The Hindu Business Line : In mathematics there is no ignorabimus

Mathematics is essentially `an activity of production and solution of problems,' writes Piergiorgio Odifreddi in The Mathematical Century, from Universities Press. Problems can be `easy or difficult, superficial or deep, theoretical or practical, pure or applied.' Their supply is inexhaustible, and solutions are often `the source of new problems'.

The book brings together `the 30 greatest problems of the last 100 years' such as the Lebesgue Measure and Steinitz classification of fields, Von Neumann's Minimax Theorem and Chomsky's language classification, Shannon's analysis of the chess game and the Mandelbrot Set.

It is hard to gauge the difficulty of a problem before having seen its solution, writes the author. "Nevertheless, mathematicians consider that the problems they formulate are not only solvable but that they will be, sooner or later, actually solved." He cites a quote of David Hilbert thus: "The conviction that every problem has a solution is a powerful incentive for the researcher. In our hearts we hear the perpetual call: there is a problem, let us find the solution. And it can be found through the use of reason alone, because in mathematics there is no ignorabimus." Yet, there can be problems that appear interesting or solvable but turn out to be disappointing or unsolvable, cautions Odifreddi.

In the list of `open problems' that he lists is `Complexity theory: The P = NP problem'. This is explained through a simple example. "To verify that a given telephone number is actually the number of a certain person is easy, for it is sufficient to look up the person's name and number in the telephone book. But to find the person whose number is a given number is difficult, for it requires the exhaustive search through the entire telephone book." That is, if you don't use a database search on a computer. The theory finds application, among others, in cryptography.

There are thousands of theoretical and practical problems that belong to the class NP, says the author. A famous example is the travelling salesman problem: "Given a map with cities connected by roads, find a path of minimal length that visits each city exactly once."

On math topics that you would love to revisit!

Tailpiece

"If I can book my train ticket at the ATM, can I also..."

"Cancel the ticket?"

"No, check my email and play Solitaire at the machine?"

http://BookPeek.blogspot.com

D. Murali