Re: Mixing OO and DB
Date: Sat, 16 Feb 2008 19:53:43 +0100
Message-ID: <1wwp02oxjb5md.16uke8wywvucx$.dlg_at_40tude.net>
On Sat, 16 Feb 2008 08:57:59 -0800 (PST), Marshall wrote:
> On Feb 16, 3:39 am, "Dmitry A. Kazakov" <mail..._at_dmitry-kazakov.de>
> wrote:
>> On Fri, 15 Feb 2008 17:49:38 -0800 (PST), Marshall wrote: >> >>> Mathematically, a circle value is a set of points, and an ellipse >>> value is a set of points. >> >> Geometry does not operate "values".
>
> All of mathematics operates on values. In fact it is pretty
> much devoid of anything like a mutable variable construct as
> used in imperative programming.
>
>> It deals with plain circles and ellipses.
>
> These are usually understood to be sets of points.
> Is there some reason you don't think sets are values?
Why should they be anything but sets?
(In case you think that I want to claim that circle values should be mutable, I don't.)
>> In a computational system that models geometrical >> objects a circle value serves as a model of circle.
>
> What is the point of such complexity? "Circle" and
> "circle value" denote the same thing. So you're
> saying a circle is a model of a circle.
The point is that a computational model is sufficiently weaker than geometry. Circle is an uncountable set, and the set of circles is even so. For this reason no computational model can be equivalent to geometry and thus the claim that circle value is circle is mathematically invalid. Because there is no bijection between the set of all circle values and the set of all circles.
>>> Mathematically, we can describe a set of points >>> with an equation, and the equation that describes ellipses reduces >>> to the equation that describes circles in certain cases. Do you want >>> to say that two different appearances of the same equation may be >>> different depending on what they were reduced from? That dog >>> won't hunt. >> >> It can also define a circle using its center and radius, or as a conic >> section, or via complex exponent, or by an uncountable number of other >> ways. All these definitions are said equivalent. It has nothing to do with >> being "same value".
>
> Whatever equation you use, that equation will reduce to
> an equation that specifies a circle in some cases. So now
> you have arrived at the same equation from two different
> directions. Do you still object to the "same value" idea after
> we have the same equation?
No, because the presumption is wrong, which is that you could prove equivalence. What you can do in mathematics, it is not always possible in the computational model due to the limitations of. I.g. P(v1=v2) might be incomputable. Even if it were computable, that could be technically impossible to do, like for NP problems. For this reason when dealing with models, we use named equivalence. Observe also that a circle value in fact refers to a set of circles due to, again limitations of the models of R. But the finite set of sets of circles is not the set of circles.
> Every circle is an ellipse. This fact might be inconvenient
> you one wishes to view every bit of mathematics from
> within a specific type theory that cannot capture this.
Yes, the theory has the power making it possible to finite state machines. Find a better hardware...
[ OT, it really amazing that humans can operate things like geometry. Either our brains contain incomputable elements or else there is something wrong in our understanding of mathematics.]
> But again, I love type theory. I just happen to think it's usually
> done in a way that is substantially too complicated.
Any constrained problem is more complex. x^n + y^n = z^n is not a challenge when x,y,z is from R.
-- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.deReceived on Sat Feb 16 2008 - 19:53:43 CET