The topic of the That’s Maths column ( TM026 ) in the Irish Times this week is the surprising and delightful way in which mathematics developed for its own sake turns out to be eminently suited for solving practical problems.

Symbiosis between pure and applied mathematics

The power of mathematics is astonishing. Time and again, mathematical theories developed without any application in mind have later proved ideal for solving practical problems. There is a vital symbiosis between pure and applied mathematics. Practical problems that require solutions provide a strong stimulus for the development of new methods and techniques. And abstract mathematics, developed purely for its inherent interest and elegance, frequently turns out to be ideally suited for novel applications.

William Rowan Hamilton, Bernhard Riemann, Albert Einstein and Eugene Wigner.

Quaternions and video games

The idea of quaternions came to William Rowan Hamilton in a flash of insight as he walked along the Royal  Canal bank in Dublin. Quaternions are four-dimensional numbers useful for solving problems in mechanics and optics. Fifty years after Hamilton’s discovery, they had become completely eclipsed and vector calculus held sway in physical applications.

For a century, quaternions seemed little more than a historical footnote. But in the past decade or two they have risen like the phoenix: they have applications in astronautics, robotics and theoretical mechanics. They are also proving indispensable in computer visualisation. Does this matter? Yes: the computer gaming industry is worth \$100 billion!

Another example is knot theory. Mathematicians abhor loose ends: a mathematical knot is like a physical one with the ends joined so that it cannot be untied. In the 1860s, Lord Kelvin tried to model atoms as knotted tubes of ether. This led nowhere, but today knot theory is a key element in DNA sequencing. It is also vital in string theory and loop quantum gravity, two candidates for a unified theory of relativity and quantum mechanics.

Tensors and relativity

In 1854, Bernhard Riemann described geometry in curved spaces of many dimensions. More than fifty years later, the tensor calculus that describes this geometry provided precisely the building-blocks that Einstein needed for his general theory of relativity. Einstein had a deep appreciation of the power of equations, and asked: “How can it be that mathematics … is so admirably appropriate to the object of reality?”

There are numerous other examples where mathematics has been of unanticipated value in applications. Group theory was developed in the early 1800s to examine the solvability of polynomial equations. In the 1960s physicists used a particular group, SU(3), to represent subatomic particles called hadrons and predicted the existence of two new particles. Functional analysis blossomed when it was used as a foundation for quantum mechanics. Number theory is proving essential for modern cryptography. Probability, originally of interest for gambling, is now the bedrock of the insurance industry. And the trigonometric series developed by Fourier are vital in just about every area of engineering.

A daft and dangerous idea

Funding agencies demand of researchers that they specify in advance the nature of discoveries yet to be made, and the impact on society of those discoveries. This is a daft and dangerous idea. It is impossible to anticipate the potential value of new mathematics, and vital for our future wellbeing to foster research in this area.

The success of mathematics in physics inspired the physicist Eugene Wigner to write of The Unreasonable Effectiveness of Mathematics in the Physical Sciences. The future will tell if mathematics will prove equally effective in other areas of science, but it is already indispensable in biology, genetics and medicine.