The Extent of Math Discovery and Invention in Ancient Greece

Maths is probably best defined by Richard Feynman as the process of looking for patterns [1]. These patterns can be numbers and shapes as well as the relationships between them and has lead scholars to consider the very nature of maths itself. Is it invented or discovered? Whilst “invented” implies that maths is a tool designed by humans to understand the universe, the latter term “discovered” implies that maths is an absolute truth, the primary cause that dictates everything that follows on from it. This essay will attempt to untangle some of the issues that are involved in arriving at some kind of answer to this question, with a specific focus on the ways of knowing (WOKS). In order to do so we will look quite closely at a a few examples of how the maths is formulated and the scope over which it operates. If maths is discovered it suggests an uncovering of a truth that exists beyond the human experience; whilst the idea that maths is invented points to a subjective truth. This question of truth in maths is directly related to the invented vs discovered paradigm.

To what extent is reason the primary WOK in establishing maths? It can be argued strongly that reason is by far the most important WOK in maths. However, in order for reason to work, we may have to embrace Platonism in full. Plato held that numbers are as real as the things they represent and the simple fact that they can be imagined means that they exist and this is the fundamental axiom or predicate of all of the branches of pure maths. Once the existence of numbers is established, a formalistic or logistic approach is required for further development which is essentially the position that Aristotle would have taken. Provided the axioms are solid, all subsequent theories and proofs can be formulated and are the natural conclusion to a set of logical deductions and inductions. [5].

A good example of the application of reason is in the development of Pythagoras theorem for example. When considering a right angled triangle, the sum of the squares of the sides are equal to the square of the hypotenuse. [4] The original development of this theorem required logical thought beyond the existence of simple numbers and works every single time everywhere it has been tested and used. It has been proven countless times all over the world since ancient Greece and there is no reason to think it will not work anywhere in the universe. We could very seriously consider that Pythagoras theorem has been discovered, through reason, but based on platonism such is it’s accuracy and utility.

However, reason alone it would seem, is not sufficient and that we must use a strong dose of imagination in order to drive things forwards. This is illustrated by probably one of the most stunning examples of development in maths by Isaac Newton and his writing of the book “Philosophiæ Naturalis Principia Mathematica” [3] Newton asked the question; “does the moon fall?”, and in order to answer that question, he wrote calculus. This is undoubtedly one of the greatest academic achievements of all time, both in science as well as in maths. Once Newton had established the initial axioms, it can be argued that all subsequent relationships were not just deduced by pure logical reasoning, but by an incredible imagination as well. This formalistic or logistic approach is essentially to the process, but with a strong dose of creative imagination added to the mix. Provided the axioms are solid, all subsequent theories and proofs can be formulated and are the natural conclusion to a set of logical deductions and inductions, provided the creator has sufficient imagination to envisage the route through. [5]. Newton clearly had tremendous imagination in his development of calculus, but it was the combination with rigorous logical reasoning that led to calculus. The combination of reason with imagination is an extremely powerful tool it would seem. At the time of Newton however, the consensus in the scientific and mathematical communities would all converge on Calculus as an invented tool. A very accurate and useful tool but a tool nonetheless. It would take Albert Einstein to change all that.

Albert Einstein was probably the first famous scientist to take the “discovered” viewpoint to an extreme. He postulated that the mathematics of calculus was more than just a useful tool and decided to apply it to a further exploration of gravity, assuming that maths was the underlying property that was driving everything. The final result was general and special relativity. [2] The significance of Einstein's results cannot be understated even if they were somewhat controversial at the time, with the shocking conclusion that time itself is not constant and that the reason it is not constant is because space is curved. Calculus describes curves. Perfectly. Space is curved. Perfectly. It is impossible not to feel that something truly fundamental is at work here and strongly points to this particular piece of maths as being discovered. During the last 100 years there have been a disproportionate amount of scientific experiments conducted that have aimed squarely at falsifying Einstein's theories and every single experiment has only provided more evidence, and to an increasing degree of accuracy that he was right. The most recent of which being the discovery of gravitational waves in 2015. [6] Space is curved. Space itself, Space-time to be more specific is calculus and is probably the strongest indication we have that maths is indeed discovered.

An interesting perspective here then, is the reliability of pythagoras which requires flat surfaces. If space is indeed curved, is pythagoras reliable? Perhaps Pythagoras is not a discovery if we consider curved space, Pythagoras breaks down instantly and is hence an invention that only works on a flat plane. Perhaps Pythagoras has always been considered as a useful invention. However, Einstein’s discovery led to one of the most fundamental paradigm shifts in the history of the physical sciences. Before Einstein maths was considered as nothing more than a useful tool in science but after his formulation of special and general relativity, maths became much more than that with maths being central to the development of new theories. The fact that maths predicted something was cause to take the theories seriously. Faith in maths as being discovered grew exponentially and was applied to fields in physics with enormous success, particularly quantum mechanics.

The platonic objects in maths are real simply because they can be imagined. We cannot see them, feel them or use any of our sensory perceptions to interact with them although we can of course use maths to represent real objects. In this sense maths requires faith both in the existence of the numbers and also in mathematical systems for them to work. We can imagine an infinite number of pythagorean triangles all floating around in an abstract existentialism; faith being the required element for them to be real for example. Faith is the realm of mathematicians who would call themselves the intuitionists and to this group we can add theoretical physicists. [5] There lies within an inherent faith in both the predicates and the outcomes. Whilst a mathematician can test the proof a large number of times for a large number of possibilities, we can never actually reach infinity, and hence require faith in the maths leading up to that singularity.

But blind faith in the mathematical process is not the answer either. The maths in Quantum mechanics predicts an infinite number of parallel Universes for example and as far as we know, these do not actually exist. There is no evidence. Schrodinger's postulates state that particles hold all simultaneous states of the different properties until they are observed and obliged to take sides, perfectly metaphorized in the Schrodinger's cat analoge where the cat is both dead and alive at the same time. Clearly that is not possible. Likewise string theory predicts multiple dimensions, none of which have ever been observed so it would certainly appear that without subjecting the findings of these theories, based in pure maths, to the rigours of scientific falsification we cannot fully trust their reliability as absolute truths that exist beyond the purely abstract platonic world. Faith alone in pure maths can lead us to some very doubtful conclusions indeed. I can also attest to this difficulty in maths with my own research into quantum mechanics. The formulae that we developed to describe the phenomena we were observing were at best, only partially successful. [7]

An interesting perspective on faith in the reality of maths is that this ultimately appears very similar to theological predicates concerning the existence of god. Assuming numbers exist in their own right, then proofs can be established, following rigorous logic in the pure maths world, not unlike the set of theological constructs that can be built by a non dissimilar logic in the creation of religious thought systems based on the assumption of the existence of god. It seems to me that both require a large element of faith in establishing the primary cause. Perhaps a religious person would be more inclined to adopt a faith based position in maths if faith forms a large part of their religious leanings.

Conclusion

So, is maths discovered or invented? It seems to me, that it doesn’t actually matter in order to be able to use and develop maths. Whether maths is invented or discovered is not important in terms of its use. In the case of it being invented, as long as the answers are solid, then they can be used in exactly the same way as if it were discovered. It makes no difference to us. It seems that the answer to this question lies more in the realm of the scientific method than in the mathematical one. Once the axioms are accepted and rigorous logic is applied, a host of outcomes are indeed discovered but the relevance of these discoveries has to be contrasted with empirical evidence that can only be found through the scientific method. Whilst pure maths can undoubtedly lead to greater discoveries of more pure maths the question as to whether this represents our real, physical universe still remain. Some maths would indeed appear to be discovered (calculus - describes space-time) and the predictions have very real consequences, but other maths would appear to be invented (Pythagoras - describes right angled triangles only on flat planes), and only have limited scope, allowing us to use the maths successfully but only under some circumstances. There are clear limitations to the meaning of maths if science is applied to the process and there is certainly a case that can be argued that at least some maths is discovered whilst others is invented.