On 12 Jun 2016, at 01:04, Brent Meeker wrote:
On 6/11/2016 3:44 PM, John Clark wrote:
On Sat, Jun 11, 2016 at 3:03 PM, Brent Meeker
<[email protected]> wrote:
> There are no "incorrect calculations".
2+2=5
If you programmed a Turing machine to start with "2" and "2" on it's
tape and print out "5" it just means it didn't compute the sum of
2 and 2. If you program your computer to print out "2+2=5" the
computer will still do a correct computation. It's just your
interpretation of the output as applying to something other than
what the computer did that is incorrect. The computer still
executed your program correctly.
That is so true. A debugging technic in Prolog illustrates this.
Indeed, Eyud Shapiro wrote a prolog program capable of debugging a
program when the program is corrected when some of its output is false
on some input. The program is so powerful (on a class of programs)
that it can be used to fully synthesize any program (of that class) by
debugging the ... empty program. So you run the empty program, and
each time it is wrong you correct it , and this of course, *relatively
to what you want to synthetize!
In fact a program is never wrong or correct, nor is a theorem. It is
wrong or correct relatively to a semantics. Incompleteness shows that
all universal machine is unable to describe fully its own semantics,
nor even to prove there is one. (Gödel's completeness theorem says
roughly that a mechanical belief system is consistent if and only if
it has a semantic).
Now, if you write a program for a universal dovetailer, you can make
it short and convince you that there is no bug. Then, when running, it
will generate all pieces of codes, and run them correctly. Of course,
it will generate all "buggy" version of programs as well, but "buggy"
is in the eyes of the one who has a goal in mind, and that will be a
notion as much relative to the generated "self-aware programs" than
to us.
Bruno
> It's just a universal Turing machine that runs all one step
programs, all two step programs, etc. Some programs stop. Some
programs fall into infinite loops. Some just keep computing.
And some are consistent with the Peano postulates and some
are not, those that aren't physicists have no use for because they
can attach no meaning to them.
Actually physicists often use continuum mathematics, which are not
consistent with Peano axioms, e.g. every number has a divisor.
It is consistent with arithmetic. It just do not conncerne the natural
numbers. But with the FPI, numbers can expect some continuous
observable or, at the least, a random oracle.
Mathematicians could start with 2+2=5 as an axiom and build some
form of arithmetic from that, it would be a pretty silly thing to
do but it wouldn't surprise me if some mathematician had actually
done it. And that's the trouble with mathematicians, sometimes when
they drift higher and higher into the stratosphere they start to
sound like Minnie Mouse on helium. Physicist are bound by
something, observational facts, but mathematicians have no such
bound so sometimes they end up moving in all directions and going
nowhere.
> These are all abstract processes that "exist" in the
mathematical sense.
What sense is that?
For every integer x there exists a successor of x, S(x). There
exist infinitely man prime integers. In every continuous mapping of
a compact convex set into itself there exists a point that is mapped
into itself.
Yes, it is the sense more or less captured by the usual inference rule
managing the quantifier (for all, it exists) in predicate logic.
> There is no sense in which they can be correct or
incorrect.
What about non-sense?
You mean what about something that is not a computation, not an
implementation of an algorithm?
The physical reality can implement, apparently, non-sensical being
(relatively to some standard sense, say), but arithmetic is similar in
that respect. That is why we are confronted to a measure problem,
indeed. Gleason theorem + Everett solves the measure problem, I think,
and we have only more work to do for getting a similar solution for
arithmetic. Advantage: thanks to the difference between G and G*,
inherited by the observable, we get an explanation of both 1p plural
physics (quanta) and the 1p singular and private physics (like when
that's hurting, consciousness, sensations, qualia).
Bruno
Brent
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