Re: On Formal IS-A definition

On May 11, 6:02 am, David BL <davi..._at_iinet.net.au> wrote:
> 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:
> > > 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.
>
> > 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.

I have written down a set of imperative statements for logic's sake! If you cannot understand something as simple as the above then you cannot understand anything let alone lambda calculus, FOL, nor any other form of writing. I have to conclude at this point that you are just being obtuse for the sake of "saving face" or trolling or some other lameness.

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

Wrong. There are variables in that code irrespective of any reference to any computation machine. Now we see that you are confusing implementation (abstract or physical) of evaluation with the expressions they evaluate.

> If you were thinking implicitly about this executing machine
> when making your previous assertions we agree there are variables.

LOL. No, I wasn't thinking implicitly about an "executing machine" because unlike you I'm not confusing expressions with evaluation.

> However I got the impression you were claiming there were
> variables irrespective of the computational machine.

Yes that is what I claim and have claimed again.

> That is ludicrous given that they are intimately tied to the
> state of the computational machine.

They are only so intimately tied in your confused brain. You are pulling in yet more distractions from the facts at hand:

1. a variable is a symbol ...
2. there are sets of symbols
3. there are sets of variables because they are symbols

Now it's as if you have even forgotten about FOL variables that you ranted so much about. Are FOL variables "intimately tied to the state of the computational machine"?? Of course not! No more and no less that are the variables of any abstract syntax such as the above.

> > 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).

What part of "a variable is a symbol" do you not understand?

> 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".

LOL "really can appear". The only thing unusual are your bizarre attempts to redefine the word "variable" to contradict generations of mathematics and computer science. Actually, it is not so unusual. It happens all the time when people let pride get in the way of rationality and refuse to admit they were mistaken. You are singularly susceptible to that fault lately.

How about this for a coupe de grace, Stanford is world renown as one of the top universities for philosophy, mathematics, logic, epistemology, etc related fields. And they maintain one of the most accurate, detailed, and comprehensive online "encyclopedias" of logic. Read the following:

   http://plato.stanford.edu/entries/algebra/#Free
   http://plato.stanford.edu/entries/logic-combinatory/#3.1
   http://plato.stanford.edu/entries/consequence-algebraic/#2

as just a few examples of the "set of variables" concept and that exact language used at by this world renown institution. Now, your head is chopped off. It is time to bow out, man up and just admit you were mistaken and stop wasting time.

> 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.

*sigh* this is getting sad. It is not a can of worms in this case, it is the simple "subset" meaning of "is a". Variables are a subset of symbols. And by the way, as far as I can see you are alone in your inability to comprehend "a variable is a symbol ..."

   http://mathworld.wolfram.com/Variable.html
   http://en.wikipedia.org/wiki/Variable_(mathematics)
   http://plato.stanford.edu/entries/pm-notation/#2

> 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.

Evidently you place little importance on the consensus of the entire educated community and value only your own ignorant opinions and cannot emotionally bear to admit a simple mistake (in which there is no shame by the way).

> 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.

Nobody here cares what /your personal/ notion of variable is. We care about the consensus definition of the community of logicians and mathematicians etc. Read the above authoritative links (not the wikipedia links by the way) and educate yourself. Gain more bearing and see me again when you are worthy and grown up enough to just admit you made a mistake. It's no big deal, trust me.

In fact, you should relish being wrong just as I do for it means you will learn something. Just as Capablanca relished defeat for it came so rarely and he learned much from every failure.

KHD Received on Tue May 11 2010 - 16:27:28 CEST