Re: 3DLDF

[Frank Heckenbach]

Hans Aberg wrote:

> At 23:13 +0100 2004/08/14, Frank Heckenbach wrote:
> > Obviously you think that C (or the C standard) is God and
> >anything it claims must be true. Well, talk as you like, but don't
> >claim mathematics for your support. C may be a good assembler-like
> >language, but it's not really famous for its mathematical concepts.
> 
> I was merely happily surprised that the C/C++ folks seems to have thought
> this through more carefully than say the guys who made FORTRAN and Pascal.

No, it's just a different paradigm. C wants to specify the internal
representation (floating point), while Pascal (not sure about
FORTRAN) wants to provide a real type (i.e., a type of some real
numbers) without specifying the internal representation.

> >> I think I pointed out that the set of real numbers is in math, as well in
> >> computers, when done correctly, developed using an axiomatic system.
> >
> >And so are integers. What does this have to do with the terminology?
> 
> I have lost you here.

Same here. You brought up this topic of axiomatic systems. I don't
know how it's related to the discussion.

> >> The use of float and int in say C is is more or less correct, as one
> >> speficies which values can be used, and all those values are potentially
> >> reachable.
> >
> >Not in C (where those terms are used).
> 
> Sure, check the header <limits.h>.

This header states the limits. Numbers outside the bounds are not
reachable. Of course, this depends on the C implementation, platform
etc., and for any integer or floating point number there can
potentially be a C implementation that supports this number.
Similarly, for any given real number, there can potentially be a
Pascal implementation that supports this number. (Even if
transcendent, it could just define this particular number as a
special symbol.) So, according to this definition, all real values
are potentially reachable. (With a big emphasis on potentially, of
course.)

> >> This is not the case of real as thought as representation of
> >> mathematical real nimbers, as many real numbers are not potentially not
> >> reachable (as say pi or e).
> >
> >Not in a floating-point representation, but with other
> >representations. So using "real" for a type of (some) real numbers
> >that does not specify the internal representations still seems
> >reasonable.
> 
> One would expect that if expressible real numbers to be contained in the
> real type, not just a subset of some rational numbers with nonstandard
> arithmetic operations, if you know want to regard floats as real numbers.
> It is hard to think of the floats of some kind as reals, as they are not
> even associative under + and *.

Just as little as C `int' values satisfy basic arithmetic
properties, so it's hard to think of them as integral numbers?

IMHO it's possible to think of them as such in both cases, as long
as you're aware of the restrictions, which applies to many other
areas of programming.

> >> >Then usual computer languages could only...
> >>
> >> Usual computer languages such as Haskell <http://haskell.org/> has
> >> potentially infinite types, for example Integer. In fact the Haskell lists
> >> are also (potentially) infinite, so it is for example possible to define a
> >> list of all the Fibonacci numbers.
> >
> >Yes, but still it can only represent a tiny fraction of all real
> >numbers, or of all lists of integer numbers etc.
> 
> You can only do that in pure math as well, as you are limited by expressing
> them via a metamathematical system, which is only potentially countable
> infinite. This is essentially a special case of the Skolem "paradox".

But mathematicians don't claim they can write down every real
number, and don't make a distinction between those they can and
can't write down (except in special situations, e.g. when dealing
with computability etc., but not generally when using numbers or
proving theorems about real numbers).

Frank

-- 
Frank Heckenbach, address@hidden
http://fjf.gnu.de/
GnuPG and PGP keys: http://fjf.gnu.de/plan (7977168E)