Theory of everything

Theory of everything is the ultimate theory of physics , that should describe all physical aspects and occurrences in a consistent manner. Such theory , if it is found, needs to unify all fundamental forces of nature. String theory is the best candidate for theory of everything. It is a theoretical framework that have merged quantum mechanics with theory of relativity. According to string theory universe has nine space dimensions and one time dimension. All the dimensions other than three are curled up and tiny so that we can not directly perceive them. Besides this crucial feature, the principles of string theory say that what appears to be elementary particles are tiny vibrating pieces of strings. Somehow if we can work out resonant or vibrational pattern of the piece of string we will be able to explain the observed properties of the elementary particles. The conclusion is that the universe is a cosmic symphony of strings resonating through ten dimensional hyperspace. What are we ? We are nothing but the melodies. We are nothing but cosmic music played on the same cosmic strings or membranes.
General theory of relativity theory of quanta

General theory of relativity + theory of quanta = Everything

General advice

Readers from all over the world , have you a feeling that you are bad at mathematics? Do not fear mathematics. Once I was also a victim. You can now conquer mathematics easily. The secret is behind logic. Logical ideas alone can define mathematics. I will state russell's paradox first. If you understand this you can safely assume that you have a good mathematical sense .
Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!
This was Russell famous paradox. All mathematics are basically logical deductions. You can follow my website for more logical analysis of mathematics.

Mathematics

Mathematics is the language of nature. It is hard to define mathematics . In general it is the study of change, numbers and shapes. Mathematics consists of various branches like arithmetic, geometry, topology, calculus and many more. Science can not progress without mathematics. This is so obvious fact that needs not be mentioned. Mathematics , in a certain sense , an exact science. We all accept the mathematical theorems and truth because we can prove those. we all accept 2+2 = 4. Mathematics is always flawless. That is what mathematicians thought every time. But Kurt Godel gave a death blow to mathematics by claiming that no mathematical system is complete. Axiomatic systems are necessary incomplete. That means there will always be some statement within the system , whose truth can not be proven using the system. This was very frustrating idea for the mathematical society and no one has yet been able to refute Godel's claim. This is known as the Godel's first incompleteness theorem. This theorem is kind of related to the "liar paradox" of usual language. Suppose there is a sentence " he is lying". Now if the sentence is true then when he says that he is lying , he is indeed lying. That means he is telling the truth. The sentence's truth can not be determined as we come to a contradictory situation. There are other similar paradoxes explained here.
"Mathematics has a better sense than the common sense"
There is hardly any branch of engineering and science where mathematics does not apply. Mathematics is a sense you never have. So nobody is gifted with mathematical ability. You can not be good at it without practising it. But it seems surprising that our mind which is so well organized can not adapt with mathematics everytime. Or may be it is a wrong idea.
"pure mathematics is a poetry of logical ideas"
Without speaking about numbers the importance of mathematics can not be described. To describe numbers or define numbers we need to know how to count. To count , in turn, will need three steps : 1) the idea of many-ness 2) one-to-one mapping and 3) sequentially speak or say the number words
First we need to have the idea of many-ness. We need to answer the question of how many? How many objects do we a collection have? Then we map each objects to other objects with one-to-one relation.
Mapping: mapping is assigning each element of a set to exactly one element of another set.
When we assign an element , say our finger to each element of a collection , we speak one for one finger , two for two fingers and so on to last one which will be the cardinal number of our collection or set. Then we give a symbol for the number if we like. This symbol will represent that particular number.
The invention of calculus has changed the world. Calculus was invented by Leibniz and Newton at the same time although there is priority dispute about who invented it first. Application of calculus is everywhere. Where there is change there is calculus. Where there is curvature there is calculus. Both the theory of relativity and quantum mechanics were developed using calculus. Everybody knows that the idea of calculus is rooted in the notion of limit. When we speak about something approaching another one or some point approaching another point , the notion of limit appears. There is precise definition of limit. You can think of it as a scheme to manipulate infinite number of small quantities. We know we can sum infinite number of terms in a geometric series, which converges. Same thing happens in case of limit. One quantity converges to another and in the end they become equal. This is the core idea of limit and calculus.
integration
Real numbers are all the numbers that can be traced on number line. It includes both rational and irrational numbers. Real number line contains no gap or it is continuous. Irrational number was defined in order to make the real number line complete. It is the set R. It is extended from - (infinity) to +(infinity) including zero. Real numbers are also called measurable numbers. It includes the set of integers too. Calculus is certainly established using the concepts of real numbers. Cantor first proved that the set of real numbers is non denumerable. It can not not put in one-to-one correspondence with natural numbers(1,2, 3, 4, 5,6 ......). Real numbers has cardinality 2^(ℵ0) , which is called cantorian continuum. Cantor also proposed that there is no set which cardinality lies strictly between that and ℵ(0). This is called continuum hypotheses. These two infinite numbers of continuum hypotheses play exactly the same role as integers 0 and 1. There is no integer in between 0 and 1. Cantor knew that this was the case but he never came up with a proof. Later Kurt Godel showed that this hypotheses is consistent with axiomatic set theory.
Pythagoras law has made a great impact on science and mathematics. This law holds for any right angle triangle in Euclidean geometry. The law says square of the hypotenuse is equal to sum of squares of other two sides. This law not only bears the fact of geometry but also laid the foundation of many scientific theories like theory of relativity and electromagnetism. Pythagoras law also states that there is a 3-tuple of integers which satisfy it. In another words there are infinite number of solutions of Pythagoras law. More specifically , there are infinite solutions for x, y and z in integers which represent the three sides of any right angle triangle. This brings us to Fermet's last theorem. The theorem states that there are no solution for x, y and z in integers for power greater or equal to 3. Proof of Farmet's last theorem took much time to be discovered. This proof involves number theory.
Gauss was perhaps one of the greatest mathematicians of all time. He was first to study the curved surfaces. What we call plane is particular cases of 2-dimensional surfaces or manifolds. In physics he contributed also. We all know Gauss's law of electromagnetism. Gauss was very talented in his early childhood. One day his teachers asked everybody in his class to add all the numbers from 1 to 100. Gauss calculated it more quickly than others within seconds. He just did it by adding two numbers and multiplying it by 50. How ? 50 pairs of numbers have the same summation like 1+99 =100 , 2+ 98 = 100, and so on.. So the total sum is 50*100 = 5000. Gauss also developed many theorems which are still very usefull.
"Mathematics gives you wings"
Roger Penrose is the world's leading mathematical physicist. He collaborated with Hawking and proved that time had a beginning. The proof concerns the postulates of the theory of relativity. He is a mathematician too. He is specialized in recreational mathematics. He makes many complicated mathematical problems seem easy by graphical representation. Mathematics, according to him, has certain platonic existence. There are three kinds of worlds :one is physical world, second is mental world and the third is platonic world. Mathematical objects and forms exist in platonic world. Mathematical objects or forms are numbers, triangles, circles, sets and many more. There is no perfect circle in real world. We can not see perfect flat space anywhere in the universe. This kind of pure mathematical or geometric objects exist only in the platonic realm.
Our mental world consists of all our thoughts and abstract percepts. It has no connection to physical reality. Our mental world contains particular or percepts. Universals are certain concepts which can exist without our mind. Mental world does not necessarily make logic and mathematics dependent on it.
Physical world is the materialistic world that we inhabit. It is the whole universe or multiverse-as we may call it.

carton
So what is the most complicated topic in mathematics? Is it topology or geometry or calculus? or is it linear algebra or statistics? It is none. Mathematics builds upon itself. If you want to understand calculus you need to understand limit and functions. If you need to understand topology you need to understand set theory. It is like a chain. If one link is missing the whole system is broken. We do not understand mathematics because of these missing links. Lot of things is dependent on definitions. We can define something when there are notions which have a certain relation to some term , which is itself one of the said notions. Why is physics hard to understand? Physics has lots of notation and symbols which scares us away. You can talk about general relativity. In relativity all boils down to 21 numbers that Riemann curvature tensor assigns at each event in spacetime. In classical electromagnetism at each event in spacetime there are 16 numbers that electromagnetic field tensor assigns. How does these numbers creep in? I guess you have to have some ideas about those theories. Take another example of quantum field theory. This theory treats particles as excitation of the fields. Everything in QFT can be thought as perturbation in an infinite number of sets of infinite number of tiny springs. Particles come from energy that the fields represent. In quantum electrodynamics there are three functions which give three different numbers describing all interaction between matter and radiation. I guesss I have to say that Pythagoras was right all along when he said that numbers rule the universe. If you know only calculus well and have some basic ideas of linear algebra you can understand a lot of advanced theoretical physics theories.
"If the multiverse theory is correct then there is someone exactly like you (doppleganger) which is reading this sentence in another universe!!"
"When you got nothing , you got nothing to loose.." If you like my writing just click on the google ads on this page or by visiting this page

Chaos Theory

Some systems are very sensitive to initial conditions. A small change in the initial conditions can create a huge disturbances in the system. Such system is known as chaotic system. For example, a flapping of the butterfly wings in New York can cause a hurricane in London. This is known as butterfly effect. Mathematics of such system is very complicated. To analyze chaotic system chaos theory has been developed. The dynamics governing such chaotic system is also very complex. We can not predict the behaviour of such system as exact initial condition is always unknown.


Miscellaneous pages
Algebra   |  coordinate geometry   |  Topology for dummies   |  Symmetry   |   Bertrand Russell philosophy   |  Sir issac newton   |   Statistical mechanics   |  Albert Einstein  |   Artificial intelligence   |  Leonhard Euler   |  Carl Friedrich Gauss   |  Number theory   |   what is science   |  complex analysis   |  perihelion of mercury  |   string theory explained  |  Calculus for dummies   |  Medical Science definition   |  Mathematical universe   |  Weyl's theory
Complex analysis
quantum mechanics and elementary physics |  Mathematics (Advanced)

"The best thing about science is that you can apply it to your life"

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

heisenberg's equation of motion
Proof
In quantum mechanics operators are replacements for classical quantities like position and momentum. They are some kind of functions too. There are seven ideas that shook the universe. These are
1. Newtonian mechanics
2. Energy and entropy
3. relativity
3. quantum theory
4. conservation principles
5. copernican astronomy
6. symmetries
7. Theory of everything

Real numbers


Real number is the basis of mathematics and theoretical physics. What is real number? A real number is nothing but a class of rational numberss. The class of rational numbers less than 1/2 is a real number. This was the consequence of the theory of real number. The irrationals were found to be some series of rational numbers, which has no rational limit nor infinite limit. And real numbers are the totality of rationals and irrationals. This was the idea behind defining real number as the class of rational numbers. The whole theory deserves another page , which will be made very soon.

Physics of everything


equations of everything physics


equations of physics


equations of physics


equations of physics

Love formula
equations of physics


equations of physics

Irrational numbers are those numbers which can not be expressed as a ratio between two integers. The first irrational number to be confronted by the Greeks was the square root of 2. Irrational numbers are still regarded as real numbers but they are not commensurate to any quantity. So they are in-commensurate numbers. Greek found that if the length of square figure is one then its diagonal is the square root of 2. They were puzzled by this weird number and its properties.

The diagonal of this square can not be multiple of any integer. On the other hand you can see that on a number line square root 2 can be represented as the distance which is the same as the diagonal of the square. All this was very troublesome for mathematics and mathematician in the ancient time. However we will proceed to the proof of the irrationality of the square root of two (2) :

Suppose root 2 is equal to the ratio of m to n. So

root 2 = m/n , if we square it we get 2n^2 = m^2 . That means that m^2 is an even number. Now let m = 2p so m^2 = 4p^2 . Thus we have 4p^2 = 2n^2 where p is the half of m. Hence 2p^2 = n^2 , and therefore n/p will also be equal to square root of 2. But we can repeat the argument : if n = 2q, p/q will also be the square root of 2 , and so on , through an unending series of numbers each of which is half of its predecessors. But this is clearly impossible, if we divide a number continuously by two 2 we must reach an odd number after a finite number of steps. Or we may put the argument in this way by assuming that the m/n we start with is in its lowest terms; in that case m and n can not both be even as we saw they were supposed to be.

So we must conclude that square root of 2 can not be represented as a ratio of two numbers. If we do we come to a contradiction as we have shown.

square root of two(2) can be expressed using circular and hyperbolic angle too. Here is an example .

circular and hyperbolic angle u

Where the symbol ^ is the exponential.

Real numbers and irrational number:

square root 2 as the diagonal of an square

As stated, no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This problem was purely geometrical but the same problem arose as regards the solution of equations, though here it took a wider form, since it involved complex numbers.

It is clear that fractions can be found which approach nearer and nearer to having their square equal to 2. We can form an ascending series of fractions all of which have their squares less than 2, but differing from 2 in their later members by less than any assigned amount. This is to say, suppose I assign some small amount in advance, say one-billionth, it will be found that all the terms of our series after certain one, say, the tenth, have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have reached sooner or later a term in the series, say, twentieth, after which all terms would have had squares differing from 2 by less than this still smaller amount. If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-an-so many places, exactly fulfils the above conditions. We can equally form an descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to the later terms of the series, and differing , sooner or later, by less than any assigned amount. In this way we seem to be drawing a cordon around the square root of 2, and , it seems difficult to prove that, it can permanently escape us. But , it is not by this method that we will actually reach the square root of 2.

DEDEKIND’S CUT AND IRRATIONAL NUMBER:

If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that , among those whose squares are not less than 2 , all have their squares greater than 2. There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. There is no lower limit short of zero to the difference between the numbers whose square is little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one class, and there is no minimum to the other. Between these two classes , where root 2 ought to be there, there is actually nothing. We have not actually caught root 2. The cordon, though we have drawn as tight as possible , has been drawn in the wrong place and has not caught 2 .

The above method of dividing all the terms of a series into two classes, of which the one wholly precedes the other, was brought into prominence by Dedekind, and is therefore called a “Dedekind cut” . With respect to what happens at the point of section, there are four possibilities.

If we divide all ratios into two classes such that all of the terms of one class is greater than all the terms of the other then there may arise four case: 1) there may be a minimum to upper the upper section and a maximum to the lower section 2) there may be a maximum to the one and no minimum to the other 3) there may be no maximum to the one , but a minimum to the other 4) there may be neither a maximum to the one nor a minimum to the other. Of these four cases, the first can be illustrated by any series in which there are consecutive terms: in the series of integers, the lower class may contain a maximum number n and the upper class can contain a minimum n+1 . The second of these cases will be illustrated in the series of ratios if we take as our lower section all the ratios up to and including 1, and in our upper section all the ratios greater than 1. The third case is illustrated if we take for our lower section all ratios from 1 upward (including 1 itself) . The fourth case, as we have seen , is illustrated if we put in our lower section all the ratios whose square is less than 2 , and in our upper section all the ratios whose square is greater than 2.

We may neglect the first of our four cases, since it only arises in series where there are consecutive terms. In the second of our four cases, we say that the maximum of the lower section is the lower limit of the upper section, or of any set of terms chosen out of the upper section in such a way that no term of the upper section is the upper limit of the lower section, or of any set of terms chosen out of the lower section is after all of them. In the fourth case, we say that there is a “gap”: neither the upper section or lower section has a limit or last term. In this case, we may say that we have an irrational section, for the class of ratios which do not have limit corresponds to irrationals.

What delayed the true theory of irrational was a erroneous belief that there must be “limits” of series of ratios. The notion of “limit” is of the utmost importance , and before proceeding further it will be well to define it.

A term x is said to be an “upper limit” of a class s with respect to a relation P if (1) s has no maximum in P , (2) every member of s which belongs to the field of P precedes x , (3) every member of the field of P which precedes x precedes some member of s . ( by “precedes ” we mean “has the relation P to”).

This presupposes the following definition of “maximum” : –

A term x is said to be maximum of a class s with respect to a relation P if x is a member of s and of the field P and does not have the relation P to any other member of s.

These definitions do not demand that the terms to which they are applied should be quantitative. For example, given a series of moments of time arranged by earlier and later, their maximum (if any) will be the last of the moments.; but if they are arranged by later and earlier, their maximum (if any) will be the first of the moments.

The “minimum” of a class with respect to P is its maximum with respect to the converse of P; and the “lower limit” with respect to P is the “upper limit” with respect to the converse of P.

The notions of limit and maximum do not essentially demand that the relation in respect to which they are defined should be serial, but they have few important applications except to cases when the relation is serial and quasi-serial. A notion which is often important is the notion “upper limit or maximum” to which we may give the name “upper boundary”. Thus the “upper boundary ” of a set of terms chosen out of a series is their last member if they have one, but , if not it is the first term after all of them, if there is such a term. If there is neither a maximum nor a limit, there is no upper boundary. The “lower boundary” is the lower limit or minimum. Readers at this moment, should be aware that limit and maximum are not the same concept or the same thing.

Reverting to the four kinds of Dedekind section, we see that in the case of the first three kinds each section has a boundary( upper or lower as the case may be), while in the fourth kind neither has a boundary. It is also clear that, whenever the lower section has an upper boundary , the upper section has a lower boundary. In the second and third cases, the two boundaries are identical; in the first , they are consecutive terms of the series.

A series is called “Dedekindian” when every section has a boundary, upper or lower as the case may be.

We have seen that the series of ratios in order of magnitude is not Dedekindian.

From the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving that there was no rational limit to the ratios whose square is less than 2 , they allowed themselves to “postulate” an irrational limit, which was to fill the Dedekindian gap. Dedekind, in the above mentioned work, set up the axiom that the gap must always be filled, i.e that every section must have a boundary . It is for this reason that series where his axiom is verified are called “Dedekindian.” But there are an infinite number of series for which it is not verified.

It is clear that an irrational Dedekind cut in some way “represents” an irrational. But this is not true to say that an irrational must be the limit of a set of ratios. Just as ratios whose denominator is 1 is not identical with integers, so those rational numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios. We have to define a new kind of numbers called “real numbers” of which some will be rational and some irrationals. Those that are rationals correspond to ratios, in the same kind of way in which the ration n/1 corresponds to the integer n; but they are not the same as ratios. In order to decide what they are to be , let us observer that an irrational is represented by an irrational cut, and a cut is represented by its lower section. Let us confine ourselves to cuts in which the lower section has no maximum; in this case we call the lower section a “segment”. Then those segments that correspond to ratios are those that consist of all ratios less than a certain ratio that they correspond to, which is their boundary; while those that represent irrationals are those that have no boundary. Segments , both those that have boundaries and those that do not, are such that , of any two pertaining to one series one is the part of the other. In that case all those segments can be arranged by the relation of whole and part. A series in which there are Dedekind gaps, i.e in which there are segments that have no boundary, will give rise to more segments than its terms, since each term will define a segment having that term for boundary , and the segments without boundaries will be extra.

We are now in a position to define a real number and an irrational number.

A real number is a segment of the series of ratios in order of magnitude.

An ‘irrational number” is a segment of the series of ratios which has no boundary.

A “rational real number ” is a segment of the series of ratios which has a boundary.

Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance , is the class of proper fractions.

In this way we come up with a new theory of real numbers.

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