Is maths invented or discovered?  One of the most interesting questions to investigate with regards to maths ToK is the relationship between maths and reality. Why does maths describe reality? Are the mathematical equations of Newton and Einstein inventions to describe reality, or did they exist prior to their discovery? If equations exist independent of discovery, then where do they exist and in what form? The following discussion on maths and reality is not intended to be a model ToK essay – but simply a resource to help a discussion on this topic. At the end of the piece I have included a number of other interesting ToK maths questions for potential discussion.

Mathematics and Reality

We live in a mathematical universe. Mathematics describes the reality we see, the reality that we can’t, and the reality that we suppose. Mathematical models describe everything from the orbital path of Jupiter’s moons, to the flight of a football through the air, from the spiral pattern of a shell to the evolution of honey bee hives, from the chaotic nature of weather, to the expansion of the universe.

But why should maths describe reality? Why should there be an equation linking energy and mass, or one predicting the decay of a radioactive atom or one even linking three sides of a triangle? We take the amazing predictive powers of mathematics for granted, and yet these questions lead onto one the most fundamental questions of all – is mathematics a human invention, created to understand the universe, or do we simply discover the equations of mathematics, which are themselves woven into the fabric of reality?

The Second Law of Motion which links force, mass and acceleration, drawn up by Sir Isaac Newton in 1687, works just as well on the surface of Mars as it does on Earth. Einstein’s equations explaining the warping of space time by gravity apply in galaxies light years away from our own. Heisenberg’s uncertainty principle, which limits the information we can know simultaneously about a subatomic particle applied as well in the post Big Bang universe of 13.7 billion years ago as it does today. When such mathematical laws are discovered they do not simply describe reality from a human perspective, but a more fundamental, objective reality independent of human observation completely.

Anthropic reasoning

Anthropic reasoning could account for two of the greatest mysteries of modern science – why the universe seems so fine-tuned for life and the “unreasonable effectiveness of mathematics” in describing reality.

The predictive power of mathematics might itself be necessary for the development of any advanced civilisation. If we lived in a universe in which mathematics did not describe reality – i.e. one in which we could not use the predictive mathematical models either explicitly or implicitly then where would mankind currently be?

At the core of mathematical models are an ability to predict the consequences of actions in the natural world. A hunter gatherer on the African savannah is implicitly using a parabolic flight model when throwing a spear, if mathematical models do not describe reality, then such interactions are inherently unpredictable – and the evolutionary premium on higher cognition which has driven human progress would have been significantly diminished. Our civilisation, our progress, our technology is all founded on the mathematical models that allow us to understand and shape the world around us.

Anthropic reasoning requires that the act of conscious questioning itself is taken into account. In other words, it is certain that we would live in a universe both fine-tuned for mathematics and fine-tuned for life because if our universe was not, we would not be an advanced civilisation able to consider the question in the first place.

This reasoning does however require that we simply accept what appear to be the vanishingly small probabilities that such a universe would be created by chance. For example, Martin Rees, in his book, “Just Six Numbers” looks at six mathematical constants which were they to alter even slightly would create a universe which could not support life.

Whilst tossing a coin and getting 20 heads in a row is unbelievably unlikely, if you repeatedly do this millions of times, then such an occurrence becomes practically assured. Therefore using this mathematical logic, any vanishingly small probabilities can be resolved. The universe is the way that we observe it, precisely because it is a universe taken from the set of all universes in which we can observe it.

Mathematics as reality

An even more intriguing possibility is that maths doesn’t merely describe reality – but that maths itself is the reality. When we view a website, what we are actually viewing is the manifestation of the website source code – which provides all the rules that govern how that page looks and acts. The source code does not simply describe the page, but it is what generates the page in the first place – it is the underlying reality that underpins what we observe. Using this same reasoning could explain why our continued search for a Theory of Everything continually discovers new mathematical formulae to explain the universe – because what we are discovering is part of the universal source code, written in mathematics.

MIT physicist Max Tegemark, describes this view as “radical Platonism.” Plato contended that there exists a perfect circle – in the world of ideas – which every circle drawn on Earth is a mere imitation of. Radical Platonism takes this idea further with the argument that all mathematical structures really exist – in physical space. Therefore there is a mathematical structure isomorphic to our own universe – and that is the universe we live in.

Whilst this may seems rather far fetched, it is worth noting that in quantum mechanics it is difficult to distinguish between mathematical equations and reality. It is already clear that mathematical equations -wave functions – describe reality at the subatomic level. At this level the spatial existence of particles is described not in terms of classical co-ordinates, but in terms of a probability density function. What is still not clear after decades of debate is whether this wave function merely describes reality (e.g. the Copenhagen interpretation), or if this wave function itself is what really exists (e.g. the Many Worlds interpretation). The latter interpretation would necessitate that at its fundamental level mathematical equations are indeed reality.

It is clear that there is a remarkable relationship between mathematics and reality, indeed this relationship is one of the most fundamental mystery in science. We live in a mathematical universe. Whether that is because of nothing more than a statistical fluke, or because of the necessary condition that advanced civilisations require mathematical models or because the universe itself is a mathematical structure is still a long way from being resolved. But simply asking the question, “Why these equations and not others?” takes us on a fantastic journey to the very bounds of human imagination.

Other Maths ToK questions for discussion

Mathematics and the world
(1) We can use mathematics successfully to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?
(2) Why is it that some mathematicians and students of mathematics feel that mathematics is in some sense “already there” to be discovered?
(3) In the light of the questions above, is mathematics invented or discovered?
(4) Mathematicians marvel at some of the deep connections between disparate parts of their subject. Is this evidence for a simple underlying mathematical reality?
(5) Some educational systems make a distinction between pure mathematics and applied mathematics. Does this reflect a fundamental difference in approach to mathematical knowledge?

Mathematics and knowledge claims
(1) What do mathematicians mean by mathematical proof, and how does it differ from good reasons in other areas of knowledge?
(2) Are all mathematical statements either true or false?
(3) Can a mathematical statement be true before it has been proven?
(4) It has been argued that we come to know the number 3 through examples such as three oranges or three cups. Does this support the independent existence of the number 3 and, by extension, numbers in general? If so, what of numbers such as 0, -1, i (the square root of -1) and a trillion? If not, in what sense do numbers exist?
(5) In the light of the question above, why might it be said that mathematics makes true claims about non-existent objects?
(6) In what sense might chaos (non-linear dynamical systems) theory suggest a limit to the applicability of mathematics to the real world?

Mathematics and the knower
(1) Can mathematics be characterized as a universal language?
(2) Why is it that mathematics is considered to be of different value in different cultures?
(3) How would you account for the following features that seem to belong particularly to mathematics: some people learn it very easily and outperform their peers by years; some people find it almost impossible to learn, however hard they try; most outstanding mathematicians supposedly achieve their best work before they reach the age of 30?
(4) Are there aspects of mathematics that one can choose whether or not to believe?
(5) How do we choose the axioms underlying mathematics? Is this an act of faith?
(6) Do the terms “beauty” or “elegance” have a role in mathematical thought?
(7) Is there a correlation between mathematical ability and intelligence?
(8) How have technological innovations, such as developments in computing, affected the nature and practice of mathematics?