Mathematics was originally conceived, and is still taught in schools today, as a means of explaining processes in the real world, and determining how things will happen in the future. However, at a reasonably high level, it is seemingly the study of things with little or no relation to the real world, but may appear to attempt to pass this off as somehow informing us about the Universe. At a glance, especially when one looks into the very highest branches of the subject, there seems to be no point of connection with reality. Indeed one might think an entirely different Universe might be required to make sense of it at all.
However, the deeper one looks into the subject, the more one realises that mathematics is relevant to everything, simply because you can describe every single thing in the Universe with numbers. It is a way of proving theorems with logic, and at its highest level may prove impractical or even impossible for many computers to handle (for instance, try telling a computer to divide by zero).
Much of mathematics is reliant on signs and symbols that show how things relate to each other. For instance 2<5 means 'two is less than five'. The level of symbolism becomes far greater than this simple example when tackling more advanced mathematics, and is often reminiscent of hieroglyphs.
Phenomena in nature can be described and predicted with the help of mathematics. Mathematical objects, invented decades ago and apparently useless, are used today to forecast events taking place in things as small as atoms and as big as hurricanes. Quantum physics, meteorology, economic theory, chemistry and all respectable sciences could not exist without mathematics.
However, despite its reliance on logic, there are contradictions within the subject. The following question became famous as Russell's antinomy1: Does the set of all sets, that are not containing themselves, contain itself?
Answering yes or no leads to a contradiction. After decades of hard thinking modern mathematics avoids this contradiction by declaring this question illegal.
The Relevance of Mathematics
Many people believe traditional mathematics is becoming an irrelevance in the modern age.
As mentioned previously, mathematics was originally conceived as a means of explaining processes in the real world, and determining cause and effect. However, the basis upon which the real world uses of mathematics is often seen as eroding away. Compared to the processing power of computers, traditional mathematical approaches often seem hopelessly inadequate. Examples of 'traditional approaches' include the rigorous solving of differential equations, the use of abstractions so that only the simplest cases can be solved, and so on.
In these days of calculators and computers, one might think mental arithmetic should be pushed into a niche area, becoming a subject for academic purists, in a similar way to Latin or Ancient Greek.
Of course, we will always have a need for basic mathematics in everyday use – working out change in a shop, division of articles, and the like, but there are other examples where a more in-depth practical understanding of the subject can come in useful.
Engineers often appreciate having been taught how to do things the 'traditional' way, finding times when it would have otherwise taken at least twice as long to get a computer to do a problem that could be solved on paper. Lecturers often advise that one needs to know the 'traditional' ways in order to tell if the computer gives you a 'sensible' answer. Computers are not the infallible machines they are made out to be.
Also, as mentioned, some questions concerning mathematics cannot and will never be answered by a computer, such as many of those relating to paradoxes and 'illegal' definitions.