Invented or Discovered?

A philosophical question about math that has been asked since the times of the ancient Greeks (and possibly even before then) is whether mathematics is discovered or is it invented by man. People seem to think it has to be one or the other, but what if it is actually both?

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Math is just a language, and like any other language that uses words to describe something (strings of symbols), math also uses symbols. Written language was developed both independently and simultaneously in ancient times. One person got an idea to use a written symbol to represent a tangible object. Sometimes multiple people got this same idea independently of each other, and other times a person would see this writing, it would spark the idea in their heads, and they would go on to develop their own written language. The same language was not developed by different people, rather each person used different symbols to represent different words. (Guns, Germs and Steel- Jared Diamond Chapter 12) The same can be said for math. Calculus was developed simultaneously, but independently by Isaac Newton and Gottfried Leibniz. Both developed different ways of doing calculus and each way gave the same results. Other times mathematicians have relied on the work of others to further their results.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

The fact that math has been developed independently and yet yielded the same results shows that math is discovered. Math is the language used to describe the natural world and as long as the world exists someone can, at any time, develop a language to describe it. It may not be the same math that we use today (the Babylonians used an arithmetic system very different from our modern one), but it would still yield the same results. Given enough time, one would think, they would eventually be able to build the same skyscrapers and the same rocket ships that we have.

On the other hand, math was invented. We invent the symbols and decide what they represent; we invent the axioms and the particular system that we use. Newton invented infinitesimals in the use of calculus while Leibniz invented his own notation for calculus. The ancient Egyptians invented a different way to calculate the area of a circle than the one we use today. (A History of Mathematics- Uta c. Merzbach and Carl B. Boyer) Math does not exist without someone to invent the symbols we use to describe it.

Many people ask, and for good reason, if this question is even important, and it just may be. What if the concept of zero or negative numbers were never invented? Without these simple concepts would we still be able to build the same skyscrapers and rocket ships? It is possible that someone could have invented a concept similar to these but using different concepts? It is even possible that someone may have invented a way around them so we could avoid them altogether and this new invention could have even lead to a much simplified math system.

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