cat-dist@mta.ca writes:
My point exactly. The elements don't matter, so why should elementhood be taken as the fundamental relation of all of mathematics? Equality doesn't matter either, but equivalence relations do.
Really? Try telling that to people who produce computer verified correctness proofs for communication protocols. I'm sure they'll be pleased to know that their proofs should undergo a very significant increase in size and obscurity to satisfy what appears to be little more than aesthetic prejudice. Equality matters in some contexts, it doesn't matter in others. In some logics equality is definable rather than explicit, in which case this point doesn't appear to make much sense. Trying to be perscriptive about this---or any other---foundational point seems to me about as useful as (and just as perverse as) perscriptive linguistics. Given the known relationships between e.g. topos theory, higher order logics and various set-theories, is arguing about which is the correct / best / most relevant foundations of mathematics really that different from arguing about whether the Russell naturals or the von Neumann naturals are the correct / best / most relevant natural numbers? [Of course, this is fairly independant from questions such as how to present the natural numbers] Ralph (who is quite pleased to be talking to someone who isn't a dogmatic platonist the-universe-is-a-model-of-ZF type). P.S. any sarcasm isn't intended to be personal.