What is mathematics?
Responses will vary. "Numbers," someone will say, others may reply with, "Doing things with numbers and letters and other symbols," whilst pragmatists will contend that "Mathematics is a necessary tool for day to day living."
The philosophers among us will avow that mathematics is replete with drama and emotion and an end in itself.
Any functional definition must acknowledge that mathematics is a language comprising an alphabet (numbers, symbols and mathematical operators such as + and x), and rules for sentence construction (mathematical properties such as 1 x 0 = 0).
To the non-mathematician, some things seem magic, but to the mathematician it is simply a case of following the rules. To demonstrate this point, consider the following set of instructions. Think of any number, double it, add fourteen to your result and take half of your answer. Finally, subtract the number you started with. I predict that your answer will be the same as the number of days in one week.
It is not a coincidence that the language of mathematics was first known as natural philosophy and practised by philosophers to explain a myriad of observations. Their approach was more akin to describing events rather than the adoption of symbolic structure. Some extremists among them had even sought to prove the existence of God! Today, their legacy is evident in a branch of mathematics known as mathematical logic, except that verbal descriptions have been replaced by a vast array of symbols anyone skilled in short-hand will drool over!
Once we accept that mathematics is a language, the rest easily follows. Just as an infant learns to talk by first mastering basic sounds and phrases, someone studying mathematics will first need to master the building blocks on which mathematics is built. These of course are the digits 0,1,2,3,4,5,6,7,8,9 and the basic operations of addition, subtraction, multiplication and division.
However, numbers in themselves are meaningless until we give them meaning, until we give them a purpose and make them work for us to unlock doors to a world of adventure and discovery. Mathematicians use the language of mathematics to produce works of elegance and beauty to rival the great works of Shakespeare.
Think of Einstein's formula E=mc^2. This shows the equivalence between the mass (m) of an object and its energy (E), with the square of the speed of light (c^2) holding it all together. Who could have envisaged -certainly not Einstein- that this formula has major applications in nuclear development?
Notwithstanding a necessary level of inflexibility a study of mathematics entails, much can be still be done using the tools we already have. When Peter is asked what he did on Sunday morning, he may say, "I read the newspaper." When John, Emeritus Professor of English Literature, is asked the same question, he may respond with, "I perused the daily chronicle of contemporary events." Both Peter and John did the same thing on Sunday morning, but Peter is obviously the clearer of the two thinkers.
Mathematics requires us to structure our information clearly, concisely and unambiguously so that another mathematician knows exactly what is happening. For instance, as a child, Karl Friedrich Gauss, a famous mathematician of the nineteenth century, was instructed by his teacher to add the whole numbers from 1 to 100. Dismissing the laborious and time consuming computational process that his class mates sheepishly adopted, Gauss paired numbers from either end of the sequence; 1+100, 2+99, 3+98, ,50+51 to obtain 50 lots of 101, that is, 5050. Gauss avoided considerable computational drudgery and sowed the seeds for what was to become the theory underpinning the sum of an arithmetic series.
To learn mathematics is to learn a second language. It may not be easy at first, but the effort is well worth it if you wish to be rewarded with the poetic cadence afforded by proving that the left hand side of the equation is equal to the right hand side.