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While many people dismiss the relevance of maths beyond high school, mathematics exists in the physical world and has the capacity to help us think more intuitively about the world around us, writes James Franklin. He explains why Australia needs more mathematicians, and why the discipline should be taught more broadly.

‘Another day's gone by and I didn't use algebra once.’

A philosophy of mathematics that connects it strongly with the real world gives a clearer insight into why everyone needs numeracy as well as quantitative and statistical understanding.

It's a statement that's made its way onto t-shirts, perfect for the innumerate who are keen to show off their smugness. It’s also a joke that reflects a broader view of mathematics as just symbol-shuffling by rules, as if mathematics is no more than a heap of equations, formulas and methods used for passing exams.

The view that mathematics consists of just symbols and rules is called nominalism in the philosophy of mathematics, meaning mere names. The philosophy of mathematics is an important subject because getting the right view of what mathematics is really about has consequences for how mathematics is taught, how it is understood, how it inspires or discourages prospective students and how experts in other areas evaluate quantitative information.

The opposite of nominalism is realism, the philosophical theory that mathematics is really about some aspects of the world. Here there is a puzzle: it is easy to say what a science like biology is about (living things and their properties), but it is harder to put a finger on what sort of properties are mathematical properties. What aspect of reality do mathematicians study?

The Sydney school of the philosophy of mathematics has a suggestion. To understand it, consider this thought experiment: imagine the Earth before there were humans to do mathematics and write formulas. There were dinosaurs large and small, trees, volcanoes, flowing rivers and winds. Were there, in that world, any properties of a mathematical nature? That is, were there, among the properties of the real things in that world some that we would recognise as mathematical?

Related: Maths as a gateway to learning

There were many such properties—symmetry, for one. Like most animals, the dinosaurs had approximate bilateral symmetry. The trees and volcanoes had an approximate circular symmetry with random elements—seen from above, they look much the same when rotated around their axis. The same goes for the dinosaurs’ eggs.

Symmetry is a fundamentally mathematical property, and a major branch of pure mathematics—group theory—is devoted to classifying its kinds. When symmetry is realised in physical things, it is often very obvious to perception.

Other mathematical properties that existed in the pre-human world are flow and continuity (of rivers, for example); discrete patterns such as grasses coming in clumps; and ratios (for example, large dinosaurs are more ponderous than small ones because the weight ratio of big to small dinosaurs is much more than their muscle cross-section ratio).

The philosophy of mathematics advanced by the Sydney school takes its cue from examples like these. Its theory is that mathematics is the science of the quantitative and structural properties of things—properties like ratio, symmetry, continuity and pattern. Those properties can exist not just in an abstract world of numbers or sets but in the very world we live in. It is a philosophy focussed on applied mathematics.

A philosophy of mathematics that connects it strongly with the real world gives a clearer insight into why everyone needs numeracy as well as quantitative and statistical understanding. The custom of having the country run by people with humanistic skills and law degrees is potentially a recipe for disaster because interpreting quantitative data is a skill quite unlike constructing a plausible legal or academic case.

A typical concept in statistical interpretation is survivorship bias. Smokers look around at other smokers and think, at least subliminally, ‘All those smokers look healthy enough to me; surely smoking can’t really be as dangerous as they say.’

The problem with that reasoning is that the smokers that are seen are the surviving ones. They are indeed healthy enough but the ones you don’t see are distinctly unwell, or worse. This demonstrates a ‘survivorship bias’ in the evidence: only the survivors are there to be observed. The mathematical concept is an essential tool for understanding the data.

Related: The invention of modern science: maths

The University of Technology in Sydney has recently made a courageous decision that will require all of its first-year students to complete a course on quantitative understanding. There will be some student resistance but it is a much-needed move to create graduates with an adequate breadth of view. Hopefully all Australian law schools will follow by requiring a course in statistical interpretation.

A proper understanding of mathematics can also help us think more intuitively about the world around us. To get to the truth in natural sciences like physics and biology, there is no choice but to get out and observe what happens to be true. Swans can be white or black or blue—there is no way of knowing which they are without observation. Mathematics is not like that. It just needs hard thinking, and the results are true—and must be true—in this world and in all possible worlds.

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A simple example, consider six crosses arranged in two rows of three, one row directly above the other. They can also be thought of as three columns of two:

The two rows of three are the same crosses as the three columns of two that is, 2 × 3 = 3 × 2. You not only notice that 2 × 3 is 3 × 2, you also understand why it must be so.

From the philosophical point of view, this sort of knowledge by intuition in mathematics is a hopeful model of intuitive knowledge in general. Perhaps we can understand ethical truths in the same way.

As in mathematics, we seem to have powerful intuitions that some things are evil. The success of intuition in mathematics in giving us access to real world truths suggests at least that there is nothing in principle wrong with such intuitions as a way of knowing.

Does maths matter? Listen to the full segment on The Philosopher's Zone.

James Franklin is a Professor in the School of Mathematics and Statistics at the University of New South Wales.

The simplest questions often have the most complex answers. The Philosopher's Zone is your guide through the strange thickets of logic, metaphysics and ethics.