Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:
Conjecture: Definitions are invented. Theorems are discovered.
The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented.
The physical world has always provided inspiration for mathematicians. Arithmetic, geometry and algebra all emerged from the practical need to understand the world and to manage practical affairs. But mathematicians have extended these subjects far beyond any immediate needs.
Thus, from arithmetic we get number theory which investigates questions unrelated – at least initially – to any physical context, and abstract algebra continues to evolve unconstrained by any practical goals.
A pure mathematician working in, say, number theory is free to invent a new kind of number. Suppose he or she defines a “twoless number” as any number whose standard decimal expression does not contain the numeral 2. So far, this is pure invention, or creation if you like.
But now that the definition is fixed, the properties of these new numbers await discovery, not invention. For example, here is a theorem:
Theorem: The set of twoless numbers is infinite.
Theorem: There exist sequences of consecutive numbers of arbitrary length devoid of twoless numbers.
The proofs of these theorems are trivial.
The definition having been made, the mathematician is free to decide where to look, but not free to choose what to find there. The process of definition is a creative act, or an act of pure invention. But the deduction of the consequences of that definition is an act of discovery. Those consequences are “out there” as a result of the definition and, of course, the pre-existing theory of numbers.
Was calculus invented or discovered? Newton, driven to model the mechanics of the physical world, introduced his fluxions. One might say that he discovered fluxions, or that he discovered that the fluxions that he had invented were useful to his purpose. Leibniz was interested in how mathematical functions vary, and invented techniques to study this question, using them to make discoveries.
Newton and Leibniz were free to choose their definitions and their choices were creative or inventive. The definitions of derivative and integral having been made, the Fundamental Theorem of Calculus that relates them is an automatic, although far from obvious, consequence, awaiting discovery.
Some definitions are rich and productive whereas others are sterile. Twoless numbers are likely to be a barren area of study! The choices of Newton and Leibniz are abundantly fruitful. A wealth of wonderful results follow from them; a whole new world to discover.
The short answer to the question “Invention or discovery?” is “Yes!”
Mathematics is an intricate entanglement or convolution of invention and discovery. Arguments have been advanced in support of the conjecture posed at the start of this article, but they hardly constitute a rigorous proof.
Moreover, if the conjecture is proved, it becomes a theorem containing a self-reference, fertile ground for paradox. If it turns out to be true, it will be a discovery. If not, it is pure invention.