The nature of mathematics

Mathematics is unique in that it can be both tangible, something you can touch, and yet also completely abstract. For example, addition can be demonstrated with something as tangible as a handful of pebbles, but at the same time the sum 2+2=4 is generalised statement that can be related to any object, or it can even be a completely abstract expression that has no physical embodiment.

Much of the history of mathematics has involved its development from the concrete to the abstract. For the ancient Greeks, mathematics was a very practical subject, with geometry (the study of shapes) as its basis. A variable (x or y, for example) was represented as a length, the square of that variable as an area, and its cube as a volume. However, such a pragmatic approach gave the Greeks issues when it came to dealing with ideas that fell outside this paradigm, such as negative numbers.

Over the course of the intervening millennia, mathematics has become more abstract in form, and therefore more flexible. But it doesn’t mean that its applications are any less practical. Even when an idea is pursued on a purely theoretical basis, it can eventually find its way into everyday usage. A good example is Joseph Fourier, a French mathematician (1768-1830), who worked with infinite serious of trigonometric functions. Yes, it’s as complex as it sounds, and during his lifetime this subject was purely theoretical in nature, a mathematical puzzle to be solved, seemingly for its own sake. However, many years later the foundations he laid form the basis of analogue-digital conversion, the technique that’s used to turn analogue sound waves into digital CDs.

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