Re: Proposal: 6NF

Jan Hidders wrote:
> vc wrote:
> > Jan Hidders wrote:
> > > vc wrote:
> > >I know that this set and the tuple are often identified in common
> > > terminology, but this is in some sense sloppy and it is this sloppyness
> >
> > It's not sloppy but is a common way of talking about this sort of
> > things depending of course on what exactly you want to say.
>
> Agreed, but I would claim that here, i.e., in the context of type
> theory and database theory, we do want to say different things than
> usual, and therefore this definition is here too sloppy,

There is nothing sloppy or ambiguos about the words 'integers' or 'the set of integers'. On the other hand, for example the first-order PA 'specification' of the natural numbers is ambiguous as there are many models, in fact infinitely many besides the standard one.

>
> > > that makes it in this case harder to see what the relationship is
> > > between the subtype=subset interpretation and inheritance between
> > > abstract data types.
> >
> > Not harder, just impossible ;) regardles of whether you talk about
> > c.r's and fileld or just integers. and real numbers.
>
> Well, impossible for you perhaps, others seem to have less trouble. ;-)

It may be so perhaps because they never asked themselves for example what to do about addition of the odd numbers as a subset (and therefore a subtype) of the natural numbers. Yet, according to their definition the subset of odd numbers is a subtype of natural numbers.

>
> > >This, however, allows us to construct Z
> > > and Q such that Z is a subset of Q while still respecting the axioms of
> > > both algebraic structures. I'm arguing that this gives us an intuitive
> > > and consistent interpretation of what these sets mean along with an
> > > easy to understand semantics of subtyping that ensures LSP.
> >
> > Alas, as long as Z satisfies the commutative ring axioms without the
> > multiplicative inverses ( such inverses if available would immediatey
> > make it a field) and Q the field axioms, there is little hope that the
> > LSP would also be satisfied -- of course you can construct 'my_Q'
> > without the division operation but that would be cheating.
>
> The other way around, I'm constructing a my_Z as a subset of my_Q that
> I presumably constructed earlier. That it's possible to define the
> operations so that the axioms are satisified and LSP holds seems rather
> obivous, and such definitions already exist in the literature, so why
> you would claim that this is not possible is beyond me.

It is not possible simply because what you would have constructed with your set of axioms is not what is known from our primary/secondary school years as integers/rationals. You constructed something different which you are free to do of course, just do not call the things integers/rationals. It's still unclear what you would have done with division in Q -> Z, or subtraction in Z->N, or addition in N->Odd, the operations neither of which can be 'constructed' in a meaningful way in their respective settings.

Also, it's unclear how you are going to construct Q without having constructed Z before, but let's suppose for the argument's sake you've done just that.

>
> > It might also be interesting to note that paying so much attention to
> > the carrier set is contrary to the algebraic data type spirit -- ADTs
> > emphasize structural properties or in simple words operations rather
> > than what the underlying set might be. That is another argument
> > against the 'subtype as subset' school of thought.
>
> ADT's don't necessarily make sense in the context of database theory /
> data modelling. Their point was to specify a data type while hiding
> implentation details, which makes sense when you are programming, but
> not in a data modelling context which is about how data is going to be
> represented. Hiding that represenation doesn't make sense because that
> is what you were supposed to be specifying.

OK.

>
> > >So, just to
> > > be clear on this, what I'm *not* saying is that this is necessarily so,
> > > i.e., we can just as well define constructions for Z and Q such that
> > > they are disjoint.
> >
> > As I said before, we may define anything we like, but we can hardly
> > claim that what we define is Q and Z if they do not possess the
> > expected properties.
>
> But the constructions together with the operations that are defined
> along with them, *do* satisfy the expected properties.

How would Q *without* division or Z *with* division (since that's what you would have to do with at least one of them in order for the LSP to work) satisfy what one expects from Q and Z in math ?

If Q and Z are what they are, the LSP does not hold, if the LSP works because you mangled Q and Z in such a way as to satisfy the LSP, then Q and Z are something else but surely not what they are in math.

>Hence the claim
> that they correctly model the concerned sets.
>

> > > > No, it's not based on that. Although, a typical construction of
> > > > reals makes 2.0 and 2 quite distinct, we can pretend that they are the
> > > > same ("up to the isomorphism").
> > >
> > > So where is that distinction in the definition of the surreal numbers?
> >
> > I'd rather not talk about the Conway construction in order to avoid
> > unnecessarily complicating the R/Q/Z simple stuff even more.
>
> Fair enough, although I insist that the record shows that you ignored
> possible counter evidence. :-)

There some foundational issues with surreal numbers, e.g. it's unclear whether or not some notion of natural numbers can be avoided when constructing surreals.

>So let's consider the construction where
> I construct Q as equivalence classes over their representations as
> strings (possibly including a fraction notation such as a/b and decimal
> notation), along with the usual operations, and Z as the subset of the
> classes that contain a representation that consists of a sign followed
> by a finite number of digits. Also here 2.0 and 2 are not distinct.

The above does not look like a plausible attempt to constuct rationals.  You'd want to supply the rules the strings from which you obtain you equivalence classes obey. You might discover then, at best, that the rules such strings obey are the same as the ones the integers do and come to a conclusion that in fact you are talking about integers disguised as strings and are actually guilty of the sin of circularity.  So what are those rules the string representations might obey for, if there are no rules (axioms), how would you derive anything from just the strings ? Besides, the notion of string relies on the more primitive notion of the natural number (sequence).

>
> > > I assume you think that an example of a problematic expression would be
> > > (2.0 / 3.0) where I could subtitute the 2.0 and 3.0 with their integer
> > > counterparts 2 and 3 and obtain (2 / 3) which would not have a defined
> > > result, right?
> > >
> > > Well, no. Since it is assumed that the integers are a subset of the
> > > reals the result of / is defined also for all the integers.
> >
> > So what is the result of 2/3 ?
>
> The number that is represented by the fraction 2/3.

Fractions do not belong to integers, that's why we use the name 'integers', therefore '/' is not an integer operation much in the same way as '-' is not a natural number operation, or '+' is not an operation of the odd numbers.

I think those who talk about separating operations from, for example, the set of natural numbers (and getting what, a set of objects without any properties ?), have been over-exposed to the dark force of excessive formalization. When we think of the natural numbers informally as pebbles or notions alowing us to count sheep for instance, addition is an intrinsic property of any such system: if you cannot combine your pebbles, then you are dealing with anything but not the natural numbers.

Likewise, any formal imitation of the the natural numbers, be it any structure obeying the PA or von Neumann numerals or whatever, also has addition as a direct consequence of the formal system rules (axioms), its not a sort of afterthought or some kind of house addition independent from the original rules.

>
> -- Jan Hidders
Received on Wed Oct 18 2006 - 18:09:11 CEST