(June 6, 2020 at 6:33 pm)Belacqua Wrote:(June 6, 2020 at 11:32 am)polymath257 Wrote: 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.
But there are any number of math questions that can't be tested empirically. Negative numbers, irrational numbers, imaginary numbers, infinite sets, numbers larger than the quantity of particles in the universe.
By declaring a priori that only empirical things are true, and math is always and only a language to talk about that, you are again begging the question. Karl Popper and Roland Penrose disagree with you that numbers are only descriptors of physical states.
Popper's Three Worlds system shows how non-physical entities also have a kind of existence, not testable through empirical means. They exist, but they have no extension or location.
Actually, negative numbers and complex (imaginary) numbers are very often used in our physical descriptions. For example, the interference properties of waves use negative numbers and quantum mechanics is a theory that uses complex numbers for the wave functions.
And, once again, math is not testable because *no* language is testable. it is either useful or not. And, in the case of math, a good part is, in fact, useful, but far from all of it.
You seem to be of the opinion that there is only one possible mathematics or logic. And that is quite far from being the case. In fact, after Godel, we know that *every* axiomatic system that is strong enough to talk about the natural numbers *has* to have undecidable questions. And, every such undecidable question splits math into at least two versions: one where the statement is true and the other where it is false.
I can even describe, if you wish, a number of those undecidable questions.
Popper was quite brilliant at one time, but he seems to have fallen off the back of the truck since he started thinking about consciousness. He is usually regarded as somewhat beyond his years now.
The three worlds mythology doesn't help at all.
