Re: all the incompleteness proofs are worthless untill...

On Mar 24, 2:30 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 24, 2:42 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:

Okay. I stand corrected. Please pardon my lack of mastery
of the local terminology.

Would you accept instead "equivalent to a formal system" the
way, say, cellular automata are equivalent to Turing machines?

Personally, I'd be disinclined unless I were given a reasonable
definition of 'equivalent'. And, to be fair, I grant that that
wouldn't be a formal mathematical definition, but rather a
philosophical or informal one.

A fair and principled response! (Not that I expected anything
else.) Since I'm speaking specifically about computation,
I suppose I am referring specifically to Turing equivalence,
which is, as you predicted, not a formal mathematical term.

Equivalence is formally defined in the familiar sense that the
functions Turing computable are the recursive functions are the
functions computable by register machines etc. So it's hard to see
that the drippings of an ice cream cone (no matter how delicious the
flavor) is equivalent to Turing computability.

I beg your pardon; I never said they were, although I can see
how you got that impression.

My point, and it is only a small one, and of little or no day-to-day
significance, is that ... ugh. My sore throat is worsening, and
what little mental acuity I normally possess is flying away.
Physical objects can perform computations. Clearly they
do not do so in any convenient fashion. My example was
the brain; it was Aatu who introduced ice cream. I propose
we blame his penchant for introducing dessert into the discussion
at odd times--what can his cholesterol count be I wonder?
(Although I suppose I must share the blame for jumping for
the offering when he drove enticingly by in his slow white truck
with the ice cream jingle coming out the speakers.)


Then, okay, but my point is that that is not the sense taken in
mathematical logic.

Accepted.


In practice, yes, assuming that the universe is bounded in size.
But in principle, I would say no. In fact, the very idea kind of
makes me itchy, because it starts to feel like WM and his
largest natural number. I am very much of the philosophical
bent to want to say "assuming we stay within resource
constraints" in a footnote in the introduction, as it were,
and never mention it again. Otherwise we have to qualify
just *everything*. The successor function on the natural
numbers is only total if we stay within resource constraints,
for example.

Hmm, I come to the opposite standard disclaimer, e.g.: Turing
computability is not bound by physical constraints. This notion, and
generalized, is quite ordinary, as seen in such ordinary textbooks as
'Computability and Logic' by Boolos, Burgess and Jeffrey.

Interesting.


I would suppose because it suits the kind of study we do in
mathematical logic in which everything we talk about is itself a
mathematical object. Though we might be able to render an ice cream
cone or the physical universe in some sense as "mathematical objects"
they are not mathematical objects in the sense of natural numbers,
tuples, functions, etc. And for the ordinary purposes of mathematical
logic there doesn't seem to be much advantage to including physical
objects as formal systems when we can discuss formal systems in the
sense of certain kinds of mathematical objects.

I do see at least one advantage: mechanization. But again, this is
likely to be a matter of one's personal background. As my profession
is "mechanizing" other people's business processes, I have a
very pedestrian view on mathematics and logic, at least by default.

By "mechanization" I mean specifically computer programs.


I don't know what you mean by "compute the theorem". What is
computable is whether a given sequence of formulas is or is not a
proof in certain systems. And in certain ordinary systems, there are
sequences of formulas that end with the theorem that there are
uncomputable numbers, and we can compute that thosse sequences are
proofs.

This is just what I was referring to.


Marshall
.

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