Re: Storing data and code in a Db with LISP-like interface

Dmitry A. Kazakov wrote:
> On 6 May 2006 16:18:03 -0700, vc wrote:
>
> > Dmitry A. Kazakov wrote:
> > [...]
> >>>> Neither it states *how* to construct union or intersection.
> >>>
> >>> That is correct, but the union axiom has a clear and intuitive
> >>> constructive interpretation that the powerset axiom lacks.
> >>
> >> It has a quite intuitive interpretation when sets are finite.
> >
> > Go ahead and provide one in the first-order language.
>
> Why should I? Is the above the definition of "intuitive"?

As is (or should be) well-known the modern set theory language is a first-order language, The powerset axiom is formulated in such language. You claimed that you can interpret the axiom constructively,  presumably without changing the language just as one could do with the union axiom. Are you admitting that you cannot do that ?

>
> > [...]
> >>. What about (a, b, c, d ) <--> { a, b, c, d } ?
> >
> > What about it ?
>
> Indeed. Can you transpose a table?

Please define "transpose a table".

>
> >> I don't care about construction of a set, if I can denote the set.
> >
> > If you do not care about constructing a structure, how do you intend
> > to use something that you have not constructed in any programming
> > language ?
> >
> >> Example:
> >> 1.3 of R.
> >
> > And this shows what exactly ?
>
> It shows that 1.3 is a much better abstraction level for dealing with R
> than Dedekind cuts traced down to ZF.

How is 1.3 an abstraction of anything ? It's (presumably) just one real number. More interestingly, what alternative real number construction, other than Dedekind's cuts, do you have in mind ?

>
> >>pi is also
> >> not an approximation, it is pi. I am free to choose a representation where
> >> pi were exact.
> >
> > I'd be curious to see such representation.
>
> enum R { Pi }; // See assembler listing for representation of Pi.

Are you claiming that you've discovered an algorithm computing the exact value of Pi in some assembly language ? Could you share your findings ?

>
> If you had an oscilloscope and some old computer with ferromagnetic coils,
> you could also directly see its representation in the memory. Fascinating,
> isn't it?

Are you claiming that "some old computer with ferromagnetic coils" can compute the exact value of Pi ? Fascinating indeed.

[...]

> >>>> Any finite subset of, in any language that supports ADTs.
> >>>
> >>> Go ahead and oblige us with an example.
> >>
> >> Take bit strings, Algorithms for +, -, * can be found in any text book on
> >> computer design.
> >
> > Could you elaborate ? I am not sure what you are trying to say here.
> > You were supposed to construct integers as I recall.
>
> For further reading see "CPU", "Turing Machine", "bignum."

How do the words ""CPU", "Turing Machine", "bignum." show how to construct integers ? Are you unable to substantiate your claim that "any language that supports ADTs." can construct integers ?

>
> >>> [...]
> >>>>>>>> In mathematics you can go either way. Is integer
> >>>>>>>> number rational? How different pairs (1,1),(6,6) can both be 1?
> >>>>>>>
> >>>>>>> You are confused, amigo. In the secondary school algebra, one learns
> >>>>>>> that an integer number ain't no rational.
> >>>>>>
> >>>>>> That depends on construction, they could well be. In the secondary school
> >>>>>> one learns that this does *not* matter.
> >>>>>
> >>>>> What does not matter ? What alternative rationals do you have in mind ?
> >>>>
> >>>> I can add new elements to Z instead of constructing a set of pairs and then
> >>>> choosing subsets there.
> >>>
> >>> What's that supposed to mean ?
> >>
> >> Take Z. Add elements denoted as "1/p", for each p, prime number. Postulate
> >> them as multiplicative inverses for p. Add all: 1/pq, where p>=q, prime.
> >> Postulate. Do same with 3, 4, 5, etc primes. When finished start to add
> >> elements which are "multiplies" of new elements to primes. Use only
> >> multiplies of primes not used in the denominator. Add an additive inverse
> >> for each new element. Done.
> >
> > That is funny. Regardless of what you are doing with integers, you
> > cannot do that as you do not have integers yet.
>
> Your question was about construction Q in which Z were a proper subset. I
> presented such.

You did not. You took Z did whatever and then said 'hey presto now integers are a subset of rationals'. Of course they are, it's a trivial observation. You were supposed to construct rationals from scratch and then get your integers. What you did is called cheating.

>
> > You said that you'd
> > 'inherit' integers from rationals, so rationals should be your
> > primordial soup presumably. What's the point of inheriting integers
> > from rationals if you already have them ? Besides you did not
> > construct integers either. Please retry if you can.
>
> It seems that you don't understand difference between mathematical objects
> and their computer models. CPU doesn't have Z, Q or R. It has a lot of
> silicon, copper and stuff like that...

Fancy that ... How can one then add one and one with all that stuff ?

Whether or not I understand the difference is unimportant. Your CPU strawman is also irrelevant as you've claimed that you can construct integers using some language 'ADTs', not 'CPU'.

>
> >> Not necessarily. You need not to inherit each implementation. In
> >> particular, you don't need to inherit data representation, because it is
> >> not "is-a" anymore. Both can have completely different representations. You
> >> can even have rational abstract (i.e. no representation at all).
> >
> > If you have 'abstract rationals' whtever that is, they are not really
> > rationals one came to expect from one's school years,
>
> That's for sure. Those rationals were built out of chalk, washed away long
> time ago...
>
> > they are surely
> > something else by virtue of being unusable since they are 'abstract'.
>
> Yep, they are, as much as abstract algebra.

Are quite sure you know what you are talking about ? What your other strawman of 'abstract algebra' has got to do with your claim that you can construct integers from 'ADTs' ? I undestand that you cannot produce such construction, is that so ?

>
> --
> Regards,
> Dmitry A. Kazakov
> http://www.dmitry-kazakov.de
Received on Mon May 08 2006 - 01:56:24 CEST