Indeed, from the axioms of addition, which are really just based upon common experience.
There are other modes of addition, for example addition modulo 4, in which that wouldn't necessarily be true.
And it all becomes a lot less clear when you ask still very simple questions like,
Does x^n + y^n ever = z^n, for n>2?
Obviously the answer is either yes or no, but whether or not you can prove it with mathematics is a different matter (in this case you actually can determine the answer is no, but there are other problems which you can't).
The world of math is exactly the same. There is only one world of mathematics, and it exists without language. You can use different languages to describe it (decimal, hex, binary, etc...) but they only make sense if they are coherent throughout the whole world of mathematics. You say you can throw random symbols together and they would make a coherent system...it doesn't work like that. I can say "asjdh jkahsldkah jvhkjlhl" and tell you it makes sense in another language, but it doesn't mean it does. Just like you can spout gibberish in language, you can spout gibberish in math. Just like a dog is a dog no matter what the name, adding the idea of one to the idea of one will create the idea of two. Changing the symbols used to represent that just changes how it looks on paper.
I'm afraid this isn't true.
Mathematics is essentially the 'language'. To speak of langauge describing the world of mathematics is not true; mathematics is the language which describes various aspects of our reality. Whatever rules you come up with for what you are allowed to do in mathematics, there will always be some statements which are true, but unprovable. Mathematics is not only a language; it is also a limited language.
I didn't say anything about new systems of maths being 'random'. There are still precisely defined rules; it's just that the rules are different.
For example, one consistent mathematical system is that of Euclidean geometry, which has six (I think) axioms, and then builds upon them. This system, as with all mathematical systems, will be limited. There will be some isolated geometric facts which are true but you can't prove with Euclid's rules. Another consistent system is hyperbolic geometry, which uses different, and mutually exclusive axioms, and will establish new facts within that system which are completely wrong in the other.
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