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Muhammed Zafar Iqbal book list

Muhammed Zafar Iqbal is a Bangladeshi Physicist. He and Hasibul Ahsan are collaborating to find the unified field theory of physics. They are known to each other for long time.
Muhammed Zafar Iqbal

Pascal’s triangle

A binomial expression is one which contains two terms connected by a plus or minus sign. Thus (p+q), (a+x)2, (2x+y)3 are examples of binomial expressions. Expanding (a + x)n for integer values of n from 0 to 6 gives the results as shown at the bottom of the page. From these results the following patterns emerge: (i) ‘a’decreases in powermoving from left to right. (ii) ‘x’increases in power moving from left to right. (iii) The coefficients of each term of the expansions are symmetrical about the middle coefficient when n is even and symmetrical about the two middle coefficients when n is odd. (iv) The coefficients are shown separately in Table 7.1 and this arrangement is known as Pascal’s triangle. A coefficient of a term may be obtained by adding the two adjacent coefficients immediately above in the previous row. This is shown by the triangles in Table 7.1, where, for example, 1 + 3 = 4, 10 + 5 = 15


Muhammed Zafar Iqbal has written many books on quantum mechnics and thoery of special relativity. He has not written any on general relativity. I asked him why he had not written anything on GR. He actually excelled at experimental physics and he is not much expert at theoretical physics. I have read many other books of him. He must be regraded as a man of science , who works for his own motherland in spite of possessing a great prospect abroad. He realizes that our nation need more scientists rather than engineers and doctors. He also acknowledges us in his book several times. But my opinion is that he should be writing more theory related books than science fictions.
Kolpo, a 13-year-old reader, requested for new science fiction at a stall in the Amar Ekhushey Book Fair. He told Dhaka Tribune that he was waiting for his copy of the new science fiction book by Prof Muhammed Zafar Iqbal. “I like his stories,” said Kolpo. “I forget that what I am reading is in fact not realistic.” Sanjida Hossain is a lecturer at Brac University. She, too, came to buy her copy of Zafar Iqbal’s new science fiction at the book fair. Titled “Tratina”, the book has seen bestseller in the book fair until now. According to its publisher, Somoy Prokashon, it has already sold 30,000 copies. “Within the first 10 days, more than 5,000 copies of ‘Tratina’ were sold,” said Mahmud Piyash, a salesman at Somoy Prokashon. However, Piyash added that they have had to approach visitors to let them know about publications from other, new science fiction authors. Somoy Prokashon published two more science fiction thrillers this year at the book fair- “Turash” by Biswajit Das and “Beyond the Man-Made Universe” by Tasruzzaman Babu. Another publication house, Anannya, has published five sci-fi thrillers this year. One of them is Tanvir Rana Mustafiz’s sci-fi collection “Science Fiction Shomogro”, featuring stories about galaxies, aliens and exotic animals. A salesman of Anannya said that keeping pace with visitors’ demands, they have also published translations of different works of science fictions. Oitijjhyo, another leading publication house, has published a sci-fi translation of “Norby: The Mixed-Up Robot”, the first in the series of 11 books by Janet and Isaac Asimov and translated by Khandaker Istiaque Mahmud. Samiul Alam, a sci-fi lover from Narayanganj, opined that most of the local science fiction books are teen thrillers rather than actual science fiction. “The sci-fi novels that I have been going through in the Ekushey book fair are mostly teen thrillers,” said Samiul. “That is why I am inclined to go for translations. But Seba Prokashoni is the best place for me. I am going to hit its stall right after I am done visiting others.”

Bigyani Sofdor Alir Moha Moha Abisker

A man was very lonely and he used to spend most of his time alone. This is a story of remote place in bangladesh where he lives. He suddenly started questioning everything. All the big questions like where had he come from? , how did the universe begin ?, can time run backwards? can we go to our past? and many other philosophical and scientific questions came into his mind. He started buying books on consmology, relativity, quantum mehcanics and mathematical and philosophical books. He stays alone in his room and looks outside to find something special. He has a computer which he keeps switched on for long time. He searches the web day and night to find something new. Suddenly a thought came in his mind . Why do we think that everything exists? What is its connection to our memory? He had learned about theory of relativity and quantum theory in the meantime after reading a lot of books. Now he tries to combine these two theories to find the ultimate theory of the universe. No scientist had been able to find the grand unified theory. Maxwell had told that scientists can not find the ulitmate secret of the nature because what we are trying to find is very part of uurselves. It is apparent that he was mentioning our mind.
{not consistent}
He now thinks about what might have gone wrong. Why are scientists not finding the ultimate theory of nature? May be physics will not help us. So we need some philoshophical theory. This kind of thinking keeps him always busy to find something new. If physics were the answer then it would been found long ago. String theory has not been verified yet. He thinks what if memory creates reality. There is no past and future , which Einstein taught us. Nothing happened in our past. We think that something happened because our mind gives us false impression of the past. It just made us recollect certain memory in our mind. Memory is not necessarily the past events. It is like the informations stored in the hard disk of computer. Physical world is just an illusion. He is now very thrilled. This also seems to correspond to the simulation theory. Simulation theory is the theory that says we are nothing but simulation inside a computer. But it is not the same as simulation. Nobody is provinding these memories into our mind. He thinks that our conventional theory of nervous system is wrong. We see and taste not because our senses send electrical impulse to our brain to process. We feel and hear not because of this electrical signal. There is something wrong with this theory that medical science promotes. Electricity and electons are too mysterious to be seen or percieved with bare senses. Why would nature build such a system to process its own electricity? There must be other explanation.

Theory of relativity

He now thinks about general theory of relativity and its connection with our body and mind. It says events are prime or fundamental entities in the universe. What we call mind is the group of events connected with certain relation. He thinks this could be a clue to the mystery. An event is a collection of four numbers or coordinates (x1, x2, x3, t). So mind is composed of all the events that are of particular type ( space-like). What we see , hear, taste or feel are events happening inside our minds. The physical world is continuous with the events in our mind. For every event happening to a piece of matter there is a corresponding event in our mind. These events are simultanuous. We see not because light comes to our eye but there are simultaneous events. This also corresponds to the fact that there is no existence of light between emission and absorption. The case with other senses is the same as the case of seeing. Thus we can dispense with the electricity based theory of our reality and everything ???

Special relativity

Special theory of relativity is concerned with time dilation and relativistic mass increase. A set of equations completely determines the laws of special relativity. Muhammed Zafar iqbal seems to have explained the mathematics concerning coordinate transformations and energy- mass equivalence. But he has forgotten some useful informations which laid the foundation of relativity. I now briefly explain this :
All velocities are relative. If we can compare or make relation between two reference frames we can account for all the facts about relativistic effects. So Einstein used two reference frames whose relative velocity we know. Many other facts like why we need four coordinates instead of three and constancy of velocity of light are very vaguely explained in his books. You can vist this page and understand the theory easily.
Muhammed Zafar Iqbal
The most basic thing about relativity is that we must add a more dimension of time t to usual three dimensions of space x, y, z to describe perfectly something happening.

Zafar Iqbal's mathematics

Muhammed zafar Iqbal has written many books on mathematics. He has written about geometry and algebra and also calculus. He has developed Bangladeshi group of international mathematics olimpiad. But I do not like his style at all. He has retained the old tradition of teaching and instructing. Our country is not getting something new. Where are the fundamentals of mathematics gone? Should we not be teaching our children what is mathematics and what is its purposes. We can not make our children interested in learning unless we make the subject pleasant. I do not think the education system is totally wrong but it is fundamentally weak and old-fashioned. Mathematics should start with few basic principles. For example theory of natural numbers can form the foundation of the whole branch of algebra and analysis. Geometry can be reduced to few axioms and propositions. For example Euclid's geometry is concerned with twenty propositions from which other theorems and corollaries can be developed. Muhammed Zafar Iqbal does not stress on the fundamentals that are needed to study the true nature of mathematics. This seems to be the same case with our academic institutes. Let me say a few things about difference between logic and mathematics.
The distinction of mathematics from logic is very arbitrary, but if a distinction is desired , it may be made as follows. Logic consists of the presmisses of mathematics, together with all other propositions which are concerned exclusively with logical constants of the premisses which asserts formal implications containing variables, together with such of the premisses themselves as have these marks. Thus some of the presmisses of mathematics , e.g the principle of the syllogism, "if p inplisq and q implied r, then p implies r," will belong to mathematics , while others, such as "implication is a relation" will belong to logic but not to mathematics. But for the desire to adhere to usage , we might identify mathematics and logic, and define either as the class of propositions containing only variables and logical constants; but respect for tradition leads me rather to adhere to the above distinction, while recognizing that certain propositions belong to both sciences.

Muhammed Zafar Iqbal book lists


Some useful arithmetic operations are :

Muhammed Zafar Iqbal

Least Square fitting

This method was invented by Carl Friedrich Gauss. Least square method enable us to define a relationship between various data. Given a set of data we can approximate a curve that will best predict the relationship. Mathematically it will be as follows:

Muhammed Zafar Iqbal books list
The correct mathematical analysis involves differentiation. The problem is to find the curve or straight line that approximate the best fitting. First we start with a straight line having slope m1 and constant m2. The next step is to differentiate the difference R(n) with respect to m1 and m2. We now have two equations with two unknown \ variables to solve. This is always possible. :
Muhammed Zafar Iqbal books list
This method is also known as linear regression analysis. After finding the parameters we come up with a curve like this one.
Muhammed Zafar Iqbal books list
The linear regression model will be , mathematically ,

Muhammed Zafar Iqbal books list

Gaussian distribution

Gaussian distribution is the curve that has special properties. It is also called the "bell curve". This special curve has many applications in statistics and mathematical physics. The general form of such gaussian curve is as follows:

Muhammed Zafar Iqbal book list

Geometry equation

Geometry and algebra help us calculate many useful properties of solids and two dimensional figures. We can use the theorems of algebra and geometry to calculate area of a trapezium, parallelogram and others. Some general useful formulas are :

Muhammed Zafar Iqbal book list
If we are given three sides of a triangle we can calculate the are of a triangle. If we know the height and a side we can calculate the area of a parallelogram as given in the figure.
Some elementary geometry formulas are given in a single package :

Muhammed Zafar Iqbal books list

Great Circle

Great circle is the shortest route on the surface of a sphere. It the curve spanning a plane that goes through the center of the sphere. If you hold a rubber string tout on the surface of a sphere it will lie along great circle. The distance between any two points on it will be shortest. The equation of great circle is :

Muhammed Zafar Iqbal books list
Where σ and &lamda; are coordinates representing longitudes and latitudes in angles. r is the radius of the sphere.

First and second Fundamental form

If you want to understand general theory of relativity you will encounter first and second fundamental form which Gauss discovered long ago. It is related to curved surfaces. The distance in curved surface is defined through Gauss's first fundamental form. It has the following form :

Muhammed Zafar Iqbal book list

Set theory

Set theory was developed by Goerge Cantor. He first gave the definition of set. A set is a collection of well defined objects. Some useful properties of set are given:

Muhammed Zafar Iqbal book list
This formula mentioned can be generalized to more than two or three sets as follows:

Muhammed Zafar Iqbal book list

Newton-Rapshon method

Newton-Rapshon method is a process to find root of some equation by iteration method.
Muhammed Zafar Iqbal book list

Rules of signs

"Rule of signs" is a special rule for determining number of roots that a polynomial equation can have. It is a very simple rule indeed

Muhammed Zafar Iqbal book list

Muhammed Zafar Iqbal book list


Hyperfunctions are generalizations of functions as a jump from one holomorphic function to another at a boundary. A hyperfunction on the real line can be concieved of "as the difference between one holomorphic function on the upper- half plane and one holomorphic function on the lower half-plane. That is , a hyperfunction is specified by a pair (f, g) where f is the holomorphic function on the upper half plane and g is the holomorphic function on the lower half plane.
Informally , a hyper function on the real line can be given as the difference f-g so that adding a same holomorphic function to both f and g would not change the result f-g.
As an example Heaviside step function can be represented as this pair of functions.

Muhammed Zafar Iqbal book list

Integration formula

Integration is the most useful method in mathematical physics. It is a branch of calculus. Its appication can be seen in theory of relativity , quantum field theory and string theory. It is quite ubiquitous now. There are some basic rules of integrating hyperbolic functions. Here are they:

Muhammed Zafar Iqbal books list
Hyperbolic functions are defined in this way:

Muhammed Zafar Iqbal books list
Some useful properties of definite integral :
Muhammed Zafar Iqbal books list
Integration by parts is an useful method in evaluating complicated integral. Here is the formula along with other methods:

Muhammed Zafar Iqbal books list
Many integrations are done by method of substitution. Here are some such methods:

Muhammed Zafar Iqbal books list

Cauchy's Integration formula

Cauchy was a great mathematician and he first gave a rigorous definition of limit. His theorem of complex integration is very well known to us. He also developed the theorem of repeated integration.

Muhammed Zafar Iqbal books list

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Calculus of variation

Calculus of variation is perhaps the most useful topic in differential geometry. It is used to find the maximum and minumum value of a function and even an inflection point where a curve changes its concavity. A summary of calculus of variation can be put in a figure like this :

Muhammed Zafar Iqbal books list

Differential geometry

Geometry in theory of general relativity is not constant. Hence it is called differential geometry. Differential geometry is the study of surfaces and curved spaces. It is entirely different geometry and it involves a concepts like tensors. Without tensors we can not study such geometry. In three dimensions we can construct a model of such geometry as follows:

Muhammed Zafar Iqbal books list
Normal curvature is defined in this way:

Muhammed Zafar Iqbal books list
Here E , F , G, L, M and N are the functions of coordinates u and v . Differential geometry is so vast a subject that it can not explained here briefly. But these are the basic ideas.

Theory of Relativity

General relativity says that spacetime curvature creates gravity. Russell's exposition on general relativity is perhaps most exciting than others. He was able to explain it to the mass people in layman terms at the time when few people understood it. I now recite one of his explanations , which is related to relativistic momentum:
It was found that when an object moves, momentum in a given direction is equal to the invariant mass multiplied by the component of velocity in the given direction. Invariant mass is the mass measured in the rest frame of the object. There is another mass which is measured in a reference frame other than the object's frame. This mass increases as the speed of the object increases. When we substitute distance travelled in unit time by distance travelled in per unit interval of spacetime, the momentum in any given direction remains the same. For ordinary velocity this is a slight change as the distance traveled is very less (the interval is almost equal to the time lapse between the events). And instead of relative mass we take the proper or rest mass . These two changes decreases the mass but increases but the velocity. As a result the momentum remains the same. But invariant mass is replaced by the mass of the object. It seems that the object possesses another kind of momentum. This is the momentum which equals the mass of the object multipled by the time traversed when traveling an unit interval.
Mathematics of general theory of relativity is hard. It is full of tensor equations and ccalculus. Here is an example of the complexity of equations:
Muhammed Zafar Iqbal books list
In theory of relativity mass is equivalent to time. Mass of a particle makes it a pefect clock. This can be infered from the equations of Einstein and Planck.
We know that energy is equivalent to mass by this relation E = mc (squared) . Again Planck law tells us that energy is equivalent to frequency E = h v where v is frequency. So
hv = mc (squared) or v = [ mc(squared)]/ h . This is the frequency of a particle with mass m. Every particle ticks at a rate specified by this relation. For massless particle like photon there is no time. This is the argument given by Roger Penrose. He has developed CCC ( conformal cyclic cosmology) using this hypothesis.

Russell's account

The Nature of the problem
Apart from pure mathematics , the most advanced of the science is physics. Certain parts of theoretical physics have reached the point which makes it possible to exhibit a logical chain from certain assumed premisses to consequences apparently very remote, by means of purely mathematical deductions. This is true especially of everything that belongs to the general theory of relativity. It cannot be said that physics as a whole has yet reached this stage, since quantum phenomena, and the existence of electrons and protons, remains, for the moment, brute facts. But perhaps this state of affairs will not last long ; it is not chimerical to hope that a unified treatment of the whole of physics may be possible before many years have passed.
In spite, however, of the extraordinary successes of physics considered as a science , the philosophical outcome is much less clear that it seemed to be when less was known. The purpose here is to discuss what is meant by the "philosophical outcome" of physics, and what methods exist for determining its nature.
There are three kinds of questions which we may ask concerning physics or, indeed, concerning any science. The first is: What is its logical structure, considered as a deductive system? What ways exist of defining the entities of physics and deducing the propositions from an initial apparatus of entities and propositions? This is a problem in pure mathematics , for which, in its fundamental portions, mathematical logic is the proper instrument. It is not quite correct to speak, as we did just now, of "initial entities and propositions". What we really have to begin with , in this treatment, is hyppotheses containing variables. In geometry, this procedure has become familiar. Instead of "axioms" supposed to be "true" we have the hypothesis that a set of enities (otherwise undefined) has certain enumerated properties. We proceed to prove that such a set of entities has the properties which constitute the propositions of Euclidean geometry, or of whatever other geometry may be occupying our attention. Generally it will be possible to choose many different sets of initial hypotheses which will all yield the same body of propositions; the choice between these sets is logically irrelevant, and can be guided only by aesthetic considerations. There is , however, considerable utility in the discovery of a few simple hypotheses which will yield the whole of some deductive system, since it enables us to know what tests are necessary and sufficient in deciding whether some given set of entities satisfies the deductive system. Morever, the word "entities" which we have been using, is too narrow if used with metaphysical implication. The "entities" concerned may , in a given application of a deductive system, be complicated logical structures. Of this we have examples in pure mathematics in the definitions of cardinal numbers, ratios, real numbers, etc. We must be prepared for the possibility of a similar result in physics, in the definition of a "point" of spacetime and even in the definition of an electron or a proton.

Muhammed Zafar Iqbal
The logical analysis of a deductive system is not such a definite and limited undertaking as it appears at first sight. This is due to the circumstances just mentioned-namely , that what we took at first as primitive entities may be replaced by complicated logical structures. As this circumstance has an important bearing upon the philosophy of physics, it will be worth while to illustrate its effect by examples from other fields.
One of the best examples is the theory of finite integers. Weistrass and others had shown that the whole of analysis was reducible to propositions about finite integers, when Peano showed that these proposistions involving three undefined ideas. The five initial propositions might be regarded as assigning certain properties to the group of three undefined ideas, the properties in question, every propostition of arithmethic and analysis is true of this triad, provided the interpretation appropriate to this triad is adopted. But it appeared further that there is one such triad corresponding to each infinite series x1, x2, x3, x4,... xn,.. , in which there is just one term corresponding to each finite integer. Such series can be defined without mentioning integers. Any such series could be taken, instead of the series of finite integers, as the basis for our arithmetic and analysis. Every proposition of arithmetic and analysis will remain true for any such series, but for each series it will be a different proposition from what it is for any other series.

"It is your life and it is ending one minute at a time.."

Take, in illustration, some simple proposition of arithematic, say: "The sum of the first n odd numbers is n(squared). Suppose we wish to interpret this proposition as applying to the progression x(0), x(1), x(2),... x(n),.. In this progression , let R be the relation of each term to its successor. Then "odd numbers" will mean "terms having to x1 a relation which is a power of R(squared)" where R(squared) is the relation of an x to the next x but one. We can now define R(xn) as meaning that power of R which relates x0 to xn , and we can further define xm + xn as meaning that x to which xm has the relation R(xn). This decides the interpretation of "the sum of the first n odd numbers". To define n(squared), it will be best to define multiplication. We have defined R(xn) ; consider the relation formed by the relative product of the converse of R together with R(xn); its cube relates x3 to x(3n) etc. Any power of this relation can be shown to be equivalent to a certain power of the converse of R multplied relatively by a certain power of R(xn). These is thus one power of this relation which is equivalent to moving backward from xm to x0 , and then forward; the term to which the forward movement takes us is defined as xmXxn. Thus we can now interpret xn(squared). It will be found that the proposition from which we started is true with this interpretation.
It follows from the above that , if we start from Peano's undefined ideas and initial propositions, arithmetic and analysis are not concerned with definite logical objects called numbers, but with the terms of any progression. We may call the terms of any progression 0,1,2,3.., in which case, with a suitable interpretation of + and X , all the propositions of arithmetic will be true of this terms. Thus 0,1,2,3,4,... become "variables". To make them constants , we must choose some one definite progression; the natural one to choose is the progression of finite cardinal numbers as defined by Frege. What were, in Peano's methods, primitive terms are thus replaced by logical structures, concerning which it is necessary to prove that they satisfy Peano's five primitive propositions. This process is essential in connecting arithmetic with pure logic. We shall find that a process similar in some respects, though very different in others, is required for connecting physics with perception.
The general process of which the above is an instance will be called the process of interpretation. It frequently happens that we have a deductive mathematical system, starting from hypotheses concerning undefined objects, and that we have reason to believe that there are objects fulfilling these hypotheses, although, initially , we are unable to point out any such objects with certainty. Usually , in such cases, although many different sets of objects are abstractly available as fulfilling the hypotheses, there is one such set which is much more important than the others. In the above instance, this set was the cardinal numbers. The substitution of such a set for the undefined objects is "interpretation" . This process is essential in discovering the philosophical import of physics.

Muhammed Zafar Iqbal

"the best teacher is your previous mistake."

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The difference between an importantand an unimportant interpretation may be made clear by the case of geometry. Any geometry , Euclidean or non-Euclidean, in which every point has co-ordinates which are real numbers, can be interpreted as applying to a system of sets of real numbers-i.e a point can be taken to be the series of its co-ordinates. This interpretation is legitimate, and is convenient when we are studying geometry as a branch of pure mathematics. But it is not the important interpretation. Geometry is important, unlike arithmetic and analysis, because it can be interpreted so as to be part of applied mathematics- in fact, so as to be part of physics. It is this interpretation which is really interesting one, and we cannot therefore rest content with the interpretation which makes geometry part of the study of real numbers, and so , ultimately , part of the study of finite integers. Geomtery, as we shall consider it in the present work, will be always treated as part of physics, and will be regarded as dealing with objects which are not either mere varibles or definable in purely logical terms. We shall not regard a geometry as satisfactorily interpreted until its initial objects have been defined in terms of entities forming part of the empirical world, as opposed to the world of logical necessity. It is , of course, possible, and even likely, that various different geometries , which would be incompatible if applied to the same set of objects, may all be applicable to the empirical world by means of different interpretations.
So far , we have been considering the logical analysis of physics. But in relation to the interpretation of geometry we have already been brought into contact with a very different problem- namely , that of the application of physics to the empirical world. This is , of course, the vital problem; althought physics can be pursued as pure mathematics, it is not as pure mathematics that physics is important. What is to be said about logical analysis of physics is therefore only a necessary preliminary to our main theme. The laws of physics are believed to be at least approximately true, although thery are not logical necessary; the evidence for them is empirical. All empirical evidence consists, in the last analysis, of perceptions; thus the world of physics must be , in some sense, continuous with the world of our perceptions, since it it the latter which supplies the evidence for the laws of physics. In the time of Galileo, this fact did not seem to raise any very difficult problems, since the world of physics had not yet become so abstract and remote as subsequent research has made it. But already in the philosophy of Descartes the modern problem is implicit and with Berkeley it became implicit. The problem arises because the world of physics is , prima facie, so different from the world of perception that it is difficult to see how the one can afford evidence for the other; moreover , physics and physiology themselves seem to give very accurate grounds for supposing that perception cannot give very accurate information as to the external world, and this weaken the props upon which the y are built.

"We are all the singing and dancing crap of the world.."

This difficulty has led , especially in the works of Dr Whitehead, to a new interpretation of physics , which is to make the world of matter less remote from the world of our experience. The principle s which inspire Dr Whitehead's work appear to me essential to a right solution of the problem , although in the detail I should sometimes incline to a somewhat more conservative attitude. We may state the problem abstractly as follows:
The evidence for the truth of physics is that perceptions occur as the laws of physics would lead us to expect-e.g we see an eclipse when the astronomers say there will be an eclipse. But physics itself never says anything about perceptions; it does not say that we shall see an eclipse , but says something about the sun and the moon. The passage from what physics asserts to the expected perception is left vague and causal; it has none of the mathematical precision belonging to physics itself. We must therefore find an interpretation of physics which gives a due place to perceptions; if not, we have no right to appeal to the empirical evidence.
This problem has two parts: to assimilate the physical world to the world of perceptions and to assimilate the world of perceptions to the physical world. Physics must be interpreted in a way which tends towards idealism and perception in a way which tends towards meterialism. I believe that matter is less material and mind less mental, than is commonly supposed , and that , when this is realised, the difficulties raised by Berkley disappear. Some of the difficulties raised by Hume, it is true have not yet been disposed of; but they concern scientific method in general, more particularly induction. On these matters I do not propose to say anything in the present volume, which will throughout assume the general validity of scientific method properly conducted.
The problems whihc arise in attempting to bridge the gulf between physics( as commonly interpreted) and perception are of two kinds. There is first epistomological problem: what facts and entities do we know of that are relevant to physics, and may serve as its empirical foundation? This demands a discussion of what, exactly, is to be learnt from a perception , and also of the generally assumed physical causation of perceptions-e.g by light-waves or sound-waves. In connection with this latter question, it is necessary to consider how far, and in what way, a perception can be supposed to resemble its external cause, or, at least, to allow inferences as to characterstics of that cause. This, in turn, demands a careful consideration of causal laws, which , however, is in any case a necessary part of the philosophical analysis of physics. Throughout this inquiry , we are asking ourselves what grounds exists for supposing that physics is "true". But the meaning of this question requires some elucidation in connection with what has already been said about interpretation.
Apart altogether from the general philosophical problem of the meaning of "truth" there is a certain degree of vagueness about the question whether physics is "true" . In the narrowest sense, we may say that physics is "true" if we have the perceptions which it leads us to expect. In this sense, a solipsist might say that physics is true; for although he would suppose that the sun and moon , for instance , are merely certain series of perceptions of his own, yet these perceptions could be forseen by assuming generally recieved laws of astronomy. So, for example, Liebniz says:
"Although the whole of this life were said to be nothing but a dream , and the visible world nothing but a phantasm , I should call this dream or phanthasm real enough , if, using reason well , we were never deceived by it".
A man who, without being a solipshist , believes that whatever is real is mental, need have no difficulty in declaring that physics is "true" in the above sense, and may even go further, and allow the truth of physics in a much wider sense. This wider sense , which I regard as the more important, is as follows: Given physics as a deductive system, derived from certain hypotheses as to undefined terms, do there exist particulars or logical structures composed of particulars, which satisfy these hypotheses? If the answer is in the affirmative , then physics is completely "true". We shall find , if I am not mistaken , that no conclusive reason can be given for a fully affirmative answer , but that such an answer emerges naturally if we adopt the view that all our perceptions are causally related to antecedents which may not be perceptions. This is the view of common sense and has always been, at least in practice, the view of physicists. We start, in physics, with a vague mass of common-sense beliefs , which we can subject to progressive refinements without destroying the truth of physics ( in our present sense of truth); but if we attempt , like Descartes, to doubt all common-sense beliefs, we shall be unable to demonstrate that any absurdity results from the rejaction of the above hypothesis as to the causes of perceptions, and we shall therefore be left uncertain as to whether physics is "true" or not. In these circumstances, it would seem to be a matter of individual taste whether we attempt or reject that may be called the realist hypothesis.
The epistomological problem, which we have just been stating in outline, will occupy us latter. There is still something to talk about ontology-i.e with the question: what are the ultimate existents in terms of which physics is true ( assuming that there are such) ? And what is their general nature.

Physics major subjects

Physics is a very vast subject. Classical physics includes all the concepts and ideas prior to Issac Newton and Einstein. Modern physics includes quantum mechanics and relativity.
all physics equations
The rest of the advanced physics topics are basically combination of quantum mechanics and relativity. Quantum mechanics have deep philosophical consequences. In classical physics electromagnetism plays a vital role. In electromagnetism magentic field created by a moving charge obeys a rule named Flemming right hand rule.

all physics equations
Viscosity is the measure of fluid's resistance to flow. If you know the viscosity of milk you can calculate how long it will take for milk to pass through the pores of cookies. This is the equation :

all physics equations


Cosmology is the universe as a whole. It describes the motion of planets and other massive ojects. It also deals with the evolution and formation of stars , galaxies. After the discovery of Einstein's general relativity cosmology gained momentum. Einstein's formulation of gravity helped scientists study the shape and size of the universe as a whole. However , general relativity breaks down when describing certain phenomena like black hole singularity and big bang. But it gives an impeccable and precise structures of our solar system and the rest of the universe.

cosmology equations
The first equation is the field equation of Einstein. It describes the basic interplay between space, time , matter and energy. Friedman developed the metric from this equation which is the quantity ds^2. Friedman analysed the time-time component of field equation and showed that the universe may be expanding. The scale factor a(t) is some parameter which is a function of time. Hubble parameter H is just a function of this scale factor and its derivative.
The second equation of Friedman comes from the other component field equations.
The theory of general relativity can be summarised in the following way:
cosmology equations
Spacetime curvature creates gravity.

Inflationary cosmology

According to the models of inflationary cosmology our universe underwent a rapid expansion right after big bang. This rapid expansion of space was exponentially accelarated. What caused this exponential expansion is a matter of controversy. According to Penrose it is the dark energy ( the lamda term of Einstein's field equation). Particle physicists believe a hypothetical field called inflaton field caused this inflation. The expansion of the universe can be explained by de Sitter Space which has a metric :

inflationary cosmology equations
In this expanding universe model we are like the two dimensional bugs on a surface of sphere. It is better to use the ballon analogy. The ballon inflates and inflates as time passes. Galaxies and everything on it recedes from each other as the surface of the ballon increases.
inflationary cosmology equations
Observations have proved that our universe is indeed expanding. As the universe explands the light that comes from distant galaxies becomes red-shifted.
redshifts of light

The big bag model survived at last due to the discovery CMB ( cosmic background radiation). CMB is a relic from the big bang of early universe. As space expanded the energy of photon decreased and radiation cooled down. Two scientists were given "Noble prize " for the discovery of the radiation that originated in the big bang.

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To explain the inflationary model of the universe we need some basics of quantum mechanics and vacuum. A vacuum according to the principles of quantum mechanics is not nothing but full of short-lived particle and anti-paricle pairs. So vacuum has some energy state. To be precise it contains the lowest energy. Zero-point energy is more appropriate term for this vacuum state.
Exponential expansion model suffered from some problems like horizon problem and flatness problem. Alan Guth proposed the inflationary model in order to avoid such problems. At the initial phase of big bang the universe was very tiny. So quantum effects were natural to exist. The universe apparently appeared from nothing. A false vacuum were created. False vacuum is the high energy state of vacuum. It is not stable.

false vacuum

The energy density of false vacuum is less the the true vacuum or ground state. The inflation field can be correlated with the vacuum energy density. But we can see there is a barrier between these two phases which prevents the inflation to happen classically. But quantum tunnelling can make it possible as the initial universe was a tiny object which obeys quantum laws.

inflation field
As the graph suggest the higgs mechanism starts when the inflation ends and matter starts to be created. The energy density of vacuum remains constant throughout the inflation phase. So universe can arise from a pure quantum fluctuation. A more mathematical picture can be concluded :

inflation field
This also suggest a new kind of theory called eternal inflation.
Eternal inflation is a theory of multi-verse. In this model, the inflation starts and never ends creating lots of pocket universes inside it. Each universe in this nirvana is a bubble forming out of nothing. This bubble forming is called "bubbles nucleation". But what put the bang in the hot big bang ? The same phenomena can be explained in another way as follows
The aswer must be the transition phase that sarts at the end of a cosmic inflation. During the inflation universe is filled with energy that is inherent in the fabric of space itself. It is usually not known how long the universe endures it but we can think of it as a ball rolling across a surface of blocks , all held together by their mutual tension. As the ball rolls over the blocks, it pushes them down. But most of the places are so sturdy that the ball passes without knocking anything out. Sometimes when the ball rolls over a weak spot it can plunge through the blocks and causing a cascade where they all fall down. When the ball and the blocks fall, inflation ends and a hot big bang occurs .
inflationary cosmology and big bang

inflationary cosmology and big bang

Pendulum physics

A pendulum is a peridic motion. Tick tok tick tock is the proper designation of a pendulum. Pendulum has a equation which can be expressed with differential equation.
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The angular momentum is the moment of inertia multiplied angular velocity. Angular accelaration is the average change of angular velocity :
cosmology equations
We can use double derivative to compute volume of some solid object as follows :

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Dirichlet's box principle

This principle states that if there are n+1 things and n boxes then at least one box will contain more than one things when distributing those things among the boxes. A proof can also be given. This can be prooved using proof by contradiction as follow:
Suppose we distribute exactly one things to one box. Then there will be a total of n things in n boxes. So n+1-n = 1 thing will be left over. This is a contradiction as we have to distribute n+1 things. One box must contain more than one things. This is as simple as that.
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Heat transfer

Heat is a form of energy. It is the measure of how much energy a hot body contains. Although it is believed that it is entirely different form of energy, it is neverthless an energy that arises due to the random motion of atoms and molecules in matter. It is studied in the subject that is called the kinetic theory of heat. However heat is transferred from one body to another in three different ways:
1. conduction: It is a heat transfer process in which heat is transferred by direct contact of one body with another. The atoms in one body vibrates and when one body is put into contact with another body the atoms impart their kinetic energy to other atoms.
2. Convection: It is a heat transfer process in which heat is transferred by the mass motion of the atoms from one part of a body to another part.
3. Radiation: It is the process in which heat is transferred in the form of radiation. The heat from the sun comes to the earth in the form of radiation.
heat transfer equations
General equation of heat transfer in any matter can be expressed with a second order differential equation.

heat transfer equations

"with a few symbols on a page you can describe a wealth of physical phenomena.."

A clock

A clock's reading can be represented with mathematics like :

mathematics equations


Statistics is concerned with quantities like average, standard deviation , mean. Statistics deal with a lot of data at once. Here are the equations of statistics in a single package:

mathematics equations
Statistics tells us how many people will die or how many children will be born in a year in a specific country. Besides it has a lot of applications in finance, marketing and accounting. Business persons rely on statistical facts every time to run their business. In physics, expecially in quantum mechanics , statistics plays a vital role too. It is a scheme to interpret common properties of a collection of data.
Mean is the average of a given number of data. Mode is the value that appears most often in a given number of data. Here is the
mathematics equations


Mapping and function have the same meaning in mathematics. Mapping assigns each and every element of a set to one or several elements in another set. An example of mapping is given below :

mathematics function
Various types of mapping can be represented with a signle diagram :

mathematics function
A function should have this property :

mathematics function
A function is actually defined through one-to-one mapping which maps each element of a set to exactly one element in another set.
With this notion of mapping a useful theorem can be constructed :
Given a set of X number of elements and another set of Y number of elements , there are X^Y( X to the power Y) number of possible functions. For more analysis about functions in detail you can follow this page.
There are many theorems and definitions concerning mapping.

mathematics function
A fixed point is the point which is mapped to itself. That is to say, if x is a point then f(x) = x .

Zermelo - Fraenkel set theory

Zermelo - Fraenkel set theory is an axiomatic system that was proposed to remove paradoxes like "Russell's Paradox" . The axioms are as follows:
mathematics function
The last one of these is the axiom of choice. Axiom of choice formally states that given a collection of non-empty sets there is at least one set which contains exactly one representative from each one of them even if the collection is infinite.


Ratio is a relation between two numbers. There are some basic rules for manipulating ratios as follows::

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Algebra cheat sheet.

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Qoutient space

The qoutient of two numbers can be found by dividing one number with the other. But we can not divide one vector space by another in this way. But there is an analogy using which we can find qoutient of two vector space. We find qoutient space of vector space V by subspace N by collpasing down N to zero. The qoutient space is denoted as V/N or (read V mod N)
Formally, Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this defintion it is apparent that all elements of N are related to zero vector. More precisely all vectors in N are getting mapped to equivalence class of zero vector.
So formally the qoutient of V and subspace W is the space V/W = { x+ W ; x belongs to V } ;

Pullback metric

There is a concept in string theory called pullback metric. Pullback metric is a metric function but with additional concept with pullback.
Suppose that you have two spaces X and Y, a metric d on Y, and a function f:X→Y. The pullback metric is the following metric on X: (f∗d)(x(1),x(2))=d(f(x(1)),f(x(2)));x(1),x(2)∈ X
Two manifolds and corresponding maps define the pullback metric

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Direct Sum

Direct sum is used to define summation of two mathematical modules like vector spaces. It is a method by which several modules are combined into a sigle larger module. Suppose given two vector space V and W such that (v1, w1) + (v2, w2) = (v1+ v2 , w1 + w2 ) where v1, v2 belong to V and w1, w2 belong to W and
α(v, w) = (αv, αw) α is any element of a field K; Then direct sum of two modules V and W is denoted by the symbol

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Quantum Field Theory

Feynman's sum over histories   |   S-matrix   !   quantum field theory

Theory of relativity

Relativity made simple   |   Special theory of relativity   |   General theory of relativity   |   Tensor calculus  |   Hamiltonian mechanics   |   Field equation  |   Perihelion of mercury|   Geodesic distance

Quantum mechanics

Schrodinger equation   |   Matrix mechanics   |  Dirac equation

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Conformal cyclic cosmology

Conformal cyclic cosmology is a cosmological model that says that big bang was not the actual beginning of our uiverse. Big bang was one of many phases of our cyclic universe. Each phase has a begining and an end. The end of one phase can be represented as the beginning or big bang of a new phase. This idea seems crazy but Roger Penrose has advocated his theory with proper mathematical logic and empirical evidences seem to support his idea. Penrose termed each phase aeon. How can end of one aeon can be interpreted as the beginning of another aeon? His idea was simple. Conformal mapping can make this happen. Conformal mapping is a kind of map which preserves angle but not distance. The end of one aeon actually expands the universe to infinite size. But this infinity can be squashed into finite size by this conformal mapping. It is a mathematical trick. Our universe in this cosmological model begins with big bang and then undergoes an exponential expansion. So we have a diagram like a cone which shows as time increases the space in all direction increases. At some stage universe reaches maximum size which can be transformed into finite size by conformal mapping. So what is the justification behind this conformal scaling ? When universe expands and grows larger and larger , the most

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boring era begins. Everything cools off and even the black holes start getting aged . They become tiny by radiating photons. All the black holes become exinct by throwing all their mass in the form of radiation. All that is left are the photons and radiations. These move at the speed of light. Photons are very hard to bore because they do not feel time. For a massless particle scale does not matter. Big is actually equivalent to small when there is no mass around. So it is very natural and legitimate to apply conformal scaling at the boundary of the infinite universe and make it an early stage of the next expansion. We can envisage a surface and look behind the surface to see the previous aeon. This is finite space now. Another problem arises when doing this kind of conformal scaling. How can the maximum entropy when the universe expands into infinite size becomes minimum at the instant of big bang ? Roger Penrose has simple explanation for this.
Big bang was a very hot and ordered event of the universe. It had a very low entropy. We know black holes also have entropy and it has a temperature. We define entropy in terms of lost heat. So black holes increase the entropy of the universe. Also a larger black hole has a larger entropy due to its large surface area. That is the consequence of Hawking equation of black hole's entropy. So black holes raise the entropy of the universe after enough time has been passed. But we can view entropy in another way in terms of infomation. Entropy is the measure of information. Black holes contain a lot of information. As they become exinct by radiating way energy information is lost and at later stage of an aeon universe entropy comes to a minimum. Thus there seems to be no problem with the entropy although it seemed at first time.

Conformal mapping

Conformal mapping preserves angle but not distance. Suppose you have a triangle and you apply conformal mapping to it. The size of the triangle will change but not the angle between its sides.
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The following proof will suffice :

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Quantum Cosmology

Quantum cosmology is a hypothetical theory about the universe described with quantum mechanics. It is in a sense theory of quantum gravity. Schrodinger wave function ψ is applied to the entire universe in such a framework. Stephen Hawking proposed no-boundary proposal of the universe. In this model of quantum cosmology our univerese has no singularity like big bang in the beginning. The notion of time did not exist prior to big bang. Asking of what came before the big bang is like asking what is at the south of south pole of the earth. Our earth has no boundary as south pole is an ordinary point like others.
So there is no boundary in our initial universe. The boundary condition of our universe is that there is no boundary. This was the special feature of our initial universe.

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With this assumption at hand there arise a lot of interesting features of the universe. The wave function ψ describes the state of a quantum particle. We can come up with no-boundary wave function of the entire universe. No boundary wave function can be expressed using the sum-over-approach of quantum mechanics. Such a no-boundary wave function do not contain a single history but multiple histories. Each history describes a possible universe with special kind of laws like ours.

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The no-boundary wave function gives an amplitude for different configurations of h(ij) and Φ (potential). Each set of values of h(ij) and φ represent a different universe. Path integral needs to be evaluated over all geometries which have no-boundary. This potential energy φ probably powered the cosmic inflation.
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The south pole is shown at the figure above which denotes the no boundary condition of our universe. This kind of representation or geometry is called "shuttlecock" geometry in Hawking and Hartle model. At the south pole the uiverse is Euclidean in the repect that all the four dimensions were spacial dimensions. Time behaved like a space dimension. Due to rapid inflation the universe expanded and Lorentzian evolution took place when space and time started to behave differently.

Rule of law

Skoll the wolf who shall scare the Moon
Till he flies to the Wood-of-Woe:
Hati the wolf, Hridvitnir’s kin,
Who shall pursue the sun.
—“GRIMNISMAL,” The Elder Edda
N VIKING MYTHOLOGY, Skoll and Hati chase the sun and the moon. When the wolves catch either one, there is an eclipse. When this happens, the people on earth rush to rescue the sun or moon by making as much noise as they can in hopes of scaring off the wolves. There are similar myths in other cultures. But after a time people must have noticed that the sun and moon soon emerged from the eclipse regardless of whether they ran around screaming and banging on things. After a time they must also have noticed that the eclipses didn’t just happen at random: They occurred in regular patterns that repeated themselves. These patterns were most obvious for eclipses of the moon and enabled the ancient Babylonians to predict lunar eclipses fairly accurately even though they didn’t realize that they were caused by the earth blocking the light of the sun. Eclipses of the sun were more difficult to predict because they are visible only in a corridor on the earth about 30 miles wide. Still, once grasped, the patterns made it clear the eclipses were not dependent on the arbitrary whims of supernatural beings, but rather governed by laws.
Despite some early success predicting the motion of celestial bodies, most events in nature appeared to our ancestors to be impossible to predict. Volcanoes, earthquakes, storms, pestilences, and ingrown toenails all seemed to occur without obvious cause or pattern. In ancient times it was natural to ascribe the violent acts of nature to a pantheon of mischievous or malevolent deities. Calamities were often taken as a sign that we had somehow offended the gods. For example, in about 5600 BC the Mount Mazama volcano in Oregon erupted, raining rock and burning ash for years, and leading to the many years of rainfall that eventually filled the volcanic crater today called Crater Lake. The Klamath Indians of Oregon have a legend that faithfully matches every geologic detail of the event but adds a bit of drama by portraying a human as the cause of the catastrophe. The human capacity for guilt is such that people can always find ways to blame themselves. As the legend goes, Llao, the chief of the Below World, falls in love with the beautiful human daughter of a Klamath chief. She spurns him, and in revenge Llao tries to destroy the Klamath with fire. Luckily, according to the legend, Skell, the chief of the Above World, pities the humans and does battle with his underworld counterpart. Eventually Llao, injured, falls back
inside Mount Mazama, leaving a huge hole, the crater that eventually filled with water. Ignorance of nature’s ways led people in ancient times to invent gods to lord it over every aspect of human life. There were gods of love and war; of the sun, earth, and sky; of the oceans and rivers; of rain and thunderstorms; even of earthquakes and volcanoes. When the gods were pleased, mankind was treated to good weather, peace, and freedom from natural disaster and disease. When they were displeased, there came drought, war, pestilence, and epidemics. Since the connection of cause and effect in nature was invisible to their eyes, these gods appeared inscrutable, and people at their mercy. But with Thales of Miletus (ca. 624 BC– ca. 546 BC) about 2,600 years ago, that began to change. The idea arose that nature follows consistent principles that could be deciphered. And so began the long process of replacing the notion of the reign of gods with the concept of a universe that is governed by laws of nature, and created according to a blueprint we could someday learn to read. Viewed on the timeline of human history, scientific inquiry is a very new endeavor. Our species, Homo sapiens, originated in sub-Saharan Africa around 200,000 BC. Written language dates back only to about 7000 BC, the product of societies centered around the cultivation of grain. (Some of the oldest written inscriptions concern the daily ration of beer allowed to each citizen.) The earliest written records from the legendary civilization of ancient Greece date back to the ninth century BC, but the height of that civilization, the “classical period,” came several hundred years later, beginning a little before 500 BC. According to Aristotle (384 BC–322 BC), it was around that time that Thales first developed the idea that the world can be understood, that the complex happenings around us could be reduced to simpler principles and elucidated without resorting to mythical or theological explanations. Thales is credited with the first prediction of a solar eclipse in 585 BC, though the great precision of his prediction was probably a lucky guess. He was a shadowy figure who left behind no writings of his own. His home was one of the intellectual centers in a region called Ionia, which was colonized by the Greeks and exerted an influence that eventually reached from Turkey as far west as Italy. Ionian science was an endeavor marked by a strong interest in uncovering fundamental laws to explain natural phenomena, a tremendous milestone in the history of human ideas. Their approach was rational and in many cases led to conclusions surprisingly similar to
what our more sophisticated methods have led us to believe today. It represented a grand beginning. But over the centuries much of Ionian science would be forgotten—only to be rediscovered or reinvented, sometimes more than once. According to legend, the first mathematical formulation of what we might today call a law of nature dates back to an Ionian named Pythagoras (ca. 580 BC–ca. 490 BC), famous for the theorem named after him: that the square of the hypotenuse (longest side) of a right triangle equals the sum of the squares of the other two sides. Pythagoras is said to have discovered the numerical relationship between the length of the strings used in musical instruments and the harmonic combinations of the sounds. In today’s language we would describe that relationship by saying that the frequency—the number of vibrations per second—of a string vibrating under fixed tension is inversely proportional to the length of the string. From the practical point of view, this explains why bass guitars must have longer strings than ordinary guitars. Pythagoras probably did not really discover this—he also did not discover the theorem that bears his name—but there is evidence that some relation between string length and pitch was known in his day. If so, one could call that simple mathematical formula the first instance of what we now know as theoretical physics.

What is reality

FEW YEARS AGO the city council of Monza, Italy, barred pet owners from keeping goldfish in curved goldfish bowls. The measure’s sponsor explained the measure in part by saying that it is cruel to keep a fish in a bowl with curved sides because, gazing out, the fish would have a distorted view of reality. But how do we know we have the true, undistorted picture of reality? Might not we ourselves also be inside some big goldfish bowl and have our vision distorted by an enormous lens? The goldfish’s picture of reality is different from ours, but can we be sure it is less real? The goldfish view is not the same as our own, but goldfish could still formulate scientific laws governing the motion of the objects they observe outside their bowl. For example, due to the distortion, a freely moving object that we would observe to move in a straight line would be observed by the goldfish to move along a curved path. Nevertheless, the goldfish could formulate scientific laws from their distorted frame of reference that would always hold true and that would enable them to make predictions about the future motion of objects outside the bowl. Their laws would be more complicated than the laws in our frame, but simplicity is a matter of taste. If a goldfish formulated such a theory, we would have to admit the goldfish’s view as a valid picture of reality. A famous example of different pictures of reality is the model introduced around AD 150 by Ptolemy (ca. 85—ca. 165) to describe the motion of the celestial bodies. Ptolemy published his work in a thirteen-book treatise usually known under its Arabic title, Almagest. The Almagest begins by explaining reasons for thinking that the earth is spherical, motionless, positioned at the center of the universe, and negligibly small in comparison to the distance of the heavens. Despite Aristarchus’s heliocentric model, these beliefs had been held by most educated Greeks at least since the time of Aristotle, who believed for mystical reasons that the earth should be at the center of the universe. In Ptolemy’s model the earth stood still at the center and the planets and the stars moved around it in complicated orbits involving epicycles, like wheels on wheels.
This model seemed natural because we don’t feel the earth under our feet moving (except in earthquakes or moments of passion). Later European learning was based on the Greek sources that had been passed down, so that the ideas of Aristotle and Ptolemy became the basis for much of Western thought. Ptolemy’s model of the cosmos was adopted by the Catholic Church and held as official doctrine for fourteen hundred years. It was not until 1543 that an alternative model was put forward by Copernicus in his book De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published only in the year of his death (though he had worked on his theory for several decades). Copernicus, like Aristarchus some seventeen centuries earlier, described a world in which the sun was at rest and the planets revolved around it in circular orbits. Though the idea wasn’t new, its revival was met with passionate resistance. The Copernican model was held to contradict the Bible, which was interpreted as saying that the planets moved around the earth, even though the
Bible never clearly stated that. In fact, at the time the Bible was written people believed the earth was flat. The Copernican model led to a furious debate as to whether the earth was at rest, culminating in Galileo’s trial for heresy in 1633 for advocating the Copernican model, and for thinking “that one may hold and defend as probable an opinion after it has been declared and defined contrary to the Holy Scripture.” He was found guilty, confined to house arrest for the rest of his life, and forced to recant. He is said to have muttered under his breath “Eppur si muove,” “But still it moves.” In 1992 the Roman Catholic Church finally acknowledged that it had been wrong to condemn Galileo. So which is real, the Ptolemaic or Copernican system? Although it is not uncommon for people to say that Copernicus proved Ptolemy wrong, that is not true. As in the case of our normal view versus that of the goldfish, one can use either picture as a model of the universe, for our observations of the heavens can be explained by assuming either the earth or the sun to be at rest. Despite its role in philosophical debates over the nature of our universe, the real advantage of the Copernican system is simply that the equations of motion are much simpler in the frame of reference in which the sun is at rest. A different kind of alternative reality occurs in the science fiction film The Matrix, in which the human race is unknowingly living in a simulated virtual reality created by intelligent computers to keep them pacified and content while the computers suck their bioelectrical energy (whatever that is). Maybe this is not so far-fetched, because many people prefer to spend their time in the simulated reality of websites such as Second Life. How do we know we are not just characters in a computer-generated soap opera? If we lived in a synthetic imaginary world, events would not necessarily have any logic or consistency or obey any laws. The aliens in control might find it more interesting or amusing to see our reactions, for example, if the full moon split in half, or everyone in the world on a diet developed an uncontrollable craving for banana cream pie. But if the aliens did enforce consistent laws, there is no way we could tell there was another reality behind the simulated one. It would be easy to call the world the aliens live in the “real” one and the synthetic world a “false” one. But if—like us—the beings in the simulated world could not gaze into their universe from the outside, there would be no reason for them to doubt their own pictures of reality. This is a modern version of the idea that we are all figments of someone else’s dream.
These examples bring us to a conclusion that will be important in this book: There is no picture- or theory-independent concept of reality. Instead we will adopt a view that we will call modeldependent realism: the idea that a physical theory or world picture is a model (generally of a mathematical nature) and a set of rules that connect the elements of the model to observations. This provides a framework with which to interpret modern science. Philosophers from Plato onward have argued over the years about the nature of reality. Classical science is based on the belief that there exists a real external world whose properties are definite and independent of the observer who perceives them. According to classical science, certain objects exist and have physical properties, such as speed and mass, that have well-defined values. In this view our theories are attempts to describe those objects and their properties, and our measurements and perceptions correspond to them. Both observer and observed are parts of a world that has an objective existence, and any distinction between them has no meaningful significance. In other words, if you see a herd of zebras fighting for a spot in the parking garage, it is because there really is a herd of zebras fighting for a spot in the parking garage. All other observers who look will measure the same properties, and the herd will have those properties whether anyone observes them or not. In philosophy that belief is called realism. Though realism may be a tempting viewpoint, as we’ll see later, what we know about modern
physics makes it a difficult one to defend. For example, according to the principles of quantum physics, which is an accurate description of nature, a particle has neither a definite position nor a definite velocity unless and until those quantities are measured by an observer. It is therefore not correct to say that a measurement gives a certain result because the quantity being measured had that value at the time of the measurement. In fact, in some cases individual objects don’t even have an independent existence but rather exist only as part of an ensemble of many. And if a theory called the holographic principle proves correct, we and our four-dimensional world may be shadows on the boundary of a larger, five-dimensional space-time. In that case, our status in the universe is analogous to that of the goldfish. Strict realists often argue that the proof that scientific theories represent reality lies in their success. But different theories can successfully describe the same phenomenon through disparate conceptual frameworks. In fact, many scientific theories that had proven successful were later replaced by other, equally successful theories based on wholly new concepts of reality. Traditionally those who didn’t accept realism have been called anti-realists. Anti-realists suppose a distinction between empirical knowledge and theoretical knowledge. They typically argue that observation and experiment are meaningful but that theories are no more than useful instruments that do not embody any deeper truths underlying the observed phenomena. Some anti-realists have even wanted to restrict science to things that can be observed. For that reason, many in the nineteenth century rejected the idea of atoms on the grounds that we would never see one. George Berkeley (1685–1753) even went as far as to say that nothing exists except the mind and its ideas. When a friend remarked to English author and lexicographer Dr. Samuel Johnson (1709–1784) that Berkeley’s claim could not possibly be refuted, Johnson is said to have responded by walking over to a large stone, kicking it, and proclaiming, “I refute it thus.” Of course the pain Dr. Johnson experienced in his foot was also an idea in his mind, so he wasn’t really refuting Berkeley’s ideas. But his act did illustrate the view of philosopher David Hume (1711–1776), who wrote that although we have no rational grounds for believing in an objective reality, we also have no choice but to act as if it is true. Model-dependent realism short-circuits all this argument and discussion between the realist and anti-realist schools of thought
According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation. If there are two models that both agree with observation, like the goldfish’s picture and ours, then one cannot say that one is more real than another. One can use whichever model is more convenient in the situation under consideration. For example, if one were inside the bowl, the goldfish’s picture would be useful, but for those outside, it would be very awkward to describe events from a distant galaxy in the frame of a bowl on earth, especially because the bowl would be moving as the earth orbits the sun and spins on its axis. We make models in science, but we also make them in everyday life. Model-dependent realism applies not only to scientific models but also to the conscious and subconscious mental models we all create in order to interpret and understand the everyday world. There is no way to remove the observer—us—from our perception of the world, which is created through our sensory processing and through the way we think and reason. Our perception—and hence the observations upon which our theories are based—is not direct, but rather is shaped by a kind of lens, the interpretive structure of our human brains.
Model-dependent realism corresponds to the way we perceive objects. In vision, one’s brain receives a series of signals down the optic nerve. Those signals do not constitute the sort of image
you would accept on your television. There is a blind spot where the optic nerve attaches to the retina, and the only part of your field of vision with good resolution is a narrow area of about 1 degree of visual angle around the retina’s center, an area the width of your thumb when held at arm’s length. And so the raw data sent to the brain are like a badly pixilated picture with a hole in it. Fortunately, the human brain processes that data, combining the input from both eyes, filling in gaps on the assumption that the visual properties of neighboring locations are similar and interpolating. Moreover, it reads a two-dimensional array of data from the retina and creates from it the impression of three-dimensional space. The brain, in other words, builds a mental picture or model.
The brain is so good at model building that if people are fitted with glasses that turn the images in their eyes upside down, their brains, after a time, change the model so that they again see things the right way up. If the glasses are then removed, they see the world upside down for a while, then again adapt. This shows that what one means when one says “I see a chair” is merely that one has used the light scattered by the chair to build a mental image or model of the chair. If the model is upside down, with luck one’s brain will correct it before one tries to sit on the chair. Another problem that model-dependent realism solves, or at least avoids, is the meaning of existence. How do I know that a table still exists if I go out of the room and can’t see it? What does it mean to say that things we can’t see, such as electrons or quarks—the particles that are said to make up the proton and neutron—exist? One could have a model in which the table disappears when I leave the room and reappears in the same position when I come back, but that would be awkward, and what if something happened when I was out, like the ceiling falling in? How, under the table-disappears-when-I-leave-the-room model, could I account for the fact that the next time I enter, the table reappears broken, under the debris of the ceiling? The model in which the table stays put is much simpler and agrees with observation. That is all one can ask. In the case of subatomic particles that we can’t see, electrons are a useful model that explains observations like tracks in a cloud chamber and the spots of light on a television tube, as well as many other phenomena. It is said that the electron was discovered in 1897 by British physicist J. J. Thomson at the Cavendish Laboratory at Cambridge University. He was experimenting with currents of electricity inside empty glass tubes, a phenomenon known as cathode rays. His experiments led him to the bold conclusion that the mysterious rays were composed of minuscule “corpuscles” that were material constituents of atoms, which were then thought to be the indivisible fundamental unit of matter. Thomson did not “see” an electron, nor was his speculation directly or unambiguously demonstrated by his experiments. But the model has proved crucial in applications from fundamental science to engineering, and today all physicists believe in electrons, even though you cannot see them.
Quarks, which we also cannot see, are a model to explain the properties of the protons and neutrons in the nucleus of an atom. Though protons and neutrons are said to be made of quarks, we will never observe a quark because the binding force between quarks increases with separation, and hence isolated, free quarks cannot exist in nature. Instead, they always occur in groups of three (protons and neutrons), or in pairings of a quark and an anti-quark (pi mesons), and behave as if they were joined by rubber bands.
The question of whether it makes sense to say quarks really exist if you can never isolate one was a controversial issue in the years after the quark model was first proposed. The idea that certain
particles were made of different combinations of a few sub-subnuclear particles provided an organizing principle that yielded a simple and attractive explanation for their properties. But although physicists were accustomed to accepting particles that were only inferred to exist from statistical blips in data pertaining to the scattering of other particles, the idea of assigning reality to a particle that might be, in principle, unobservable was too much for many physicists. Over the years, however, as the quark model led to more and more correct predictions, that opposition faded. It is certainly possible that some alien beings with seventeen arms, infrared eyes, and a habit of blowing clotted cream out their ears would make the same experimental observations that we do, but describe them without quarks. Nevertheless, according to model-dependent realism, quarks exist in a model that agrees with our observations of how subnuclear particles behave. Model-dependent realism can provide a framework to discuss questions such as: If the world was created a finite time ago, what happened before that? An early Christian philosopher, St. Augustine (354–430), said that the answer was not that God was preparing hell for people who ask such questions, but that time was a property of the world that God created and that time did not exist before the creation, which he believed had occurred not that long ago. That is one possible model, which is favored by those who maintain that the account given in Genesis is literally true even though the world contains fossil and other evidence that makes it look much older. (Were they put there to fool us?) One can also have a different model, in which time continues back 13.7 billion years to the big bang. The model that explains the most about our present observations, including the historical and geological evidence, is the best representation we have of the past. The second model can explain the fossil and radioactive records and the fact that we receive light from galaxies millions of light-years from us, and so this model—the big bang theory—is more useful than the first. Still, neither model can be said to be more real than the other.
Some people support a model in which time goes back even further than the big bang. It is not yet clear whether a model in which time continued back beyond the big bang would be better at explaining present observations because it seems the laws of the evolution of the universe may break down at the big bang. If they do, it would make no sense to create a model that encompasses time before the big bang, because what existed then would have no observable consequences for the present, and so we might as well stick with the idea that the big bang was the creation of the world. A model is a good model if it:
1. Is elegant
2. Contains few arbitrary or adjustable elements
3. Agrees with and explains all existing observations
4. Makes detailed predictions about future observations that can disprove or falsify the model
if they are not borne out.
For example, Aristotle’s theory that the world was made of four elements, earth, air, fire, and water, and that objects acted to fulfill their purpose was elegant and didn’t contain adjustable elements. But in many cases it didn’t make definite predictions, and when it did, the predictions weren’t always in agreement with observation. One of these predictions was that heavier objects should fall faster because their purpose is to fall. Nobody seemed to have thought that it was important to test this until Galileo. There is a story that he tested it by dropping weights from the Leaning Tower of Pisa. This is probably apocryphal, but we do know he rolled different weights down an inclined plane and observed that they all gathered speed at the same rate, contrary to Aristotle’s prediction. The above criteria are obviously subjective. Elegance, for example, is not something easily measured, but it is highly prized among scientists because laws of nature are meant to economically compress a number of particular cases into one simple formula. Elegance refers to the form of a theory, but it is closely related to a lack of adjustable elements, since a theory jammed with fudge factors is not very elegant. To paraphrase Einstein, a theory should be as simple as possible, but not simpler. Ptolemy added epicycles to the circular orbits of the heavenly bodies in order that his model might accurately describe their motion. The model could have been made more accurate by adding epicycles to the epicycles, or even epicycles to those. Though added complexity could make the model more accurate, scientists view a model that is contorted to match a specific set of observations as unsatisfying, more of a catalog of data than a theory likely to embody any useful principle. We’ll see in Chapter 5 that many people view the “standard model,” which describes the interactions of the elementary particles of nature, as inelegant. That model is far more successful than Ptolemy’s epicycles. It predicted the existence of several new particles before they were observed, and described the outcome of numerous experiments over several decades to great precision. But it contains dozens of adjustable parameters whose values must be fixed to match observations, rather than being determined by the theory itself. As for the fourth point, scientists are always impressed when new and stunning predictions prove correct. On the other hand, when a model is found lacking, a common reaction is to say that the
experiment was wrong. If that doesn’t prove to be the case, people still often don’t abandon the model but instead attempt to save it through modifications. Although physicists are indeed tenacious in their attempts to rescue theories they admire, the tendency to modify a theory fades to the degree that the alterations become artificial or cumbersome, and therefore “inelegant.” If the modifications needed to accommodate new observations become too baroque, it signals the need for a new model. One example of an old model that gave way under the weight of new observations was the idea of a static universe. In the 1920s, most physicists believed that the universe was static, or unchanging in size. Then, in 1929, Edwin Hubble published his observations showing that the universe is expanding. But Hubble did not directly observe the universe expanding. He observed the light emitted by galaxies. That light carries a characteristic signature, or spectrum, based on each galaxy’s composition, which changes by a known amount if the galaxy is moving relative to us. Therefore, by analyzing the spectra of distant galaxies, Hubble was able to determine their velocities. He had expected to find as many galaxies moving away from us as moving toward us. Instead he found that nearly all galaxies were moving away from us, and the farther away they were, the faster they were moving. Hubble concluded that the universe is expanding, but others, trying to hold on to the earlier model, attempted to explain his observations within the context of the static universe. For example, Caltech physicist Fritz Zwicky suggested that for some yet unknown reason light might slowly lose energy as it travels great distances. This decrease in energy would correspond to a change in the light’s spectrum, which Zwicky suggested could mimic Hubble’s observations. For decades after Hubble, many scientists continued to hold on to the steady-state theory. But the most natural model was Hubble’s, that of an expanding universe, and it has come to be the accepted one. In our quest to find the laws that govern the universe we have formulated a number of theories or models, such as the four-element theory, the Ptolemaic model, the phlogiston theory, the big bang theory, and so on. With each theory or model, our concepts of reality and of the fundamental constituents of the universe have changed. For example, consider the theory of light. Newton thought that light was made up of little particles or corpuscles. This would explain why light travels in straight lines, and Newton also used it to explain why light is bent or refracted when it passes from one medium to another, such as from air to glass or air to water.
The corpuscle theory could not, however, be used to explain a phenomenon that Newton himself observed, which is known as Newton’s rings. Place a lens on a flat reflecting plate and illuminate it with light of a single color, such as a sodium light. Looking down from above, one will see a series of light and dark rings centered on where the lens touches the surface. This would be difficult to explain with the particle theory of light, but it can be accounted for in the wave theory. According to the wave theory of light, the light and dark rings are caused by a phenomenon called interference. A wave, such as a water wave, consists of a series of crests and troughs. When waves collide, if those crests and troughs happen to correspond, they reinforce each other, yielding a larger wave. That is called constructive interference. In that case the waves are said to be “in phase.” At the other extreme, when the waves meet, the crests of one wave might coincide with the troughs of the other. In that case the waves cancel each other and are said to be “out of phase.” That situation is called destructive interference.
In Newton’s rings the bright rings are located at distances from the center where the separation between the lens and the reflecting plate is such that the wave reflected from the lens differs from the wave reflected from the plate by an integral (1, 2, 3,…) number of wavelengths, creating constructive interference. (A wavelength is the distance between one crest or trough of a wave and the next.) The dark rings, on the other hand, are located at distances from the center where the separation between the two reflected waves is a half-integral (½, 1½, 2½,…) number of wavelengths, causing destructive interference—the wave reflected from the lens cancels the wave reflected from the plate.
In the nineteenth century, this was taken as confirming the wave theory of light and showing that the particle theory was wrong. However, early in the twentieth century Einstein showed that the photoelectric effect (now used in television and digital cameras) could be explained by a particle or quantum of light striking an atom and knocking out an electron. Thus light behaves as both particle and wave. The concept of waves probably entered human thought because people watched the ocean, or a puddle after a pebble fell into it. In fact, if you have ever dropped two pebbles into a puddle, you have probably seen interference at work, as in the picture above. Other liquids were observed to behave in a similar fashion, except perhaps wine if you’ve had too much. The idea of particles was familiar from rocks, pebbles, and sand. But this wave/particle duality—the idea that an object could be described as either a particle or a wave—is as foreign to everyday experience as is the idea that you can drink a chunk of sandstone.
Dualities like this—situations in which two very different theories accurately describe the same phenomenon—are consistent with model-dependent realism. Each theory can describe and explain certain properties, and neither theory can be said to be better or more real than the other. Regarding the laws that govern the universe, what we can say is this: There seems to be no single mathematical model or theory that can describe every aspect of the universe. Instead, as mentioned in the opening chapter, there seems to be the network of theories called M-theory. Each theory in the M-theory network is good at describing phenomena within a certain range. Wherever their ranges overlap, the various theories in the network agree, so they can all be said to be parts of the same theory. But no single theory within the network can describe every aspect of the universe— all the forces of nature, the particles that feel those forces, and the framework of space and time in which it all plays out. Though this situation does not fulfill the traditional physicists’ dream of a single unified theory, it is acceptable within the framework of model-dependent realism. We will discuss duality and M-theory further in Chapter 5, but before that we turn to a fundamental principle upon which our modern view of nature is based: quantum theory, and in
particular, the approach to quantum theory called alternative histories. In that view, the universe does not have just a single existence or history, but rather every possible version of the universe exists simultaneously in what is called a quantum superposition. That may sound as outrageous as the theory in which the table disappears whenever we leave the room, but in this case the theory has passed every experimental test to which it has ever been subjected.

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Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
perihelion of mercury by Feynman
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