Saturday, May 26, 2007

The Mathematical Century

Note: I shall devote this entire post to Mathematics.

I have been reading a book - "The Mathematical Century: The 30 Greatest Problems of the Last 100 Years", written by Italian mathematician Piergiorgio Odifreddi. The one I have been reading is a translated version (from Italian to English). The Mathematical Century provides interesting insight into the development and history of Mathematics in the past decade. By looking at some of the most famous problems in mathematics (eg Fermat's last theorem, Abel's Impossibility Theorem), Piergiorgio Odifreddi traces the development of mathematics since ancient times, from the days of the Geeks, to Pythagoreas and then to modern day mathematics, the progress of which has been largely influenced by Hilbert's problems and perhaps the unification of which by the Bourbaki group.

The book is written in prose-style, free of technical details (this refers to equations, symbols, but does not refer to mathematical terms). (However, some of the ideas presented are rather abstract.) Odifreddi starts by discussing the four main philosophical foundations of mathematics in the 1900s, mainly - Sets (ZFC, etc), Structures, Categories and Functions.

While I admit that I could only understand less than 40% of what was written in the book, the book provided insightful breadth, (rather than depth). Indeed, modern day mathematics is no longer limited to abstract algebra or topology. With the birth of the PC, computer-assisted proofs have become acceptable (eg the Four Colour Theorem) and the P vs NP complexity problem is indisputably one of the most open problem in computer science and mathematics.

Overall, Odifreddi offers a sagacious bird's eye view on mathematics from the perspective of mathematical problems.

Anyone looking forward to appreciating/understanding mathematics should also read A Mathematician's Apology (PDF), an essay written by G. H. Hardy in 1940. It is an essay that is not only reflective, but personal as well. In it, Hardy made one of his most famous observations:
No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.
While one may agree or disagree with the above statement, the essay certainly proffers intriguing views.