Is Mathematics Invented or Discovered? This Answer Might Surprise You

Is mathematics invented or discovered? I will share an insight about one of the oldest philosophical riddles of mathematics. And the answer might surprise you.

Whichever side you were on, it's good news and bad news at the same time:

1. "Invented" is the winner.
2. "Invented" is probably not what you thought it was.

As a matter of fact, it is nothing new. It is simply a rephrasing of principles you might already have known, either consciously or unconsciously.

The answer to the riddle is simply that the question contains a hidden ambiguity. And as you probably know, if your definition of a symbol is ambiguous (i.e. you use one symbol with two meanings, or one symbol as the alias of another), you might get two results: either a stop in error (contradiction), or an infinite loop (undecidability).

With ambiguous definitions, you cannot process that proposition "Is mathematics invented or discovered", because it would be mathematically undecidable.

Of course, someone will object, that's a a philosophical question, outside of mathematics, so it's a lost cause...

And yet, we might find a somewhat sensible way to reason mathematically about this philosophical question about mathematics.

To solve this issue we need to disambiguate: we need to split a few things in our mind. Here is a rough avenue.

First (and this is implicit in the statement of the question), we must posit a duality between the mind and reality. Let's consider them as two universes, or two sets: the Subjective (internal universe), and Objective (external universe). The Subjective universe contains ideas, the Objective universe contains things. In physics the elements of the Objective universe are called bodies, and we agree (more or less) that they are composed of particles.

In the Objective universe, discovery is the fact of getting to know, consciously, that something is objectively there. It could be an object, or a law. To discover means "to remove the veil from something"

In the Subjective universe, you can generate any idea you want, and this is the act of creation. The Objective (physical) universe does not afford such freedom; it seems to operate under certain stable laws, that cannot changed on a whim. The Law of Gravity is to be obeyed, and Mother Nature makes a better job at exerting general surveillance and ruthlessly enforcing it, than the best secret police force in a totalitarian state. What you can do in defiance, is only two things: to perceive what is there, and to act in an effort to make things go your way, within the Law.

Mathematics is symbol processing, so where does it belong to? To the Subjective universe. Of course the paper, the pen, the computer, the blackboard, printed page or monitor to visualize results are real: they belong to the (Objective set). They are three-dimensional, you can feel them. But the concepts of Mathematics? You cannot touch them, you cannot feel them, you cannot perceive them. The only way you can perceive them, is by creating them first. In your mind. It is both odd and interesting that you can communicate about them to another person; but only to a person who has created them first. In their mind. You use reality to communicate about Mathematics (voice, paper, etc.).

Mathematics uses symbols as its raw material: they are ideas, elements of the Subjective universe, but with a very special, interesting property. Indeed, what is a symbol, such as the letter A, the digit 1, or an emoji, etc.? It is common knowledge that it has both form and meaning. We can state this in the following way: it is a pair of elements, one in the Objective universe (a form), and one in the Subjective universe (an idea). The technical term that embodies this duality is ideogram: literally, an idea (ideo-) materialized as a mark (-gram). It's quite simple: think of these pairs (idea, object) as stitches between two surfaces (the Objective and the Subjective). The idea part of the symbol has a "ghost" in the objective universe, and the physical part of the symbol has a "ghost" in the objective universe. You can conceive the symbol is a stitch between the two parts; or say that the idea and the thing are a stable reflection of each other.

What is reasoning, in a mathematical sense? It is clearly an operation of the Subjective universe. If you look carefully at it, it is transformation of symbols, thanks to general rules that are stable, as well as constraints you chose to apply to your specific problem. It is the rules and constraints that allow reasoning. Reasoning in the mathematical sense, should run somewhat parallel to the evolution of the Objective universe, to be useful. By being useful, we mean solving problems.

Everybody knows, that mathematics is about solving problems. A problem is reflected in the subjective universe; but ultimately is in the objective universe, otherwise it wouldn't be a problem.

If you you use arithmetics to count units of sheep or quantities of grain, it has some use, only to the degree that it allows you to predict, or decide about, the physical (objective) universe. Perhaps, the detour might be very circuitous; but in the end, it will have to have a reflection in the physical universe. Even the most abstract theory, such as topology (the study of relation between objects) is useful to the degree that it solves problems in the objective universe. And solving problems is necessary for survival, in plants, animals and human beings.

A good metaphor of the two-universe idea is: *Mathematics is a superb map-maker (Subjective), but it never walks the terrain (Objective) *. A map gives you the places of trees, houses, streets. But they are symbols for trees, houses, streets. The map has no idea of what they are; it cannot experience them, feel them, know them. Yes, one might argue, but what if you enriched the map? Add pictures, temperatures, humidity, etc. You would have just enriched your symbols, and it's still symbols. You could go to infinity, it would just be symbols, reducible to 0s and 1s. Symbols. Always symbols. That's all it is and will be. Always and forever.

The map makes sense only thanks to the little stitches. The coordinates must point to an actual location in the Objective universe. The temperature measurements must corresponds to particles in motion, the humidity to the presence of water molecules, etc.

You could say that the map is a physical object actually since it needs to be on paper or a screen to be perceived. Or, is it? No it's just little blobs of color (call them dots, or pixels, if you wish). It is you, thanks to your symbolic processing engine of the mind, who are (consciously or unconsciously) parsing those patterns of dots, isolating them into symbols, and arranging them in a data structure. You can delegate that process, partly, to a machine, but that's still the same process.

All right, then what about reasoning? It is the act of observing and, with symbolic reasoning, arriving a new, useful conclusion about reality, to solve problems in order to survive. The subtle thing about reasoning is that it required a mathematical process, i.e. in involved the manipulation of these stitched ideas called "symbols". So in order to arrive at a useful, problem-solving conclusion, you had to make sure that the stitches of your symbols were still attached to their counterparts in the physical universe.

In mathematical terms, you had achieved a first transform from the Objective universe to the Subjective Universe, and then a second transform from the Subjective univers to the Objective one. In the simple example, you had four sheep in one place, and three in another. You did not have to put them all together physically in one place and count them. You could transform the problem in 4 + 3 = 7, then do a transform and no realize that you have seven sheep in the physical universe.

Perhaps you might have understood my point just with the above, but let's add one more concept: convergent evolution, a process that evidently applies to all living beings, including human beings. What is it? Convergent evolution, is the rule that two living beings (tree, animals) even completely unrelated genetically, culturally, etc., attempting to solve a similar problem will eventually tend to similar, optimal solutions.

A clear example is that completely unrelated animals, separated by millions of years (or possibly galaxies) will arrive at similar forms, in they find themselves in similar conditions. An example is carcinogenesis where "crab-like" creatures repeatedly evolve in shallow coastal waters. That's because that body-plan is fairly optimal for those conditions. Another example is the striking similarity of shape between sharks and dolphins; the reason has to do with hydrodynamics (the forces involved in an aquatic environment) and ultimately the geometry of the objective, tridimensional universe.

And geometry is the hint here. Mathematics will converge toward laws of the physical universe (or accurately it will be able to project them on laws of the physical universe), because of the fundamental axiom that it must solve problems of the physical universe. That's convergent evolution, which is simply that the space of optimal solutions is finite, within a stated set of constraints.

Mathematics is a subjective space of possible problem‑solving structures,with problems originating in the objective universe.

That means that mathematics is undoubtedly created; that much is obvious. But is constrained by the stitches of its symboles with the objective universe, which must continue to hold. Hence the rules and constraints that it evolves are led to go into a certain direction, toward certain optimal decisions, which have some relation with the laws of the physical universe.

That explains why mathematics can support experimental sciences by confirming measurements and predicting them. That's why mathematical creations sometimes lead to actual discoveries in the objective universe.

The answer is extremely simple:

Mathematics is created and always created; but not in any way because it carries with it symbols of the object universe. That's why it leads to discoveries in the objective universe.

And that is the process that is defined invention. Invention is not pure creation, and it is not pure discovery.

It comes from Latin in- "toward", and -venire "to come". That's not pure, imaginative creation. That's following a map, to arrive at the destination. And all journeys in Mathematics eventually lead to the objective universe.

The answer is: Mathematics is _both: creation and discovery; and connecting the two with mathematical reasoning is called invention.