Re: What databases have taught me
Date: Sat, 01 Jul 2006 19:43:52 GMT
Keith H Duggar wrote:
> Ok, let me see if I can summarize, as it appears to me, the
> philosophical difference between your, my (and Marshall?),
> and Dmitry's viewpoints.
I disagree that the differences are philosophical.
> 1) All of us agree that there is an unlimited number of
> consistent operations O that can be defined on a set S.
I am not sure exactly what you mean by 'consistent'. What would make 'substring' consistent or inconsistent as an operation on an integer data type?
> 2) All of us agree that mathematicians and others often
> distinguish a subset OA of O as 'axioms'.
> Let us call OD = O - OA 'derived' operations. And let us
> call a subset OS of OD 'scoped' operations which are
> operations that are currently "in scope".
I find your terms confused. The set OA is useful for deriving S and not for deriving O. Calling a set of operations 'derived' does not make them derivable. The set O - OA contains many operations one cannot derive from OA.
I suggest your use of the term 'derived' reveals a preconception that lies at the root of our disagreement.
I agree that you say that, and I think you say that because you assume one can derive all of O from OA.
Yes, I say that.
I have Dmitry killfiled so I don't know all of what he says. That seems like a reasonable description based on what I have seen others excerpt, though.
>>>>What is important is not which operations we consider >>>>fundamental and which we consider auxiliary. What is >>>>important is all of those operations exist and we can >>>>communicate them to each other. >>> >>>Not important? Well that is a large part of mathematics, >>>finding sets of operations (the smaller the better) we >>>consider fundamental and from which we can derive other >>>operations. >> >>Axiomatization is arbitrary. Mathematicians also spend >>time trying to find completely different sets of axioms >>for the same things and then trying to prove the >>equivalence between them.
> It may be arbitrary (if by arbitrary you meant the "subject
> to judgment" and not "without reason") but it is obviously
> valuable. No?
I am not sure value is a concept that has meaning in pure mathematics. Mathematicians did quite well without axiomatization for several millenia. It certainly proved useful in the 20th century for exploring different philosophies of mathematics and allowed Goedel a reason to express his incompleteness theorem.
Because computing science, in a sense, is the process of building abstract machines and formalisms, the fact that one can axiomatize is very useful. The fact that one can arbitrarily choose different axiomatizations is useful too. Received on Sat Jul 01 2006 - 21:43:52 CEST