Abstract maths far from pure indulgence

20:19, Sep 10 2012

A re those wacky abstract mathematicians who dwell in universities wasting their time and our money with their seemingly impractical deliberations?

A review of the surreal topics being researched by pure mathematicians quickly has the non-mathematician wondering what practical purpose this work would ever have.

History, however, shows us that it is frequently the case when some aspect of pure mathematics is being researched just out of curiosity, much later the result finds a perfect application. This is often in an industry that didn't even exist when the mathematics was being formulated. An example is in the design of computers.

Computers comprise a large number of switches, each of which can only either turn a signal 'on' (represented by a 1) or turn it 'off' (represented by a 0). That is the fundamental currency of any computer. Clearly, the problem is how do you go from a large number of switches, each only capable of on/off actions, to building a machine that can process vast amounts of information and perform arithmetical operations?

The answer is that the design of computers required three fundamental mathematical concepts to make this leap. Each was worked out decades, and in two cases hundreds of years, before the advent of the electronic digital computer.

The first piece of mathematics required to make a computer work is called logarithms. A computer cannot directly multiply numbers, it can only add and subtract. However, adding the logarithms of the numbers is the same as multiplying the numbers. So if you convert the numbers to logarithms all you have to do if you want to multiply the numbers is add the logarithms. Adding is something computers are good at. The point is that logarithms were invented in the late 16th century by a Scottish mathematician, John Napier, long before anyone knew what an electronic computer was, let alone had any use for it.


The second piece of mathematics required to design computers is called binary arithmetic. The on/off signals of computers is a binary operation. Somehow decimal numbers fed into the computer have to be converted to a system of on/off signals, a binary system, so that the computer can work with them. Binary arithmetic was invented in the 17th century by the great German mathematician and philosopher Gottfried Wilhelm Leibniz (1646-1716). He constructed a very early mechanical calculator but it was to be hundreds of years before his binary arithmetic was applied to the electronic switching of modern digital computers.

The third piece of mathematics essential to computer design is called Boolean logic, named after the Englishman George Boole (1815-1864). He devised an alternative to elementary algebra. This work was the result of research Boole was conducting on 'the laws of thought'. But today Boolean logic is probably the most important mathematical tool used in computers because it lies at the heart of how the on/off bank of switches can be translated into signals that can, in turn, be translated into the mathematical logic required to process binary information.

Another example of mathematics, performed as an intellectual curiosity, was that of non-Euclidean geometry. A key contributor to this was Carl Friedrich Gauss (1777-1855), arguably the greatest mathematician. He famously refused to allow any of his six children to enter the mathematics and science professions for fear of 'lowering the family name'.

Non-Euclidean geometry was taken a step further by Bernhard Riemann (1826-1866) in what is now referred to as Riemannian geometry. This is a geometry in more than three dimensions; again at the time, it was an intellectual curiosity with no immediate application. That was until Marcel Grossman introduced it to Albert Einstein. Grossman discussed Einstein's work on gravity with him and instantly saw a beautiful application for the dormant mathematics of Riemannian geometry.

As a result of these conversations and lessons given by Grossman, Riemannian geometry forms the mathematical basis of Einstein's general theory of relativity. Einstein provided the application but it was Riemann, 50 years earlier, who provided the mathematical basis.

It is probable that if it was not for the intellectual musing of mathematicians like Napier, Leibniz, Boole, Gauss and Riemann, the progress of human civilisation would have been slower and could even have taken a completely different turn.

Perhaps the seemingly pointless topics of abstract mathematicians conducted by pure mathematicians is not a waste of time after all.

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