Is Math Invented Or Discovered: a Long-lasting Rhetorical Dilemma

Is math invented or discovered? This apparently simple question I want shortly analyse in this essay. Such a question has created a dispute between mathematiciens since mathematics existed. Some say that mathematics is totally invented, some others claim that it is mostly discovered while a last group says that it’s a mixture of both.

These are the three different types of view on this question. The two first ones have been associated with schools of thought called Formalism and Platonism respectively. The third school regroups ideas of both Formalism and Platonism, as it mixes them somehow. The Platonism side was initiated by Plato and his famous statement that says that, “all mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them”.  He believed that we humans discovered math and he only considered it as a spiritual way to get to the knowledge that already existed. On the Formalism side, Einstein and many other mathematicians like David Hilbert or Georg Cantor believed that mathematics was an invented set of tools. Famous philosophers like Aristotle, claimed that, “mathematics did not exist as an independent entity. Mathematics is simply the outcome of logical operations based on syllogisms, and the search starts with axioms”. Axioms are statements that you assume to be true without having actual proof for them, for instance, if a=b and b=c then a must equal c. This example for Einstein is completely invented as there is no actual proof of it: it couldn’t have been discovered.

The last school believes in mathematics being both invented and discovered. A German professor of mathematics in the 19th century stated: “God created the natural numbers, all else is the work of man”. This can be seen for example through integers. Positive integers have always been there, they have a meaning in nature. On the other side negative integers have only been an invention of humans, they have no actual purpose in nature. Most mathematicians argue that numbers may exist physically, but mathematical statements don’t. For them invention and discovery both play a crucial role.

As visible in the short analysis above, particularly bright people tried to answer the question of whether mathematics was invented or discovered. As it is very difficult to state with 100% certainty that it is one or the other, It’s highly likely that this debate will continue. Often, in these kinds of situations, the truth lies somewhere in between. As a consequence I personally would support the third school of thinking.