In article <[EMAIL PROTECTED]>, Xah Lee <[EMAIL PROTECTED]> wrote: >In hindsight analysis, such language behavior forces the programer to >fuse mathematical or algorithmic ideas with implementation details. A >easy way to see this, is to ask yourself: how come in mathematics >there's no such thing as "addresses/pointers/references".
There is. Each variable in predicate calculas is a reference. No matter how large the formulae, a change in a variable is, in mathematics, immediately propagated to all occurances of the variable (potentially changing the references of other variables). If the predicate calculas variables were not equivilent to references, then the use of the variable in a formula would have to be a non-propogating copy. and a change to the original value whence not be reflected in all parts of the formula and would not change what the other variables referenced. Consider for example the proof of Goedel's Incompleteness theorem, which involves constructing a formula with a free variable, and constructing the numeric encoding of that formula, and then substituting the numeric encoding in as the value of the free variable, thus ending up with a number that is "talking about" iteelf. The process of the proof is *definitely* one of "reference" to a value in the earlier stages, with the formula being "evaluated" at a later point -- very much like compiling a program and then feeding the compiled program as input to itelf. You cannot do it without a reference, because you need to have the entire number available as data at the time you start evaluating the mathematical formula. -- Ceci, ce n'est pas une idée.
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