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Keith H Duggar wrote:
> Bob Badour wrote:
>
>>I am speaking of data type as a concept. The integer data >>type has an unlimited number of operations defined on it. >>In most contexts, only a tiny subset of them are in scope.
You are not being clear. You are using integers to mean both a set and a data type without specifying which you mean.
Without any operations, the set is just a bunch of symbols. To manipulate those symbols, one must have operations.
You
> and I can communicate using integers, prove theorems about
> integers, think about integers, etc by appealing only to a
> very small subset of those unlimited operations. So of what
> relevance are the remaining unmentioned or undiscovered
> operations to our discussion and thinking?
The relevance is they are there for our use any time we want them. Without any operations, we cannot do anything meaningful with the set of integers. We cannot prove theorems about them at all.
It is only after we formally specify which operations we are using that we can do anything useful.
The word 'data' implies representation suitable for machine manipulation. Without operations, symbols are useless.
> And suppose we define integers in the usual set theoretic
> or algebraic ways. Why can we not treat other operations as
> simply derived or auxiliary?
All operations can be derived or auxiliary. If we removed all the derivable operations, we would have nothing left from which to derive the rest. 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.
What's more, the set of operations in the algebra is a proper subset of the operations for the data type. For example, substring is an operation defined for integers because it has two integer parameters.
How is it useful to think of
> them as defining integers?
They do not define the set of values. And without the set of values, they do not define the data type either. The data type is both the set of values and the set of operations defined on those values. Received on Fri Jun 30 2006 - 21:24:14 CDT