Mathematics : A Discovery or Invention ?

The question of discovery vs. invention of mathematics doesn't make too much sense. An invention is the discovery of a possibility. Likewise a discovery often results from an invention. Thus the invention of the telescope leads to the discovery of the moons of Jupiter. The two notions are not clearly separated, especially if the discovered possibility does not take material form, as in mathematics.

In mathematics it often happens that the same thing is invented/discovered by different people in almost identical detail. G.H. Hardy recognized the genius of Ramanujan partly because some of his extraordinary and complex formulas had also been discovered by other people.

The fact that the same complicated piece of mathematics is re-invented by different people suggests that mathematics is discovered in an even stronger sense than a mere possibility. The real mystery is why and how this happens. In other words, why is the the realm of mathematical possibilities so constrained?

Mathematics is a complex activity that humans engage in. Clearly many aspects of that are inevitably socially constructed. It is an empirical fact that the content of mathematical knowledge is often independent of the cultural context it occurs in. We might have created math for some purpose, but mathematical knowledge appears to be independent of our access to it.

An advanced alien species might have an entirely different language to describe mathematical knowledge. Our theorems may be obvious trivialities to them, and their theorems incomprehensibly complex to us, but they surely would recognize the Fibonacci numbers.

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