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
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.
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 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.
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.
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
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.
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
Quantum theory is modern physics which describes the behavior of sub-atomic particles. It was established by two great scientists , namely Albert Einstein and Max Planck. Max Planck showed that energy of a process is quantized. Any periodic process can carry energy that is integer multiple of hv [ its frequency times Planck’s constant h]. Einstein later showed that light is composed of energy packets called photon. Planck’s theory was applied by Bohr inside atom and a successful atomic model was created. Any electron in its quantum orbit can emit or absorb light of certain frequency and jump from it. Later Schrodinger was very frustrated by this weird quantum jumping of electron.
Luis De Broglie hypothesized that sub-atomic particles act like wave. He also showed exact relationship between matter wave and particle’s momentum which is mass times velocity. Every elementary particle like electron is a wave and every wave is a particle of some sort. In classical mechanics this effect is negligible due to short wavelength associated with the massive objects. In quantum world a tiny particle shows more waviness.
The last revolution was the Schrodinger’s wave equation which describes the wave function of particle in terms of probability. Schrodinger’s equation successfully explains Hydrogen and other simple atoms and shows how bizarre an atom looks. An electron does not exist in a definite place but it has only some probability to be in some place at a time. Electrons tend to exist at some place. This was the consequence of Heisenberg’s uncertainty principle. What we call electron is really a collection of radiations which try to be localized at some point in space.
Earlier Heisenberg stated his uncertainty principle which shocked everyone. In quantum world nothing has definite position and momenta. If we measure position more precisely the momentum becomes more uncertain and vice versa. We can not know both of them simultaneously with arbitrary precision. The nature has set a limit to our knowledge about it.
Secret of creation :
The beginning of our universe was a quantum event. General theory of relativity fails at this event when universe was very tiny and dense. Whenever small object is concerned quantum mechanics is necessary to describe its behavior. Spacetime is smooth and continuous on large scale. All the stars , galaxies and planets exist in a continuous spacetime manifold. What if spacetime may not be smooth in quantum level? It might have holes and gaps like the pretzels and doughnuts. The secret of the universe can only be traced in quantum level where everything is governed by chance and discontinuity. We do not know , in advance , when an electron will jump from one orbit to another. We only know if they do they will emit a photon of specific wavelength to go into another orbit. Nature, so it seems, is full of revolutionary occurrences , whose cause is never known. All we can predict is the probability of an outcome : like the chance of getting a six by throwing a six headed dice.
It is hard to reconcile quantum mechanics into the framework of general theory of relativity. There arise many mathematical problems while doing that. Quantum mechanics and general relativity become very antagonistic to each other. They do not shake hands at all. Quantum mechanics certainly holds the key to unlocking secrets of the universe. Three fundamental forces can be explained by the principles of quantum mechanics. Quantum mechanics has changed the view of our world completely when it comes to how these fundamental forces work. Electromagnetic force is the exchange of photons between two charge particles. Similarly strong nuclear and weak nuclear forces are carried by other sub-atomic particles like gluons and boson. Electromagnetic force is long range while strong and weak nuclear forces are short range where strong magnetic has the special property that when the distance in the nucleus increase the intensity of the force also increase. This has to do with another weird phenomena that quantum theory describes. This phenomena is quantum fluctuation.
Vacuum is not truly empty. It is full of activities dues to random generation of particle -antiparticle pairs all the time. This is also a consequence of Heisenberg’s uncertainty principle. The energy of vacuum can not be zero because in context of quantum mechanics zero is a precise number. Electron and positron pair can come into existence out of nothing. So vacuum is not truly passive . It is always seething with virtual particles and energy.
The greatest mystery is probably quantum entanglement which states that information can travel faster than light, which is also known as the spooky action at a distance. Einstein was deeply worried by this fact and unfortunately he could never come up with any valid counter argument to refute this. An electron when entangled can communicate instantly with its entangled pair over a larger distance; probably at the other side of the universe. Secret of creation or secrets of the universe deeply lies at the heart of quantum mechanics. I like to rephrase Nikola Tesla : If you want to find the secret of the universe think in terms of vibration , energy and frequency. In quantum mechanics all particles are vibrations or oscillations in their corresponding quantum field. Oscillation in the quantum field creates energy and that energy, in turns, create particles according to Einstein’s famous energy mass equivalence formula (E=mc^2). Einstein was a strong opponent of quantum theory throughout his whole life.
Theory of relativity throws a little light on the secrets of creation. It has some philosophical consequences and some applications can be found in causal theory of perception and analysis of mind. Mind and matter are not very different from relativistic point of view. What we call mind is certain group of events connected with specific relation. These are all the events that have space-like interval between them so any other observer can judge any of two to be simultaneous. What we call matter is also a group of time-like events. What happens when we look into another people’s eye are the events connected with one light rays. These events has light-like or zero interval. Quantum theory might be more fundamental than theory of relativity but relativity is able to describe four fundamental forces. Special theory of relativity describes three forces whereas general relativity describes one force.
Quantum theory basics :
The most important basics of quantum theory may be coherence and de-coherence. In an isolated quantum system there is quantum coherence. That is , the states remain uniform and in phase. Schrodinger’s equation describes a quantum system as an smeared out probability of finding a particle like electron in space. These probability wave evolves through time. Schrodinger’s equation also states that particle can exist at different places at the same time , which is known as quantum superposition. That is what we better know as the Schrodinger’s cat. Schrodinger cat is alive and dead at the same time unless we look at it. That means, there is coherence in the system unless we observe any such quantum system. After the system is observed, the system collapsed to one of the states of the system. This can be explained as the particular state of the Schrodinger cat , which can either be dead or alive after box is opened. Decoherence is the act of observation which collapses the system. Due to decoherence the quantum system no longer remains coherence and this happens through the interaction with the environment.