| > 1+1=2 is only true because it doesn't contradict our experiences. Not quite. In the mathematical sense, 1+1=2 is true because (a) it is a well-formed sentence in a certain formal system and (b) there exists a proof of it in that formal system. The formal system consists of a grammar for well-formed sentences as well as rules for deriving true sentences. The key point here is that the definition of "true" is part of the definition of the formal system. Unlike in philosophy, we have a clear and unambiguous definition of what "true" means, and its evaluation does not depend on properties of our physical universe.[0] So when people say that "mathematical theorems are true in all possible universes", they're well-intentioned but I would argue that they are misleading. The deeper (philosophical) truth is that the (mathematical) truth of mathematical theorems is independent of universes[1]. [0] The fact that our minds are drawn to thinking about the specific type of formal systems that are usually considered to be reasonable foundations for mathematics may well be a consequence of the properties of our physical existence.[2] However, if by some magic the definitions of formal systems studied by alien civilizations in a different physical universe were to be transmitted to us, we would be able to arrive at the same conclusions about those systems as the aliens, and vice versa. There would be no disagreement about what sentences are true in such an alien formal system. [1] Where I use the word "universe" in the physical sense and not in the sense that is found in set theory and its ilk. [2] However, I personally don't think so. I believe (though I cannot prove it) that something like the Church-Turing hypothesis also applies to foundational formal systems at least up to basic set theory (possibly minus the axiom of choice). It is conceivable that some alien civilization in a much stranger universe would consider a certain extension of our set theory as the natural choice for mathematical foundations, but they would recognize our set theory as a subset of what they're studying. |