Philosophical question: do these results give insight into the truth “out there” or are they artifacts of mathematical logic? We may be trained to reason from the axioms, using the established rules of logic. Some theorems are always true and can be proven, some are true but not provable, and others may or may not be true depending on our model. In practice, we already know many important results in a field long before anyone comes up with a list of axioms, and the axioms are chosen so that they allow one to deduce the important results. If ZFC didn’t let us prove things we knew to be true, we wouldn’t be using it. If the Peano axioms produced a single theorem that conflicted with established number-theoretic results, the axioms would be changed.
It’s almost a given that in higher math, we can’t have nice things. Much effort goes into coming up with novel concepts to get around the various pathologies: there’s the Weierstrass function and the Peano Curve and the Devil’s Staircase and the Alexander Horned Sphere, not to mention the good old Banach-Tarski paradox. Our attempts to reduce our intuitions about the outside world into a a small number of obvious statements always end with a horde of monsters showing up and exposing our ignorance. It seems that all our maps are destined to have an area labeled “Here be dragons”.
So back to the first question: are these problems of truth a property of the territory, or do they only show up when we try to draw a map?
-- Henri Poincare
Philosophical question: do these results give insight into the truth “out there” or are they artifacts of mathematical logic? We may be trained to reason from the axioms, using the established rules of logic. Some theorems are always true and can be proven, some are true but not provable, and others may or may not be true depending on our model. In practice, we already know many important results in a field long before anyone comes up with a list of axioms, and the axioms are chosen so that they allow one to deduce the important results. If ZFC didn’t let us prove things we knew to be true, we wouldn’t be using it. If the Peano axioms produced a single theorem that conflicted with established number-theoretic results, the axioms would be changed.
It’s almost a given that in higher math, we can’t have nice things. Much effort goes into coming up with novel concepts to get around the various pathologies: there’s the Weierstrass function and the Peano Curve and the Devil’s Staircase and the Alexander Horned Sphere, not to mention the good old Banach-Tarski paradox. Our attempts to reduce our intuitions about the outside world into a a small number of obvious statements always end with a horde of monsters showing up and exposing our ignorance. It seems that all our maps are destined to have an area labeled “Here be dragons”.
So back to the first question: are these problems of truth a property of the territory, or do they only show up when we try to draw a map?