“Mathematics — Discovered or Invented?” Language constraint that created a false “paradox”

There is an age-old “unanswered” question about mathematics:

Was mathematics discovered or invented?

Foreshadowing: I have the answer.

Though it is not considered a paradox, it acts like one, because mathematics is obviously both discovered and invented — yet, one cannot be both.

This question has been discussed at length by some of the brightest minds in the subject. It is considered a deep philosophical question with multiple interpretations and is central to the philosophy of mathematics.

There’s a video clip of mathematician Roger Penrose spending four minutes answering this question in an interview. Another mathematician, Edward Frenkel, spends thirteen minutes on the Lex Fridman podcast discussing this conundrum. There is a five-minute TED-Ed video discussing this.

Yet none are able to reach a definitive conclusion — there is no straightforward answer, nor a workaround, nor an explanation of why this question is so difficult. What surprised me is that even bilingual experts in mathematics and physics could not see the underlying issue.

Yet, to me, the answer is so very obvious.

Neil deGrasse Tyson comes closest to spotting the problem with this topic — a language constraint. But even he did not catch the actual error. Tyson suggests that perhaps neither “discovery” nor “invention” is a suitable word for the topic, and that the language requires a third term to account for it.

So, here's the answer:

There is an error in the question. It is a language constraint — though not a complex one, like a grammatical constraint. It is, rather, a terminology constraint, and it does not lie in the choice between “invented” or “discovered”. The issue lies in the word ‘mathematics’.

We are using one word for two different things!

There are two different mathematics, yet we are speaking of it as one:

• Math-the-Logic
• Math-the-Language

Math-the-Logic exists independently of whether we have discovered it. A great example to Math-the-Logic can be found in Euclid’s axioms.

“Things that are equal to the same thing are also equal to one another.”

This is not an invention. This was discovered. Essentially, Math-the-Logic is what is referred to in philosophy as a priori knowledge. In simple terms, a priori knowledge is knowledge that does not require experience to be known. By contrast, a posteriori knowledge requires experience — for example, common sense in deciding whether to bake, fry, or steam certain foods. For many, the answer may seem obvious. But to those who know nothing about cooking, they simply don't have the common sense for it.

Math-the-Language consists of methods for understanding and interpreting Math-the-Logic. What algebra and Calculus are complex methods invented by humans to understand the existing fundamental logic behind algebra and Calculus. The logic behind algebra and calculus exists as part of the universe and was discovered by the invention of algebra and calculus. If more advance aliens exists, they may have invented their own methods that allows them to understand and interpret the very same logic.

Math-the-Logic and Math-the-Language have always been two distinct entities. We simply never created separate terms for them — at least in the language I speak.

So the question should be:

• Is Math-the-Logic discovered or invented? Answer: Discovered.
• Is Math-the-Language discovered or invented? Answer: Invented.

These are two different things, so there is no contradiction.

One may then point out that once again blur the lines between invention and discovery. Calculus an many other Mathematical methods were invented multiple times by different people independent and without knowledge of each other. Does this mean that calculus were discovered and not invented. Not exactly, since the bow and arrow, pottery, agriculture, blowgun, debatably the wheel, as well as the telephone, were parallel inventions by different people or civilisations, independent and without knowledge of each other, yet they are physical inventions. But if multiple people could invent the same thing independently of each other, something about it must be discovered. And that would be the mechanism that allows the invention to function just like the mathematical logic that allows the mathematical method to function.