Re: Motion 31 draft text V04.4, extra notes
Nate, and P1788
On 6 Apr 2012, at 17:13, Nate Hayes wrote:
>> But I believe we also want to support intervals as ranges, e.g. for
>> constraint propagation. In that case it is essential to provide for
>> unbounded intervals at Level 1.
>
> Folks, I apologize in advance for being pedantic but someone needs to
> explain this to me:
>
> Why is it "essential"?
>
> Specifically: what is a concrete example of a computer algorithm that can
> produce more meaningful results with unbounded Level 1 intervals as opposed
> to "overflown" intervals at Level 2?
This question has both a theological and a practical aspect. If someone belongs to a faith that has commandments against eating pork, they may ask me if it's "essential" to me to eat pork. (Not just Jews and Muslims, my ex-wife belongs to a Christian church that avoids pork, so for many years I didn't let ham or frankfurters in the house, for the sake of marital harmony.) Personally I think St Mark had it right when he commented on some words of Jesus (Mark 7:19) "This he said, making all meats clean". So, not "essential", I reply, but sometimes "useful". And Kate and I agreed to differ on this.
An unbounded interval is one that has an "infinity" as a bound. There is a wide range of attitudes to infinity, from that of Lee Winter, who (if I interpret rightly) believes that infinity (oo) "simply doesn't exist", to that of myself and other pure-math minded folk who believe "of course you can define oo if it's useful, what's the big deal?". My first serious use of oo was 54 years ago when I was taught projective geometry in the sixth form at the age of 17. The projective plane P has a "line at oo" with infinitely many "points at oo" on it: these ideas were defined independently by Kepler and Desargues in the early 1500s, says Wikipedia.
Even then at school, I was taught how points at oo can be expressed in terms of ordinary arithmetic, by using "homogeneous coordinates", namely P is all real triples (x,y,z), excepting (0,0,0), such that (cx,cy,cz) is counted the same as (x,y,z) for any nonzero c. A point at oo is one of the form (x,y,0). In this way one sees the infinities of projective geometry are free of contradictions -- provided ordinary arithmetic is so.
So for 54 years I have held the mathematician's view that something "exists" if you can construct it in terms of existing math concepts. That's how the "existence" of negative numbers was justified. Ditto complex numbers. I'm sure there were plenty people at the time who warned that anyone dabbling in either of these would be dragged off by someone with horns, a red cloak and a trident.
Infinities aren't "essential", but they sure are "useful". I know as well as you, Nate, that one can't compute meaningful results at Level 2 for numbers beyond the largest representable, REALMAX. But that does NOT imply
(1) X= [a,oo] actually *means* [a,B] for some large finite B; nor
(2) it is always computationally useful to replace X by [a, REALMAX].
Either of these may be true in the context of a given algorithm. But infinity means infinity. If a user of a constraint propagation code, or a mathematical programming code, gives the bounds on some variable x as "0 to oo", they MEAN "there is no upper bound on x".
We COULD do banking without negative numbers: probably we used to, with two separate books, one for credit and one for debit balances. We COULD do electrical engineering without complex numbers. We COULD write Maxwell's equations as Maxwell did, without vector notation. We COULD do without oo. But let's do things conveniently. What's the big deal?
This is all separate from the issue of whether 1788 should give a way to *distinguish* a genuinely unbounded interval from an "overflowed" one that is known to be bounded but has a bound outside the representable range.
Regards
John Pryce