Maths: Invented or Discovered?

Are mathematicians inventing mathematics, or are we simply discovering the workings of the universe?

The mathematical language, just like any other language, has been invented in order to express our thoughts in a universal way, thus facilitating communication between scientists. We have invented concepts – numbers, shapes, sets, symbols, and so on – by abstracting them from the world around us. Additionally, axioms of mathematics (statements that are regarded as being self-evidently true) have been established and in doing so we have been able to discover the complex connections among the concepts that we’ve invented, hence creating the theorems of mathematics.

Mathematics has the powerful ability to describe, explain and predict phenomena in the physical world. These phenomena have always been present – they have not been invented. For example, a right hand triangle has always had sides that obey the equation:


The mathematician, Pythagoras merely discovered this fact through observation and calculations, and formulated an equation to express this relationship using accepted mathematical language.

Many of those who argue that mathematics is invented suggest that mathematics is a human construct and without us there would be no mathematics; the reason that mathematics is suited to describing the physical world is because we’ve invented it to do just that. In contrast, the structures of mathematics are intrinsic to nature, and if we did disappear, mathematical truths would still hold.

Although this question is rather philosophical, it has always fascinated me. To me, Mathematics is a mixture of both inventions and discoveries; in order to make discoveries to understand the way our physical world works, we have to invent mathematical language that describes these phenomena.

Let me know what you think! M x



  1. I think of it as any other language. We invent the language to communicate what we see/know about the world and universe. To say mathematics is eternal, well, what it describes is. The very notion of separation between objects is relative. The Rock is only different from the Mountain by observer. If I take a piece of the mountain away by chisel, did I discover a rock? Do I invent the rock? Or am I just describing the reality I see with arbitrary notions? Math is the ultimate arbiter to me. It simply states that there is such a thing as the ability to call some thing “one”, and that another of similar thing is also “one”. The rest is assumptions we hold rigorously. At least, to me 🙂

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  2. Oh I love this question and I still don’t know the answer. But you are right it is amazing how often we “invent” a mathematical construct only to discover further down the road that it describes something in nature thousands of years old. I think for me, the answer is both. Numbers and number sets I believe were discovered/defined but not invented (Dr Hawking’s aptly titled book – God created the integers). Indeed much of mathematics is “invented” in order to describe something that already existed like you say- I am thinking Newton’s derivation of the calculus. I think for the most part you have to be right – we are inventing a language to describe what is already there. This is really analogous to language generally. The work rock doesn’t really mean anything except for the value we as human beings have attached to it so that when you say “rock” you have an image of a rock in your head. In the same way dy/dx is just an issue of notation – the differential is the rate of change, something which has always been in existence.

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