(June 6, 2020 at 11:32 am)polymath257 Wrote: It should be pointed out that even math statements like 1+1=2 need to be tested in the real world to determine *when* they apply.
For example, if you pour 1 quart of water and 1 quart of ethanol together, you do NOT get 2 quarts of mixture. You get slightly less. So, in this case, 1+1=2 is NOT a good description of reality.
Another: If you smash 1 proton against 1 other proton, it is quite possible to end up 3 protons, 1 anti-proton, and a number of pions and other particles. The description 1+1=2 is simply not a good descriptor of what happens in this case.
Another: if you take 1 rock and forcefully smash it into another rock, it is quite possible to get 3 or more rocks out at the end. Once again, 1+1=2 is not a good descriptor of what is going on.
And, in fact, those cases where some quantity (like energy) *does* work in a way where addition 'works' consistently are known as 'conservation laws' and are quite important *because* the math works for those cases.
This is a very simplistic case, but the basic idea remains: the application of math to the real world and observations needs to be tested. It is quite possible that the phenomena being studied are not well described by any particular mathematical formalization.
This is why math is a *language* for helping us to understand the world. It alone is not and cannot be definitive about anything, but needs to be tested just like any other scientific issue.
I agree that math is a language, but when it comes to equations like 1 + 1 = 2, we not only expect this to be precisely true in math but also be unconditionally true in the real world as well, even if not perfectly (in a Platonic sense). The examples you provided aren't simply "1 + 1 = 2" examples, but rather "1 + 1 + some other stuff happening = outcome other than 2". The way I see it, a "1 + 1 = 2" example in real life is one where you assume if you have one particular object and you have another object that is identical to that, then (short of other factors involved that may interfere with their interactions or whatever) we must have only two of these objects, not more not less.
