Yes I did not enter the halls of higher learning in the traditional ways. You can converse for many years, does not mean you become devoid of the lessons that spoken amongst the commentors. How is it you can think that while listening to scientists you cannot uncover the the processes they use? If I had given thirty years to study, what exactly had I studied? I am a doctor of nothing.:)

This is a torus (like a doughnut) on which several circles are located. Unlike on a Euclidean plane, on this surface it is impossible to determine which circle is inside of which, since if you go from the black circle to the blue, to the red, and to the grey, you can continuously come back to the initial black, and likewise if you go from the black to the grey, to the red, and to the blue, you can also come back to the black.

My quote at Backreaction on this and that, reveals not only part of the understanding gained through this "infinite regress," but also the understanding we have with the world around us. Some would be better served to see the image of the Klein bottle, but I wanted to show what is going on in a "abstract way" to what is happening inside of us, and at the same time, what is happening outside.



I had used the brain and head as a place of our conscious awareness within context of our environment, our bodies. The topological explanations of the numbers above, and used them in the next paragraph. There will be confusion with the colour lines, please disregard that.

While I talked of the emotive and mental realities. I included the spiritual development in the end. The way this interaction takes place, is sometimes just as the mental function(yellow). Other times, it is the emotive realization of the experience. It is coloured by our emotion(red).

While we interact with our environment, there is this turning inside out, continuously. Sometimes we may say that "1" is the emotive realization, while the number 2 is seen as a mental extension of the situation. While the areas overlap each other, an outward progression may mean that the spiritual progress is numbered 4, while the interaction of the emotive, mental and spiritual progression may be number 3. Ultimately the spiritual progression is 4 (Violet). All these colours can mix and are significant in themself. They reveal something about our very constitution.

While some may wonder how could any conceptualization ever integrate the "Synesthesia views" of the world when it sees itself presented with such a comparison? The journey of course leads to the "Colour of Gravity." Discard your body, and one will wonder about the "clear light." What it means, in the "perceptive state of existence." If one is prepared, then one shall not have "to much time on their hands" getting lost in the fog.

Plato and Aristotle, Up and Down by Kelley L. Ross, Ph.D.

Rafael has Plato pointing up and Aristotle gesturing down to indicate the difference in their metaphysics. For Plato, true existence is in the World of Forms, in relation to which this world (of Becoming) is a kind of shadow or image of the higher reality. Aristotle, on the other hand, regards individual objects in this world as "primary substance" and dismisses Plato's Forms -- except for God as a pure actuality, without matter.

However, when it comes to ethics and politics, the gestures should be reversed. Plato, like Socrates, believed that to do the good without error, one must know what the good is. Thus, we get the dramatic moment in the Republic where Plato says that philosophers, who have escaped from the Cave and come to understand the higher reality, must be forced to return to this world and rule, so that their wisdom can benefit the state. Aristotle, on the other hand, says that the "good" is simply the goal of various particular activities, without one meaning in Plato's sense. The particular activities of most human affairs involve phronésis, "practical wisdom." This is not sophía, true wisdom, for Aristotle, which involves the theoretical knowledge of the highest things, i.e. the gods, the heavens, and God.

Thus, for philosophy, Aristotle should point up and would represent a contemplative attitude that was certainly more congenial to religious practices in the Middle Ages. By the same token, Aristotle's contribution to what we now think of as science was hampered by his lack of interest in mathematics. Although Aristotle in general had a more empirical and experimental attitude than Plato, modern science did not come into its own until Plato's Pythagorean confidence in the mathematical nature of the world returned with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a theory of opposites that is now only of historical interest, rejected Plato's attempt to match the Platonic Solids with the elements -- while Plato's expectations are realized in mineralogy and crystallography, where the Platonic Solids occur naturally.

Therefore, caution is in order when comparing the meaning of the metaphysics of Plato and Aristotle with its significance for their attitudes towards ethics, politics, and science. Indeed, if the opposite of wisdom is, not ignorance, but folly, then Socrates and Plato certainly started off with the better insight.

It is good that you go to the top of the page of the linked quotes of Kelley L. Ross. You must know that I developed this site without really understanding the extent Mr. Ross had taken this issue. There is much that is familiar, and with him, an opposing view too.

See:

Induction and Deduction
Intuitively Balanced: Induction and Deduction

Self Evident Truths

I am convinced that those societies [as the Indians] which live without government enjoy in their general mass an infinitely greater degree of happiness than those who live under European governments.
-- Thomas Jefferson to Edward Carrington, 1787

There is a process that I had come to recognize when I look back to historical perspective that was important to me. Alain Connes defined it in terms of mathematics for me when he said,"the basic concepts of mathematics are infinitely closer to an indecomposable element", a kind of elementary particle" of thought with a minimal amount of ambiguity in their definition."

A VIEW OF MATHEMATICS Alain CONNES

Most mathematicians adopt a pragmatic attitude and see themselves as the explorers of this mathematical world" whose existence they don't have any wish to question, and whose structure they uncover by a mixture of intuition, not so foreign from poetical desire", and of a great deal of rationality requiring intense periods of concentration.

While I myself am not versed in the number of mathematical systems out there, it was a journey to select the quotes and have these quotes direct my thinking, by comparative and correlative understandings. One had to have some core value system in order to recognize the "essential element of paradigmatic changes" that theoretical positions undergo when mathematical abstracts are introduced to creating perspective change in any system that is already in place.

Nature's Greatest Puzzle by Chris Quigg-SLAC Summer Institute August 2, 2004

While the thrust is experimental in design, there had been a move away from theoretical principle? If you would like a Michael Shermer of the science world to be represented by a man of Woitian design who thinks same, it would have been far better to listen to those who will acknowledge string theories mathematical construct for what it is.

It is a reductionistic one, and pushes perspective farther back then Steven Weinberg's first Three Minutes. That is the lesson, that while the very idea of the matter defined states of existence, along side of those massless entities through collisional processes, we are looking to describe the interactions. It is also asking for such theoretical thinking to move itself ever closer to that "indecomposable element."

Mind Maps: Mathematical Constructs

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. Arthur Miller

The journey is an arduous one when one ventures ever close to that indecomposable element. That is the very exercise one should be interested in. Again, I would have had to listen to all those rejected from the validation process, to hear them say, "why not my exercise in mathematic judgement?" Something is wrong with it. The system, and not my construct of it? Hey, I hear you, what the heck do I know:)

I had to know something Qui? I spot the very idea of Liminocentric structures to ponit out that a philosophical take is driving someone else's visions of thinking. No it is not Heliocentricity. I take this position as well because I recognize something. Okay, I am not a Garrett Lisi, whose derivations to all degrees of freedom, and the emergent process ultimately is defined in all elemental qualities to it's finality.:)

But you see this is the point of using "the vision" and thinking about what is self evident. I advocated Susskind's rubber band image of mind, after intense efforts to deduction. What value this "aha moment" when it is a leap of faith? It arose out of deeper place that talks to that structured faculty of the mind to explore the reaches within.

And I fiddled with it, I monkeyed with it. I sat in my attic, I think for two months on and off. But the first thing I could see in it, it was describing some kind of particles which had internal structure which could vibrate, which could do things, which wasn't just a point particle. And I began to realize that what was being described here was a string, an elastic string, like a rubber band, or like a rubber band cut in half. And this rubber band could not only stretch and contract, but wiggle. And marvel of marvels, it exactly agreed with this formula.

I was pretty sure at that time that I was the only one in the world who knew this. LEONARD SUSSKIND:

So you see there is a deductive process and an intuitive one that is in place that I am speaking too. It of course recognizes "the reason" with which such a deduction should take place. That it leads to to what is "self evident." From there, that time away from intense thinking allows for the limits of our reasoning, a look into the future.

Strip yourself of your Ego. Strip yourself of your name. How naked then, that any possession means little. Think then, that what comes into mind is free, and like a pebble widens as rings. While on the soapbox and standing there, you then recognize how unimportant one's name is. It is coming to see, the finer parts of one's constitution that we finally see who the person is.

See:

What a Good String Theorist Should Know?

Heralded from the 21st Century: String Theory

Winter Shangrila

As to the date stamped on the pictures these are the correct ones. Mother Nature has sent us about another ten inches since then.



As some of you know, I spent about seven months in 2007 building this house with my son and wife. We moved into it in late November, and was not able to complete some of the projects that needed to be completed.

The most obvious of course is the siding that will be started, when I have poured the aggregate walkaways. I had prepared these before winter rolled in. I could not get my finisher to come in and complete because of a large contract he had an option on that superseded my work if he obtained. He let me know of course, and fighting time with Winter coming I tried my best. A spring/summer job then.



After the foundation was done and before we started on the house construction, we brought in a 1200lb compactor to make sure that before any concrete was pour inside and out, the future walkways, this process was done. Also to build it up I brought in 2 10 yard truckloads of crush to build it up and pounded again, for levelling and to give it the height in relation to the house.

In construction, there are three main types of compactor; the plate compactor, the "jumping jack" and the road roller. The roller type compactors are used for compacting crushed rock as the base layer underneath concrete or stone foundations or slabs. The plate compactor has a large vibrating baseplate and is suited for creating a level grade, while the jumping jack compactor has a smaller foot. The jumping jack type is mainly used to compact the backfill in narrow trenches for water or gas supply pipes etc. Road rollers may also have vibrating rollers.

If you notice I had some rock work done on the pillars and the front of the house.

Cultured Stone HistoryOwens Corning Cultured Stone products originated in 1962 when brothers Garrett and Floyd Brown of Vallejo, California saw the need for a new kind of building material.

All around this house is stones of similar size, but we went and used a concrete product that looks much like those same rocks. While constructing the foundation, the plans did call for a 4 inch ledge to be in place for these rock additions, but it would have been to cumbersome to use the natural rocks and stone around here. I brought in an old Italian mason to do the work. We were very pleased.

Wildlife

Moose (Alces alces)

This was captured outside our garage door by my wife. I happened to be on the other side of the house, when I heard the dogs barking, and my wife telling me to come quick.

Over the years we've viewed many Moose on properties that we had owned, and on other lands.

Unfortunately, we have run into about four over the years with varying degrees of cost to vehicle repairs. The "plus side" is that we have not been hurt in these collisions. Were they unavoidable? Having this country life and going to work at the dark hours have made myself mostly a candidate for such incidents. Weather does dramatically place a part in this, and not being totally observant to the environment would have to be included.

Pileated Woodpecker (Dryocopus pileatus)

This picture is taken from one of the locations my wife feeds the birds. There are about four different types of birds that we have identified.

Evening Grosbeaks(Coccothraustes vespertinus)Yellow are males, females greyer.

These second two pictures are from the second location. At this second location there are other mouths that are being feed that I will put up shortly.

Pine Grosbeak(Pinicola enucleator)Females are gray with two white wing bars. Male is pink to red with two white bars

This little fellow below is a frequent visitor, along with many others who fight over territory amongst the trees. They have learnt to sometimes sit amongst the birds okay, while other times they make it their mission to keep each other from the feed.

American Red Squirrel(Tamiasciurus hudsonicus)

Understanding Nature

While people even like myself may talk about it, have we really learnt of the differences in the animal world and the characteristics to each of the species? There is something primordial about the animal per say, yet, within context of the groups that they form, how well do we understand it's culture. Would it be wrong to assign such thing as "cultured" when a civilized society would seem more apt to such an expression?

When you spend years with different animals you get to learn some of the things that are unique to them, as well as being the recipient of an animal wrath. Unbeknownst even to them, this primordial call of the wild, will inhibit even the most loyal animal, when it is ready to show it's colours for the opposite sex. I know by not being aware and walking in front of the line the animal paces, I did not realize how a bite on the side could hurt. That was my mistake for underestimating.

So to the mothering instinct of the newborn, that a mother will protect. The very land becomes it's home and being marked, will become the territory of it's babies. How close to that zone, and how much a warning given? After the babies are grown and gone what is it that leaves mothers to believe that they are still protecting their home.



That one would think Horses would show their very nature and we think the males dominant, when the very order is selected by the mare? So too, I would like to fool them when they are young and born to this world that the imprint and betrayal of my size, would have them believe at 1200lbs I could still pick them up. They are always still wary then. But then, that is history. They were sent to their rightful place up north.


See:

Our Local Wildlife

The History of Magnetic Vision

Where Hockey Started, and Horses Live

WE ARE A HEAVENLY FLOWER

AND we should consider that God gave the sovereign part of the human soul to be the divinity of each one, being that part which, as we say, dwells at the top of the body, inasmuch as we are a plant not of an earthly but of a heavenly growth, raises us from earth to our kindred who are in heaven. And in this we say truly; for the divine power suspended the head and root of us from that place where the generation of the soul first began, and thus made the whole body upright. When a man is always occupied with the cravings of desire and ambition, and is eagerly striving to satisfy them, all his thoughts must be mortal, and, as far as it is possible altogether to become such, he must be mortal every whit, because he has cherished his mortal part. But he who has been earnest in the love of knowledge and of true wisdom, and has exercised his intellect more than any other part of him, must have thoughts immortal and divine, if he attain truth, and in so far as human nature is capable of sharing in immortality, he must altogether be immortal; and since he is ever cherishing the divine power, and has the divinity within him in perfect order, he will be perfectly happy. Now there is only one way of taking care of things, and this is to give to each the food and motion which are natural to it. And the motions which are naturally akin to the divine principle within us are the thoughts and revolutions of the universe. These each man should follow, and correct the courses of the head which were corrupted at our birth, and by learning the harmonies and revolutions of the universe, should assimilate the thinking being to the thought, renewing his original nature, and having assimilated them should attain to that perfect life which the gods have set before mankind, both for the present and the future.

Plato from Timaeus, 90a-d, translated by B. Jowett

Mind Maps: Mathematical Structures?

Plato's doctrine of recollection, however, addresses such criticism by saying that souls are born with the concepts of the forms, and just have to be reminded of those concepts from back before birth, when the souls were in close contact with the forms in the Platonic heaven. Plato is thus known as one of the very first rationalists, believing as he did that humans are born with a fund of a priori knowledge, to which they have access through a process of reason or intellection — a process that critics find to be rather mysterious

What are Mind Maps?

A mind map is a diagram used to represent words, ideas, tasks or other items linked to and arranged radially around a central key word or idea. It is used to generate, visualize, structure and classify ideas, and as an aid in study, organization, problem solving, decision making, and writing.

It is an image-centered diagram that represents semantic or other connections between portions of information. By presenting these connections in a radial, non-linear graphical manner, it encourages a brainstorming approach to any given organizational task, eliminating the hurdle of initially establishing an intrinsically appropriate or relevant conceptual framework to work within.

A mind map is similar to a semantic network or cognitive map but there are no formal restrictions on the kinds of links used.

The elements are arranged intuitively according to the importance of the concepts and they are organized into groupings, branches, or areas. The uniform graphic formulation of the semantic structure of information on the method of gathering knowledge, may aid recall of existing memories.

Well straight to the point then I guess.

Bee:

The important thing about the basis of our societies is not actually its fixed structure but the way to readjust it. A bit of scientific method would be good there.

As I mentioned previously on Backreaction site and in giving subsequent information about this process. It has been a journey of my own discovery, that I would say that at the basis of reality is such a mathematical structure.

I know when this process started for me and it would not serve any purpose at this point to speak to it directly. People have their reasons for and against such a proposal as their being such a mathematical structure, so what currently leads me to say that their are these two opposing views?


Wigner’s Gift Horse By JULIE REHMEYER • Feb 1, 2008 See here for article.

Stephen Wolfram argues that the way to unlock the rest of science is to give up on mathematics and look for explanations analogous to computer code. Very simple computer programs can produce remarkably complex behavior that mimics phenomena science has had difficulty modeling, like the motion of fluids, for example. So studying the behavior of these programs may provide scientists with new insights about these phenomena. Indeed, Wolfram thinks the universe itself may be generated by a computer program simple enough to be expressed in a few lines of code. “If the laws are simple enough, if we look in the right way we’ll find them,” he says. “If they’re not, it will be tougher. The history of physics makes one pessimistic that we could ever end physics. I don’t share that pessimism.”

Tegmark believes in an extreme form of Platonism, the idea that mathematical objects exist in a sort of universe of their own. Imagine that, Tegmark says, “there’s this museum in this Platonic math space that has these mathematical objects that exists outside of space and time,” Tegmark says. “What I’m saying is that their existence is exactly the same as a physical existence, and our universe is one of these guys in the museum.”

Also worth reading is the sum of any position that would infer the stance of Plato versus anti-Plato, to help distinguish whether or not one might have something of value in terms of the question of whether Mathematics is invented or discovered.

Mathematical Platonism and its Opposites by Barry Mazur January 11, 2008. See here.

For the Platonists. One crucial consequence of the Platonic position is that it views mathematics as a project akin to physics, Platonic mathematicians being—as physicists certainly are—describers or possibly predictors—not, of course, of the physical world, but of some other more noetic entity. Mathematics—from the Platonic perspective—aims, among other things, to come up with the most faithful description of that entity.

For the Anti-Platonists. Here there are many pitfalls. A common claim, which is meant to undermine Platonic leanings, is to introduce into the discussion the theme of mathematics as a human, and culturally dependent pursuit and to think that one is actually conversing about the topic at hand.

Mapping the interaction from a scientific point of view?

As I read through the article I had previous insights while reading through Sir Roger Penrose's lecture on the Extended Physical WorldView. While I myself had picked the title, it would have been nicer to show the very image on the start of that lecture.



This is a important statement I am making below because it distinguishes between where we think we might be going in terms of computer technologies versus what will always remain within the human domain.

So on the one hand one might think about technologies in the 21st Century and wonder if computer technology can ever reach the status of Consciousness with which the "synaptic event" could include images, all the while it would include all the history to that point?

While it is never clear to me about the origins of the universe, it had some relation in my mind to what first allowed any soul's expression. While I had shown the relation to the synaptic event, there had to be a place created for such an expression, to be fortunate and validated.

Do I know what plan for every individual is, of course not, but that you choose such an expression is self evident. There is much to the word, "self evident" that remains to be explored within context of this site, and of value, in the iconic image of Raphael's expression with Plato and Aristotle at it's centre.

Now, neuronic networking is supposedly the platform computer technologies can take in their designs, but what true aspect of the emergent process could ever define the human being and one's potential? The information that could enter such an synaptic event within your own thinking mind?

So the process is one of self discovery. About processes within your own self that allow one to possibly develop the new mathematics that speak directly to the very unfolding of the universe?

The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner

The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

[3 M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: "All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting" (p 188).]

What can arise from any person, to have defined that it reaches back into the forms. That it chooses to manifest "as the person" you have become? What lies dormant, that you awake to it in such a way that it will manifest in all that you do, and become part of the "history of recollection" that you unfold in this life, and have learnt these things before? You dream?

I, Robot:

....signs of new life emerge as images photonically flicker in the new logic forming apparatus....

I had a dream....

We might like to think that computers are capable, while the very idea of the "image" holds such vasts amount of information. This is not a new idea from an historical perspective if one ever thought to consider the alchemists of our early science.

How would you contain all the probability and outcomes, with ever looking beyond space and time, to realize that the "heavens" in some way, meet the earth. Manifest within you? Can find re-birthing through you? Inner/outer become one.

Phase Transitions and the Unparticles

Another Odd Thing About Unparticle PhysicsHoward Georgi

The peculiar propagator of scale invariant unparticles has phases that produce unusual patterns of interference with standard model processes. We illustrate some of these effects in e+e− → µ+µ−.

To be a bit more precise, every space that feels "real" has associated with it a sense of distance between any two points. On a line segment like the Koch coastline, we arbitrarily chose the length of one side of the first iterate as a unit length. On the Euclidean coordinate plane the distance between any two points is given by the Pythagorean theorem

Fractals have a special property that underpins the odd features of unparticles. They are patterns that are "scale invariant". In the case of Britain, the ins and outs of the coastline contain smaller jagged features that in turn contain smaller ins and outs and so on. This characteristic means that these unparticles don't change appearance when viewed at different scales- which is very different from objects we're familiar with.

Wrapping the Energetic Particles to the Cosmos

The Origin of the Highest Energy Particles
Date: 2008-01-09 14:00:00

Abstract:

After almost a century of observations, the ultra-high energy sky has finally displayed an anisotropic distribution. A significant correlation between the arrival directions of ultra-high cosmic rays measured by the Pierre Auger Observatory and the distribution of nearby active galactic nuclei signals the dawn of particle astronomy. These historic results have important implications to both astrophysics and particle physics.

See:http://pirsa.org/08010000/

Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

As I pounder the very basis of my thoughts about geometry based on the very fabric of our thinking minds, it has alway been a reductionist one in my mind, that the truth of the reality would a geometrical one.



The emergence of Maxwell's equations had to be included in the development of GR? Any Gaussian interpretation necessary, so that the the UV coordinates were well understood from that perspective as well. This would be inclusive in the approach to the developments of GR. As a hobbyist myself of the history of science, along with the developments of today, I might seem less then adequate in the adventure, I persevere.




On the Hypotheses which lie at the Bases of Geometry.
Bernhard Riemann
Translated by William Kingdon Clifford

[Nature, Vol. VIII. Nos. 183, 184, pp. 14--17, 36, 37.]

It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.

From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space - the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.

For me the education comes, when I myself am lured by interest into a history spoken to by Stefan and Bee of Backreaction. The "way of thought" that preceded the advent of General Relativity.


Einstein urged astronomers to measure the effect of gravity on starlight, as in this 1913 letter to the American G.E. Hale. They could not respond until the First World War ended.

Translation of letter from Einstein's to the American G.E. Hale by Stefan of BACKREACTION

Zurich, 14 October 1913

Highly esteemed colleague,

a simple theoretical consideration makes it plausible to assume that light rays will experience a deviation in a gravitational field.

[Grav. field] [Light ray]

At the rim of the Sun, this deflection should amount to 0.84" and decrease as 1/R (R = [strike]Sonnenradius[/strike] distance from the centre of the Sun).

[Earth] [Sun]

Thus, it would be of utter interest to know up to which proximity to the Sun bright fixed stars can be seen using the strongest magnification in plain daylight (without eclipse).

Fast Forward to an Effect

Bending light around a massive object from a distant source. The orange arrows show the apparent position of the background source. The white arrows show the path of the light from the true position of the source.

The fact that this does not happen when gravitational lensing applies is due to the distinction between the straight lines imagined by Euclidean intuition and the geodesics of space-time. In fact, just as distances and lengths in special relativity can be defined in terms of the motion of electromagnetic radiation in a vacuum, so can the notion of a straight geodesic in general relativity.

To me, gravitational lensing is a cumulative affair that such a geometry borne into mind, could have passed the postulates of Euclid, and found their way to leaving a "indelible impression" that the resources of the mind in a simple system intuits.

Einstein, in the paragraph below makes this clear as he ponders his relationship with Newton and the move to thinking about Poincaré.

The move to non-euclidean geometries assumes where Euclid leaves off, the basis of Spacetime begins. So such a statement as, where there is no gravitational field, the spacetime is flat should be followed by, an euclidean, physical constant of a straight line=C?

Einstein:

I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity. ... In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry. ... If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincare:--Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity. Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable ... to retain Euclidean geometry. For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. (33-4)

It is never easy for me to see how I could have moved from what was Euclid's postulates, to have graduated to my "sense of things" to have adopted this, "new way of seeing" that is also accumulative to the inclusion of gravity as a concept relevant to all aspects of the way in which one can see reality.

See:

On the Hypothese at the foundations of Geometry

Gravity and Electromagnetism?

"The Confrontation between General Relativity and Experiment" by Clifford M. Will

The Founder of Probabilty Theory?

I want to understand what makes the world tick. Einstein said he wanted to know what was on God's mind when he made the world. I don't think he was a religious man, but I know what he means.Lenny Susskind

Pierre de Fermat IPA: [pjɛːʁ dəfɛʁ'ma] (August 17, 1601 – January 12, 1665)

With Blaise Pascal, Pierre de Fermat became the founders of the theory of probability.

A Short History of Probability

"A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.