Month: March 2013

Mathematics – more than the sums of its parts !

Posted on by Abhishek

leonardo

Mathematics is the language of nature. It is the form of expression through which nature reveals her secrets to us. Mathematics is more than numbers and equations, just as poetry is more than letters and alphabets. It is the harmony of the universe.

Maths ‘used’ to be such a beautiful subject during the pre – university and schooling days. Gone are those days when we felt the pure ecstasy while writing – “HENCE PROVED” at the end of a theorem. During engineering, we had to study different kinds of mathematical concepts. Some studied just for the heck of it, while most did for the sake of scoring ( me included !). It was an easy way to up your aggregate. Study and practice some standard problems, and you were sure to score high number of marks. We never even bothered to find out what they are used for and how !!

Luckily for me, the taste of mathematics did not die off. After I came over to do my Masters, I noticed some of the same ‘old’ mathematical concepts crop up again.  This time I decided to explore them in detail, find out about the brains behind them or the history that was responsible for their existence. This journey took me on the path of the of genesis of computer science, and as it turns out, mathematics is at the very heart of it.

To emphasize this fact, let us consider some of the greatest computer scientists of all time :

  • Charles Babbage, “Father of the computer” , was a top mathematician at Peterhouse, Cambridge. He used extensive mathematics in the development of Analytical Engine and Difference Engines. These engines were later used in modern computers. He was also the professor of Mathematics at Cambridge. 
  • Computer Science owes a lot to cryptography, and the very man behind it was the great Alan Turing. It was his elite mathematical skills which made him stand out as an expert crypto – analyst during the second world war. And the very race to build advance cryptanalysis systems drove him to suggest his ‘Turing model of computation’. His vast mathematical skills allowed him to contribute towards artificial intelligence. No doubt, he is aptly known as the Father of Computer Science.
  • Edsger Dijkstra, the man behind the famous Dijsktra’s algorithm in finding out the shortest paths from a single source in a graph, was a champion of mathematics. His chief interest was formal verification.
  • Donald Knuth, the Father of Analysis of Algorithms, got his Phd In Mathematics.
  • Dennis Richite, the creator is the C programming language, graduated from Harvard with degrees in Physics and Applied Mathematics.
  • Vint Cerf, one of the founding fathers of the Internet, was a Mathematics graduate.
  • Ron Rivest, Adi Shamir and Leonard Adelman – the inventors of the famous RSA Algorithm, are all mathematicians, and it was their pure mathematical prowess that makes encryption look so easy and keeps our communication secure over the internet.
  • Last but not the least, Sergey Brin and Larry Page – the brains behind Google’s Page Rank Algorithms, used advanced mathematical concepts in graph theory and matrix evaluation to give you the results you desire. In particular, Sergey Brin, studied mathematics along with computer science during his Bachelor’s. Larry Page considered the idea exploring the mathematical properties of the World Wide Web as his dissertation theme.

The world of Computer Science is full of mathematicians, who have used their mathematical prowess to create wonders in the computer world. Then why is that we take such a lethargic approach to mathematics during our bachelor’s studies? Why is that, mathematical is treated in such an disorganized and uninterested field of study at the undergraduate level in India?  Hardly few of us even know why were are supposed to study it, and where would we be applying them. Some of my friends used to even complain about ordinary differential equations, as to why study them when we would never use them in our daily life.

Even today, when we go out into the industry, a mathematician is at the heart of everything for an engineer. Suppose I were to develop a routing algorithm, it has to be validated my a mathematician and certified to be stable in nature before I can publish it. Before I can design a router, I need to understand buffer management, which falls in queuing systems, a branch of probability and statistics.

Also, a mathematical proof is far more powerful and rigorous than a scientific proof. I put to put across the following text from the great book written by Simon Singh on Fermat’s Last Theorem. The article can be obtained at http://www.nytimes.com/books/first/s/singh-fermat.html. Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. On the other hand,  the scientific theory can never be proved to the same absolute level of a mathematical theorem: It is merely considered highly likely based on the evidence available. So-called scientific proof relies on observation and perception, both of which are fallible and provide only approximations to the truth.

Scientific proof is inevitably fickle and shoddy. On the other hand, mathematical proof is absolute and devoid of doubt. Pythagoras died confident in the knowledge that his theorem, which was true in 500 B.C., would remain true for eternity. Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it “beyond all reasonable doubt.” On the other hand, mathematics does not rely on evidence from fallible experimentation, but it is built on infallible logic.

To demonstrate this, let us consider the problem of the “mutilated chessboard”

Mutilated_checkerboard_problem

We have a chessboard with the two opposing corners removed, so that there are only 62 squares remaining. Now we take 31 dominoes shaped such that each domino covers exactly two squares. The question is: Is it possible to arrange the 31 dominoes so that they cover all the 62 squares on the chessboard?

There are two approaches to the problem:

(1) The scientific approach

The scientist would try to solve the problem by experimenting, and after trying out a few dozen arrangements would discover that they all fail. Eventually the scientist believes that there is enough evidence to say that the board cannot be covered. However, the scientist can never be sure that this is truly the case, because there might be some arrangement that has not been tried that might do the trick. There are millions of different arrangements, and it is possible to explore only a small fraction of them. The conclusion that the task is impossible is a theory based on experiment, but the scientist will have to live with the prospect that one day the theory may be overturned.

(2) The mathematical approach

The mathematician tries to answer the question by developing a logical argument that will derive a conclusion that is undoubtedly correct and that will remain unchallenged forever. One such argument is the following:

  • The corners that were removed from the chessboard were both white. Therefore there are now 32 black squares and only 30 white squares.
  • Each domino covers two neighboring squares, and neighboring squares are always different in color, i.e., one black and one white.
  • Therefore, no matter how they are arranged, the first 30 dominoes laid on the board must cover 30 white squares and 30 black squares.
  • Consequently, this will always leave you with one domino and two black squares remaining.
  • But remember, all dominoes cover two neighboring squares, and neighboring squares are opposite in color. However, the two squares remaining are the same color and so they cannot both be covered by the one remaining domino. Therefore, covering the board is impossible!

This proof shows that every possible arrangement of dominoes will fail to cover the mutilated chessboard. Similarly Pythagoras constructed a proof that shows that every possible right-angled triangle will obey his theorem.

When mathematics is so fundamental and vital to our survival and future of engineering, why does our curriculum neglect it? Very few of us know the importance and applications of linear algebra, very few can appreciate the Bayes Theorem, hardly anyone knows the applications of Runge Kutta Methods, the list goes on and on. Today, when I look back, my heart bleeds to the fact that, once what was supposed to be our core competency, is killed by the very discipline of study that stands on its pillars and looks towards it for support. Today, I feel so constrained by my lack of mathematical ability. I want to study Computer Performance and Modelling – but it requires Probability, and I am weak in it. I am interested in Numerical Analysis – but I hardly remember any of the pre-requisite material, because all I did was study it only for that exam, without understanding it. I am a computer networks guy, yet I don’t have Graphy Theory as one of my subjects of study – ask why, because I am scared of the mathematical background the subject demands ! If this continues to be our sad story, I wonder how will we ever make worthwhile contributions and innovations in the days to come!

Time to think – probably mathematically, as to where things went wrong !

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