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"You do not start with axioms - you start with proofs. Only when you have found a bunch of related proofs, can you come up with axioms. You end with axioms." U
by al2o3cr 12y ago
"You do not start with axioms - you start with proofs. Only when you have found a bunch of related proofs, can you come up with axioms. You end with axioms."
Um, what? You have no FACTS to do proofs with without axioms; they are the fundamental statements one assumes to be true to define a logical system.
Somebody who insists that they need "multisorted algebras" while misunderstanding a fundamental definition in mathematical logic is exceptionally confused...
- v64 12y agoI interpreted this to be a top down vs bottom up argument. In mathematics, we certainly have ZFC as an axiom system now, but mathematics was performed for thousands of years without a "stable" axiomatic base. For example, calculus was developed before the axiomatic delta-epsilon definition for limits was created. The math was developed first, and it was only after the fact that mathematicians went back and set the theory on a stable theoretical footing. Therefore, I believe Stepanov suggests a bottom up approach: You write an algorithm that performs your work first, and then once you have that, you can refactor the code into classes that encapsulate the various parts of your existing algorithm.
- innguest 12y agoThe axioms were found much much later than a lot of the work had already been done. When Euclid wrote down his list of axioms do you think he was laying the ground for work yet to be done? No, the axioms were the synthesis he came up with by observing the work that had been done before by others.