Re: Another Reason Why Collatz is Unprovable


Craig Feinstein wrote:
mensanator@xxxxxxxxxxx wrote:
Craig Feinstein wrote:
mensanator@xxxxxxxxxxx wrote:
Craig Feinstein wrote:
T.H. Ray wrote:


Hilbert's ideas have had a great influence on
modern
mathematics, even
though they are wrong.<<

Wrong? I can only assume you mean the formalist
program that Hilbert (and Frege, Russell and
Whitehead
among others) meant to pursue before Godel
published
his proof.

Yes, that is what I was talking about. Hilbert's
formalist program only
produces facts which are "trivial", i.e., follow
logically from
well-accepted axioms. Because of this, modern
mathematics research has
become dull, like some kind of sophisticated logic
puzzle. It's never
going to produce anything surprising, as nothing
surprising ever comes
from logic. When one takes mathematics to a higher
level, trying to
understand numbers as they are from computational
experiments, one
finds that mathematics is interesting and one sees
things which are
surprising. (In other words, math is interesting;
logic is
uninteresting.) Wolfram's way of doing it in his book
NKS is ultimately
how mathematics should be done in the computer age,
in my humble
opinion. Another person who also has some interesting
writing on this
subject is Rudy Rucker.<<

Do you think logicians find their research uninteresting?

Let's put it this way. Doing logic puzzles (which is basically what
logicians do with their time, although their puzzles are much more
sophisticated than the recreational variety) is a challenging and
enjoyable activity for many people. But it's never going to produce
anything surprising, since you are basically processing and rehashing
information that you already have.

Computational experimental mathematics *does* produce things which are
surprising, for instance fractals. For "most" computational
experiments, you can't predict where the computations will lead without
actually performing the computations themselves. (This is Rudy Rucker's
reformulation of Wolfram's principle of computational equivalence.)

Collatz is an example of this with a twist: In this case, the
computations always seem to lead to one, yet it is still impossible to
predict this with absolute certainty without performing the
computations themselves.

Absolutely wrong. In my paper

Blueprint for Failure:
How to Construct a Counterexample to the Collatz Conjecture

which has just been published at

<http://home.versatel.nl/galien8> see Volume 5 (2006)

I demonstrate that the list of counterexamples based on
Factor Congruence is empty for any 3n+C system where
C is a power of 3 (including C=1).

Although this does not prove the conjecture, it is an example
of predicting with absolute certainty that does NOT involve
performing the computations themselves. It is an example of
a shortcut which you claim does not exist.

Congratulations on getting your paper published. And in a good journal
too. I like journals open to new ideas like theirs. Your paper doesn't
contradict any of my claims though. I never claimed that you can't
predict counterexamples without running the algorithm itself. I only
claimed that you can't predict that the algorithm will converge to one
without running the algorithm itself. Proving that there can't exist
certain types of counterexamples is not an impossible problem. It's
already been done before. See Lagarias' 1985 paper, more than twenty
years before your paper. He has a result there which says "There are no
nontrivial cycles with period length less than 275,000."

There is a fundamental difference between the two problems. The Collatz
conjecture is about proving equality, T^k(n)=1. Proving that there
can't exist certain types of counterexamples (cycles) is about proving
inequality, T^k(n) != T^m(n). Proving inequality is an easier problem
than proving equality, as you only have to show that one side of the
inequality has a characteristic that the other side does not have; you
don't have to explicitly compute both sides, arithmetically or
algebraically. (That's why many difficult problems similar to FLT turn
out to be solvable, because they are about proving inequality.)

But if I proved that EVERY set of POSSIBLE counterexamples
is empty, then wouldn't that prove the conjecture true?
So if you conceed that a proof without computation exists
for some counterexample types, why can't there be similar
proofs for all types of counterexample?

And although my paper proves that one of the sets of possible
counterexamples contains at least one element making such a
proof impossible, it doesn't rule out a proof that the set only
contains one element or that it only contains negative elements.

A partial proof without computation renders invalid your claim
that a proof must contain computation.

This argument is of the same flavor as someone claiming that since he
can bisect an angle with a straightedge and compass, it doesn't rule
out the possibility that he can trisect an angle with only a
straightedge and compass, so therefore, the proof of the theorem that
angle trisection is impossible must be an invalid proof.

Not surprisingly, you've got the analogy ass-backwards.
Angle trisection has been proved to be impossible
because it requires an infinite number of steps, just
as your computation method requires testing an infinite
number of cases. You, therefore, assume that angle
BI-section also requires an infinite number of steps
and must therefore also be impossible. You then
conclude that all divisions of angles are therefore
impossible.


The fact is that you haven't proven that *all* types of counterexamples
are impossible,

I never said I proved "all".

only *some* types of counterexamples are impossible,
which is not news at all.

Again, the point sails right over your head. The point being
that the proofs did NOT require an infinite number of steps.
For your argument that Collatz is unprovable to stand, you
must show that ALL proofs would require an infinite number
of steps.

But you can no more do that than you can prove that
bisecting an angle requires an infinite number of steps.
And if you tried to show that, you would be called a crank
because there is a FINITE proof of angle bisection.

And although my paper doesn't complete the proof of
the conjecture, it suffices to demonstrate your premise
that a proof MUST contain an infinite number of steps
is complete rubbish.

It is possible to say plenty of things about
what the Collatz sequence cannot do; however, predicting exactly where
the Collatz sequence will end up after k iterations is a
computationally irreducible problem.

You claim that, but you haven't proved it. In fact, what evidence
we have indicates just the opposite.

(Read Wolfram's book "New Kind of
Science" to learn about this concept.)

I am a Free Thinker and accept no dogma. I get out my Python
programs and actually do real research, not just parrot some
hand-waving argument I found in some controversial book.

I put no stock in your appeal to authority fallacy.


Nobody in this thread seems to get this now.

Have you ever considered why?

But I can guarantee that
not so far in the future, every serious mathematician will understand
this principle.

Everyone already understands the principle. What everyone
objects to is your inappropriate application to this problem.


Craig




Your whole paper is just a house of cards and I've just
pulled the tablecloth out from under it.

Now, that is surprising! A mathematics
discovery for which there is no logical explanation!


Godel, of course, was a logician. Tarski, as well.

If one thinks of mathematical logic as simply a set
of formalist rules, one might as well think of
arithmetic as accounting.

Computing doesn't hold much fascination for me, though
I appreciate it very much, because I hate to
calculate. Similarly, logical advances shorten some
paths and create new ones. In the end, I think we can
agree that we're all on our way to the same destination.

Tom


"Wrong" is not a particularly good choice of
words, I think. One can be wrong in the proof
strategy
one chooses, to the extent that the strategy is
shown
ineffective. Hilbert is influential, however,
because
he was right, in principle, about what mathematics
is
and what it can do. He saw far past the obvious --
I
find the Hilbert space much more of a "heaven" than
Hilbert claimed that Cantor created.


My own path leads to the independence of
language
and meaning. As Popper said, "I do not believe
in belief." I think, as did he, that what will
be
left when belief has been rendered entirely
irrelevant to a mathematical result, is
something
like Tarski's correspondence theory of truth.
Something
very naturally graspable ...

Can you give any references to this idea?<<


The Stanford Encyclopedia does a good job of
putting it
in negative terms:


http://www.seop.leeds.ac.uk/archives/spr2004/entries/t
ruth-correspondence/#1

When I speak of the correspondence theory of truth,
however, I mean as it was rehabilitated by Karl
Popper
(in Objective Knowledge: an evolutionary approach)
for
scientific application, sans philosophical baggage.
This
view goes by the broad name of metaphysical
realism, in
which verisimilitude ("truth-likeness") takes the
place
of the Platonic idea of truth.

Tom

Thank you Tom, I'll check out that link.

Craig


.

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