Re: On Formal IS-A definition

On May 11, 11:00 am, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
> On May 10, 7:29 pm, David BL <davi..._at_iinet.net.au> wrote:
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> > On May 10, 2:06 pm, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
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> > > On May 8, 9:44 pm, David BL <davi..._at_iinet.net.au> wrote:
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> > > > On May 9, 2:34 am, Nilone <rea..._at_gmail.com> wrote:
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> > > > > On May 8, 7:11 am, David BL <davi..._at_iinet.net.au> wrote:
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> > > > > > 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.
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> > > 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.
>
> I almost don't have time to refute these inane flimflam
> "objections". But this one is so lame and easy as to nearly
> answer itself:
>
> var X = 0 ;
> var Y = 5 ;
> X = Y ;
> Y = 0 ;

I have no idea what point you think you've made.

Now there are some variables associated with that imperative code when it executes on some computational machine.

If you were thinking implicitly about this executing machine when making your previous assertions we agree there are variables. However I got the impression you were claiming there were variables irrespective of the computational machine. That is ludicrous given that they are intimately tied to the state of the computational machine.

> which exactly reflects the binding relation above. Do you
> still not comprehend? { X, Y } is a set of variables. For
> gods sake just google "set of variables" and see that the
> world of mathematics is replete with this concept.

Some people say "set of variables" when what they actually mean is a set of symbols, and there is an intention for those symbols to be used as variables (e.g. a summation index).

In order to appeal to authority we need a respected logician or set theorist who will comment on whether variables really can appear in mathematical sets. The question would have to be posed carefully because it is unusual to consider variables to be part of one's "ontology".

Your whole argument depends on the meaning of "is a" in the natural language sentence "a variable is a symbol". I hate to break the news, but you should be aware that "is a" is a big can of worms.

Evidently you place more importance on your vague intuition of "is a" than the ridiculous outcome that necessarily follows your kind of faulty reasoning : i.e. that a variable is a value, and variables never change. Given that your notion of variable doesn't match mine at all, I have no doubt that we each will be invoking the Principle of Incoherence. Received on Tue May 11 2010 - 12:02:35 CEST