book: The Pleasures of Counting ~ T.W Korner

One of the many maths books on the Cambridge reading list for prospective mathematicians. I'm just going to highjack this blog to list down my thoughts and to copy some really cool and interesting bits from this great tome of knowledge.

[Only just finished the first Part (4chapters) so i'll edit this post as I read more of the book] To begin with, the first few chapters of PoC relate to the abstraction of real world events and simplification into models. It begins with the early examples of the spreading of Cholera in 19th century Britain and then moves onto the development of radar and other methods of detection in the Battle of the Atlantic during the Second World War where German submarines tried to take down Allied convoys carrying supplies from America to Britain. A grand topic was Operation Research; the application of maths and sciences in refining tactics and improving use of current resources in order to increase efficiency, as opposed to simply building new gadgetry. The special thing about Operation Research is that it requires a definite ability to think laterally. Simple things such as changing the paint scheme of fighter planes to reduce their visibility to submarines is probably not what you would expect of a mathematician's solution to a problem. Also contained within the first part are little nuggets of comedy: throughout the military there are apparently completely obsolete positions. For example, during an operational research investigation into the firing of artillery, it was found that in addition to the crew of five who manned the cannon, there would always be a sixth soldier who stood at attention beside the cannon without any other apparent function. Instruction manuals past and present gave no indication to as to the job of the sixth soldier and questioning serving and retired officers also proved equally unhelpful. It was not until they questioned a Boer Veteran that the mystery was solved. "The sixth soldier holds the horses."

Furthermore, Korner provides some philosophical commentary on the use of mathematics: "Mathematics does not enable us to make moral choices and overenthusiastic use of mathematics can obscure the moral choices we must make." While not completely related, this quote brings to mind the recklessness and over-reliance on cold mathematics by the traders at Long Term Capital Management [talked about in my last post on: When Genius Failed].

Also posed is probably the reason why I wish to do mathematics in the future: Although the scientists behind the Operational Research during the Battle of the Atlantic were all highly intelligent and academic, the mathematics they used were relatively simplistic. One might wonder why they did not use high-powered mathematics to solve such problems. However, a beautiful analogy given by Korner is that: "Just as it is unprofitable to use precision tools in stone quarrying, so it is unprofitable to use delicate mathematical techniques on imprecise data." Thus in the context of a war, where data is not completely accurate, simplistic methods must be used to produce good rough estimates. Then why have such a highly intelligent and probably over-qualified group to solve such problems using simpler techniques? Korner postulates the answer as the difference between 'knowledge' and 'competence'. The 8 year old child may find multiplication hard but addition easy whilst the 14 year old student can breeze through multiplication but finds algebraic manipulation a mystery. At 18 years old, calculus is difficult but algebraic manipulation presents few problems. By the end of university, multi-dimensional calculus becomes a problem but the simple one dimensional calculus of school becomes an old friend. An even better analogy is of learning a new language. When one is trying to learn a foreign language, even buying a train ticket becomes an arduous and emotional task but for the fluent speaker, they can concentrate on the thoughts and not worry about the language in which to express them. So it is with mathematicians and mathematics. Mathematics becomes a familiar language and so mathematicians can feel free to concentrate on the essentials of their problem. Thus with modern firms looking for graduates in numerate degrees such as maths, physics and engineerying; they are not looking for a genius to use their knowledge Galois theory or particle physics to make them money, but simply to have a competence and confidence at lower levels indicated by a knowledge of higher levels.

Finally, the author addresses the readers as females. While I have experience this before in the form of D&D rulebooks where the Hero is female, it is stil a strange feeling. Interesting to note a lack of chauvinism in academia and I feel I must do more research on this and I'll post more when I find out and also read the second part of this book.