# How was Mathematics developed and created

Mathematics is profoundly effective at describing the world around us. It is the language physicists use to formulate theories about our universe, neurologists use to model our brain, and economists use to model the stock market. This naturally leads us to ask the question why mathematics is so effective at describing our universe – a question asked many times before by a number of great minds.

The universe appears to have been designed by a pure mathematician.

– James Jean (1877 -1946)

How is it possible that mathematics, a product of human thought that is independent of experience fits so excellently to objects of reality?

– Albert Einstein (1874 -1955)

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand, nor deserve.

– Eugene Wigner (1902 -1995)

To understand these peoples fascination with this enigma, we have to understand the origins of mathematical reasoning and their relationship with logic, before we start speculating the relationship between mathematics and nature itself. I would like to note at this point that the (very brief) history of scientists and philosophers I will be providing is tailored to the context of this article, and that I knowingly, and purposefully omit a lot of information central to understanding these characters’ work, and role in history.

Fermat’s last theorem; a mathematical statement which has been known to be true since it had been conjectured in 1637. It simply says that the equation:
can not be satisfied for any three values, provided the value of n is greater than 2. However, finding a proof for this theorem turned out to be an immense challenge, one which was only overcome in 1994 by mathematician Andrew Wiles – over 350 years after it had been formulated! So at which point can it be said that the theorem was a truth? In 1937? In 1994? or was is true all along? The Platonists answer would be that is was true all the time – and furthermore, that this objective truth resides in the realm mentioned above. To the Platonists this reality is as valid as the universe around us. What is important here is that this is the first time the explicit belief is expressed that mathematics is a fundamental ingredient of this universe, which exists outside of the human experience. This is a very profound insight, considering the period of human history this originates from – this is before physics had made so many ground breaking discoveries, based on mathematical modelling, and before humans discovered the universal applicability of mathematics. Since antiquity mathematics has been associated with the perfect, divine – but nevertheless as real as anything else in this universe.

The great philosopher and mathematician Rene Descartes (1596 – 1650) invented/discovered what we now know as the Cartesian system of coordinates – which intimately linked geometry and algebra to be two sides of the same coin. This allowed mathematicians the algebraically analyse the world around them (which, naturally, is full of geometrical shapes), which lead to immense breakthroughs which would otherwise not have been possible. Even though Isaac Newton despised Descartes (so much, in fact, that he sometimes refused to write his name!), and tried formulating his laws of motion without the Cartesian system – he was not able to, having to eventually concede that the Cartesian coordinate system was the simplest, most logical way to map the physical space. However this was still a time of alchemy and mysticism, and thus the power of mathematics at explaining the world around us was still seen as being inevitably linked with the divine.

This ‘divine’ link is part of the reason Euclidean geometry (as derived from Euclids axioms) was largely unquestioned for more than a thousand years, since it was defined around 300BC by Euclid of Alexandria. Euclidean geometry is based on 10 postulates (statements which were taken to be indisputably true) from which he sought to prove a large number of geometrical propositions on the basis of logical deductions. The first four postulates were very easy to understand, for instance the first one read:

Between any two points a straight line maybe be drawn”.

In contrast, the fifth axiom was considerably less self–evident and slightly more complicated:

If two lines lying in a plane intersect a third line in such a way that the sum of the internal angles on one side is less than the two right angles, the two lines inevitably will intersect each other if extended sufficiently on that side.

Below is Euclid’s 5 postulates are visually represented.

Whilst nobody doubted its validity, it lacked the compelling simplicity of the other axioms. In fact, I am not convinced that Euclid himself was entirely happy with his fifth postulate; in his book The Elements the proofs for his first 28 postulates do not make use of it. Over the years many had tried to deduce an explicit proof for this postulate without any success, forcing the mathematical community to reconsider this divine status given to Euclidean geometry in the case that it did not hold true. In the Nineteenth century the breakthrough finally occurred, and people realized that by choosing an axiom different from Euclid’s fifth resulted in an entirely different, but equally valid geometry. This was huge. For millennia Euclidean geometry was the solid foundation on which all of mathematics and even nature itself was based.

The first to publish on the topic of this new, non- Euclidean geometry was Russian mathematician Nikolai Lobachevsky (1792 -1856). In his work, Lobachevsky laid out geometrical relationships on a hyperbolic surface – which simply means that instead of calculating relationships on a flat surface, they were calculated on a curved surface called a hyperbole. Different geometries can be seen in Figure 3 below.

However, Lobachevsky’s work went largely unnoticed because he published his findings in a rather obscure journal. Independently, Hungarian mathematician Janos Bolyai (1802 -1860) made the same findings.

I have found things so magnificent that I was astounded…. I have created a different new world out of nothing.

– Janos Bolyai

The existence of non – Euclidean geometries had been anticipated, and worked on by the great Carl Friedrich Gauss (1777 – 1855). Gauss is widely regarded as one of the most influential mathematicians of all time, yet he feared that publishing this radically new geometry would be seen as philosophical heresy by Kantian philosophers of his time- who held firm the belief that geometry is somehow linked to the divine. Gauss sent a letter to Janos’ father about his son’s work, expressing his thoughts:

…Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind for the last thirty or thirty five years. …I have put little on paper, my intention was not to let it be published during my lifetime. On the other hand it was my idea to write down this later so at least it should not perish with me. It is therefore a pleasant surprise that I am spared this trouble…

– Carl Friedrich Gauss.

This was one of many instances in history, in which a breakthrough in mathematics and/or physics had occurred completely independently at the same time. Another fantastic example is that Calculus was invented/discovered by Isaac Newton (1642 – 1726) at the same time as it was being developed, independently, by Gottfried Leibniz (1646 – 1716) in Germany. It seems as though certain breakthroughs in our understanding of the world seem almost inevitable- leading many to believe that we are indeed discovering mathematics – that mathematics is embedded in nature. However, what the advent on non-Euclidean geometry showed to many, is that axioms could apparently be chosen to give rise to many different mathematical geometries. This no longer made mathematics seem as this divine, perfect model nature was based on. Instead, it had become rather arbitrary, and susceptible to human manipulation. Surely this meant that mathematics was being invented, rather than discovered?

For a long time these non–Euclidean geometries were treated as non-physical, amusing curiosities. It wasn’t until Albert Einstein made use of these geometries over one hundred years later, in his theory of general relativity that non–Euclidean geometry was shown to have physical meaning. Besides the obvious fact that non–Euclidean geometry applies on the earths surface (which is round, not flat) it turns out that the fabric of the universe itself has a shape – it is not flat. This is something nobody would have anticipated when Riemann published his works in the 1800s, which would then come in so useful for Einstein about one hundred years later.

This brings me onto one of the most curious aspects of mathematics; the fact that quite often theorems or, as Platonists would say, ‘objective truths’ are discovered in mathematics long before they are made use of by scientists non–Euclidean geometry being only one example of a mathematical ‘toy’ turning out to be useful for describing the physical world over one hundred years after its initial publication. Another fascinating example is Knot Theory. This was developed 1771. in order to support a now disproven theory about the structure of matter. According to it, atoms were tightly knotted tubes of ether – a mysterious substance which has now also been shown to not exist. The variety of chemical elements could therefore be accounted for by the variety of different knots in the ether. This sparked a serious interest in classifying knots – which knots are possible, and how many are there. Even though this aforementioned theory of atoms was quickly shown to be false- mathematicians remained interested in knots for no other reason than curiosity. This lead to a very rich understanding of knots, and a lot of conjectures based on them. Imagine, then, the delight when it was found that knot theory is key to understanding fundamental processes involving the molecules of life – DNA. DNA consists of two strands intertwined heavily- and therefore any empirical understanding how DNA makes copies of itself will make use of Knot theory – a mathematical curiosity from the 1700s. Furthermore, in the 1960s Knot theory found applications in string theory – a very mathematical and abstract branch of physics which postulates tiny strings to be at the center of all matter.

There are countless examples of old mathematical models being put to use to understand the physical world – most of them in quantum mechanics. This relationship between the physical world and mathematics is described by Physicist Mario Livio as the “passive role of mathematics” at describing nature. He distinguishes this from the “active” role which mathematics also plays; which is the fact that physicists can develop mathematical models of a system in order to understand it. These models, in turn, are so successful that it makes us question whether we are inventing, or discovering them. It is the reason Physicist Eugene Wigner published a paper entitled “the unreasonable effectiveness of mathematics in the natural sciences” in which he argues that mathematics is inherent to nature. Other modern physicists such as Max Tegmark go even further and claim that the universe itself is maths, and lays out a good argument for it (which I do not want to discuss here because I think this article is long enough as it is). An interesting point of view is voiced by mathematician Sir Michael Atiyah, in which he takes into account our brains evolutionary predisposition to make sense of the world around it:

If one views the brain in its evolutionary context then the mysterious success of mathematics in the physical sciences is at least partially explained. The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, this is well suited for its purpose.

– Sir Michael Atiyah

However Atiyah does recognise that this explanation does not address the “passive” effectiveness of mathematics discussed above. Even though natural selection could explain why we cope with physical phenomena on the human scale, it could not explain by mathematics successfully deals with all scales – from atoms to galaxies. Richard Hamming (1915 – 1998) believed that this could be explained by the fact that humans select and continuously improve mathematics, to fit a given situation. What some might call an evolution and human selection of mathematical ideas – a large number of ideas are spawned, but only the ones fit for describing the world survive. Even though I do believe there is truth to this – I do not see it sufficient to explain the “unreasonable effectiveness of mathematics in the natural sciences”.