Re: Another Reason Why Collatz is Unprovable


Craig Feinstein wrote:
Proginoskes wrote:
Craig Feinstein wrote:
T.H. Ray wrote:

T.H. Ray wrote:
Craig Feinstein wrote:


My proof that Collatz is unprovable proves that all methods including
proof by induction and proof by contradiction are useless for trying to
prove the Collatz Conjecture. The people who think that my proof is
invalid because it doesn't explicitly mention these types of techniques
by name are talking just like the people who lived a little after
Galois' time and said that his impossibility proof for finding roots of
5th degree polynomials was invalid because it didn't explicitly
consider every possible formula in terms of elementary arithmetic
operations for the roots of a general 5th degree polynomial, but he
only considered formulas which fit into his limited model of possible
formulas. The only difference between these two proofs is that my
result is simple and obvious and can be easily understood with a little
common sense, while Galois' proof requires a sophisticated brilliant
argument which requires a lot of time to understand.<<

Don't you mean Abel's Impossibility Theorem (which is
consistent with Galois theory)?

Thank you for pointing this out to me. I found this:

http://en.wikipedia.org/wiki/Abel-Ruffini_theorem
http://en.wikipedia.org/wiki/Galois_theory

I didn't know that this was known before Galois. I remember learning it
differently. You learn something everyday. So I take my analogy back
and replace Galois' name with Ruffini, since he was first.<<

Speaking for myself, I am partial to W.R. Hamilton's
commentary on Abel's result. His 1839 paper to the
Royal Irish Academy is very compact and
eloquent, IMO.


The depth of that result lies in demarcating arithmetic from analysis. I
can't see that it has anything to do with proof strategy,
as you imply. Proof methods and calculating methods
are not identical. Even Chaitin, the most enamored of
computing power that I know, acknowledges the role of
philosophy ("digital philosophy") in proofs.

Do you, like Kronecker, think that arithmetic is all
there is to mathematics?


Feinstein:

If I had lived in Kronecker's time, I would have probably sided with
him in the debate against Cantor - I don't think actual infinities have
any place in mathematics. (Only potential infinities, as Aristotle
distinguished them.) However, now that the genie is out of the bottle,
there's nothing I can do to stop it.

I would agree with Kronecker that "G-d created
numbers. Everything else is the work of man."

Craig

Does that mean that G-d didn't create analysis? :-)

In my opinion, Satan created Math Analysis.

After going through Abstract Algebra for the first time, I thought it
was a "communist plot" (starting with the part involving Cayley
subgroups, I think).

You were right. Where do you think the Rubik's Cube came from? :-)


There are three branches of
math: Applied, pure, and impure. Mathematical Analysis is an example of
impure math :-)

Where does Graph Theory fall?

Pure math, as long as it doesn't involve infinite graphs.

There are lots of people who would say that it's partially applied. The
Internet that this response is travelling over is a network, which is
one of the basic applications of graph theory.

Of course, that depends on your definition of "pure math".

--- Christopher Heckman

Any
mathematics in which the focus of the study is understand some infinite
object is an impure math. See
http://groups.google.com/group/sci.math/browse_frm/thread/2c82eae5ce4b4535/2669c6a889930b2e?lnk=st&q=cafeinst+infinity+doesn't+exist&rnum=1&hl=en#2669c6a889930b2e
for a parody I wrote discussing this matter.

Craig


--- Christopher Heckman

(Kronecker's actual statement is that G-d created the
integers, or the natural numbers ...)

Creation is a very risky business, though. It often
misbehaves.

Absolutely. But that's what makes it interesting.


Tom

.

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