When abstract thinking becomes useful in practice

Professor Hendrik Van Maldeghem, of Ghent University's Department of Mathematics, describes why the abstract nature of maths helps make it a useful, practical discipline…

Sometimes there is a perfect word in another language to describe a feeling or idea that cannot be adequately expressed in one's own language – like the word 'voilà' in French, the undertone of which cannot be caught by any English expression. The same is true in mathematics. For example, sometimes one is unable to solve a geometric problem, but once 'translated' into algebra, the solution becomes apparent. The comparison with languages is not coincidental; mathematics is really a language for science in general. It provides a way to express scientific ideas and rules, to put down certain laws of nature in formulae, and to derive new laws from old ones by deduction, algebraic manipulation and logic.

In this way, mathematics is very important for our society. But how come one can even translate things within mathematics? And what makes mathematics so universal that it can serve all kinds of science? Paradoxically, it is precisely the high level of abstraction that renders mathematics so useful and practical. This can be explained using a very basic example.

Everybody knows how to add numbers. But a number, I would argue, is already a very abstract notion. If you add five and three, then you get eight, but eight what? In fact, children start learning adding numbers by assigning certain things to it, such as apples. So, five apples plus three apples make eight apples. No matter which 'things' you take, you always end up with the same number in front of it. So, since the 'things' are apparently not important for the addition, we can leave them out and only keep the numbers. That's abstraction: you do not know what you are talking about, but you can apply it to anything you like. Even to other mathematical fields.

Nevertheless, a little care is needed. Indeed, the subtraction 4-7=-3 can be applied to degrees Celsius, or to different currencies (where a minus sign indicates a debt), but not to our apples above. The moral of this is that one has to check carefully which theory to use for a certain practical problem. If the weather forecast goes wrong, it is not because the mathematical model behind it is deficient, but rather because the model is not perfectly suitable for the complexity of our atmosphere.

A similar but more involved idea applies to symmetries. For example, the 24 rotations of a cube form an algebraic structure called a group. The study of this group reveals properties of both the cube and the octahedron, since the latter admits exactly the same abstract group as rotations.

Of course, not every abstract theory in mathematics immediately makes its way to practical applications. Generally, mathematicians do not care about immediate applicability – they just want to know. Each day, mathematicians generate hundreds of pages of new theories, and it is debatable how efficient this is. Why would society pay mathematicians to deliver abstract theories, the majority of which rarely make it to the real world? Would it not be better to steer the research to improve the probability of applicability of abstract research?

The answer is no. Simply because there is no good (mathematical) model to know which theories will lead to good applications. The logical consequence would be to pursue only the successful ideas of the moment. This, however, leads to narrowing the field; it may be successful for a while, but in the long run there will not be enough variations of mathematics left to apply. The theory of prime numbers that we nowadays need so badly in cryptography was developed in the 19th Century; without that base, our cryptosystems would not be at the level of where they are now.

So we need pure and abstract mathematics, for now, and for future generations – all for practical reasons.


This article originally appeared on Publicservice.co.uk: Abstract thinking