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Keith H Duggar wrote:
> Bob Badour wrote:
>
>>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. >>> >>>What I don't get is why said unlimited number of >>>operations defined /on/ integers are the definition /of/ >>>integers. >> >>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.
But they are there regardless whether you use them.
>>>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.
But the point is the set of values is not the data type. The data type is both the set of values and the set of operations. Making up a new operation does not alter the data type because that operation was always there even if never previously expressed.
>>It is only after we formally specify which operations we >>are using that we can do anything useful.
And what I am saying is we don't really define anything that wasn't already there and what we choose as defining operations are quite arbitrary. The data type existed before we defined anything just as the values exist before we ever express them.
Consider for a moment that an infinite number of rational numbers will never be directly expressed by a human being. Does that make those numbers any less members of the set of values of rational numbers?
Likewise, there are an infinite number of operations on rational numbers that no human will ever express. Does that make those operations any less members of the set of operations of the data type?
>>The word 'data' implies representation suitable for >>machine manipulation. Without operations, symbols are >>useless.
I really don't see your point. I can do all sorts of useful things with the set of integer values, addition and subtraction. That doesn't make modulo any less an operation on the integer data type.
>>>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.
But one could just as easily choose a different set of axioms to derive those operations instead. In fact, any set of axioms from which one could derive those axioms is equivalent to those axioms. Thus, in the end, there is no useful distinction between any set of operations and any other set of operations as far as concerns the definition of a data type.
>>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.
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.
>>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.
Those axioms might be all that is required to describe the set of values. However, the data type is more than the set of values. Received on Sat Jul 01 2006 - 12:45:55 CDT