Applied mathematics is the use of maths to address questions and solve problems outside maths itself.
Counting money, designing rockets and vaccines, analysing internet traffic and predicting the weather all involve maths. But why does this work? Why is maths so successful in describing physical reality? How is it that the world can be understood mathematically?
This question has puzzled many deep thinkers. Einstein addressed it repeatedly, once asking “How can it be that mathematics, being after all a product of human thought, independent of experience, is so admirably appropriate to the objects of reality?” His conclusion was that the comprehensibility of the world is one of its great eternal mysteries.
The universe is profoundly mathematical in nature, and we do not really understand why. Physicist Eugene Wigner looked at the question in an essay entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. It seems astounding that we can describe the behaviour of physical systems with great precision and use that description to predict their future evolution.
The ancient Greeks were interested in mathematics primarily as a philosophical pursuit. The real value of the subject was its logical structure and consistency and its inherent elegance. Applications were few and subordinate to the pure subject. Galileo was the first to devise a mathematical treatment of physical systems.
He wrote that the book of nature is written in the language of mathematics, and anyone ignorant of that language “is wandering in a dark labyrinth”. Those inclined to brag about their mathematical ineptitude might usefully reflect upon this.
Calculus
Towards the end of the 17th century, Newton and Leibniz formulated calculus, the branch of mathematics dealing with change. This was a powerful extension of Galileo’s language. Calculus is the Latin word for a small stone, like those used to aid calculations. It is ironic that small stones, or calculi, were the downfall of both Newton and Leibniz. A bladder stone caused Newton’s demise while a kidney stone led to the death of Leibniz.
Calculus gave us a means of understanding the motion of heavenly bodies and of mechanical systems on Earth. Its predictive power is illustrated by Maxwell's equations for electricity and magnetism. Using these equations, Maxwell predicted the existence of radio waves, which were generated and detected experimentally by Heinrich Hertz about 20 years later.
Nature of mathematics
Attitudes of professional mathematicians to the nature of mathematics cover a wide spectrum. Formalists view it as a mechanistic process, deducing theorems from initial assumptions, always following clear logical steps. To the formalist, mathematics is not an abstract representation of reality, but is more like a game with clearly defined rules but no deep underlying meaning.
In contrast, the Platonic view holds that mathematical concepts are eternal and unchanging. The inspiration was Plato’s Theory of Ideas: the everyday world we know is just a shadow of ultimate reality. This in turn was preceded by the Pythagorean view that reality is essentially numerical in nature. “All is Number” was their motto.
A recent book, Our Mathematical Universe, by Swedish cosmologist Max Tegmark, maintains the universe is not just well described by mathematics; it is mathematics.
However, commentators have objected that Tegmark’s arguments depend on a controversial many-worlds interpretation of quantum mechanics and that physical theories must be based on experimental evidence, and not on metaphysical reasoning. We still have much to learn about the ultimate nature of reality.
Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics, University College Dublin. He blogs atthatsmaths.com