Sunday, October 21, 2007

The Magic of Numbers 2



Of Numbers, Directions, Definitions, Solutions and Dissolutions

Sometimes I decidedly feel like Alice in Lewis Carroll’s famous and oft-quoted Alice’s Adventures in Wonderland who said: “Cheshire-puss,” she began rather timidly…”Would you tell me please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. “I don’t much care where,” said Alice. “Then, it doesn’t matter which way you go,” said the Cat.

There is also the famous Kerry man joke about some passing motorist who asked a farmer from this county where a particular place was and he got the rather witty reply, “I would not start from here!” Well I suppose this jocose story illustrates by the contrary that the only possible starting point is from where we are. Also the way we may choose to go may be blocked here or there which may cause us to take certain detours to get to our destination.

Everything has a history, mathematics included. Thank God or whatever power of nature has put us here, conscious and thinking, on the planet earth because the general non-academic reader like myself is always generations behind in his reading and understanding of the complexities of modern specialities in any subject.

Words are tightly defined in any subject, especially in the sciences and in mathematics which could possibly be called the language of the former. Let’s look at a few definitions.

(a) Function: The mathematical meaning of this word is expressed most simply as a table. I remember a great teacher and mathematician Br Martin Collins, C.F.C. teaching us what a function was and how to draw such a table and the curves associated with these tables. Such a table, he told us, gives the relation between two variable quantities when the value of one variable quantity is determined by the value of another. Thus, one variable quantity may express the years from 1828 when the school I was taught in was founded (O’Connell Schools, North Richmond Street) to 2006 and the other the number of pupils on the rolls from year to year. We also learned to draw the resulting graphs or curves and learned all about the X and Y axes.

(b) Transcendental Numbers: Having studied philosophy and theology as well as mathematics at University I am well acquainted with this word and its various meanings in the different subject areas. In theology the substantive “transcendence” refers to the Divine Being or God as does the adjective “transcendent” which can also be used substantively. Then, in philosophy, I remember having to wade through Kant’s elaboration of the “transcendental argument” in reply to scepticism and the empiricism of Hume. I’ve forgotten all of the complexities involved here but appreciate its particular philosophical use and nuance. Then I read some little of the spiritual works of Emerson and Thoreau who wrote after the American Civil war - these Transcendentalists saw themselves as a generation of people struggling to define spirituality and religion in a way that took into account the new understandings their age made available after the “corpse-cold” (Emerson’s term) rational religion that came after the Enlightenment.

Then we come to mathematics and we find that the word “transcendental” in mathematics has not the meaning that it has in philosophy. A mathematician would say that π = 3.141592 65… is transcendental because it is not the root of any algebraic equation with integer coefficients. When a circle’s diameter is 1 its circumference is π. In other words Pi or π is the ratio of a circle's circumference to its diameter in Euclidean geometry. Another way of phrasing this is to say π = c/d where c = length of the circumference and d = length of the diameter. It is also interesting to note that while the value of pi has been computed to more than a trillion (10 to the power of 12) digits, practical science and engineering will rarely require more than 10 decimal places. Thanfully!

So much for a few numbers and definitions and for the seeming lack of direction in our lives. However, it has well been said that “necessity is the mother of invention.” Without numbers or mathematics or sums or measurements we would be at a considerable loss. The whole world about us like buildings, bridges, water pipes, electricity, central heating, cars and all machinery and technology would not exist without the beauty and eccentricity and indeed complexity of numbers. All these things are the results or fruits or solutions to practical problems which day to day living forced humankind to engage with.

With reference to my rather eccentric and complex title I have indeed referred to all the words therein contained except “dissolution.” Mathematics is an imaginative science and like all constructs of the imagination is a powerful tool for construction. However, it can also be used for destruction as the Atom Bombs that were dropped on Hiroshima and Nagasaki so potently illustrated. Science and the mathematical method thereof can lead also to dissolution. The imagination is indeed powerful, but does any work of the imagination exist without the imagination or mind(s) that conceive them? And minds, of course as history has taught us to our detriment, are as often evil as they are good in their intent. If the imagination can conceive of beauty it can also conceive of ugliness; if good then also evil. Can any science ever be morally neutral? My title says it all – I am so confused!