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Earlier quoted context omitted.
It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.
Don't they, though? The path of an orbit in vacuum should follow pi, unless coordinates are discrete (which they might be, I suppose)
The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…
The etymology of invent has the terms 'contrived' and 'discover' baked in. if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover, we find that it is rooted in the ability to 'compare' and 'imagine'. From this we can then formulate the opening statement.
We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.
But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.
Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.
In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.
But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.
So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.
The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…
It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.
The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…
It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.
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.
I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe th…
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe th…
It impacts whether mathematics should be patentable, since patents generally covers inventions, but not discoveries.
The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discover…
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".