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Comment by VHRanger | Hacker News Reader

I'd say mathematical concepts are "discovered" rather than "invented". Even something like complex numbers.

Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new words, they're invented. They wouldn't be there without us, as they are, for the most part, artifacts of our cognition.

If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?

That's a pretty hard question to answer, and I think impossible to answer definitively (at least unless we met a civilization that did arrive at different concepts). It's kinda like the allegory of the cave; it's hard to envision another way of looking at the world, but that doesn't necessarily mean it's impossible for there to be one.

It’s a good question to think about.

A constant refrain of mine is that our brains have convinced us that they’re universal understanders, but we really don’t know that to be true.

Imagine the difficulty of dealing with aliens who have a different mapping of the physical universe, different than our mathematics, which is both true but literally and physically incomprehensible to us.

Their observations of nature and their ability to predict stuff should be consistent with ours.

They might not use the same mathematical tools or the same physical models, but they should make the same predictions. That is, we might not be able to understand their theory of gravity, but whatever their theory is, it has to be able to make predictions about orbits, black holes, etc.

I don't think we could assume much about the mathematical concepts they use beyond that.

I’d expect other civilizations to have also invented rockets, microwave ovens, radio communications, and more. Does another civilization arriving at the same thing as us mean it isn’t an invention?

Their mathematics will most likely have fundamental differences from ours that have led them to solve problems we haven't noticed. They will be appalled at how we have missed such basic features of the universe. And we will be appalled at some things they haven't noticed.

That's self-evidently untrue. The properties that make a circle would be true regardless of whether a human ever set eyes on a perfect circle. Us identifying those properties is an act of discovery via research. Codifying those mathematical truths into a written notation is the only component of the process that could really be called invention.

No human has ever set eyes on a perfect circle, because it (most likely) doesn't exist in nature. As such, I would argue that a perfect circle is a concept (or a model, if you will) that we've invented to make it easier for us to deal with an imperfect real world. I would not call identifying properties of such a model an act of discovery: one can come up with any model, no matter how far from reality, and use some set of axioms to identify its properties, but none of it will make said model real or fundamental in any way. The best we can hope for is that the model will be useful for making predictions about the real world.

It's not self-evidently untrue, this is an incredibly complex philosophical question with no easy "right answer". There is also a spectrum of positions about this.

The Stanford Encyclopedia of Philosophy is a good resource for reading about this topic: https://plato.stanford.edu/entries/platonism-mathematics/#Ob...

Edit: another good link: https://plato.stanford.edu/entries/nominalism-mathematics/

Well it really depends on how far you push the definition. There are inherent properties that can be discovered, but the way they're calculated and described is purely arbitrary. You need to get the same result of an area of a circle in the end, but how you get there is invented. Far more feasible and evident for complex stuff than the basics of course.

As shown in the video, the depressed quadratic was basically solved by 3 people in 3 different ways, with today's description and definition being different from that too.

But we still chose the labels. We chose what the elevate to the level of a mathematical object. Heck, even the idea of "what is true" is not universal in mathematics. Intuitionist and classical logic have different ideas of what. it means.

I wonder if one can place mathematical concepts on a spectrum from discovery to invention? To me, the pythagorean theorem feels much more like a discovery of a "hidden" eternal truth that was once beyond our grasp. But to me, complex numbers seem more like a notational choice, more akin to an invention. To put it another way, I would expect aliens to have the pythagorean theorem, but perhaps different notation for complex numbers, like representing them with matrices and computing over them with linear algebra. Such a representation would reduce the distinction of the "imaginary axis" which is really not that special.

I may be naive; I haven't worked or thought with complex numbers since completing my basic education.

Definitions are invented, theorems are discovered.

The Pythagorean theorem is a discovery about Euclidean geometry. But in order for that discovery to be meaningful, one must first invent Euclidean geometry, or at least something sufficiently similar to Euclidean geometry.

If you are not familiar, this is part of a very long-running discussion in the philosophy of mathematics: https://plato.stanford.edu/entries/philosophy-mathematics/#F... (For context on the intro to that section, Platonism is, roughly speaking, the idea that mathematical objects truly exist and mathematicians are discovering them)

I wasnt, thanks!

I'd say especially complex numbers. Few things seem more natural once you get used to them and see how they make everything fall into place.

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