The Uncertainty Principle

At a time when Einstein had gained international recognition, quantum theory culminated in the late 1920’s statement of the Uncertainty Principle, which says that the more precisely the position of a particle is determined, the less precisely the momentum is known in this instant, and vice versa. The above phrasing of the principle is a succinct version of the mathematically precise uncertainty relation that Heisenberg published in 1927. Since the momentum of a particle is the product of its mass and velocity, the principle is sometimes stated differently, however, its meaning remains the same: The act of measuring one magnitude of a particle, be it its mass, its velocity, or its position, causes the other magnitudes to blur. This is not due to imprecise measurements. Technology is advanced enough to hypothetically yield correct measurements. The blurring of these magnitudes is a fundamental property of nature.

The uncertainty relation describes the "blur" between the measurable quantities of a particle in mathematical terms. Like much of the math in quantum theory, it is not for the faint of heart, which is to say it is completely unintelligible to most people. Therefore we restrict ourselves to a brief account on the underlying ideas and how they developed into the "Copenhagen Interpretation", which Niels Bohr and Werner Heisenberg jointly elaborated as a complete and consistent view of quantum mechanics (the Copenhagen Interpretation refers to Bohr's place of birth).

Around 1925 there were two competing mathematical theories that both attempted to explain electron orbits. Matrix mechanics developed by Heisenberg interprets the electron as a particle with quantum behaviour. It is based on sophisticated matrix computations, which introduce discontinuities and quantum jumps. In contrast, wave mechanics developed by Erwin Schrödinger interprets the electron as an energy wave. Because wave mechanics entails more familiar concepts and equations, it quickly gained popularity among scientists.

Schrödinger and Heisenberg were no too fond of each other's competing works. Schrödinger says about matrix mechanics: "I knew of [Heisenberg's] theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualisability." Heisenberg's comment on wave mechanics was: "The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it. [...] What Schrödinger writes about the visualisability of his theory 'is probably not quite right,' in other words it's crap."

The Copenhagen Interpretation.

Despite the differences, Schrödinger published a proof in 1926, which showed that the results of matrix and wave mechanics are equivalent; they were in fact the same theory. According to the Copenhagen Interpretation, the wave and particle pictures of the atom, or the visual and causal representations, are "complementary" to each other. That is, they are mutually exclusive, yet jointly essential for a complete description of quantum events. Obviously in an experiment in the everyday world an object cannot be both a wave and a particle at the same time; it must be either one or the other, depending on the situation. In later refinements of this interpretation, the wave function of the unobserved object is a mixture of both, the wave and particle pictures, until the experimenter chooses what to observe in a given experiment.

The notion of the observer becoming a part of the observed system is fundamentally new in physics. In quantum physics, the observer is no longer external and neutral, but through the act of measurement he becomes himself a part of observed reality. This marks the end of the neutrality of the experimenter. It also has huge implications on the epistemology of science: certain facts are no longer objectifiable in quantum theory. If in an exact science, such as physics, the outcome of an experiment depends on the view of the observer, then what does this imply for other fields of human knowledge? It would seem that in any faculty of science, there are different interpretations of the same phenomena. More often than occasionally, these interpretations are in conflict with each other. Does this mean that ultimate truth is unknowable?

The results of quantum theory, and particularly of Heisenberg's work, left scientists puzzled. Many felt that quantum theory had somehow "missed the point". At least Albert Einstein did so. He was an outspoken critic of quantum mechanics and is often quoted on his comment regarding the Uncertainty Principle: "The Old One (God) doesn't play dice." He also said: "I like to believe that the moon is still there even if we don't look at it." In particular, Einstein was convinced that electrons do have definite orbits, even if we cannot observe them.

Is the moon still there when nobody is looking at it?

The two philosophies seem incompatible at first. Heisenberg is in good company with famous contenders of idealistic positions, such Plato, Schopenhauer, and Husserl, but so is Albert Einstein. If we take Heisenberg's view for granted, strict causality is broken, or better: the past and future events of particles are indeterminate. One cannot calculate the precise future motion of a particle, but only a range of possibilities. Physics loses its grip. The dream of physicists, to be able to predict any future event in the universe based on its present state, meets its certain death.

If we regard reality as that which can be observed by all, we have to find that there is no objective movement of an electron around the nucleus. This viewpoint would imply that reality is created by the observer; in other words: if we take Heisenberg literally, the moon is not there when nobody is looking at it. However, we must consider the possibility that there is a subatomic reality independent of observation and that the electron may have an actual trajectory which cannot be measured. The moon may be there after all. This conflict is the philosophical essence of the Uncertainty Principle.

Relativity and quantum theory are inconsonant up to the present day, despite great efforts in creating a unified theory capable of accommodating both views. After having published his papers on Relativity, Einstein dedicated the rest of his life to working on such a unified field theory, yet without success. The physicists who followed his lead developed a new model called string theory during the 1970s and 1980s. String theory was successful to some extent in providing a mathematical model that integrates the strong and the weak nuclear forces, electromagnetism, and gravitation. In spite of this, it cannot yet be called a breakthrough, because (1) the theory has not been corroborated thoroughly by observational evidence; and (2) there is not one, but five competing string theories. The latter point has recently been addressed by M-theory, a theory that unites existing string theories in 11 dimensions.

The Zen of Quantum Theory.

We shall leave the problem of theoretical unification to the physicists and instead briefly consider a philosophical unification of Relativity and quantum theory. Is this possible? Contemplating the subatomic realm seems like a Zen exercise. The nuclear reality embodies duality and multiplicity, such as is evident in the complicated structure of atoms and particles. It transgresses the narrow world of opposites. We have to realise that in spite of the different parts and components, the subatomic world in actuality is an undivided whole, where the boundary between the observer and the observed is blurred. Object and subject have become inseparable, spatial and temporal detachment is an illusion. When the American physicist J.R. Oppenheimer (1902-1967) describes the structure of probability clouds, he almost sounds like a Zen Master: "If we ask, whether the position of the electron remains the same, we have to say no. If we ask, whether the position of an electron changes with the course of time, we have to say no. If we ask, whether the electron is in a state of rest, we have to say no. If we ask, whether the electron is in motion, we have to say no."

excerpt taken from http://www.thebigview.com/spacetime/uncertainty.html 

The Good Life

The Good Life : By Bertrand Russell

There have been at different times and among different people many varying conceptions of the good life. To some extent the differences were amenable to argument; this was when men differed as to the means to achieve a given end. Some think that prison is a good way to preventing crime; others hold that education would be better. A difference of this sort can be decided by sufficient evidence. But some differences cannot be tested in this way. Tolstoy condemned all war; others have held the life of a soldier doing battle for the right to be very noble. Here there was probably involved a real difference as to ends. Those who praised the soldier usually considered the punishment of sinners a good thing in itself; Tolstoy did not think so. On such a matter no argument is possible. I cannot, therefore, prove that my view of the good life is right; I can only state my view and hope that as many as possible will agree. My view is this: The good life is one inspired by love and guided by knowledge.

Knowledge and love are both indefinitely extensible; therefore, however good a life may be, a better life can be imagined. Neither love without knowledge no knowledge without love can produce a good life. I the Middle Ages, when pestilence appeared in a country, holy men advised the population to assemble in churches and pray for deliverance; the result was that the infection spread with extraordinary rapidity among the crowded masses of supplicants. This is an example of love without knowledge. The late war afforded an example of knowledge without love. In each case, the result was death on a large scale.

Although both love and knowledge are necessary, love is in a sense more fundamental, since it will lead intelligent people to seek knowledge, in order to find out how to benefit those whom they love. But if people are not intelligent, they will be content to believe what they have been told and may do harm in spite of the most genuine benevolence. Medicine affords, perhaps, the best example of what I mean. An able physician is more useful to a patient than the most devoted friend, and progress in medical knowledge does more for the health of the community than ill-informed philanthropy. Nevertheless, an element of benevolence is essential even here if any but the rich are to profit by scientific discoveries.

Love is a word which covers a variety of feelings; I have used it purposely, as I wish to include them all. Love as an emotion-which is what I am speaking about, for love "on principle" does not seem to me genuine- moves between two poles: on one side, pure delight in contemplation; on the other, pure benevolence. Where inanimate objects are concerned, delight alone enters in; we cannot feel benevolence toward a landscape or a sonata. This type of enjoyment is presumably the source of art. It is stronger, as a rule, in very young children than in adults, who are apt to view objects in a utilitarian spirit. It plays a large part in our feelings toward human beings, some of whom have charm and some the reverse, when considered simply as objects of aesthetic contemplation.

The opposite pole of love is pure benevolence. Men have sacrificed their lives to helping lepers; in such a case the love felt cannot have had any element of aesthetic delight. Parental affection, as a rule, is accompanied by pleasure in the child's appearance but remains strong when this element is wholly absent. It would seem odd to call a mother's interest in a sick child "benevolence," because we are in a habit of using this word to describe pale emotion nine parts humbug. But it is difficult to find any other word to describe the desire for another persons welfare. It is a fact that a desire of this sort may reach a degree of strength in the case of parental feeling. In other cases it is far less intense; indeed it would seem likely that all altruistic emotion is a sort of overflow of parental feeling, or sometimes a sublimation of it. For want of a better word, I shall call this emotion "benevolence." But I want to make it clear that I am speaking of an emotion, not a principle, and that I do not include in it any feeling of superiority such as is associated with the word. The word sympathy expresses part of what I mean but leaves out the element of activity that I wish to include.

Love at its fullest is an indissoluble combination of the two elements, delight and well-wishing. The pleasure of a parent in a beautiful and successful child combines both elements; so does sex love at its best. But in sex love, benevolence will only exist where there is secure possession, since otherwise jealousy will destroy it, while perhaps actually increasing the delight in contemplation. Delight without well-wishing may be cruel; well-wishing without delight tends to become cold and a little superior. A person who wishes to be loved wishes to be the object of a love containing both elements, except in cases of extreme weakness, such as infancy and severe illness. In these cases benevolence may be all that s desired. Conversely, in cases of extreme strength, admiration is more desired than benevolence: this is the state of mind of potentates and famous beauties. We only desire other people's good wishes in proportion as we feel ourselves in need of help or in danger of harm from them. At least, that would seem to be the biological logic of the situation, but it is not quite true of life. We desire affection in order to escape from the feeling of loneliness, in order to be, as we say, "understood." This is a matter of sympathy, not merely of benevolence; the person whose affection is satisfactory to us must not merely wish us well but must know in what our happiness consists. But this belongs to the other element of the good life- namely, knowledge.

In a perfect world, every sentient being would be to every other the object of the fullest love, compounded of delight, benevolence, and understanding inextricably blended. It does not follow that, in this actual world, we ought to attempt to have such feelings toward all the sentient beings who we encounter. There are many in whom we cannot feel delight, because they are disgusting; if we were to do violence to our nature by trying to see beauties in them, we should merely blunt our susceptibilities to what we naturally find beautiful. Not to mention human beings, there are fleas and bugs and lice. We should have to be as hard pressed as the Ancient Mariners before we could feel delight in contemplating these creatures. Some saints, it is true, have called them "pearls of God," but what these men delighted in was the opportunity of displaying their own sanctity.

Benevolence is easier to extend widely, but even benevolence has its limits. If a man wished to marry a lady, we should not think the better of him for withdrawing if he found that someone else also wished to marry her: we should regard this as a fair field of competition. Yet his feelings toward a rival cannot be wholly benevolent. I think that in all descriptions of the good life here on Earth we must assume a certain basis of animal vitality and animal instinct; without this, life becomes tame and uninteresting. Civilization should be something added to this, not substituted for it; the ascetic saint and the detached sage fail in this respect to be complete human beings. A small number of them may enrich a community; but a world composed of them would die of boredom.

These considerations lead a certain emphasis on the element of delight as an ingredient in the best love. Delight, in this actual word, is unavoidable selective and prevents us from having the same feelings toward all mankind. When conflict arises between delight and benevolence, they must, as a rule, be decided by a compromise, not by complete surrender of either. Instinct has its rights, and if we do violence to it beyond a point it takes vengeance in subtle ways. Therefore in aiming at a good life the limits of human possibilities must be borne in mind. Here again, however, we are brought back to the necessity of knowledge.

When I speak of knowledge as an ingredient of the good life, I am not thinking of ethical knowledge but of scientific knowledge and knowledge of particular facts. I do not think there is, strictly speaking, such a thing as ethical knowledge. If we desire to achieve some end, knowledge may show us the means, and this knowledge may loosely pass as ethical. But I do not believe that we can decide what sort of conduct is right and wrong except by reference to its probable consequences. Given an end to be achieved, it is a question for science to discover how to achieve it. All moral rules must be tested by examining whether they tend to realize ends we desire. I say ends we desire, not ends that we ought to desire. What we "ought" to desire is merely what someone else wishes us to desire. Usually it is what the authorities wish us to desire-parents, schoolmasters, policeman, and judges. If you say to me, "You ought to do so-and-so," the motive power of your remark lies in my desire for your approval-together, possibly, with rewards or punishments attached to your approval or disapproval. Since all behavior springs from desire, it is clear that ethical notions can have no importance except as they influence desire. They do this through the desire for approval and the fear of disapproval. These are powerful social forces, and we shall naturally endeavor to win them to our side if we wish to realize any social purpose. When I say that the morality of conduct is to be judged by its probable consequences, I mean that I desire to see approval given to behavior likely to realize social purposes which we desire, and disapproval to opposite behavior. At present this is not done; there are certain traditional rules according to which approval and disapproval are meted out quite regardless of consequences.

The superfluity of theoretical ethics is obvious in simple cases. Suppose, for instance, your child is ill. Love makes you wish to cure it, and science tells you how to do so. There is not an intermediate stage of ethical theory, where it is demonstrated that your child had better be cured. Your act springs directly from desire for an end, together with knowledge of means. This is equally true of all acts, whether good or bad. The ends differ, and the knowledge is more adequate in some cases than in others. But there is no conceivable way of making people do things they do not wish to do. What is possible is to alter their desires by a system of rewards and penalties, among which social approval and disapproval are not the least potent. The question for the legislative moralist is, therefore: How shall this system of rewards and punishments be arranged so as to secure the maximum of what is desired by the legislative authority? If I say that the legislative authority has bad desires, I mean merely that its desires conflict with some section of the community to which I belong. Outside human desires there is no moral standard.

Thus, what distinguishes ethics from science is not any special kind of knowledge but merely desire. The knowledge required in ethics is exactly like the knowledge elsewhere; what is peculiar is that certain ends are desired, and that right conduct is what conduces them. Of course, if the definition of right conduct is to make a wide appeal, the ends must be such as large sections of mankind desire. If I defined right conduct as that which increases my own income, readers would disagree. The whole effectiveness of an ethical argument lies in its scientific part, i.e, in the proof that one kind of conduct, rather than some other, is a means to an end which is widely desired. I distinguish, however, between ethical argument and ethical education. The latter consists in strengthening certain desires and weakening others.

We can now explain more exactly the purport of the definition of the good life with which this section began. When I said that the good life consists of love guided by knowledge, the desire which prompted me was the desire to live such a life as far as possible, and to see others living it; and the logical content of the statement is that, in a community where men live in this way, more desires will be satisfied than one where there is less love or less knowledge. I do not mean that such a life is "virtuous" or that its opposite is "sinful," for these are conceptions which seem to me to have no scientific justification.

String Theory

We live in a wonderfully complex universe, and we are curious about it by nature. Time and again we have wondered--- why are we here? Where did we and the world come from? What is the world made of? It is our privilege to live in a time when enormous progress has been made towards finding some of the answers. String theory is our most recent attempt to answer the last (and part of the second) question.

So, what is the world made of? Ordinary matter is made of atoms, which are in turn made of just three basic components: electrons whirling around a nucleus composed of neutrons and protons. The electron is a truly fundamental particle (it is one of a family of particles known as leptons), but neutrons and protons are made of smaller particles, known as quarks. Quarks are, as far as we know, truly elementary.

Our current knowledge about the subatomic composition of the universe is summarized in what is known as the Standard Model of particle physics. It describes both the fundamental building blocks out of which the world is made, and the forces through which these blocks interact. There are twelve basic building blocks. Six of these are quarks--- they go by the interesting names of up, down, charm, strange, bottom and top. (A proton, for instance, is made of two up quarks and one down quark.) The other six are leptons--- these include the electron and its two heavier siblings, the muon and the tauon, as well as three neutrinos.

There are four fundamental forces in the universe: gravity, electromagnetism, and the weak and strong nuclear forces. Each of these is produced by fundamental particles that act as carriers of the force. The most familiar of these is the photon, a particle of light, which is the mediator of electromagnetic forces. (This means that, for instance, a magnet attracts a nail because both objects exchange photons.) The graviton is the particle associated with gravity. The strong force is carried by eight particles known as gluons. Finally, the weak force is transmitted by three particles, the W+, the W- , and the Z.

The behavior of all of these particles and forces is described with impeccable precision by the Standard Model, with one notable exception: gravity. For technical reasons, the gravitational force, the most familiar in our every day lives, has proven very difficult to describe microscopically. This has been for many years one of the most important problems in theoretical physics-- to formulate a quantum theory of gravity.

In the last few decades, string theory has emerged as the most promising candidate for a microscopic theory of gravity. And it is infinitely more ambitious than that: it attempts to provide a complete, unified, and consistent description of the fundamental structure of our universe. (For this reason it is sometimes, quite arrogantly, called a 'Theory of Everything').

The essential idea behind string theory is this: all of the different 'fundamental ' particles of the Standard Model are really just different manifestations of one basic object: a string. How can that be? Well, we would ordinarily picture an electron, for instance, as a point with no internal structure. A point cannot do anything but move. But, if string theory is correct, then under an extremely powerful 'microscope' we would realize that the electron is not really a point, but a tiny loop of string. A string can do something aside from moving--- it can oscillate in different ways. If it oscillates a certain way, then from a distance, unable to tell it is really a string, we see an electron. But if it oscillates some other way, well, then we call it a photon, or a quark, or a ... you get the idea. So, if string theory is correct, the entire world is made of strings!

Perhaps the most remarkable thing about string theory is that such a simple idea works--- it is possible to derive (an extension of) the Standard Model (which has been verified experimentally with incredible precision) from a theory of strings. But it should also be said that, to date, there is no direct experimental evidence that string theory itself is the correct description of Nature. This is mostly due to the fact that string theory is still under development. We know bits and pieces of it, but we do not yet see the whole picture, and we are therefore unable to make definite predictions. In recent years many exciting developments have taken place, radically improving our understanding of what the theory is.

excerpt came from http://www.nuclecu.unam.mx/~alberto/physics/string.html

Einstein's Relativity

The notion of relativity is not as revolutionary as many believe. In fact, spatial relativity is part of our everyday experience. Spatial relativity, also called Galilean relativity in honour of Galileo who first formulated the concept of relative motion, is often confused with Einstein's theories. Galileo simply described the fact that an observer in motion sees things differently from a stationary observer, because he has a different spatial coordinate system, or "reference frame" in Relativity speak. It might sound more complicated than it actually is. Consider the following example:

Galilean relativity: the train example (courtesy of Stephen Hawking).

Two people riding on a train from New York to San Francisco play a game of ping-pong in the sport compartment of the train. Lets say, the train moves at 100 km per hour (= 27.8 m/s) and the two players hit the ball at a speed of two meters per second. In the reference frame of the players, the ball moves back and forth at this particular speed. For a stationary observer standing beside the railroad, however, things look quite different. In his reference frame the ball moves at 29.8 m/s when it is played forward in the direction where the train is heading, while it moves at 25.8 m/s in the same direction when it is played backwards. Thus he doesn't see the ball moving backward at all, but always moving towards San Francisco. For an observer in outer space, things look again totally different because of the Earth's rotation, which is opposite to the train's movement; therefore the outer space observer always sees the ball moving East.

Einstein's new concept of relativity.

Einstein's Relativity differs from classical relativity, because of the way he looked at time. Before Einstein, people thought time to be absolute,

Isaac Newton

"Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space. Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another"

which is to say that one big clock measures the time for the entire universe. Consequently one hour on Earth would be one hour on Mars, or one hour in another galaxy. However, there was a problem with this concept. In an absolute time frame the speed of light cannot be constant. Roemer found that the speed of light is finite and has a certain, quantifiable velocity (usually abbreviated with "c"), which at first implies Galilean relativity. This would mean that while the Earth rotates at a velocity of v, light emitted in the direction of the Earth rotation must be c + v, while light emitted in the opposite direction would travel at c - v, relative to an outside observer.

In 1881, A. Michelson conducted an experiment which proved that this is not the case. With the help of an apparatus that allowed measuring minute differences in the speed of light by changes in the resulting interference patterns, Michelson observed that the speed of light is always the same. No changes whatsoever. The experiment has been repeated later with greater precision by Michelson and E.W. Morley.

Special Relativity published in 1905.

Numerous attempts were made at reconciling these discrepancies, yet they were all unsuccessful, until Einstein solved the dilemma with his famous paper On the Electrodynamics of Moving Bodies in 1905, in which he developed his Special Relativity Theory. Special Relativity is an extremely elegant construct that deals with things moving near or at the speed of light. Surprisingly, the new concept of space and time that arises from Relativity is based only on two simple postulates: 1. The laws of physics are the same in all inertial (=non-accelerating) reference frames, and 2. The speed of light in free space is constant.

It is a matter of common experience that one can describe the position of a point in space by three numbers, or coordinates. For the purpose of explaining the relativistic model, Einstein added time as a fourth component to the coordinate system, and the resulting construct is called spacetime. Just as there is an infinite number of 3-D reference frames in Galilean relativity, there is an infinite number of 4-D spacetime reference frames in Einstein's theory. This is to say that Einstein put an end to absolute time. The revolutionary insight lies in the conclusion that the flow of time in the universe does indeed differ depending on one's reference frame

In his usual humble way, Einstein explained how he reinvented physics: "I sometimes ask myself how it came about that I was the one to develop the theory of Relativity. The reason, I think, is that a normal adult stops to think about problems of space and time. These are things which he has thought about as a child. But my intellectual development was retarded, as a result of which I began to wonder about space and time only when I had already grown up." On Relativity, he said: "Relativity teaches us the connection between the different descriptions of one and the same reality."

This view of Relativity, that there are different realities, has been picked up unanimously by the public, and hence, has taken on a far greater meaning than that of the original scientific theory, the focus of which was -strictly speaking- on mechanics and electrodynamics. This astonishing success was at least in part due to Einstein's personality. He understood himself as a philosopher as much as a scientist, and he was ready to discuss philosophical issues at any time, particularly matters involving Relativity. The philosophical aspect of Relativity forced people to think differently about the universe. Suddenly, the cosmos was not a God-created clockwork anymore, but a totality of disparate realities with the same basic natural laws.

An outstanding feature of Special Relativity is its mass-energy relation, which is expressed in the well-known formula: E=mc².

Einstein derived this relation in an attempt to reconcile Maxwell's electromagnetic theory with the conservation of energy and momentum. Maxwell said that light carries a momentum, which is to say that a wave carries an amount of energy. Due to the principle of conservation of momentum, if a body emits energy in the form of radiation, the body loses an equivalent amount of mass that is given by E/c². This describes the relation between energy and mass.

According to the conservation principle, in a closed system the sum of mass and its energy equivalent is always the same. The mass-energy relation tells us that any change in the energy level of an object necessarily involves a change in the object's mass and vice-versa. The most dramatic consequences of this law are observed in nature, for example in nuclear fission and fusion processes, in which stars like the Sun emit energy and lose mass. The same law also applies to the forces set free in the detonation of an atomic bomb.

General Relativity published in 1916.

Eleven years after On the Electrodynamics of Moving Bodies, Einstein published his second groundbreaking work on General Relativity, which continues and expands the original theory. A preeminent feature of General Relativity is its view of gravitation. Einstein held that the forces of acceleration and gravity are equivalent. Again, the single premise that General Relativity is based on is surprisingly simple. It states that all physical laws can be formulated so as to be valid for any observer, regardless of the observer's motion. Consequently, due to the equivalence of acceleration and gravitation, in an accelerated reference frame, observations are equivalent to those in a uniform gravitational field.

This led Einstein to redefine the concept of space itself. In contrast to the Euclidean space in which Newton’s laws apply, he proposed that space itself might be curved. The curvature of space, or better spacetime, is due to massive objects in it, such as the sun, which warp space around their gravitational centre. In such a space, the motion of objects can be described in terms of geometry rather than in terms of external forces. For example, a planet orbiting the Sun can be thought of as moving along a "straight" trajectory in a curved space that is bent around the Sun.

excerpt taken from http://www.thebigview.com/spacetime/relativity.html

Euclidian-Non Euclidean Geometry

Euclid of Alexandria was a 3rd century B.C. mathematician, most famous for his series of thirteen books known as the Elements, which was the most thorough description of geometrical theorems and principles that the world would see for more than two thousand years (perhaps even to this very day). Euclid began his grand work by stating five central “axioms,” and using these axioms, he proved one theorem after another, each more complex, practically inventing the principles of modern geometry.

Most of Euclid’s basic axioms are rather obvious and seem to be self-evident, such as his second axiom: “A straight line segment can be extended indefinitely in a straight line.” There are very few mathematicians who would think to dispute this, nor would they most of his other axioms.

The first problems with Euclid’s axioms did not begin to arise in a legitimate fashion until the 19th century, most notably in the works of Charles Friedrich Gauss and Janos Bolyai (who had a complex and competitive working relationship which is worth learning about in its own right) and their attempts to shed new light on the single disputed axiom of Euclid’s Elements, particularly by attempting to find a proof for Euclid's most controversial axiom.

The Parallel Postulate

Euclid’s fifth axiom, or “The Parallel Postulate” is rather simple in essence, though it has caused mathematicians no end of trouble. Simply put, the parallel postulate says that one may discover if two lines are truly parallel in the following way:

A third line is drawn which intersects both lines in question, then, by measuring the angles created by the intersection of these lines one can determine if they are parallel. Only if the two “outside” angles (and, consequently, the “inside” angles as well) created by this third line add up to exactly 180 degrees are the two lines parallel. If they add up to more or less than this, the lines are clearly going to converge at one point or another, and are therefore not parallel.

To anyone who has taken some geometry in school, this sounds on the surface to be just as reasonable as Euclid’s other axioms. So where does the trouble come from?

The Basics of Non-Euclidean Geometry

The trouble stems from the fact that Euclid’s geometry was designed to work in only specific cases – that is, on flat surfaces of two dimensions or in standard three dimensional space with straight lines. His geometry (specifically his fifth axiom) does not hold true in other circumstances.

Take, to use a classic example, a standard world globe. As the lines of longitude cross the equator, one can measure the angle thus created and, via the parallel postulate conclude that, indeed, these lines are parallel, as the angles add up to 180 degrees. Of course this is not the case, however, as these lines do, in fact, meet, both at the north and the south poles. Thus, in the case of a curved surface, one needs a non-Euclidean geometry. Another form of the same example says that on the surface of a sphere, the interior angles of a triangle need not add up to 180 degrees. In fact, a triangle on the surface of the Earth can be made using three right angles – a clear divergence from standard geometry.

The Importance of Geometries

While this all may seem rather trivial to those who do not focus their energies on mathematical studies, it surely does have great importance to the world, both in a mathematical sense, and in the realm of physical sciences.

Mathematically, non-Euclidean geometry has opened up an entirely new realm of possibilities, as such mathematicians as Gauss and Bolyai (as already mentioned) worked feverishly to develop new theorems based on curved geometrical systems.

In physics, the examples are even more clear. A fundamental feature of Einstein’s General theory of Relativity was the fact that the four dimensions of Space-Time which make up the entire universe around us are not “flat” or “smooth” as physicists had before assumed, but was rather “curved,” “twisted,” and “warped,” by the masses of objects. Thus, when developing the mathematics for his theory, he was forced to rely on non-Euclidean works in order to describe his findings. One of the titles of a chapter in Einstein’s book “Relativity: The Special and the General Theory” (Chapter 25) was called, simply, “Gaussian Co-ordinates,” and uses the work of Gauss to develop a new coordinate system for use in his theory.

In the plainest terms, non-Euclidean geometry took something that was rather simple and straightforward (Euclidean Geometry) and made it endlessly more difficult.

With this in mind, it’s sometimes hard to tell if these mathematicians should be thanked or cursed.

excerpt taken from http://math.suite101.com/article.cfm/euclidean_v_noneuclidean_geometry

Free Will

Free Will - in 4 acts

Free Will - Rene Descartes

I now have no difficulty directing my thought away from things that can be imagined to things that can be grasped only by the understanding and are wholly separate from matter…

I acknowledge that it is impossible for God ever to deceive me…

Next I experience that there is in me a certain faculty of judgment, which, like everything else that is in me, I undoubtedly received from God. And since he does not wish to deceive me, he assuredly has not given me the sort of faculty with which I could ever make a mistake, when I use it properly

No doubt regarding this matter would remain, but for the fact that it seems to follow from this seems to follow from this that I em never capable of making a mistake…

But once I turn my attention back on myself, I nevertheless experience that I am subject to countless errors. As I seek a cause of these errors, I notice that passing before me is not only a real and positive idea of God…but also, as it were, a certain negative idea of nothingness…and that I have been so constituted as a kind of middle ground between God and nothingness…

I make mistakes because the faculty of judging the truth, which I got from God, is not, in my case, infinite

…the will is the chief basis for my understanding that I bear a certain image and likeness of God

…the power of willing, which I got from God, is not, taken by itself, the cause of my errors, for is it most ample as well as perfect in its kind. Nor is my power of understanding the cause of my errors. For since I got my power of understanding from God, whatever I understand I doubtless understand rights, and it is impossible for me to be deceived in this. What then is the source of my errors? They are owing simply to the fact that, since the will extends further than the intellect, I do not contain the will within the same boundaries; rather, I also extend it to things I do not understand.

I should never judge anything that I do not clearly and distinctly understand.

…what I must do to attain truth…if I pay enough attention to all the things that I perfectly understand, and separate them off from the rest, which I apprehend more confusedly and more obscurely.

Free Will - Bertrand Russell

The free-will question consequently remains just where it was. Whatever may be thought about it as a matter of ultimate metaphysics, it is quite clear that nobody believes it in practice. Everyone has always believed that it is possible to train character; everyone has always known that alcohol or opium will have a certain effect on behaviour. The apostle of free will maintains that a man can by will power avoid getting drunk, but he does not maintain that when drunk a man can say "British Constitution" as clearly as if he were sober. And everybody who has ever had to do with children knows that a suitable diet does more to make them virtuous than the most eloquent preaching in the world. The one effect that the free- will doctrine has in practice is to prevent people from following out such common-sense knowledge to its rational conclusion. When a man acts in ways that annoy us we wish to think him wicked, and we refuse to face the fact that his annoying behaviour is a result of antecedent causes which, if you follow them long enough, will take you beyond the moment of his birth and therefore to events for which he cannot be held responsible by any stretch of imagination.

Free Will - David Hume

It may be said, for instance, that, if voluntary actions be subjected to the same laws of necessity with the operations of matter, there is a continued chain of necessary causes, pre-ordained and pre-determined, reaching from the original cause of all to every single volition of every human creature… . The ultimate Author of all our volitions is the Creator of the world, who first bestowed motion on this immense machine, and placed all beings in that particular position, whence every subsequent event, by an inevitable necessity, must result. Human action, therefore, either can have no moral turpitude at all, as proceeding from so good a cause; or if they have any turpitude, they must involve our Creator in the same guilt, while he is acknowledged to be their ultimate cause and author.

Free Will - Frank Tipler

Indeterminism is a property of all quantum cosmological theories for which the universal wave function includes in its domain the set of all compact four-dimensional manifolds. Thus, indeterminism holds both in the Hartle-Hawking quantum cosmology and in the quantum Omega Point Theory. However, it may be merely an epistemological, and not an ontological, indeterminism in the Hartle-Hawking cosmology.⁷

Although it has been shown that the human nervous system can use nonrelativistic quantum mechanical uncertainty to randomize, it does not follow that it can access the quantum gravity regime. As I pointed out above, true ontological free will requires quantum gravity uncertainty, because there is a deterministic equation controlling nonrelativistic "uncertainty." There are two ways in which the human nervous system might be able to access the quantum gravity regime in the randomization process. The first is a mechanism suggested by Penrose, who in effect points out that, if a substantial portion of the brain were to act as if it were in a coherent quantum state, it might be able to amplify a signal from the Planck scale up to the macroscopic level. The known amplification power of the nervous system —amplification of a single photon energy to nerve pulse energies constitutes a magnification of 10²⁰—is insufficient by a factor of 10⁸, so Penrose's proposal is speculative, to say the least. The second possibility is that the randomizer may use vacuum fluctuations inside the brain. A system which is capable of detecting single photons is certainly sensitive enough. One of the most important unsolved problems in particle physics is accounting for the magnitude of the vacuum energy density. If the fluctuations in topology are neglected, the calculated value is too high by a factor of about 10⁵⁴. The most popular method of resolving this problem is to include the topological fluctuations: some calculations indicate that these can cancel out the factor of 10⁵⁴. But if this is the cancellation mechanism, then the residual fluctuations in the vacuum energy density would necessarily reflect quantum gravity uncertainties, and thus a randomizer based on the fluctuations would be ontologically indeterministic. A state transition of the human brain in this case would be totally unpredictable. In this situation, we would have ontological free will.⁸

Immanual Kant - What is Right?

METAPHYSICAL FOUNDATIONS OF THE THEORY OF RIGHT

What is Right?

This question may be said to be about as embarrassing to the jurist as the well-known question, "What is truth?" is to the logician. It is all the more so, if, on reflection, he strives to avoid tautology in his reply and recognise the fact that a reference to what holds true merely of the laws of some one country at a particular time is not a solution of the general problem thus proposed. It is quite easy to state what may be right in particular cases (quid sit juris), as being what the laws of a certain place and of a certain time say or may have said; but it is much more difficult to determine whether what they have enacted is right in itself, and to lay down a universal criterion by which right and wrong in general, and what is just and unjust, may be recognised. All this may remain entirely hidden even from the practical jurist until he abandon his empirical principles for a time and search in the pure reason for the sources of such judgements, in order to lay a real foundation for actual positive legislation. In this search, his empirical laws may, indeed, furnish him with excellent guidance; but a merely empirical system that is void of rational principles is, like the wooden head in the fable of Phaedrus, fine enough in appearance, but unfortunately it wants brain.

1. The conception of right -- as referring to a corresponding obligation which is the moral aspect of it -- in the first place, has regard only to the external and practical relation of one person to another, in so far as they can have influence upon each other, immediately or mediately, by their actions as facts.

2. In the second place, the conception of right does not indicate the relation of the action of an individual to the wish or the mere desire of another, as in acts of benevolence or of unkindness, but only the relation of his free action to the freedom of action of the other.

3. And, in the third place, in this reciprocal relation of voluntary actions, the conception of right does not take into consideration the matter of the matter of the act of will in so far as the end which any one may have in view in willing it is concerned. In other words, it is not asked in a question of right whether any one on buying goods for his own business realizes a profit by the transaction or not; but only the form of the transaction is taken into account, in considering the relation of the mutual acts of will. Acts of will or voluntary choice are thus regarded only in so far as they are free, and as to whether the action of one can harmonize with the freedom of another, according to a universal law.

Right, therefore, comprehends the whole of the conditions under which the voluntary actions of any one person can be harmonized in reality with the voluntary actions of every other person, according to a universal law of freedom.

Universal Principle of Right

"Every action is right which in itself, or in the maxim on which it proceeds, is such that it can coexist along with the freedom of the will of each and all in action, according to a universal law."

If, then, my action or my condition generally can coexist with the freedom of every other, according to a universal law, any one does me a wrong who hinders me in the performance of this action, or in the maintenance of this condition. For such a hindrance or obstruction cannot coexist with freedom according to universal laws.

It follows also that it cannot be demanded as a matter of right, that this universal principle of all maxims shall itself be adopted as my maxim, that is, that I shall make it the maxim of my actions. For any one may be free, although his freedom is entirely indifferent to me, or even if I wished in my heart to infringe it, so long as I do not actually violate that freedom by my external action. Ethics, however, as distinguished from jurisprudence, imposes upon me the obligation to make the fulfillment of right a maxim of my conduct.

The universal law of right may then be expressed thus: "Act externally in such a manner that the free exercise of thy will may be able to coexist with the freedom of all others, according to a universal law." This is undoubtedly a law which imposes obligation upon me; but it does not at all imply and still less command that I ought, merely on account of this obligation, to limit my freedom to these very conditions. Reason in this connection says only that it is restricted thus far by its idea, and may be likewise thus limited in fact by others; and it lays this down as a postulate which is not capable of further proof. As the object in view is not to teach virtue, but to explain what right is, thus far the law of right, as thus laid down, may not and should not be represented as a motive-principle of action.

Right is Conjoined With the Title or Authority to Compel

The resistance which is opposed to any hindrance of an effect is in reality a furtherance of this effect and is in accordance with its accomplishment. Now, everything that is wrong is a hindrance of freedom, according to universal laws; and compulsion or constraint of any kind is a hindrance or resistance made to freedom. Consequently, if a certain exercise of freedom is itself a hindrance of the freedom that is according to universal laws, it is wrong; and the compulsion of constraint which is opposed to it is right, as being a hindering of a hindrance of freedom, and as being in accord with the freedom which exists in accordance with universal laws. Hence, according to the logical principle of contradiction, all right is accompanied with an implied title or warrant to bring compulsion to bear on any one who may violate it in fact.

Strict Right May Be Also Represented as the Possibility
of a Universal Reciprocal Compulsion in Harmony with
the Freedom of All According to Universal Laws

This proposition means the right is not to be regarded as composed of two different elements -- obligation according to a law, and a title on the part of one who has bound another by his own free choice to compel him to perform. But it imports that the conception of right may be viewed as consisting immediately in the possibility of a universal reciprocal compulsion, in harmony with the freedom of all. As right in general has for its object only what is external in actions, strict right, as that with which nothing ethical is intermingled, requires no other motives of action than those that are merely external; for it is then pure right and is unmixed with any prescriptions of virtue. A strict right, then, in the exact sense of the term, is that which alone can be called wholly external. Now such right is founded, no doubt, upon the consciousness of the obligation of every individual according to the law; but if it is to be pure as such, it neither may nor should refer to this consciousness as a motive by which to determine the free act of the will. For this purpose, however, it founds upon the principle of the possibility of an external compulsion, such as may coexist with the freedom of every one according to universal laws. Accordingly, then, where it is said that a creditor has a right to demand from a debtor the payment of his debt, this does not mean merely that he can bring him to feel in his mind that reason obliges him to do this; but it means that he can apply an external compulsion to force any such one so to pay, and that this compulsion is quite consistent with the freedom of all, including the parties in question, according to a universal law. Right and the title to compel, thus indicate the same thing.

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There is Only One Innate Right: The Birthright of Freedom

Freedom is independence of the compulsory will of another; and in so far as it can coexist with the freedom of all according to a universal law, it is the one sole original, inborn right belonging to every man in virtue of his humanity. There is, indeed, an innate equality belonging to every man which consists in his right to be independent of being bound by others to anything more than that to which he may also reciprocally bind them. It is, consequently, the inborn quality of every man in virtue of which he ought to be his own master by right (sui juris). There is, also, the natural quality of justness attributable to a man as naturally of unimpeachable right (justi), because be has done no wrong to any one prior to his own juridical actions. And, further, there is also the innate right of common action on the part of every man, so that he may do towards others what does not infringe their rights or take away anything that is theirs unless they are willing to appropriate it; such merely to communicate thought, to narrate anything, or to promise something whether truly and honestly, or untruly and dishonestly (veriloquim aut falsiloquim), for it rests entirely upon these others whether they will believe or trust in it or not. But all these rights or titles are already included in the principle of innate freedom, and are not really distinguished from it, even as dividing members under a higher species of right.