(March 21, 2021 at 10:15 pm)Angrboda Wrote:(March 21, 2021 at 7:47 pm)polymath257 Wrote: And do the other statements I made also 'reflect metaphysical truths'?
Does it 'reflect a metaphysical truth' to say that a certain chess position allows mate in 3 moves? Because, when it comes down to it, mathematical 'truths' are in the same category as that statement.
We can make a rook move like a bishop, or a queen like a pawn. Mathematical objects are not thusly mutable. 2 + 2 = 4 can only be changed by changing the entire system, but then you've just created an isomorphism.
ETA: The rules of chess which describe mating in three moves constitute a system which is entangled with the laws of logic and mathematics, so your question isn't very informative.
Making a rook move like a bishop is violating the rules that were set up, sort of like writing 2+2=5. If you change the rules, you get different results.
We have certain rules in mathematics. They are called the axioms. But, in contrast to what Euclid thought, those axioms are NOT 'intuitively obvious rules of thought'. Other systems of axioms are possible and are studied in math,
For example, there is a system where 1+1=1. In that system, we still have 2+2=4, but 2=1 and 4=1. Furthermore, that system is quite useful when studying logic (it corresponds to 'or'). There are also *many* systems where addition or multiplication are not commutative (the answer may depend on the order in which you do things).
We *choose* particular mathematical systems to help us model things in the universe. The system for chess is not very useful in that regard, but the system involving differential equations is. That doesn't mean that differential calculus is more than our language for describing things.
