Mathematics - a game!

In the spirit of the title of this blog, and as I have too many things I have or want to do, I'll put down here just rough items laying out my line of thought:

1. In order to show that there is no "reality" behind mathematics, i.e., that mathematical objects are not "real" in any sense that gives them existence independent of human thinkers, I argue that mathematics is practically identical to chess: A set of invented rules, whose consequences we are explore.
2. This refutes the argument that solely from the way of talking that we "discover" new things in mathematics, it would follow that these things have an independent existence: We also discover new openings in chess, new moves, and new universal truths (like "two rooks and a king can win against a sole king").
3. Critical thought 1: What about the concepts of "truth" and "proof," which play such an important role in mathematics? I want to argue that "truth of a statement" in math can be handled as "a position can be won" in chess: I.e., as a nontrivial mapping of a complex subset of all possible situations to a two-element set { T, F }. Of course, in chess, there is also a draw, whereas for truth, we typically assume the law of the excluded middle. However, one can either modify chess a little so that every position maps to { W, L }; or one considers non-standard logics in math.
4. Critical thought 2: Math, as opposed to chess, has a usable mapping to many real-world items and problems: Rational  numbers map nicely to lengths, areas and volumes exist in math as well as in real life etc. So there!--math must be something that is "much more real" than chess. I think this is not true: First, the chess rule set and its consequences is a "theoretical construct" like math, which happily maps to real world chess pieces and boards--with similar problems, e.g., that the mapping is not completely two-way (there is no mapping for pieces that straddle the boundaries of squares etc.). Still, math seems so much more "usable in unexpected ways" than the "chess model," which only maps to "real chess."
5. Critical thought 3: Chess, or at least chess theory, seems to be simply a subset of mathematics. So it is not really possible to show anything interesting about mathematics by recurring to chess. Hmm. Interesting. I do not at all think that chess is a subset of mathematics; at most, it overlaps with mathematics ... but the arguing gets tricky here.
6. ...etc.