There is an apocryphal tale of the French mathematicians who published collectively under the name Nicolas Bourbaki, who apparently wrote 200 pages on the number one alone (Bergamini, 1972). I cannot say I believe this story completely, having heard similar anecdotes of topics broad (psychology) and narrow (the modern Italian historical novel). The experts shake their heads and chuckle wryly at the naivety of asking for a definition. Thus forewarned, I shall not fall into this trap: I shall say only what mathematics means to me. Mathematics is, to me, the science of number; its uses, both practical and theoretical (Courant, Robbins & Stewart, 1996); and associated notations. I will explain my take on these in reverse order. Notations I do not regard as especially important: Shakespeare is still Shakespeare, whether in the original Elizabethan English, translated into Norwegian, or performed in British Sign Language. It is unfortunate that notation is often the first thing people think of on hearing the word ‘mathematics’. Perhaps more effort is needed in schools to describe this as a code, shorthand, or language.
The uses of mathematics I regard as practical, meaning applied or descriptive, or theoretical. The practical element is relatively easy to explain: a shepherd counts his sheep to keep track of them, plan grazing, and so on; that number is also a handy descriptor for others, rather than taking them to the hills to see for themselves! This practical element also elucidate processes and phenomena which cannot be observed or comprehended directly, such as the detection of orientation in vision (Anderson, 2001) or the origin of the universe. The phrase ‘theoretical uses’ is something of an oxymoron – perhaps another synonym of ‘use’ would be better here – service, practice, exercise… However, I am inclined to think that theory – the study of a subject purely for the subject’s sake, as it were – is a ‘use’ whose day is not yet come. Reading a popular science book on mathematics some years ago, I was amazed (briefly, and I really shouldn’t have been. It won’t happen again.) that so many ‘pointless’ theories were discovered to have practical applications, some discovered long after the theory was first promulgated (Stewart, 1998). A true application may appear later, or suggest itself immediately: the effort is not wasted.
I define number very broadly and inclusively: the integer 6, a triangle, or the algebraic notation x2, are to me number. Similarly, the word ‘happy’, Hals’ Laughing Cavalier, and a smiley icon convey the same emotional information – in informational, pictorial/geometric, and symbolic forms. Where does number come from? While I don’t want to come over all Pythagorean, I feel that number is present in the structure and ordering of the Universe. Not as a separate and discoverable entity in itself, but as our code for the otherwise incomprehensible and ineffable. It should not have surprised me that the Fibonacci Sequence can be seen in the different petal arrangements on flowers – I should have been asking how could petal arrangements be modelled, and what are the biological processes underlying the Fibonacci Sequence’s fit to the data. God may or may not play dice, but is almost certainly a mathematician.
References
Anderson, R. (2001). Detection and Representation of Oriented Contours in Human Vision. Ph.D. Thesis, University of Birmingham.
Bergamini, D. (ed.) (1972). Mathematics. 3rd ed., Netherlands: Time-Life International.
Courant, R., Robbins, H., & Stewart, I. (1996). What is mathematics?: an elementary approach to ideas and methods. 2nd ed., Oxford University Press Inc.
Devlin, K., (2000) The Maths Gene, Why Everyone Has It, But Most People Don’t Use It. London: Weidenfeld & Nicolson
Richer, É. (no date). PlanetMath [online]. [cited 12th January 2010]. http://planetmath.org/encyclopedia/NicolasBourbaki.html
Stewart, I. (1998). Nature’s Numbers: Discovering Order and Pattern in the Universe.London: Phoenix.
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