Re: A little knowledge is a dangerous thing - THE HALTING PROOF
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 6 May 2005 09:26:52 -0700
Charlie-Boo wrote:
> But note exactly what he is saying.
Believe me, I did. It is incredibly silly.
Here it is again:
Bhupinder Singh Anand wrote:
> A closer look at the arguments will show that,
> formally, the possibility that there is always a
> non-algorithmic, but effective, way, of assigning
> a natural number to any given real number cannot
> be ruled out.
The issue here is not even so much the algorithmicity
here as it is what kind of purpose such an assignment
could possibly have or serve.
Obviously, one possible assignment is both
effective and algorithmic, namely, to every real,
assign the natnum 0. But that is boring. Another
possible assignment is, to each(different)real, assign
a DIFFERENT natnum. But that is just plain impossible
(without even CARING about effectiveness), purely for reasons
of cardinality. Any assignment Bhup might want to make lies
somewhere between these two extremes of trivially and impossibly
difficult, but as we pass beyond the "algorithmically" difficult
ones, I don't see why anybody would think we would run into
some "effectively" difficult ones, before arriving at the
"impossibly" difficult ones.
> The key word is "non-algorithmic",
> and also perhaps "given".
NO, the KEY word is EFFECTIVE.
> I can see at least 3 possibilities:
>
> 1. There is a way ("effective") that is not an algorithm.
Right. Bhup invites us to think that this might be possible.
Church and Turing invite us to think that this is a
contradiction in terms. In either case, until somebody
comes up with a definition OTHER-than-Church&Turing's-
proposed-one of "effective", Church and Turing are winning
and Bhup is losing. A bad theory always beats no theory
at all.
> Your statement that there is no formal definition
> of effective, I think, makes his point easier to make:
No, it makes his point incoherent.
How can he allege that something "effective, but non-
algorithmic" exists, if he can't define "effective"?
> How can you say there is no
> "non-algorithm" that works if you can't
> even say what an algorithm is?
WE CAN! WE HAVE! OR at least Church and Turing have!
THEY say that an algorithm is a Turing Machine!
Anybody who wants to say it is anything else needs
TO BRING ONE BACK ALIVE (i.e. one that the general
community of observers will concede to be "effective"
AND that is clearly NOT TM-emulatable). WE KNOW what
"algorithm" means. Bhup DOESN'T KNOW what "effective"
means. Unless he agrees with us that it means "algorithmic".
> But that is a little like "belling the cat".
> The bell in this case is
> a non-algorithm that is effective.
> Or perhaps he is saying
Oh, shut up. He is right here; he is talking;
we can all see what he is saying.
> "It might exist, but, not being an algorithm,
> we'll never be able to describe it."
Well, there are PLENTY of functions like that.
But is any of them EFFECTIVEly computable????
If you can't even DESCRIBE it, how can you allege
that can EFFECTIVEly evaluate it????
> 2. There is a different way for each number,
> but no single way for all numbers.
That's just sort of inherently ridiculous.
If this is in fact the case, what is it about
this uncountable infinity of "different ways"
that MAKES EACH of them a WAY?? Are more
than some simple countable subset of these myriad
ways THEMselves "effective"????
More to the point, "effective" methods obey a sort
of compositionality; if I have an effective way of
doing 1 thing and an effective way of doing another,
then I also have (IRrespective of whether ANYbody likes
it OR NOT) an effective way of doing BOTH things, namely,
first do one, THEN do the other. Actually I have TWO effective
ways, since I could do the substasks in the other order.
Thus, if there is an effective way of assigning a different
natural to every real, then there is an effective definition
of the real whose nth place is 9 minus the nth place of
the-real-that-was-effectively-assigned n. But THAT is a real
that differs from ALL the reals that got something assigned
to them.
> 3. If he means "given" as in
> "we just have to do one, whichever one is
> given", then sure. No, NOT sure.
> The first one you give me I will call 1, the next
> one I will call 2, etc.
That does NOT count. It has to be given
IN TERMS OF THE REAL NUMBER, NOT in terms of the
temporal order in which they were presented to you.
Time doesn't exist in THIS world except while a TM is running.
> Or the fact that it is "given" means it has
> already been described finitely.
That Just Plain Obviously won't help; only a small
denumerable subset of the reals have finitary descriptions.
> I think Bhup is trying to use Turing's argument
> to his advantage. It tells us that no algorithm will do,
> but doesn't tell us there's nothing besides algorithms!
There are obviously PLENTY of things besides algorithms:
ALGORITHMS ARE FINITE and there are PLENTY of infinitary
things out there. But the point is that THOSE things all
have the property that YOU CANNOT WRAP YOUR FINITE HUMAN
BRAIN around them and you therefore in some sense CANNOT
USE them, effectively, for computing and specifying.
.
- References:
- Re: A little knowledge is a dangerous thing - THE HALTING PROOF
- From: george
- Re: A little knowledge is a dangerous thing - THE HALTING PROOF
- From: Charlie-Boo
- Re: A little knowledge is a dangerous thing - THE HALTING PROOF
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