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Is maths invented or discovered? One of the most interesting questions to investigate with regards to maths Theory of Knowledge (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 below passage is a brief introduction to some of the ideas on this topic I wrote a while back. Hopefully it will inspire some further reading!

## 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.

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

**Maths Invented or Discovered? **

The PBS Ideas Channel has just released a new video which discusses whether maths is invented by humans, or whether it is discovered (ie whether it can be said to really exist). It’s an excellent 10 minute introduction to a pretty complicated topic – and certainly accessible for students:

For those interested in more detail – here are some of the basics (some of this information is simplified from the Stanford Encyclopedia which goes into far more detail). You can also see more discussion of the topic on this site here

**1) Platonism**

The basic philosophical question in maths is ontological – ie concerned with *existence. *The Platonic school (named after Greek philosopher Plato) hold that mathematical objects can themselves be said to exist. Is there a “perfect circle” – in the realm of “ideas” upon which all circles on Earth are simply imitations? Is this circle independent of human thought? Does pi exist outside of human experience – and indeed space and time? The hard Platonists argue that mathematical structures themselves are *physically* real – and indeed that our universe may be a mathematical structure. Some other schools of mathematical philosphy include:

**2) Logicism**

Logicism seeks to reduce all of mathematics to logical thought – if all mathematics is reducible to logic does that mean that mathematics is purely an intellectual exercise? 20th Century efforts by Bertrand Russell and others to reduce mathematics to logical statements have not enjoyed much success.

**3) Intuitionism:**

“According to intuitionism, mathematics is essentially an activity of construction. The natural numbers are mental constructions, the real numbers are mental constructions, proofs and theorems are mental constructions, mathematical meaning is a mental construction… Mathematical constructions are produced by the *ideal* mathematician, i.e., abstraction is made from contingent, physical limitations of the real life mathematician. But even the ideal mathematician remains a finite being. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it.” (Paragraph from Stanford).

Mathematics therefore does not really exist in any physical sense – it is merely a construction of the mind.

**4) Fictionalism:**

“Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters. This position was first articulated in the introductory chapter of (Field 1989), and has in recent years been gaining in popularity.” (Paragraph from Stanford).

This line of thought tries to explain the amazing effectiveness of mathematics in describing the real world in a novel way – by denying that it does! The reality that we think is being described by mathematics is nothing more than fiction – there is an underlying reality which we know nothing about. Think about Nick Bostrom’s Computer Simulation argument – if we were within a computer simulation, then our mathematical laws may very well explain the computer code – but the real reality would be that which existed outside the computer.

Like this topic? Then you might also enjoy:

Is God a Mathematician? – A Michio Kaku video which looks at how mathematics can be used to model the universe.

Simulations -Traffic Jams and Asteroid Impacts – An example of the power of mathematics in modelling the real world

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams. I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers. These all have worked solutions and allow you to focus on specific topics or start general revision. This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.