Is Mathematics Invented or Discovered?

I thought it fitting to start with such a timeless quote by Galileo,

 “Mathematics is the language in which God wrote the Universe”

He was the first to really put forward the idea that Mathematics was not just a helpful tool to explain the Universe; but the fundamental mechanism by which it ticks. In some ways, he cemented the idea that Mathematics was something one discovered,  even though he might not have explicitly said it. Today we shall challenge this view, and seek to answer some rather fundamental questions in subsequent posts with regards to this statement, such as:

  • Why is Mathematics so effective at explaining the Universe?
  • Is Mathematics truly embedded in the Universe?

and finally…

  • Is Mathematics Discovered or Invented?

Now there is certainly no guarantee that this post will answer this question fully and concretely. Rather – it shall serve as a basis by which one thinks about such problems in a way not previously thought, or possibly as a way to learn more about Mathematics in a way not previously thought. Regardless to this, the question – or The Question as referred to by mathematician Barry Mazur, extends beyond the mathematics classrooms and falls upon the realm of philosophical thought. Anyone, of all ages and backgrounds shall find some interest in The Question, certainly at least for a moment. The answer to which seemingly jumping from one side to another of the debate without certitude as to what the answer to such a question might be.

“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?” – Albert Einstein

It is imperative that before one takes a stance on this debate, one must know what each side is proposing. By summarizing I am in reality simplifying many many books, hours of discussion and years of thought into a single neat paragraph.

Discovery:-

Mathematics is the underlying language of reality. Like physicists probing the world around them, mathematicians intend to unveil the truth of reality through logic and mathematical manipulation.

Invention:-

Mathematics is a human endeavor and an artifact of human civilization and culture. It is no less special than one of the many axiomatic systems it could have been based on. Given different axioms, Mathematics would still exist and function just as it did, just studying different things.

You might have encountered a new word not previously mentioned before, Axioms. These sound rather complicated but are nothing but rules by which one does Mathematics. For example, the ability to add numbers in the way that we usually add numbers is an axiom, that is, we can’t add anything that isn’t a number. At least, then we wouldn’t be calling it addition in the ‘numbers sense’.

The biggest problem one has to come to terms with when standing on the opposite side of the Discovery stance, is the fact that it feels so natural and so obviously true. Mario Livio greatly explains this phenomenon in his quote:

“Most working Mathematicians are Platonists at heart. They think that mathematics has this existence out there and we are merely discovering the truths of Mathematics.” -Mario Livio

But – being a skeptic at heart, I always caution myself whenever arguments or stances to debates seem ‘the right choice’ either because it feels natural, or because everyone seems to agree that it must be so. After all, ‘right’ is a very subjective term. Were we all not to question our own beliefs and ideas, we would be caught in an ‘idea avalanche’ of sorts where a single idea dominates all. For no reason other than that’s what’s being said in people with an authority or by a group of people en masse.

The reason why I decided to think that Mathematics is invented, is not simply because it was the only side that was against the idea that it was discovered, but because it made more logical sense for it to be so.  I might expand on this in a later post more technically what made me think about such a problem, but I’ll only flesh out the justification for it in this post.


There is no doubt that there is this indescribable feeling of elation and excitement when one gets to solve a mathematical problem. Depending on how long you’ve been working on the problem, you get this feeling of accomplishment like you do when solving a puzzle that you’ve been working hard at for hours. Certain results that seem to have nothing to do with another surprisingly are related, terms in equations cancel perfectly as if they’re meant to be and things just feel right.

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But those nasty axioms always end up ruining the fun…

The axiomatic system by which a person can do Mathematics is not a concrete thing that exists in nature. It was created by a human being, roughly 2000 years ago in Ancient Greece. His name was Euclid and the reason why he is remembered today is because he wrote the first and most influential textbook of all time – The Elements.

euclid

In this text, he wrote down all the results that the Ancient Greeks knew about mathematics, along with a proof for each statement. The only problem is…algebra wasn’t really a thing back then. So the entire book proves these statements with Geometry. Lots and lots of Geometry. Obviously, to make what he was writing justified and to also make things understandable for posterity, Euclid decided to state in the beginning of his book what are the rules of the game. One such rule is the fact that you can draw a line between any 2 points or that you can draw circles. These are the axioms.

Axioms are now a staple in modern Mathematics with every field having the rules for their respective ‘games’ as it were. Why are some axioms chosen over others? The answer is mixed.

Culture, Interest and Simplicity

These are several factors which contribute to the inclusion of an axiom or not, but it goes without saying that a particular outcome of Mathematics that is affected by ‘culture’ is already an argument against a single, pure, unalterable Mathematics. A point for the Invention argument.

If at this point in time you still don’t see axioms as being inherently man-made but rather  gifted down upon mankind by some higher being, in the next part we shall be having a closer look at the axioms of Euclid, the father of geometry. How all but one of these axioms seemed intuitive…and how its exclusion gave rise to strange and new worlds.

TO BE CONTINUED…


Resources:

Mathematician Barry Mazur on The Question

Perspective on The Question by leading Mathematicians and Physicists 


 

Follow this blog if you would like to read similar content, as well as Philosophy, Statistics and more Mathematics  :)!

Thanks for reading!

-Sinthorel

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