Re: 3DLDF

[Hans Aberg]

At 01:50 +0100 2004/08/14, Frank Heckenbach wrote:
>> >> It clearly so that in some computer language paradigms, the word
>>"real" is
>> >> used instead of the mathematically correct term "float". C/C++,
>>however, do
>> >> it correctly.
...
>> >Well, I'm a mathematician and I haven't heard the term "float" (or a
>> >German equivalent) used anywhere in mathematics except in computer
>> >programs in C etc.
>>
>> Floating point numbers are quite common in applied math, for example
>> physics.
>
>That's not the question. Your claim was that the *term "float"* was
>mathematically correct. I don't see any evidence of that.

Are you stuck on the fact that I used "float" as a short hand for "floating
point number"? Or are you claiming that floating point numbers cannot be
given a mathematical description? Or what?

>> >I suppose you refer to the fact that floating point numbers can't
>> >represent all real numbers, but neither can the integer type of C
>> >and many other languages represent all integers, etc.
>>
>> Mathematical real numbers can be represented in computers, in for example
>> theorem provers, and to some extent symbolic algebra programs.
>
>Any computer can only represent countably many numbers, i.e. "almost
>none" of the real numbers.

Yes, it is the same as in pure math; in reality one is restricted to only
potentially infinite finities. In practise, one can in a computer represent
a finitude more numbers than in pure math, because usually there fits more
symbols in a computer than on a paper (assuming that one has bothered to
put the metamath into a computer). :-)

> You can represent a different subset that
>floating point numbers (say, expressions involving pi, e and certain
>integrals), but it's still a small subset, and calling such a type
>"real" is/isn't just as justified than calling a floating-point type
>"real".

So this your statement probably declares null and void all the pure math
about real numbers, that presumably is formally expressible via a
metamathematical theory. (If one wants to implement properly implement the
set of real numbers in a computer, one does that exactly the same way as in
pure math, via a finite, but potentially infinite via substitutions,
axiomatic system.)

>> A language
>> has both the type Integer, for multiprecision integers, and Int for
>> interfacing with C "int". The types "int" and so forth in C, are called
>> "integral types", not integers, in its paradigm;
>
>AFAIK, "integral" is just the adjective to "integer".

Yes, but this difference is exploited to technically distinguish integral
type from integer number. :-)

> Since "real"
>is both a noun and an adjective, it seems just as well suited.

To denote real numbers or floating point numbers?

>(I.e., if you mean "a type of some integral numbers" instead of "the
>set of all integers", I could say just as well "a type of some real
>numbers" instead of "the set of all reals").

It turns out that axiomatic set theory is not very practical in pragmatic
computer programming languages, so therefore one is instead using a simpler
type theory. But axiomatic set theory is implementable in a computer (I am
working on such things in my theorem prover).

  Hans Aberg