Anyway, yes, ok, I'll admit that at one point I said something like "you cannot prove 1+1=2", when really what I should have said was "proving 1+1=2 is actually more difficult than simple 'common sense' tells us".
Then we are in agreement. My only problem was with the statement that one cannot prove 1+1=2. It is, indeed, somewhat more difficult than people think.
It IS possible, but I don't think it's the case. Clear now? One can acknowlege something is possible without believing it to be true. Right? It is POSSIBLE, for example, that evolution is wrong. I don't think it is though.
For the record, I don't think the Earth is flat, either, even though I acknowlege the POSSIBILITY.
... Sigmaleph's entire point was to show you why there is absolutely no way for 1+1=2 to be false. It doesn't come from any external observations that are possibly falsifiable. It's because this result comes directly from the definitions of 1, 2, + and =.
And you are basically ignoring it wholesale.
I'm not saying that mathematics are an absolutely certain science, mind you. It's always possible for a mathematical system such as Peano's integers to be self-contradicting, and therefore nonsensical. But the possible "falseness" of 1+1=2 has nothing to do with that of the theory of evolution or the theory of rotundity. 1+1=2 is false if and only if its entire abstract "universe" is invalid.
No, that was not my point. My point was that you can prove that 1+1=2. It remains
possible, in an absolute technical sense, that 1+1=/=2, and not necessarily because Peano Arithmetic is inconsistent, thanks to Cartesian Demon arguments.
That is: On occasion, people think they have proven something and they have not, having made a mistake in some step. Given this, I cannot assign a probability of literally, exactly 0 to the hypothesis that there is a demon out there who confuses me every time I look at Peano's axioms and derive S(0) + S(0) = S(S(0)), while the correct result is S(S(S(0))).
The probability is not zero, but close enough for most purposes. We routinely make statements of fact that do not have probability 0 of being wrong, because the probability is so small it is not worth the effort of bringing up (that includes every statement I've made in this post).
Or, in other words: Is 1+1=3? No. I know this, you know this, and Lighthorseman knows this. Is it possible, in the most literal interpretation of the word, that 1+1=3? Yes. But only in the most literal interpretation of the word "possible", which is generally useless in any discussion. I don't know why Lighthorseman felt the need to bring up epsilon probabilities of some facts being wrong. I don't know if it was justified (it rarely is). But the only point I was trying to make is that you can prove that 1+1=2. Lighthorseman having acknowledged that, my work here is done.
