Maths of maths
The only maths I seem to be using is Addition, Subtraction, Multiplication, Division, Averaging, Simple and Compound Interests, and elementary probability. Still trying to figure out what to do with the rest.
With Best Wishes
The only maths I seem to be using is Addition, Subtraction, Multiplication, Division, Averaging, Simple and Compound Interests, and elementary probability. Still trying to figure out what to do with the rest.
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The pigeonhole principle
The pigeonhole principle is one of those which people believe one of the most faltoo principles in Combinatories.
Imagine that 3 pigeons need to be placed into 2 pigeonholes. Can it be done? The answer is yes, but there is one catch. The catch is that no matter how the pigeons are placed, one of the pigeonholes must contain more than one pigeon.
The logic can be generalized for larger numbers. The pigeonhole principle states that if more than n pigeons are placed into n pigeonholes, some pigeonhole must contain more than one pigeon. While the principle is evident, its implications are astounding. The reason is that the principle proves the existence (or impossibility) of a particular phenomenon.
The pigeonhole principle (more generalized)
There is another version of the pigeonhole principle that comes in handy. This version is “the maximum value is at least the average value, for any non-empty finite bag of real numbers” (thanks Professor Dijkstra)
Do not let the math jargon intimidate you. The idea is intuitive. For typical data sets, the average is the “middle” value, so clearly the maximum should be at least as big. While this version sounds different, it is mathematically the same as the one stated with pigeons and pigeonholes. Let’s see how the two are connected.
Consider again the problem of stuffing pigeons into pigeonholes and consider the average. If we have more than n pigeons and n pigeonholes, then the average value of (pigeons / pigeonholes) is greater than one. This means the maximum value should also be larger than one. In other words, there has to be some value of more than one pigeons per pigeonhole. Indeed, the two versions are about the same idea.
Now see the power of ideaEvery number can be paired with another to sum to nine. In all, there are four such pairs: the numbers 1 and 8, 2 and 7, 3 and 6, and lastly 4 and 5.
Each of the five numbers belongs to one of those four pairs. By the pigeonhole principle, two of the numbers must be from the same pair–which by construction sums to 9
2. On New Years at New York’s Time Square, over 820 people will have the same birthday.
It will now be useful to use the second version “the maximum must at least be the average.”
There are roughly 300,000 attendees on New Years split over a possible 366 birthdays. The average is 300,000 / 366 = 819.7 people per birthday. The maximum must at least be the average, so there must be a birthday that at least 820 people share.
3. Imagine a certain college has 6,000 Indian students, at least one from each of the 30 states/UT. Then there must be a group of 200 students coming from same state.
Again, we invoke the second version that “the maximum must at least be the average.”
The average is 6,000 / 30 = 200 students per state. The maximum must at least be the average, so there must be a state where 200 students share in common.
Telegraph has created its list of top 10 Nobel prize winners. Last on the list is Sir Clive Granger who was an econometrician for his analysis of time series data with his colleague Rob. Being modest as all great people are, he credits all his success to the lucky breaks he got in his autobiography. He never made to Cambridge and Oxford, but pipped a lot of them to Nobel prize, he concludes
" My story ends with a recipe for success. Do not start too high on the ladder, move to a good but not top university, work hard, have a few good ideas, chose good collaborators (I had over eighty in my career), attract some excellent students, wait twenty years or so, and then retire. It worked for Rob and I."
Continuing with diagonalization
Diagonalization uses some interesting logic to state that if you postulate a statement, an contradictory statement also exists, and you cannot have an algorithm that can solve both contradictory states. But there is a catch. What if there are no contradictions in nature. What if the nature is always right, and found its solutions. Can natural numbers display alternate properties. Can we have unconstrained input set for Turing Machines. If a given state is achieved, what if an alternate state does not exists. In that case Hilbert's second principle can still be realized, and you can create a set of axioms that prove all mathematical statements.
On other implications of Halting problem being not solvable
The Halting problem is a fundamental result in Complexity theory which says that a Turing Machine cannot be created which can solve all problems that could be decided in finite time. If such a machine could be created it could be shown to be self contradictory through diagnolization principle (assuming a problem has been solved by that machine, another problem can be created with opposite results which the machine then cannot solve). This concept was used by Godel to prove his incompleness theorem. The theorem says that you cannot create a set of principles(axioms) that can prove all mathematical statements with completeness(prove all statements true), and soundness(prove all wrong statements false). He said if such a proof system were created, it would solve the halting problem, an impossibility. The implications of this approach are very large. It puts a fundamental limits on what mankind can achieve. Take a glance.
a. A society where everyone is equally rich cannot be created: To create an equal society you will have to create a perfect set of distribution laws, and if it were created, diagonalization can show it to self contradictory. I will ignore diagnolization inferences in remaining parallelizations.
b. Mankind cannot solve all problems in Science..there will always be unsolved problems. Researchers be happy.."Everything that needs to be discovered has been discovered" will never happen.
c. A pure society created on the principles of Koran, Ram Rajya or any other religious conjecture will never achieve perfect happiness for its members: If such a society were created, an equivalent Turing Machine could be created that would solve the happiness function. And through diagnolization it could be shown that an alternate happiness function exists for the same solution which would be contradictory to results already obtained, an impossibility.
d. Poverty can never be eliminated from the face of earth: If poverty were eliminated, through diagonalization it can be shown that an alternate defnition of poverty function exists, and a contradiction would appear in society.
Can you think of others :)
Unification Theories
Physics as a science has attempted to find common causes from unrelated events. A lot of effort has gone into unification attempts like the Maxwell great attempt to unify electomanetic wave equations. Was this approach important..does it have practical implications..It does..Newton unified diverse observations like apple falling on earth, and moon rotating around the earth to create the concept of gravititional forces..resulting all in advances is space technology today. Oersted in the days of gas lamp unified independent observations of magnetism and electricity, resulting in modern field of electrical engineering and specifically turbines which is the basis of all electricity we take for granted today. The Nobel prize in1959 was awarded to the attempts of unifying weak atomic currents and electromagetic interactions. This was the first time a Muslim received Nobel prize (Abdus Salam, Pakistan).
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GDP Comparisons | |||
(All figures inMillion Dollars) | |||
2006 | 2007 | Absolute GDP Growth | |
India | 886,867 | 1,089,940 | 203,073 |
Pakistan | 128,996 | 143,760 | 14,764 |
India/Pakistan Ratio | 6.875151 | 7.581664 | 13.75461 |