(March 21, 2021 at 7:40 pm)Angrboda Wrote:(March 21, 2021 at 7:21 pm)polymath257 Wrote: First, there are several different 'logics'. Aristotle produced the first list of 'laws of logic', but he was rather vague about specifics and it took later thinkers to straighten out some of his mess.
Boole gave a formal that, essentially, encompassed propositional logic, but his system didn't include phrases like 'there exists' and 'for every', which are usually seen as logical connectives.
That leads to quantifier logic. But that doesn't include aspects of equality or mathematics.
Because of the attempts to provide a logical foundation to mathematics that spanned the turn of the last century, a number of different approaches to logic were investigated. For example, intuitionist logic denies the classical 'law of excluded middle', especially as it applies to infinite systems. Other systems explicitly allow certain types of contradiction, but still manage to avoid triviality.
I would claim that the statement 2+2=4 seems obvious cause the number of definitions and assumptions required to prove it are minimal. I would bet that it is far less obvious that 29*31=899 or that a number is divisible by three if and only if the sum of its decimal digits is divisible by three.
In fact, the assumptions required to prove 2+2=4 are few enough that they would NOT be enough to show that 3 and 4 are different numbers. In essence, simply having a notion of 'next' and definitions of 2,4, and + are enough.
It wasn't the triviality of proving 2 + 2 = 4, but rather that as abstracts, the numbers seem to reflect metaphysical truths, that there's something underlying them that's ontic in a way that linguistic conventions aren't.
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.
