Atunci cnd a vizitat Universitatea din Moscova, Paul Adrien Maurice Dirac, fizician celebru i fondatorul mecanicii cuantice, precum i la Universitatea din Cambridge, a fost ntrebat despre filozofia sa in fizica si a scris pe o tabl: "legile fizicii ar trebui s aib frumusee matematic "Aceast fraz a rmas conservata pe tabl i astzi.. De fapt, profesorul Dirac tia c ecuaiile matematice foarte semnificative apar n toate lucrurile create. Chiar dac acestea sunt profund sofisticate, n acelai timp, ele pot fi explicate cu o ajutorul unor ecuatii matematice adecvate.

Fie ele fizice sau chimice, multe atribute ale vietii sunt dependente de legi matematice i aparentele lor sunt, de asemenea, de asemenea construite prin modelare cu ajutorul unor principii matematice.Factorul cel mai important aici este simetria, care este descris ca o coresponden exact i un frumos echilibru ntre prile ale unui obiect, lucrurile sunt create cu diferite atribute simetrice si cu frumusetea artistica .

Tipul de simetrie cel mai frecvent este simetrie bilateral; acest lucru creeaz un efect de oglind, care este o coresponden exact ntre dreapta i n stnga. Un obiect face o simetrie exact cu reflectarea sa n oglind. O simetrie perfect, care este foarte similara cu efectul de oglind, poate fi gsit n corpul uman. Partea stng i cea dreapt a corpului nostru sunt simetrice corespunztor. Imaginai-v o linie de demarcaie, care trece de la mijlocul fruntii, pe nas, barbie si piept n jos, putem vedea o simetrie perfect pe ambele parti ale corpului. Braele noastre, picioare, ochi, urechi, nas i buze sunt proiectate cu o simetrie bilateral. Aceleai structuri simetrice pot fi, de asemenea, vzute n cele mai multe alte creaturi. Toate mamiferele, reptile i psri sunt create simetric.

Simetria frumosului de fulg de zpad cu forma lui hexagonala lor regulat este de asemenea fenomen natural deosebit. n plus fa de acestea, exist forme n natur, care au o simetrie radiala tridimensionala. Cele mai semnificative dintre aceste forme sunt poliedre regulate. Un exemplu de astfel de poliedre sunt elementele sarii cristaline, care au structuri cubice. Pn de curnd, faptul c exist o creatur n natur, care are o form regulat poliedric, constnd din douzeci pri, era necunoscut. Cu toate acestea, atunci cnd un tip de adenovirus care provoaca infectii si hepatita la cini a fost descoperit, s-a constatat c exist o creatur cu douzeci de fee regulate n natur.

Una dintre cele mai frumoase probe de simetrie radiala in natura sunt margaretele. Structuri simetrice nu exist numai n lumea normal i n lumile micro, dar, de asemenea, pot fi gsite n lumea macro, la fel ca toate obiectele cereti uriae, Soarele, Luna, galaxii, roiuri stelare din cer. . . . toate planetele se nvrt n jurul Soarelui ntr-o manier simetric, iar galaxiile au o simetrie spiral. Este interesant faptul c structura simetric a fiinelor vii este extrem de evident n exterior, mai degrab dect pe plan intern. De exemplu, organele interne din corpul uman, cum ar fi plamanii, ficatul, stomacul i intestinele nu sunt simetrice i avem doar o inim ntr-o parte a cavitatea toracic noastre. Creatorul nu creeaz lucruri pentru un singur motiv sau scop, dimpotriv, El le creeaz pentru mai multe motive i scopuri. De exemplu, dac nu am avea doi ochi i, dac acestia nu ar fi amplasati simetric pe feele noastre, nu am putea vedea obiecte tridimensionale. n acelai mod, n cazul n care urechile noastre ar fi fost amplasate simetric pe capetele noastre, atunci vom avea mari dificulti n determinarea direciei sursei de sunete. Dac nu am avea picioarele simetrice i picioarele, nu am fi putut s meargem bine.Simetria este, de asemenea, strns legat de robustee fizic i mental. Potrivit unui studiu, femeile care sufer de o boal infecioas n timpul sarcinii sunt mai susceptibile de a avea copii cu caracteristici asimetrice. Acelai studiu arat c bebeluii cu asimetrii sunt mai sensibile la boli de inima decat ceilalti.

Un alt studiu arata ca persoanele cu dintii asimetrici sunt mai susceptibile de a avea mai microorganismele duntoare n gura lor, dect cei care au dinti simetrici. Este interesant faptul c exist exista o diferen mai mare ntre amprentele de pe mna stnga fata de cele de pe mana dreapta la oamenii predispusi la schizofrenie decat la cele ale oamenilor normali.

Simetria este un fenomen care este utilizat de ctre animale i insecte. De exemplu, un experiment a aratat ca albinele prefera florile care sunt simetrice. De fapt, florile cu forme perfect simetrice produc nectar mai mult dect cele care sunt asimetrice. ntr-un experiment, o floare simetric a fost facuta asimetrica cu ajutorul unui foarfece. Floarea a fost atractiv pentru albine cand era simetrica; dup ce a fost facuta asimetrica, floare a devenit neatractiva pentru albine, chiar dac a avut aceeai cantitate de nectar ca si nainte.

Toate aceste fapte arat c exist mult nelepciune i frumusee ascuns n simetrie pe care Creatorul a folsit-o pentru a creea toate fiinele.

Ce este o lege a naturii? Legile naturale descriu comportamentul natural si sunt, de obicei, exprimate ca ecuaii. Dumnezeul Naturii este un matematician? De ce este faptul c att de multe legi poate fi exprimat ca o ecuaie absolut? De ce nu exist legi fundamentale, la toate? Toate acestea sunt intrebari legitime pe care ni le punem.Ecuaiile care stau la baza functionarea universului sunt ntr-un anumit sens, deja "acolo", independent de existena uman inca de la inceputuri , astfel c oamenii de tiin sunt arheologi cosmici, ncercnd de a descoperi legile care au culcat ascunse inca de la crearea universului.

Dintre sutele de mii de oameni de stiinta de cercetare care au trit vreodat, foarte puini au o ecuaie important legata de numele lor. Doi oameni de stiinta care au fost adeptii descoperirii ecuaiilor fundamentale i care , de asemenea, au perceput rolul matematicii n domeniul tiinei, au fost Albert Einstein i genialul fizicianul Paul Dirac. Ambii au fost remarcabili n capacitatea lor de a gasi noi ecuaii si au fost avut convingerea c c ecuaiile fundamentale ale fizicii trebuie s fie frumoase.

Ce este frumuseea matematic? Acest lucru poate suna ciudat. Cu toate acestea, frumusetea este un cuvnt care vine uor pentru noi toi atunci cnd suntem micai de vederea unui copil care zmbeste, a panoramei unui munte sau a unei orhidee inflorite. Ce nseamn s spunem c o ecuaie este frumoasa?. Ca o mare lucrare de art, o ecuaie frumoas are printre atributele sale mai mult dect simpla atractivitate - are universalitate, simplitate i o enorma putere. Sa ne gandim la capodopere, cum ar fi ,,Mere i pere a lui Cezanne lui, domul geodezic a lui Buckminster Fuller, cantecele Ellei Fitzgerald din Manhatan sau lui AndreaBocelli la Scala din Milano . De exemplu, ecuatia E = mc2 este ea nsi extrem de puternica, simbolurile sale incapsuleaza chintesena cunotinelor care pot fi aplicate pentru fiecare conversie a energiei, de la cele din fiecare celul din fiecare lucru viu de pe Pmnt pana la cea mai ndeprtata explozie cosmic. Mai bine zis ecuatia pare a fi aici inca de la nceputul timpului

http://web.unbc.ca/~chenj/epilogue.pdf

http://www.rbhm.org.br/issues/RBHM%20-%20Festschrift/16%20-%20Tattersall%20-%20final.pdf

http://arxiv.org/pdf/hep-th/9408175.pdf

Many quotations remind us of Diracs ideas about the beauty of fundamental physical laws.For example, on a blackboard at the University of Moscow where visitors are asked to write ashort statement for posterity, Dirac wrote: A physical law must possess mathematical beauty.Elsewhere he wrote: A great deal of my work is just playing with equations and seeing whatthey give.. And nally there is the famous statement: It is more important for our equationsto be beautiful than to have them t experiment. This last statement is more extreme than Ican accept. Nevertheless, as theoretical physicists we have been privileged to encounter in oureducation and in our research equations which have simplicity and beauty and also the powerto describe the real world. It is this privilege that makes scientic life worth living, and it isthis and its close association with Dirac that suggested the title for this talk.Yet it is a title which requires some qualication at the start. First, I deliberately chose towrite SOME beautiful equations ... in full knowledge that it is only a small subset of suchequations that I will discuss, chosen because of my own particular experiences. Other theoristscould well choose an equally valid and interesting subset. In fact it is not a bad idea thatevery theorist dun certain age be required to give a lecture with the same title. This wouldbe more creative and palatable than the alternative suggestion which is that every theorist berequired to renew his/her professional license by retaking the Ph.D. qualifying exams.Second, I do not wish to be held accountable for any precise denition of terms such asmathematical beauty, simplicity, naturalness, etc. I use these terms in a completely subjectiveway which is a product of the way I have looked at physics for the nearly 30 years of myprofessional life. I believe that equations speak louder than words, and that equations bringfeelings for which the words above are roughly appropriate.Finally, I want to dispel the notion that I have chosen a presentation for my own evilpurposes. Some listeners probably anticipate that they will see equations from the work ofDirac, Einstein and other true giants. The equations of supergravity will then appear, and theaudience will be left to draw its own conclusions. I assure you that I have no such delusionsof grandeur. My career has been a mix of good years and bad years. If the good years teachgood physics, then the bad years teach humility. Both are valuable.The technical theme of this talk is that the ideas of spin, symmetry, and gauge symmetry,in particular determine the eld equations of elementary particles. There are only three gaugeprinciples which are theoretically consistent. The rst of these is the spin-1 gauge principlewhich is part of Maxwells equations and the heart and soul of the standard model. The secondis the spin-2 gauge principle as embodied in general relativity. Both theories are conrmed byexperiment. Between these is the now largely known theoretical structure of supersymmetryand the associated spin-3/2 gauge principle of supergravity. Does Nature know about this?Here, you can draw your own conclusions.This viewpoint is what led me to work on supergravity in 1976. It is view of the unicationof forces before the unication program was profoundly aected by string theory. However, Iconfess that I myself think far less about unication now than I used to. Instead I think andworry about the survival of our profession and our quest to understand the laws of elementaryparticle physics. I hope that it is not a delusion to think that this presentation may contribute in a small positive way to the survival of that quest.

http://www.fountainmagazine.com/Issue/detail/Symmetry-and-Beauty

Science Issue 48 / October - December 2004

Symmetry and BeautyBayram YerlikayaWhen visiting Moscow University, Paul Adrien Maurice Dirac, the famous physicist and the founder of Quantum Mechanics, as well as being the fifteenth Lucasian Professor of Mathematics at Cambridge University, was asked about his philosophy in physics and he wrote on a blackboard physical laws should have mathematical beauty. This phrase remains preserved on the same blackboard today. As Sir Michael Berry said at the opening of Dirac House in 1997, he showed that the simplest wave satisfying the requirements was not a simple number but consisted of four components. This seemed like to complicate matters, especially for those minds that were still reeling from the unfamiliarity of ordinary quantum mechanics. Four components! Why should anybody take Diracs theory seriously? Foremost and above all for Dirac was the fact that the logic leading to the theory was,although deeply sophisticated, in a sense beautifully simple.Much later, when someone asked him what do you think of the equation? he is said to have replied: I think that it is beautiful. In fact, Professor Dirac knew that very significant mathematical equations occur in all created things. Even though these consist of deeply sophisticated matters, at the same time they occur with a beautiful simplicity and are a clear description of the action of creation of the Eternally Besought of All. When we examine his quotation in this light, we are better able to understand what he meant.

Be they physical or chemical, many attributes of beings are dependent on mathematical laws and their appearances are also shaped along mathematical principles. When we observe creation from this standpoint, we can perceive the perfection as well as the spectacular beauty that is inherent in every being. As reflections of the Attributes of the Names of God Almighty, Jamil (The Owner of Beauty), Bari (The One Who Creates from nothing), Sani (The Maker of All) and Musawwir (The Designer), this beauty found in the external appearance of beings is dependent on more than one factor coinciding. The most important factor here is symmetry, which is described as an exact correspondence and beautiful balance among the parts of an object. Beings are created with various symmetrical attributes and with great artistic beauty.

The most common symmetry type is the bilateral symmetry; this creates a mirror effect which is an exact correspondence between the right and left sides. An object forms an exact symmetry with its reflection in the mirror. A perfect symmetry that is very similar to the mirror effect can be found in the human body. The left and right sides of our body are symmetrically corresponding. Imagine a dividing line that passes from the middle of the forehead, through nose, chin and down the chest, we can see a perfect symmetry on both sides of the body. Our arms, legs, eyes, ears, nose and lips are designed with a bilateral symmetry. The same symmetrical structures can also be seen in most other creatures. All mammals, reptiles and birds are symmetrically created.

Another type of symmetry is rotational (radial) symmetry. Imagine a metal object that is in the shape of an equilateral triangular placed on the sand. If we will rotate this object 120o around an axis that passes through its center, the new position of the object will fit exactly into its original mark left on the sand. The reason for this is that the radial symmetry for equilateral triangles is 120 degrees. In the same way, a square has a radial symmetry of 90o and a regular polygon with n number of sides has a radial symmetry of 360/n o.

The beautiful symmetry of snow flakes, with their regular hexagonal shape are a beautiful natural phenomenon. In addition to these there are shapes in nature that have a three-dimensional radial symmetry. The most significant of these shapes are regular polyhedrons. An example of such polyhedrons is the salt crystalline elements that have cubical structures. Until recently, the fact that there is a creature in nature that has a regular polyhedral shape, consisting of twenty sides, was unknown. However, when a type of adenovirus that causes infections and hepatitis in dogs was discovered, it was found that there is a creature with twenty regular sides in nature.

One of the most beautiful samples of radial symmetry in nature is the daisy. Symmetrical structures do not only exist in the normal world and in the micro worlds, but also can be found in the macro world, like all the huge celestial objects, the Sun, the Moon, galaxies, star clusters in the sky . . . . All planets move around the Sun in a symmetrical manner, whereas galaxies have a spiral symmetry. It is interesting that the symmetrical structure of living beings is overwhelmingly apparent externally, rather than internally. For example, the internal organs in the human body, like the lungs, liver, stomach and intestines are not symmetrical and we have only one heart in one side of our chest cavity. Moreover, the lobes of the brain are not symmetrical either. However, all the metabolic processes in human body function properly. Does this mean that the mathematical beauty found in our external appearance is merely for esthetical reasons? God does not create things for only one reason or purpose, on the contrary, He creates them to serve many motives and in relation with many functions. For example, if we did not have two eyes and if they were not symmetrically placed on our faces, we would not be able to see objects three-dimensionally. In the same way, if our ears were not symmetrically placed on our heads, then we would have great difficulty in determining the direction and source of sounds. If we did not have symmetrical feet and legs, we would not be able to walk well, and if our arms were not symmetrical, we would not be able to balance our bodys centre of gravity while walking. If birds did not have symmetrical wings, they would not be able to fly, and if the fins of fishes were not symmetrical, they would not be able to swim smoothly.

Symmetry is also closely related to physical and mental robustness. According to one study, women who suffer from an infectious disease during pregnancy are more likely to have babies with asymmetrical features. The same study claims that asymmetrical babies are more susceptible to heart disease than symmetrical babies.

Another study shows that people with asymmetrical teeth are more likely to have more harmful microorganisms in their mouth than those who have symmetrical teeth. It is interesting that there tends to be a greater difference between the fingerprints on the left and right hands of schizophrenic people than on those of normal people.

Symmetry is a phenomenon that is used by animals and insects. For example, an experiment showed that bees prefer flowers that are symmetrical. Actually, flowers with perfectly symmetrical shapes produce more nectar than those that are asymmetrical. In one investigation, a symmetrical flower was made asymmetrical with a pair of scissors. The flower had been attractive to bees before its shape was changed; after made asymmetrical, the flower became unattractive to bees, even though it had just the same amount of nectar as before.

All these facts reveal that there is much wisdom and beauty hidden within the symmetry that the Almighty Designer uses to shape all beings. We take symmetry for granted. To have two eyes placed equidistance and two ears on each side of the head is the norm. Anything else strikes us as strange. But if we just take a few moments to think about why our eyes are where they are, and why our ears are placed on the sides of our heads, the answer is obvious. Gods mercy is infinite; in even the simplest example of symmetry there is a reason. We should not take this world for granted, but rather use every opportunity to dwell upon and be thankful for the wonderful world that has been created for us.

References

Stewart, I. & M. Golubitsky, Fearful Symmetry, Blackwell, 1992.Rosen, J., Symmetry Discovered, Cambridge University Press, 1975.Tarasov, L., This Amazingly Symmetrical World, Mir Publishers, Moscow: 1986.

http://www.spaceandmotion.com/physics-deduce-laws-of-nature.htm

The meaning of the Natural LawsBefore looking at the details of the origins of the natural laws, it is inspiring to think about the role of the mathematical natural laws and how some scientists think about them.The Power of understanding in Equations. Great equations share with fine poetry an extraordinary power. Poetry is the most concise and highly charged form of language, just as equations are the most succinct form of understanding of the physical reality that they describe. For example, E = mc2is itself enormously powerful: its symbols encapsulate knowledge that can be applied to every energy conversion, from ones in every cell of every living thing on Earth, to the most distant cosmic explosion. Better yet, it seems to have held good since the beginning of time. The WSM tells us why.Great equations are just as rich a stimulus to the imagination as is poetry. Shakespeare could no more have foreseen the multiple meanings that readers have perceived in "Shall I compare thee to a summer's day?" than Einstein could have predicted the myriad consequences of his equations of relativity. None of this is to imply that poetry and scientific equations are the same. Every poem is written in a particular language and loses its magic in translation, whereas an equation is expressed in the universal language of mathematics: E = mc2is the same in English as it is in Urdu. But there are differences; poets seek multiple meanings and interactions between words and thoughts, whereas scientists intend their equations to convey a single, logical meaning.What is a Law of Nature? The natural laws describe natural behavior and are usually expressed as equations. Before the WSM, each law was a mathematical copy of measurements of Nature; thus it was an empirical law. An analogy popularized by the physicist Richard Feynman helps to clarify this relationship between equations and empirical laws. Imagine people watching a game of chess. If they had never been taught the rules of chess, they could work them out simply by observing how the players moved the various pieces. Now imagine that the players are not playing ordinary chess, but are moving the pieces according to a much more complicated set of rules on a hugely extended board. For the observers to be able to work out the rules of the game, they would have to watch parts of it extremely carefully, looking for patterns and any other clues they could muster. That, in essence, was the predicament of scientists before the WSM. They closely observed nature - the movements of the pieces - and tried to glean the underlying laws.It is difficult to understand Nature beyond the empirical laws of even though they can be written down conveniently as equations. But some laws, like the WSM, are not empirical because they directly describe Nature itself. In physics, Einstein's equation of general relativity gives a new meaning to gravity by equating the curvature of space to the energy density of matter at that location. Then the curvature produces the gravity force. In quantum theory, Schrdinger's equation describes the behavior of matter in the micro-world as waves, enabling an understanding of atoms and molecules that had proved impossible with older ideas. John Nash, the Nobel prize-winning mathematician who suffered from schizophrenia and is the subject of the film A Beautiful Mind, came up with equations that determined how two people ought to behave in competitive games. Economists and biologists later found that his ideas were extremely relevant to their work.The God of Nature is a Mathematician? Why is it that so many laws can be expressed as an absolute equation; i.e. that two different quantities (the equation's left and right sides) are exactly equal? Why do fundamental laws exist at all? A popular, tongue-in-cheek explanation is that God is a mathematician, an idea that unhelpfully replaces profound questions with an unverifiable proposition.The Indian-American astrophysicist Subrahmanyan Chandrasekhar probably spoke for most great theoreticians when he remarked that when he found some new fact or insight, it appeared to him to be something "that had always been there and that I had chanced to pick up". According to this view, the equations that underlie the workings of the universe are in some sense already "out there", independent of human existence, so that scientists are cosmic archaeologists, trying to unearth laws that have lain hidden since time began.Of the hundreds of thousands of research scientists who have ever lived, very few have an important equation to their name. Two scientists who were adept at discovering fundamental equations and also perceptive about the role of mathematics in science, were Albert Einstein and the brilliant theoretical physicist Paul Dirac. Both were remarkable in their ability to find new equations that were as fecund as the greatest poetry. And both were captivated by the belief that the fundamental equations of physics must be beautiful.What is Mathematical Beauty? This may sound strange. Yet beauty is a word that readily comes to all of us when we are moved by the sight of a smiling baby, a mountain vista, an exquisite orchid. What does it mean to say that an equation is beautiful? Fundamentally, it means that the equation can evoke the same rapture as other things that we describe as beautiful. Like a great work of art, a beautiful equation has among its attributes more than mere attractiveness - it has universality, simplicity, and elemental power. Think of masterpieces such as Cezanne's Apples and Pears, Buckminster Fuller's geodesic dome, Ella Fitzgerald's recording of Manhattan. They provide a presence of something monumental in conception, fundamentally pure and crafted so carefully that its power would be diminished if anything in it were changed.An additional quality of a scientific equation is that it has utilitarian beauty. It must tally with every relevant experiment and make predictions of future experiments. This aspect is akin to the beauty of a classic Swiss clock or watch that is so well made and accurate that collectors love them and enjoy watching their movement.The concept of beauty was especially important to Einstein. According to his son Hans, "He had a character more like that of an artist than of a scientist as we usually think of them. For instance, the highest praise for a good theory was not that it was correct or exact, but that it was beautiful." He once went so far as to say that "the only physical theories that we are willing to accept are the beautiful ones", taking it for granted that a good theory must agree with experiment.Dirac was even more emphatic than Einstein in his belief in mathematical beauty as a criterion for the quality of theories. In the latter part of his career, he spent much time touring the world, giving lectures on the origins of the equation that bears his name, stressing that the pursuit of beauty had always been a lodestar as well as an inspiration. During a seminar in Moscow in 1955, when asked to summarize his philosophy of physics, he wrote on the blackboard in capital letters, "Physical laws should have mathematical beauty."http://www.spaceandmotion.com/mathematical-physics/logic-truth-reality.htm

Mathematical Truths Vs Truths of Physical RealityToday's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality. (Nikola Tesla)Interestingly though, once we have a logical language which describes the mathematical relationship between objects, then we can do away with the objects and simply consider the exact logical (mathematical) relationships. Hence mathematics has a remarkable power which people did not understand, that further enhanced itsmystical aspect.Mathematics was associated with a more refined type of error. Mathematical knowledge appeared to be certain, exact, and applicable to the real world; moreover it was obtained by mere thinking, without the need of observation. Consequently, it was thought to supply an ideal, from which everyday empirical knowledge fell short. It was supposed on the basis of mathematics, that thought is superior to sense, intuition to observation. If the world of sense does not fit mathematics, so much the worse for the world of sense. ... This form of philosophy begins with Pythagoras. (Bertrand Russell)Herein lies the great weakness, and the great strength of mathematics. It is possible to evolve more and more complex relationships between things, which shed light on ideas far beyond the original relationships. Unfortunately, it is also possible that these things do not actually exist, except as evolved complex mathematical relationships.The skeptic will say: "It may well be true that this system of equations is reasonable from a logical standpoint. But this does not prove that it corresponds to nature." You are right, dear skeptic. Experience alone can decide on truth. ... Pure logical thinking cannot yield us any knowledge of the empirical world: all knowledge of reality starts from experience and ends in it.(Albert Einstein, 1954)Some things that satisfy the rules of algebra can be interesting to mathematicians even though they don't always represent a real situation. (Richard P. Feynman)From this we can conclude that there are two types of mathematical truths.i) Mathematical Truths only.ii) Mathematical Truths which also correspond to Physical Reality.An important example of a mathematical truth which is also true of physical reality is Pythagoras' theorem. This is the reason for this relationships great power, and its use in Einstein's metrics. (See:Deducing the Most Simple Science Theory of Reality)For an example of a simple mathematical truth only, let us consider the partial reflection of light by glass of varying thickness. If we assume that the light is either reflected by the front surface of the glass or the back surface of the glass, then by summing Feynman's probability arrows for both paths we can correctly calculate the probability of light reflecting from any thickness of glass.But you may rightly ask, what are 'surfaces', and how do they reflect light?And you would of course be wasting your time, because light does not reflect from the surface of glass. As Feynman writes;Thus we can get the correct answer for the probability of partial reflection by imagining (falsely) that all reflection comes from only the front and back surfaces. In this intuitively easy analysis, the 'front surface' and 'back surface' arrows are mathematical constructions that give us the right answer, whereas .... a more accurate representation of what is really going on: partial reflection is the scattering of light by electrons inside the glass. (Richard P. Feynman)This is a fundamental limitation of mathematics. It is quite possible to have a true mathematical relationship, that suggests a particular physical model, and yet the theory may be completely wrong. This makes mathematics very confusing and deceptive.I mention this because it is very important in explaining why mathematical physics is now so absurd as many of its mathematical truths have been misunderstood, which has resulted in incorrect theoretical interpretations (which is why a correct knowledge of physical reality is so important to mathematicians / mathematical physics).\http://www.mi.sanu.ac.rs/vismath/stakhov2009/mathharm.pdf1. Introduction: Diracs Principle of Mathematical Beauty and beautiful mathematical objects1.2. Mathematics. The Loss of Certainty. What is mathematics? What are its origins and history? What distinguishes mathematics from other sciences? What is the subject of mathematical research today? How does mathematics influence the development of modern scientific revolution? What is a connection of mathematics and its history with mathematical education? All these questions always were interesting for both mathematicians, and representatives of other sciences. Mathematics was always a sample of scientific strictness. It is often named Tsarina of Sciences, what is reflection of its special status in science and technology. For this reason, the occurrence of the book Mathematics. The Loss of Certainty [38], written by Morris Kline (1908-1992), Professor Emeritus of Mathematics Courant Institute of Mathematical Sciences (New York University), became a true shock for mathematicians. The book is devoted to the analysis of the crisis, in which mathematics found itself in the 20-th century as a result of its illogical development. Kline wrote: The history of mathematics is crowned with glorious achievements but also a record of calamities. The loss of truth is certainly a tragedy of the first magnitude, for truths are mans dearest possessions and a loss of even one is cause for grief. The realization that the splendid showcase of human reasoning exhibits a by no means perfect structure but one marred by shortcomings and vulnerable to the discovery of disastrous contradiction at any time is another blow to the stature of mathematics. But there are not the only grounds for distress. Grave misgivings and cause for dissension among mathematicians stem from the direction which research of the past one hundred years has taken. Most mathematicians have withdrawn from the world to concentrate on problems generated within mathematics. They have abandoned science. This change in direction is often described as the turn to pure as opposed to applied mathematics. Further we read: Science had been the life blood and sustenance of mathematics. Mathematicians were willing partners with physicists, astronomers, chemists, and engineers in the scientific enterprise. In fact, during the 17th and 18th centuries and most of the 19th, the distinction between mathematics and theoretical science was rarely noted. And many of the leading mathematicians did far greater work in astronomy, mechanics, hydrodynamics, electricity, magnetism, and elasticity than they did in mathematics proper. Mathematics was simultaneously the queen and the handmaiden of the sciences. However, according to the opinion of famous mathematicians Felix Klein, Richard Courant and many others, starting from 20-th century mathematics began to lose its deep connections with theoretical natural sciences and to concentrate its attention on its inner problems. Thus, after Felix Klein, Richard Courant and other famous mathematicians, Morris Kline asserts that the main reason of the contemporary crisis of mathematics is the severance of the relationship between mathematics and theoretical natural sciences, what is the greatest strategic mistake of the 20th century mathematics.

1.2. Diracs Principle of Mathematical Beauty By discussing the fact what mathematics are needed theoretical natural sciences, we should address to Diracs Principle of Mathematical Beauty. Recently the author has studied the contents of a public lecture: The complexity of finite sequences of zeros and units, and the geometry of finite functional spaces [39] by eminent Russian mathematician and academician Vladimir Arnold, presented before the Moscow Mathematical Society on May 13, 2006. Let us consider some of its general ideas. Arnold notes: 1. In my opinion, mathematics is simply a part of physics, that is, it is an experimental science, which discovers for mankind the most important and simple laws of nature. 2. We must begin with a beautiful mathematical theory. Dirac states: If this theory is really beautiful, then it necessarily will appear as a fine model of important physical phenomena. It is necessary to search for these phenomena to develop applications of the beautiful mathematical theory and to interpret them as predictions of new laws of physics. Thus, according to Dirac, all new physics, including relativistic and quantum, develop in this way. Paul Adrien Maurice Dirac (1902-1984)At Moscow University there is a tradition that the distinguished visiting-scientists are requested to write on a blackboard a self-chosen inscription. When Dirac visited Moscow in 1956, he wrote "A physical law must possess mathematical beauty." This inscription is the famous Principle of Mathematical Beauty that Dirac developed during his scientific life. No other modern physicist has been preoccupied with the concept of beauty more than Dirac. Thus, according to Dirac, the Principle of Mathematical Beauty is the primary criterion for a mathematical theory to be used as a model of physical phenomena. Of course, there is an element of subjectivity in the definition of the beauty" of mathematics, but the majority of mathematicians agrees that "beauty" in mathematical objects and theories nevertheless exist. Let's examine some of them, which have a direct relation to the theme of this article. 1.3. Platonic Solids. We can find the beautiful mathematical objects in Euclids Elements. As is well known, in Book XIII of his Elements Euclid stated a theory of 5 regular polyhedrons called Platonic Solids (Fig. 1). And really these remarkable geometrical figures got very wide applications in theoretical natural sciences, in particular, in crystallography (Shechtmans quasi-crystals), chemistry (fullerenes), biology and so on what is brilliant confirmation of Diracs Principle of Mathematical Beauty.Figure 1. Platonic Solids: tetrahedron, octahedron, cube, icosahedron, dodecahedronSolidele lui Platon

http://www.rbhm.org.br/issues/RBHM%20-%20Festschrift/16%20-%20Tattersall%20-%20final.pdf

1Epilogue Mathematics, Beauty and Reality: The Evolution of Scientific Theories When one explores wilderness uninhabited by human beings, it is very difficult to determine where and how to start. It is very difficult to go through forests or bushes. More importantly, it is almost impossible to know if one can get sufficient food supply along the way and what one can get at the end of the journey. Facing great uncertainty and physical difficulty, early explorers often try to find trails opened and used by animals. These trails greatly expedite the exploration of otherwise impenetrable wilderness. More importantly, there are good reasons for animals to use the trails, which often lead to resource rich destinations. When one explores intellectual wilderness uninhabited by researchers, it is very difficult to determine where or how to start. More importantly, it is almost impossible to know if one can keep a job for financial survival along the way and what one can get at the end of the journey. Facing great uncertainty and financial difficulty, early explorers of intellectual wilderness often try to find and use intellectual trails in our mind evolved through our animal ancestors. These intellectual trails greatly expedite the exploration of otherwise infinitely many leads. More importantly, there are good reasons for our animal ancestors to evolve these trails, which often help to obtain resources. There are many intellectual trails developed in the mind. We will only discuss two of them. The first is mathematics. If some decision making process is truly important and is needed again and again in life, it is highly economical that quantitative modules to be developed in the mind to expedite the process. For example, predators need routinely to assess its distance from the prey, the geometry of the terrain, the speed differential between itself and the prey, the energy cost of chasing down its prey, the probability of success of each chase and the amount of 2energy it can obtain from prey to determine whether, when and where to start a chase. There are many other sophisticated functions, such as navigating by migrating birds over long distance, that need sophisticated mathematical capabilities. Many animals need to make precise calculation of many of these quantitative problems many times in life. To reduce the cost of estimation, mathematical models must be evolved in their mind so many decision making processes can be simplified into parameter estimation and numerical computation. It is highly likely that, if some function is very important for the survival of the animal, in the process of evolution, this function will become encoded into animal mind. By incurring fixed cost of developing and maintaining such a mathematical function, it reduces the variable cost in each decision making. Therefore, it is natural that a fundamental understanding about life can be expressed as a mathematical theory. More precisely, since entropy is the only mathematical function to measure scarce resources, it is almost inevitable that a basic theory on life and social systems should be a mathematical theory based on entropy. As we have discussed in this book, low entropy, information, economic value and resources are essentially the same thing looked from different angles. Because of the fundamental importance of entropy inlife, human mind, which is an evolutionary product, thinks around entropy. This is why an entropy theory based economic theory turns out to be so simple and universal. This is also why so many people have been engaged in the discussion to apply the concept of entropy in economics for many years, despite the stern reprimand from Paul Samuelson, the most powerful authority in economics: And I may add that the sign of a crank or half-baked speculator in the social sciences is his search for something in the social system that corresponds to the physicists notion of entropy. (Samuelson, 1972, p. 450) The second intellectual trail is the sense of beauty. The geometry of beauty is the visible signal of adaptively valuable objects: safe, food-rich, explorable, learnable habitats, and fertile, healthy dates, mates, and babies. (Pinker, 1997, p. 526) More generally, the sense of beauty is an evolved intuition about resources. Long ago, Eddington noticed the relation between entropy and beauty: 3 Suppose that we were asked to arrange the following in two categories distance, mass, electric force, entropy, beauty, melody. I think there are the strongest grounds for placing entropy alongside beauty and melody, and not with the first three. Entropy is only found when the parts are viewed in association, and it is by viewing orhearing the parts that beauty and melody are discerned. All three are features of arrangement. It is a pregnant thought that one of these three associates should be able to figure as a commonplace quantity of science. The reason why this stranger can pass itself off among the aborigines of the physical world is that it is able to speak their language, viz., the language of arithmetic. It has a measure-number associated with it and so is made quite at home in physics. (Eddington, 1958 (1935), p. 105) Personally, for many years, I was deeply attracted by the beauty of stochastic processes and their deterministic representations in partial differential equations. In the end, the theory developed in this book was germinated from the theory of stochastic processes and partial differential equations. The above discussion indicates that beautiful mathematics often has deep connection with the real world. These connections, once established, are often plain and obvious. But the process of establishing the connections may be long and elusive. The understanding about the relation between information and physical entropy provides a good example. Shortly after Shannon (1948) developed the entropy theory of information, Weaver commented: Thus when one meets the concept of entropy in communication theory, he has a right to be rather excited --- a right to suspect that one has hold of something that may turn out to bebasic and important. (Shannon and Weaver, 1949, p. 13) This sense of excitement attracted a lot of attempts to apply the concept of entropy to many other areas. As it is often the case, earlier attempts to apply some promising intuition do not yield concrete results easily. In an editorial, Shannon tried to discourage the jumping on the bandwagon: Workers in other fields should realize that that the basic results of the subject are aimed at a very specific direction, a direction that is not necessarily relevant to such fields as psychology, economics, and other social sciences. Indeed, the hard core of information theory is 4essentially, a branch of mathematics, a strictly deductive system. (Shannon, 1956) Recent authority reinforces the idea that information theory has only limited connection with physical and social sciences. The efforts of physicists to link information theory more closely to statistical physics were less successful. It is true that there are mathematical similarities, and it is true that cross pollination has occurred over the years. However, the problem areas being modeled by these theories are very different, so it is likely that the coupling remains limited. In the early years after 1948, many people, particularly those in the softer sciences, were entranced by the hope of using information theory to bring some mathematical structure into their own fields. In many cases, these people did not realize the extent to which the definition of information was designed to help the communication engineer send messages rather than to help people understand the meaning of messages. In some cases, extreme claims were made about the applicability of information theory, thus embarrassing serious workers in the field. (Gallager, 2001, p. 2694) If Shannons entropy theory of information is purely a mathematical theory with little connection with the physical laws, it would be a miraclethat information defined as entropy turns out to have some magic technical properties in communication problems. However, once mathematical theories are thought to be a natural part of our evolutionary legacy, it would be natural for entropy theory of information to possess these properties. How can the independence of human volition be harmonized with the fact that we are integral parts of a universe which is subject to rigid order of natures laws? (Planck, 1933, p. 107) This question is called one of mans oldest riddles. A major insight from this theory is that human mind, shaped by natural selection, is intune with natural laws to lower the cost of information processing. Most of information we receive are processed unconsciously. It is only in rare occasions when decision making is needed, information processing5becomes conscious. And in most situations, there are no real choices any way. For example, you have the choice to eat or not to eat. But if you decide not to eat, you will be wiped out by natural selection. In a competitive world, one has to follow optimal choice, which is not really a choice, on most important decisions to avoid being selected out. In the following, we will discuss the general patterns of the evolution of scientific theories to understand the origin and process of scientific revolutions. Scientific theories are developed to reduce the cost of understanding nature, which includes human society. Costs consist of fixed cost andvariable cost. Fixed cost helps reduce variable cost. The basic set of fixed assets of a scientific theory is called paradigm (Kuhn, 1996). When the individual scientist can take a paradigm for granted, he need no longer, in his major works, attempt to build his field anew,starting from first principles and justifying the use of each concept introduced. (Kuhn, 1996, p. 20) The establishment of a paradigm, by incurring a common fixed cost, reduces variable cost in scientific development and communication. As a theory matures, its fixed assets accumulated. For a high fixed cost system, its variable cost will be very low when uncertainty is small. This is why science education is a narrow and rigid education, probably more so than any other except perhaps in orthodox theology (Kuhn, 1996, p.166). To further utilize the fixed assets that have been acquired by the scientific community, existing paradigm is being applied to broader and broader fields. For normal-scientific work, for puzzle-solving within the tradition that the textbooks define, the scientist is almost perfectly equipped. Furthermore, he is well equipped for another task as well --- the generation through normal science of significant crises. When they arise, the scientist is not, of course, equally well prepared. Even though prolonged crises are probably reflected in less rigid educational practice, scientific training is not well designed to produce the man who will easily discover a fresh approach. (Kuhn, 1996, p.166) Therefore, somebody, who appears with a new candidate for paradigm is usually a young man or one new to the field (Kuhn, 1996, p.166). The following quote about the emission of light or heat has a parallel in scientific research: 6 That the whole world is not aglow with radiation is a consequence of a competition between the discarding of energy as radiation and as heat. The products of most reactions are in such intimate contact with their surroundings that any excitation is quickly transferred to the neighboring molecules in the form of thermal motion. However, there are some reactions for which the contact is so weak that the excited state survives long enough for the relatively slow business of squeezing out a photon to occur. (Atkins, 1991, p. 206) The above quote can be directly translated into a comment about research. That the whole world is not aglow with great idea is a consequence of a competition between the dissemination of informationas great idea or as small idea. Most of us have intimate contact with our surroundings that any new idea is quickly transferred to the academic community in the form of thermal motion, or low impact research. However, there are some cases where the contact with the academic community is so weak that ideas hold long enough in the mind for the relatively slow business of squeezing out a truly fundamental theory to occur. All practicing scientists are educated in a common paradigm, which make it easy for them to communicate with each other. The developers of new paradigms, however, have no such luxury. From information theory, equivocation is high in communication when the receiver ofinformation does not share the same common background or paradigm with the sender of information. The promotion of a fundamentally new idea is in general so difficult that Wallace, the cofounder of the theory of evolution, gave much more credit to the promotion of new ideas over their creation. No one deserves either praise or blame for the ideas that comes to him. But the actions which result from our ideas may properly be so treated, because it is only by patient thought and work that new ideas, if good and true, become adapted and utilized. (George, 1964, p. 280) Because of the harsh environment to new ideas, many pioneers in scientific research were able to develop and promote new theories only by sheer perseverance. Neither by poverty, nor by incomprehension of the contemporaries who ruled over the condition of his life and work, did he allow himself to be crippled or discouraged. This is a comment aboutKeplers life from Einstein. It is also a reflection of lives of many other 7pioneers. Their struggle provides a profound testimony that information is costly and information with high value is very costly to obtain.