Well I think the point is there is no "new kind of math". There's just types of math we've discovered and what we haven't. No new math is created, just found.
I don't know what you're even trying to argue here.
We're not comparing math to reality (though there's a strong argument to be made that reality has a structure that is mathematical in nature - structural realism didn't die a scientific philosophy just because someone came up with a pithy saying), we're talking about if math is discovered or invented.
Most mathematicians would argue both - math is a language, we have created operations, axioms are proposed based on human creativity, etc., but the actual laws, patterns, etc. are discovered. Pi is going to be pi no matter if you're a human or someone else - we might represent it differently with some other number system or whatever, but that's a matter of representation, not mathematical truth.
It seems that addition (for instance) was "created" long before us.
On the other hand, it seems highly unlikely that a civilization similar to ours could "invent" an essentially different kind of mathematics (or physics, etc.)
Well, I was thinking more along the lines of, say, multiplication and division - you can handle every single equation humanity has ever come up with without either of them. It might be messy and awful and annoying, but I would say in particular these operations are invented more than discovered.
So, more properly phrased, we created some operations.
I think you're saying a pithy saying proves nothing (Voltaire), which is true; sometimes it summarises a line of argument though.
Math is a mental map which coincides with reality in useful ways. Different maps can also be useful. The models we construct are based on arbitrary axioms which we hold to be true. Different axioms could lead to different theories which are just as useful. So it isn't discovered (i.e. mapping directly to reality and waiting to be discovered), it is created.
To pick one example, adding the concept of zero changed our model/map of reality fundamentally without changing reality.
You have a minority view on this argument, though. Scientific and structural realism both reject the idea that math is just a map. You've got company with the instrumentalists and antirealists, but the majority consensus is that math is somewhere between the structure underlying the territory to all the territory.
Zero was already part of the territory. Lack of something is a very normal state in the universe. Once we added it to our understanding of math, we were discovering it, not creating it. Of course people who are scientific or structural realists would agree it didn't change reality - because reality already had it, whether we knew it or not.
If it were merely a creation, there would be no reason for two independent mathematicians to land on the same creation given some directed effort. But of course we do see that. There is an objectivity to mathematics that must be accounted for.
"Where" mathematics exists is in the abstract combinatorical space of an infinite repeating application of logical rules. This space doesn't exist in a substantive sense, but it is accessible/navigable by studying the consequences of logical rules. It is the space of possible structure.
If this space of possible structure is real, but seemingly immaterial, how does our matter brain access it?
I think we create mathematics as thought structure in our mind. We can agree on things when we create the same structures. But this structure did not exist prior to creation.
I don't know what real means; I might call it real depending on one's definition. I definitely wouldn't call it immaterial (though it's not material either). We access it by construction: apply relevant rules and discover their consequences. Two people probing this structure are equally constrained by the requirements of consistency. There is no Benacerraf-style access problem.