Is Math an Invention or a Discovery?

A few months ago I was sitting at home watching one of those shows about the universe, you know where they try to condense everything there is to know about our world into a few short episodes? This particular episode was about Isaac Newton and all of his work. In this episode they were discussing how he invented Calculus and how he forever changed the way that we understand our universe. At this point I was pretty intrigued when my boyfriend raised a question I never really gave much thought to. He said to me “Do you believe math is something we discovered or something we invented?” My immediate reaction was that math was a discovery; there is no way that we just made all of this up! After this conversation occurred I started to notice that this was a question I began to think about often, but I never really could come up with a solid answer. So I will raise the same question again, was Math invented or discovered?

Fibonacci Sequence in a sunflower. Image: Ginette, via Flickr.

Fibonacci Sequence in a sunflower. Image:
Ginette, via Flickr.

Let’s start with the discovery side of things; there are many different mathematicians who believe that math was a discovery, such as Plato and Euclid. Mathematical Platonism is “the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.”[1] This philosophical viewpoint is stating that our universe is made up entirely of math. When we begin to understand math we are allowing ourselves to understand more about how the world around us works [2]. Have you ever thought about how math occurs in nature, that there are patterns and sequences all around us? Euclid believed that nature was a physical manifestation of math [3]. Examples of mathematics in nature include honeycombs, wings of insects, shells, and flowers. We also find the opposite of patterns in nature, uniqueness. The theory that no two snowflakes are the same is an example of uniqueness occurring in nature. Another more modern theory that supports the notion that math is a discovery is the mathematical universe hypothesis, which was proposed by a cosmologist Max Tegmark. This theory states, “Our external physical reality is a mathematical structure.”[4] Basically he is saying that math is not necessarily used to describe our universe, but rather our universe is one mathematical object. I think this theory is very intriguing and would make perfect sense. It would explain why math can be applied to everything that we know.

On the other side we have the belief that math is an invention. The most common theory is that math is a completely human construct, which we made up in order to help us have a better understanding of the world around us. This theory is called the intuitionist theory. The theory is a rejection to Mathematical Platonism and states that “The truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition.”[5] Opposing the mathematical universe hypothesis is Gödel’s first incompleteness theorem. His theorem states that any theory that it has axioms can’t be consistent and complete at the same time. [6] This theory would show that math itself is like one giant loop. Every time we solve one problem based on assumptions we gain another problem that we must now base on assumptions we made from the last problem. This cycle will continue to repeat itself over and over and is inexhaustible.

Another common observation about math is how we actually carry out the process. If math were a discovery would we always have the same method for each problem. As shown in class the Egyptians had a completely different way to multiply that can be more effective than our current system of multiplication because it involves less memorization. Are our different methods for the same math problem enough to show that math is an invention? Or is it enough that we can get to the same solution, so the process isn’t as important? There is even the possibility that there are more discoveries to be made which could end our need for different methods to get to the same solution. There could be a missing link in our chain that we have to work around in order to get the solutions we need, but if we found that missing link we would only need one method to solve our mathematical problems.

In my own opinion the recurring theme of mathematics in nature is evidence enough for me to believe that math is a discovery and not an invention. With that said there are compelling arguments on both sides and it may take us years, if ever, to really prove whether or not math is a discovery or an invention

[1] http://plato.stanford.edu/entries/platonism-mathematics/

[2] http://science.howstuffworks.com/math-concepts/math4.htm

[3] https://www.youtube.com/watch?v=X_xR5Kes4Rs

[4] http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

[5] http://en.wikipedia.org/wiki/Intuitionism

[6] http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

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