Is mathematics invented or discovered

There has been a dearth of philosophical threads recently in GD, so I am going to try to correct that :slight_smile:

Summary of the debate inside spoiler tag for those not familiar with it:

Basically the debate is largely based on refuting the other side’s position.
If mathematics is invented then the main retort is why does it work so well? Why is it we can use our man-made mathematics to build bridges and launch spaceships? In fact, we have often predicted phenomena from abstract mathematics.

OTOH if mathematics is discovered, how is it discovered? What did we actually see to make us think of quaternions, or complex numbers, say?

Quite a lot of books have been written about this topic but most just compile many examples, and the arguments are not much more sophisticated than I’ve presented here.
Many mathematicians and physicists quite like the idea of mathematics being “out there” and so go for option 2. In terms of the counter-argument, in as far as any engage with it, they suggest we have some sort of “mathematical intuition” that happened to fall out of evolution because our brains were made in a mathematical universe.[/spoiler]

Here’s my view: it’s invented. It’s better to think of mathematics as a verb than a noun: it’s a toolkit for manipulation information and deriving non-obvious facts from obvious facts. Essentially formalizing and building on our reasoning power, which is obviously very useful.

But it would be trivial to define a branch of mathematics that doesn’t “work” – that gives incorrect predictions about the external universe. The reason our mathematics does not do this is that new mathematics must follow certain rules of basic logic e.g. self-consistency.
So indeed it is a fact about the external universe that it follows fundamental logic (e.g. it’s self-consistent) but that’s a much smaller claim than saying every branch of mathematics actually exists “out there”.

I am aware of the standard retort to this position: that I’m alluding to Logicism and that was soundly defeated in the early-mid 20th century. I have a response, but I’ll put it in a separate post so that the OP is not too long.

I guess I’d say the axioms are invented and the theorems are then discovered.

Mathmatical formulae are merely expressions of natural truths (the state of nature) and are therefore not inventions and cannot be patented.

As with most subjects like this the truth is in the middle and not on either edge.

But where in nature are these “natural truths”? Where is a physical example of a geometrically precise circle, for example? Not just “very very very very close to circular”, but absolutely perfect in its circularity.

Sure, we initially derive these ideal mathematical constructs from the characteristics of actual physical objects, but I think it’s skipping a pretty important step to ignore the difference between their real existence and our perfected abstractions.

(Oh look, here we are back in this cave again. Oh well, at least there’s a nice fire.)

Mathematics is about relationships and proportions. They exist. We discover these relationships, like pi. When we add or subtract, multiply, divide or find a square root, we are again merely discovering truths that exist in nature.

Here: Pi

This argument is prefectly circular. :stuck_out_tongue:

Both those abstractions are a long way from existing physically “in nature”, though. Your “state of nature” is defined so broadly that it’s hard to see how you’re differentiating between “inventing” and “discovering”.

Pi, for example, is an abstract concept of ratio between a (perfect) circle’s circumference and diameter. Nobody was ever, say, walking through the woods and found a pi.

Yes, it’s an abstract universal constant of our universe. And if intelligent lifeforms exist outside of Earth they would have discovered Pi too, but they would not have an understanding of say, President Trump, who physically exists in nature.

In what way does that make it “discovered” rather than “invented”? Language is universal to human beings, for example: does that mean that we “discovered” language rather than “invented” it? You still don’t seem to be defining this “abstract universal constant” in a way that makes any meaningful distinction between discovery and invention.

(And of course, the ratio of a circle’s circumference to its diameter depends on what kind of space you’re in: it can be different in positively or negatively curved space than in Euclidean space, for example. Is non-zero space curvature “discovered” or “invented”?)

Not sure what that has to do with your point, actually. Are you saying that the intelligent aliens couldn’t discover Trump, or couldn’t invent him?

Donald Trump is a physical thing composed of water and flesh, which are composed of molecules which are composed of atoms which are composed of electrons/protons/neutrons which are composed of gluons etc. which are composed of things which can be best described using mathematical relationships. So I don’t see a clear line of division between the abstract and the physical. Maybe there is, though. I would like to be persuaded.

Intelligent Aliens could discover trump because he exists. On the one hand they can’t invent him because he already exists. But maybe they could independently invent him.??

Well, a falling apple is the classical example…

The inside angles of a square are 90 degrees. Does this number “exist” out there as some universal constant?
One difference is this answer is basically self-evidently obvious to humans from how we define squares and radial coordinates, but why should the universe care about what’s obvious to us? And if 90 is a constant then so is an infinity of numbers (I’m not sure which “aleph”)

Let’s say aliens were to come to earth tomorrow and say that in their geometry-based language, the ratio of diameter and circumference is very intuitive, and they don’t see the point of trying to represent it in digits…are they wrong?

A few places.

  1. The trace of a rotation of any polygon in 2D space

  2. The sum of all velocity vectorsfor an object in an elliptic orbit.

  3. Euler’s Formula

Now if you claim that a circle, to be perfect, would require a plane with infinite number of points to represent you are arguing convention geometrically precise circle and not the natural effects of the math.

While I don’t know if this is your point, if it is you are begging the question as it does not change the fact that any polygon’s traced path when rotated in a 2D plane produces a circle.

Rotational invariance in a plane; or the unit circle in the complex plane being related and e^πi + 1 = 0 will arise from “natural math”

I missed the edit window

*arguing convention geometrically precise
should be

arguing for a definition that is purely convention

To repeat what I posted on the previous closed thread:

“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?”
  – Einstein

The classic paper on the subject is The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner.

A quote and a link…what’s the actual argument?
If it’s just that maths is incredibly useful in the real world…I think everyone knows that, and I mentioned it in the OP.

But why is mathematics incredibly useful in the real world? This is the whole essence of the question in the OP.

Wigner laid out many of the basic arguments and possibilities, and his paper is worth reading.

Either -
Mathematics exists as an external attribute of the universe, independent of the human mind - in which case it is discovered, not invented.

Or -
The human mind is so fundamentally a part of rest of the universe that our pure creative thought can predict how the universe behaves. As Einstein clearly put it,

“Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.”

In this case mathematics is invented, but because the mind is not a separate entity from the universe, the creations of the mind can be incredibly useful in understanding external phenomena.

This question has been debated in detail and at length for decades by top scientists and mathematicians, without any consensus emerging.

I’d say neither, rather that it is defined, refined and formalised.

A bit like how colours aren’t really invented. They are simply the result of different wavelengths of electromagnetic radiation and so every possible colour has the potential to exist. Scientific discovery allows us to clearly define, predict and produce them.

I think mathematical truths do exist, but they do not exist “in nature.”

I think some people are reluctant to saying that mathematics is discovered because of their philosophical commitment to materialism: the belief that only the material world can be said to exist or be discoverable.