Re: Operationalize orthogonality

Tony D wrote:
> Pickie wrote:
> > Booleans don't in themselves convey order.
>
> Indeed not. You could define an ordering of your own for your own
> purposes. Something like, say,
>
> greater_than(false,false) = false
> greater_than(false,true) = false
> greater_than(true,false) = true
> greater_than(true,true) = false
>
> There you are, an order defined on booleans. It's arbitrary,
> meaningless, and pretty much pointless, but it's an ordering all the
> same.
>
> [ snip ]
>
> > Individual bits can represent booleans, but where is the concept of
> > order coming from that turns a set of bits into a coded string of bits?
>
> Turn it around. Try "booleans can represent individual bits". Getting
> anywhere now ? Maybe I haven't been explicit enough for you; I consider
> a type to be a set of values (or a set of equations defining the
> values) and the operators for that type. A type generator will have to
> offer a way to define the operators for a type, as well as a way of
> describing the acceptable values for that type. By beginning with
> relations and booleans, the type generator will allow us to describe
> new types *and the operators upon them*.
>
> [ snip ]

Thanks Tony,

That's makes your position totally clear. It was that first step (defining an order) I couldn't get clear in my mind. I don't know enough to say whether you are right or wrong, but I think I can see the basis of your thinking.

>
> > So, to answer your question. Not only do I not see it, I do not
> > acknowledge it is possible. Obviously there are systems that do it,
> > but _not_ by building _solely_ on booleans and relations (even
> > theoretically).
>
> Fair enough. You're wrong, but to convince you would obviously require
> me to define a representation of integers using relations of booleans,
> and then to define the operators of arithmetic on them. It's doable,
> but will take ages (remember, "frightening degrees of circumlocution")
> and sorry, but life's too short (and mine too busy) for that.

No, don't do that! I agree that it's better to bring these in from somewhere else. Now I have some other things to think about. Just what is it that gets brought in? Why would this be better than something that looks like it's based on first principals? Received on Mon Jun 12 2006 - 00:18:59 CEST