Mathematics, a branch of science centered on the study of numbers, its interactions, and the ways it can be manipulated, is both discovered and invented. Rather, more accurately, the principles upon which math is run is discovered, but the symbols used to interpret it (and lull the author to sleep) are created to help understand the concepts uncovered, although this definition is seen as a compromise for anyone who’s entrenched in either side.
Before going further, however, it is right to start with how mathematics started and how it works. Arithmetic and geometry first arose, developed by Greek thinkers after thousands of years of refining ways to count. Important insights into these disciplines had been made by Pythagoras of Samos (ca. 570BC – ca. 496BC) and his students. Rational and irrational numbers were defined, and geometrical relationships were discovered. But more importantly there was the pioneering Pythagorean insistence on mathematical proof - a procedure based entirely on logical reasoning, by which starting from some postulates, the validity of any mathematical proposition could be unambiguously established (Burkert, 1972). As time went on, the field found greater success in explaining the world and beyond, and its results & contributions are almost beyond reproach. Along the way, the philosophy of mathematics appeared from time to time, pondering how it is able to accurately explain every quantifiable phenomena.
Discovery seems to imply that the thing in question was there beforehand, while invention implies an original concoction (Lessel, 2016). Using these definitions, we can then introduce each argument, first from the perspective of the realist, and then from the anti-realist.
To people like Eugene Wigner and Max Tegmark, mathematics has been there from the beginning, and human thought only ever reached understanding by creating the symbols upon which such innate concepts like gravity, geometry and certain sequences of numbers are understood, and can then be used to decipher greater secrets only once thought of. Mathematical realists, as they are known, see mathematics as the language the universe speaks, useful in more ways than one, and the methodologies developed by human brain power merely translate this language for people to grasp and use and understand.
Others like Immanuel Kant and Ernst Mach, however, dispute those who say mathematics is its own language, and posit that mathematics is a human language made to understand a world only experienced by people, dependent on rules and axioms that can fall apart if proven wrong, or if no intelligent being is alive to experience it. Anti-realists claim that mathematical statements have truth values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. More importantly, anti-realists see the “rules” that realists say are inherent to the world as man-made; we do not directly transcribe data from the universe, but only merely write what we hear using a special form of communication entirely different to the original words.
I think this speaks to other debates about certain dualities like order and chaos; good and evil. Order, to me, is enforced to control chaos, just as much as chaos arises when order pervades. Good responds to evil the way evil retaliates to good. Similarly, something you find is just as important as thinking of a word to call it, and finding out that gasoline burns the way it does helps propel the invention of the internal combustion engine. On the flip side, something invented requires a proof that it works—no cannon would fire if its gunners do not know how a cannon must work to be truly destructive. Mathematics, then, is both discovered and developed, and I think the line between them does not exist—they need each other to work as well as many people think it does.
In other words, mathematics is simply a way to express regularities. The question may be why there are regularities in the universe. But if one assumes that there are, then mathematics is simply a way to express them once we start to see them.
Put another way, if we encounter and begin to recognize a regularity for which no mathematics exists, we invent the mathematics necessary to formalize what we see. It’s not that mathematics is independent of the regularities we want to express. It’s our language for expressing them.
[But I’d rather not get too bogged down by philosophy. It is better to do work, or at least try to do so without falling fast asleep or getting lost.]