I just made friends with a mathematician, and apparently it is big right now whether mathematics was invented or discovered. Posed like that, i must answer this question with the first option, definitely, but i think the question implies a connection between externality and validity of knowledge, and this connection itself must be questioned.In other words, we’re mixing things up. It is a good question to ask what is external to the subject, what stays beyond the me. It is also a good question which ideas are valid, which knowledge is relevant. But there is no connection between the two and it’s very dangerous to just assume they will always go together.
One example of a knowledge that is in the world but is completely irrelevant: the exact position of each particle in a rain cloud. This information is there, in the world, but no one really cares. And there is absolutely no reason we should, either. You yourself is not 1.9 meters tall, your height is a number with infinite precision, and those extra digits are both external to the subjective (as in “truth”) and completely unimportant to understanding (as in “nonsense”).
It makes no difference to the validity of mathematics whether or not it exists in spite of any mathematicians. Mathematics will not be more valid if it can be proven to be discovered rather than invented.
In fact, i suspect the whole discussion stems from that mathematics looks like it has been invented but some people erroneously assume that it would only be valid if it was discovered, so they want to believe anyway…
Mathematics is almost like a living proof that relevance and externality are not connected: it does not conform to the world, it conforms to its own internal rules, it has to be self-consistent but it doesn’t have to be consistent with the world. I would go as far as to say that it has (or had) to vehemently refuse all obligation to conform to the world.
If the naive being-in-the-world is not enough to distinguish between valid and invalid knowledge, then how do we make this distinction? That is a much more difficult question. It is also not a question that requires a final and definite answer.
[This is dedicated to Oliver, because he is our hero!]