Mathematics - an invention or a discovery?

A friend of mine recently posed a question to me - was Mathematics “discovered” or “invented”. It is one of those questions which made me laugh at first because I thought it was a ridiculous question, but after a bit of thought, I realised its profundity (ignoble founders, I get you 😜).

Before I present the answer I gave my friend, let me explicitly define what I mean by “discover” and “invent”. If patterns and rules already exist, as a consequence of the property of the world we live in, then we humans “discover” the patterns. Examples include discovering fire, laws of motion, theory of evolution, etc. If something doesn’t exist and is brought to this world as a result of human imagination and ingenuity, then humans “invented” it. Examples include the steam engine, telegraph, telephone, computer, etc.

So, in this context, was mathematics “discovered” or “invented”? My answer was this: “I feel that math is invented. It is a formal language to describe and “discover” patterns in the world. That is, rules and patterns already exist (laws of motion, theory of evolution, etc.), and mathematics is just one of the tools we invented to find discover these rules”. The answer I provided made sense to my friend and he agreed. However, I realised that I was missing an alternative perspective as the answer sounded obvious to me. I began to search for this other perspective, which led me to several interesting articles and videos.

Let me start by fleshing out the question - did we “invent” mathematics to better describe the existing patterns in the universe or did we simply “discover” mathematics, which is the language of the universe?¹

Eugene Wigner, who received a Nobel Prize in Physics in the ’60s, wrote a famous article which he titled “the unreasonable effectiveness of mathematics in natural sciences”.² In the paper, he argued that there is something mysterious about how mathematics is able to capture the rules of the universe.

Using physics as an example, Wigner argued that a lot of the mathematics that were used to describe physical phenomena were developed by mathematicans years/decades before they were actually used to do so. The mathematicans concieved the math simply because it was interesting, and not with the intent of using it to describe physical laws. Yet, Wigner argued, these tools turned out to be integral to physics. In addition, the accuracy with which these mathematical rules described phenomena was surprising. This is why he phrases the effectiveness of mathematics as unreasonable - it is a “miracle” that these tools/concepts invented by humans can describe the universe so well.²

Wigner effectively argued that mathematics is language of the universe and we simply discovered it.³ Understandably, several scientists/intellectuals were in agreement, including Einstein:

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

On the other hand, in the article “the reasonable ineffectiveness of mathematics”⁴ Derek Abbott cogently argues that mathematics is just a tool invented by humans and that there is nothing mysterious about its effectiveness nor is it actually very effective for real-world scenarios.

His key argument is that as engineers, we understand that these elegant mathematical equations only work for idealised scenarios and do not work at all scales. As an example, he talks about the transistor whose analytical equations, derived in the 1970s for micrometer scale transistors, do not hold for the current nanometer scale transistors we use today. With an increase in compute power, engineers have moved from analytical equations to the use of numerical methods as this captures non-linearities in systems much better. Using these as evidence, he argues that mathematics is incredibly effective only in describing simple, idealised systems, but not for real-world scenarios.

Derek Abbott’s article also illuminated my implicit bias towards mathematics as inventions. Engineers are taught to use interesting mathematics as convienent tools. For example, linear systems are ubiquitous in engineering simply because they is easy to work with and not because they describe everything around us well. Delta functions, step functions, etc, which we use for system identification, are in fact ideal functions that do no exist in the real world. As engineers, we are repeatedly told the extent of limitations of these mathematical tools we use and when they break down. It is no wonder that mathematics as an invention was my default view, as it was a corollary of the outlook I had towards the world.

In the end, searching for an alternative perspective didn’t change my stance, but it did increase my perception of the depth of this question and my understanding of the cause for my stance :nerd_face:

Footnotes

1. This is described beautifully in https://www.ted.com/talks/jeff_dekofsky_is_math_discovered_or_invented ↩

2. A copy of the paper can be accessed from here - https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf ↩ ↩²

3. This is part of the Platonian school of thought, which argues that mathematics has its own existence. A long description can be found here (I haven’t read it completely though). ↩

4. You can read it - https://ieeexplore.ieee.org/document/6600840 ↩