Depending on how philosophical you want to get, it is not so clear what exists and what does not. Neither it is absolutely clear what means to be discovered.

I think that ideas can be discovered. In my opinion, theorems, or more precisely the proof of these theorems, for instance, are discovered. In the sense that they are not an invention, they "emerge" from more fundamental definitions.

You seem to be of the opinion that the only things that can be discovered are those that emerge from the "real world", whatever the real world is. That is, I guess, an acceptable interpretation, but I do not think is the only possibility.

To give a very concrete example, chess rules do not exist in the real world, but knight's tours are discovered.

On the contrary. I think that "real" and "imaginary" numbers are both discovered and neither concept has more reality than the other.

I think that even chess rules have a meaningful "existence" (in a Platonic sense) even if they are in some sense arbitrary. We can't discover "the one true rule system" but we can discover various possible systems and also theorems about them.

Why aren't real and imaginary numbers both invented?

They are necessary for the continuum.

If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.

It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.

> If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.

Then again, have you ever observed a non-computable number as a component of the value of a wave function in nature? Real numbers add a lot of cruft which can never be practically observed (due to not being computable).

One of the main reasons we lose things is that our results have been built upon the real numbers since they were easier to conceptualize and to work with, but it's possible that a lot can be recovered (maybe even everything we care about) using only the computable numbers. For instance, see https://en.wikipedia.org/wiki/Computable_analysis for some results in recovering analysis (limits, differentiation, integration, etc) using the computable numbers.

Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.

Complex numbers have the same cardinality as the reals.

That's true, but irrelevant.

Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.

I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.

I read that we can advance physics the most by research into mathematics. Sooner or later the oddest math finds application in nature, which is odd. See also: https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

Negative numbers do of cause have their application in nature (negative charge etc.). But it is possible to do correct mathematics that have no interpretation in real live. E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.

> E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.

The only trouble here is in the assigned interpretation. You do have -4 students left compared to the initial state but there is no reason to assume the initial state was 0.

If aliens ever draw a square and wonder what it's diagonal is, they'll have to face the idea of irrational numbers.

Many reals are constructible so they are accessible in the sense that it's a theory of phenomena I can act and build upon, and the theory discusses up to an infinite number of cases even if I won't build out that far.

Don't they, though? The path of an orbit in vacuum should follow pi, unless coordinates are discrete (which they might be, I suppose)

You could argue that it is only the model we construct using maths that contains pi.

The real question would be whether all simple, precise models of the concept contain something that can be identified as pi.

Pi can be a unit of one in an alternate numbering system.