The personal expression can get very far too, further than perhaps an advanced alien race may know. Supposing an alien race never discovered category theory, their mathematics could still be more advanced than ours, but perhaps more verbose, longer to write down or otherwise bulky and inelegant. The question of whether elegant mathematics is more advanced I guess becomes tricky to answer. But pedagogically and for practical application, elegance is paramount.

I think the most celebrated mathematicians are such because of their personal expression. And that "style element" also inspires people to go further than before. Grothendieck, who featured here a few weeks ago, is a case in point.

Sad thing is it seems likely that, at least from some vantage point in the universe, we're the aliens that discovered category theory.

> The same for negative numbers, for that matter.

It might be expressed differently, but the concept of addition seems very fundemental, and additive inverses (negative numbers) are a very real part of addition.

> It's silly to think that the sqaure root of negative one is something real to be discovered

If we assume the aliens also have a need for multiplication, combining it with multiplication we get polynomials, and to factor polynomials you need complex numbers.

Besides, if they think of quantum mechanics they'll surely need some way to express what we call the complex numbers too.

Even if they don't think of them as "numbers", they'll absolutely discover analogous concepts and theorems.

"additive inverses (negative numbers) are a very real part of addition"

Inverses are just an abstraction that helps you rearrange operations.

You'd think something like accounting would need negative numbers, but nope, it gets along just fine without them.

They have the concept of "minus", which is just another way of adding with a negative.

Also deficits and surpluses seem like normal accounting terms.

"They have the concept of "minus", which is just another way of adding with a negative."

You have chosen to generalize it into the anstract concept of negative numbers. That doesn't mean that negative numbers are a fundamental truth about nature to be discovered.

Think about it like functional programming. Is that discovering some natural law of computing? Or is it inventing a new way to express computation? Of course it must be the latter, because you can compute anything without the use of functions.

> It's silly to think that the sqaure root of negative one is something real to be discovered.

As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".

Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?

1. Isn't an infinite universe consistent with observations?

2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.

3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.

4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.

It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.

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.

>The same for negative numbers

Are you sure negative numbers are not "real" in the same way natural numbers are?

That would mean particles like electrons with a negative charge are not "real", too. And someone can argue that his negative bank balance is not "real".

As for complex numbers, I fail to see how they are less real than real numbers. Real numbers are points on the real axis, while complex numbers are points in R^2 plane. If we question the reality of real numbers, we can also question the reality of the plane.

Mr Euclid would frown to hear that.

While negatively charged particles are real, the 'negative' charge is a way to distinguish them from a 'positive' charge. It's a naming convention more than it is an actual negative value.

We could easily call them 'black' and 'white' particles and the system itself would stay the same

This is not true. You're losing a crucial property in your translation, namely that the negative numbers are the additive inverse of the positive numbers. In other words, it's essential for electrodynamics that a + (-a) = 0. You could add this requirement to you 'black' and 'white' terminology, but then you've just arrived at negative numbers under a different name.

Don't let the naming confuse you. The name "negative" might be a peculiar, human-specific thing, but the concepts of addition and additive inverses are not.