Re: On Formal IS-A definition

On May 10, 2:06 pm, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
> On May 8, 9:44 pm, David BL <davi..._at_iinet.net.au> wrote:
>
> > On May 9, 2:34 am, Nilone <rea..._at_gmail.com> wrote:
>
> > > On May 8, 7:11 am, David BL <davi..._at_iinet.net.au> wrote:
>
> > > > Values are immutable. Variables accessed by imperative programs are
> > > > usually mutable. Sets are values. If a set contained a variable then
> > > > it wouldn't be immutable.
>
> > > We can generalize values and variables to elements of domains, where a
> > > value is any element of a domain while a variable is an element of a
> > > domain for which a homomorphism to another domain is defined.
> > > Assigning to a variable would reduce to modification of the
> > > homomorphism, so sets containing variables would not be modified by
> > > assignment to a variable.
>
> Mathematically there is no "modification" or "mutation" nor
> any such anthropomorphic passage of time sense. A variable is a
> symbol. That symbol might have a binding. That binding is also a
> relation whose key is the variable symbol and in the case of an
> imperative interpretation if the variable is "mutable" also the
> "time" or "program counter" or similar is part of the key. For
> example the variable X might have the following binding relation
>
> S T V
> X 0 0
> X 1 0
> X 2 5
> X 3 5
> ...
>
> where S is the variable symbol, T is the "time", and V is the
> bound value. But note that nothing "changes" at T=2 from this meta
> perspective of math where "time" is just yet-another dimension.
>
> > Wrong. You can't modify a homomorphism just like you can't modify a
> > number or a set. Homomorphisms are values and are therefore
> > immutable. You have invented a homomorphism variable to hold a
> > homomorphism value. What you claimed were variables were just values
> > intended to act as inputs to a homomorphism function.
>
> Except his error is irrelevant. Variables are symbols and are
> representable by sets. Their bindings (regardless of extent ie
> dependence on "time") can be represented by relations which are
> sets just as their interpretations are relations which are sets.

I haven't seen a variable yet. All you have provided are symbol values and relation values. I'll agree there's a variable when I see one! There needs to be an imperative statement, a quantification in a formula, a lambda expression, an integral etc. Received on Tue May 11 2010 - 01:29:53 CEST