Ninth Week BeyondResearch Insight, Year 1

Welcome to my ninth week of BeyondResearch Insights! In our last Abstract Analysis session we briefly discussed a very common question about math, which I will try to tackle today: Is mathematics discovered or invented?

One of the arguments of the “discovered” camp is that the same mathematical ideas, for example numbers, have been independently developed by different cultures around the world. This either points to the fact that counting is somehow innate to human nature, which seems unlikely as for instance the Amazonian tribe Pirahã has no notion of it, or that counting exists independently of us and we simply stumbled upon it and then adopted it thanks to its usefulness in everyday life.

Precisely this usefulness and applicability of math in the real world also points towards it being discovered. After all, why do physical models, which are described mathematically of course, predict nature so well? Or as Albert Einstein put it “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Well, we might speculate that laws of nature are inherently mathematical, which is a core property of the universe, which we must discover to understand it.

Such views are often held by Platonists, among which were even some legendary mathematicians like Kurt Gödel. They believe that there exist things that are entirely abstract in nature, for example properties like something being blue or, notably for our discussion, mathematical objects like a triangle (as all real-life triangles are merely imperfect representations of the abstract idea of it). Proponents of Platonism thus argue that math is discovered because it is a concept that exists independently of our idea or knowledge of it in some abstract realm (Plato’s Realm of Ideas or Realm of Forms).

On the other hand, there are also many proponents of math being invented. They are particularly opposed to the argument above. They would claim that math models nature so well because we built it based on our experiences of the world around us and the patterns in it. This “invented” camp is held by different streams of philosophy such as empiricism and formalism, the latter of which includes famous mathematicians like David Hilbert and Georg Cantor.

Personally, I am quite undecided, so I am leaning towards the third popular opinion: math is to an extent both discovered and invented. As I understand it, math has been developed by humans thanks to our creativity and ingenuity (invented), but we have taken considerable inspiration from the natural world and the patterns in it (discovered).

What do you think? Is mathematics discovered or invented?

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