A while ago, I disageed with a poster who said something to the effect that
'a computer can deal with rational numbers, but not irrational numbers'.
I gave examples where, perhaps a computer would be able to compute that:
sqrt(2) ^ 4 = 4
(pi ^ 2) / pi = pi
My whole point was that the difference between what is computable and
what is not computable has nothing to do with our intuition as to what
kinds of numbers are being represented; computability is like 'syntactic
manipulation' of some sort of 'representation' of the objects that we
wish to compute with.
Obviously, no system of computation can work for all real numbers (for
obvious cardinality reasons; there are to many real numbers).
But, it is possible to create a system which can deal with a (countable)
subset of the real numbers. If we define a certain set of operations
for this set of numbers, and a set of relations for this set, we may find
that the functions can be computed, and the relations may be decided.
(This depend on the functions and relations that we wish to compute/
decide.)
This has nothing to do with whether the numbers are rational or irrational.
(Of course, since this set _is_ countable,
there will be an injection from this set _INTO_ the natural
numbers, and the functions and relations can be transformed into
corresponding functions and relations for natural numbers.)
My basic point is that, if I want to know what (pi^2)/pi might be, a
computer _CAN_ tell me that it is 'pi'; the fact that 'pi' intuitively
represents an irrational number has nothing to do with it.
Similarly, a computer does not have 'rational numbers' or even 'integers'
in its memory; only representaions of these.
(It just so happens that the interpretation of 'bit sequences' as
natural numbers is supported by machine instructions; but I see any such
machine instruction as performing a syntactic maniputation on the
representation; it is not intrinsicly different than a term rewriting
system for solving equations as above.)
So, we can only compute with _SOME_ of the real numbers in any computing
system, but they need not be rational numbers.
- Steve Pipkin
In article <············@topdog.cs.umbc.edu>,
Stephen <······@cs.umbc.edu> wrote:
> Obviously, no system of computation can work for all real numbers (for
> obvious cardinality reasons; there are too many real numbers).
Any time a statement is prefaced with "obviously", I view it with
suspicion :)
I don't know exactly what this statement means by "system of
computation" or what it means by "can work" or by "all real numbers".
But here are some similar statements
1. There are more integers than there are electrons in the universe,
so no system of computation can work for all integers either.
(an uninteresting observation to those of us whose consciousnesses occupy
finite space/time)
2. Consider the case of "constructive real numbers," those which can be
defined in finite terms. There is no problem in talking about them
in some ways, though you may not be entirely happy with limits on what
you can do. Like can you compare them for equality or ordering? Dunno.
3. Then consider all other real numbers. do you have to compute with
them? Maybe not. Perhaps they do not exist? (Where's the flaw in
this.... Of all the nonconstructive real numbers, those that cannot
be defined in finite terms, take the smallest one. And describe it
as "the smallest nonconstructive real number". That makes
it constructive. :) )
After you've found the flaw above, consider...
Do you want to "work" with these? What did you have in mind,
and what limit does it present to you to not being able to work with
them?
I'm not sure this has much to do with computer algebra, interesting
as it may be philosophically.
a perfectly good real number is 3*pi+4*pi^2..
--
Richard J. Fateman
·······@cs.berkeley.edu 510 642-1879
·······@peoplesparc.cs.berkeley.edu (Richard J Fateman) writes:
>3. Then consider all other real numbers. do you have to compute with
>them? Maybe not. Perhaps they do not exist? (Where's the flaw in
>this.... Of all the nonconstructive real numbers, those that cannot
>be defined in finite terms, take the smallest one. And describe it
>as "the smallest nonconstructive real number". That makes
>it constructive. :) )
>After you've found the flaw above, consider...
The flaw is that not all subsets of real numbers have a least
element. Hence, there may not be a smallest non-constructive real
number.
Torben Mogensen (·······@diku.dk)