On Jun 16, 2013, at 1:45 PM, Joe Niederberger <email@example.com> wrote:
> Ultimately, mathematics often strives towards the "proof" as the gold standard. A proof, in the most rigorous sense, is a formal object in a formal system, which is abstract in one sense (if its tokens and primitives are thought to *refer* to something else -- the "abstraction" involved is distilling just those properties of that something else one wants to encode in the formal system.) On the other hand, if the system is thought of as simply a game of symbols and rules with no external referents, its a bit of a misnomer to call it abstract, as it abstracts from *nothing* in that case -- its just a game. The same could be said for algebraic systems, if we really view them as purely formal games. I know in colloquial speech people would ordinarily refer to any symbolic game of this type to be "abstract". But from a different viewpoint, they are really *concrete*, in that the symbols and rules are rigidly fixed.
I am going to try to clarify my view. Abstraction is not a set of rules, it is a process of the mind. It is the process that takes place after the rules have been set. The rules of chess are quite finite and simple but figuring out and organizing the resultant complexity is abstraction. The rules of real numbers (a field) are even more finite than chess, and even simpler, yet the possibilities of abstraction seemingly infinite. The source (referents) of the rules are different, man made versus nature. But the resulting systems are both abstract and unsolved. As far as I am concerned, all systems that have a set of rules and are unsolved must be dealt with through abstraction.
Chess is an interesting example, because technically it is solvable and computers have solved a lot of it, at least the end game. But to a human it is unsolved and always will be because the computational requirements of the "solution" are too great. A human being will always have to deal with the system of chess through abstraction.
I am not sure what you mean by "gold standard" with regard to proofs. I thought the purpose of proofs was to prove that your abstractions follow the rules. You are using a different sense of abstract than I am. You are contrasting it with concrete, which is kind of my point about the problem with visualization. Abstract is a type of thought, not a type of object or rule.