Turing Computer

[69dodge]

About that rope and pulley computer. . .

Robert L. Ripley of Charles Fort College?
um. . .

the Pulleg Mountains?
Yes, indeed.

And doesn't "Apraphul" sound remarkably like "April fool"?
(The article appeared in Scientific American, April 1988.)

It's a good article. But its purpose is "to introduce readers to the principles of digital computing without resorting to tiny and mysterious electronic components." It's not meant to be taken literally as historical fact.

 

[SFB]

About that rope and pulley computer. . .

Robert L. Ripley of Charles Fort College?
um. . .

the Pulleg Mountains?
Yes, indeed.

And doesn't "Apraphul" sound remarkably like "April fool"?
(The article appeared in Scientific American, April 1988.)

It's a good article. But its purpose is "to introduce readers to the principles of digital computing without resorting to tiny and mysterious electronic components." It's not meant to be taken literally as historical fact.

Yes, indeed! Thanks

 

[DanishDynamite]

About that rope and pulley computer. . .

Robert L. Ripley of Charles Fort College?
um. . .

the Pulleg Mountains?
Yes, indeed.

And doesn't "Apraphul" sound remarkably like "April fool"?
(The article appeared in Scientific American, April 1988.)

It's a good article. But its purpose is "to introduce readers to the principles of digital computing without resorting to tiny and mysterious electronic components." It's not meant to be taken literally as historical fact.

Nice call, 69dodge!Guess I should read my google hits a little more closely before posting.

Still, the basic idea that a computer could be made out of toilet paper and paper clips (or whatever) still holds.

 

[xouper]

SFB: Though not digital in nature, the abacus, often considered the first computer, still predates the rope and pulley device.

Why is an abacus not digital? And why is an abacus considered a computer, since all it does is store numbers and does not do any computations?

 

[xouper]

Peter Soderqvist: ... Goldbach 's conjecture is one formally undecidable statement! Goldbach's conjecture is a truth mathematical statement, because there is no counterexample, but still unprovable because Goldbach's conjecture have no theorem!

Am I reading this right that you say Goldbach's conjecture is unprovable or undecidable? When did this happen?

 

[Dymanic]

Originally posted by xouper

Am I reading this right that you say Goldbach's conjecture is unprovable or undecidable? When did this happen?

I can't decide. It's a de-lemma.

As I see it there are three possibilities:

1) The Goldbach conjecture is false. This would require but a single counterexample to establish.

2) The Goldbach conjecture is true, and (though a proof has yet to be found) provable.

3) The Goldbach conjecture is true, but unprovably so. That is, undecidable within any known (or yet to be created) system of mathematics. (It cannot be false and undecidable.)

We also cannot distinguish a priori between 1 and 2 (i.e., the decidability is undecidable).

 

[xouper]

From http://plus.maths.org/issue2/xfile/

You can explore the Goldbach conjecture yourself with this Goldbach calculator. Simply enter an even integer, n, greater than 4 and the calculator will find all the Goldbach pairs.

<FORM action=http://plus.maths.org/issue2/xfile/goldbach.cgi method=post target="_blank">n = <INPUT value=42 name=num> (4 < n < 10,000) <INPUT type=submit value="Find pairs"></FORM>

 

[Peter Soderqvist]

A conjecture is not a theorem!
Your Goldbach calculator is a truth theorem machine, but incomplete, because the calculator is a finite tool, but all possible even integers are infinite! In short, no matter how many even integers you have tried out, there are still an infinite amount of even integers left, or untried! Goldbach conjecture is truth, but unprovable! Goldbach conjecture is undecidable from a computational point of view, since every possible even integer cannot be tried out, because; finite is weaker than infinite!

Hofstadter, Gödel, Escher, Bach
All consistent axiomatic formulations of number theory include undecidable propositions. Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...
http://www.miskatonic.org/godel.html

 

[xouper]

Peter Soderqvist: A conjecture is not a theorem!

Agreed. No one has claimed it was. Certainly not me.

Your Goldbach calculator is a truth theorem machine, but incomplete, because the calculator is a finite tool, but all possible even integers are infinite!

Quite true. I posted the calculator as a fun toy, not as a proof of anything. It only goes up to 10,000.

In short, no matter how many even integers you have tried out, there are still an infinite amount of even integers left, or untried!

Of course.

Goldbach conjecture is truth, but unprovable!

Please show the proof that Goldbach's conjecture is unprovable. Seems to me that would be at least as noteworthy as a proof of the conjecture itself.

Goldbach conjecture is undecidable from a computational point of view, since every possible even integer cannot be tried out, because; finite is too weak to compute infinite!

Agreed, but that is not the only way to prove statements about the natural numbers. There are an infinite number of primes, but this can be proven without a computer. I can also prove without a computer that all primes are of the form either 6N+1 or 6N-1, even though there are an infinite number of primes.

Please show us how you know Goldbach's conjecture cannot be proven?


Edited to add: Gödel did not prove that Goldbach's conjecture is not provable.

 

[Dymanic]

Originally posted by xouper

You can explore the Goldbach conjecture yourself with this Goldbach calculator.

That's cool.

Originally posted by Peter Soderqvist

Goldbach conjecture is truth, but unprovable!

I tend to agree that it probably is true. But you are overlooking the very real possibility that some very large counterexample exists -- as well as the likewise very real possibility that someone may eventually produce a proof.

 

[LW]

I can also prove without a computer that all primes are of the form either 6N+1 or 6N-1, even though there are an infinite number of primes.

Better make that all sufficiently large primes...

Otherwise you have two nasty counterexamples.

 

[LW]

Your Goldbach calculator is a truth theorem machine, but incomplete, because the calculator is a finite tool, but all possible even integers are infinite!

I guess that you have a typo here and you intended to write that the set of all even integers is infinite. Each individual even integer is finite.

In short, no matter how many even integers you have tried out, there are still an infinite amount of even integers left, or untried! Goldbach conjecture is truth, but unprovable! Goldbach conjecture is undecidable from a computational point of view, since every possible even integer cannot be tried out, because; finite is weaker than infinite!

As xouper already wrote, your argument doesn't hold water. Moreover, it has already been proven [I can't remember by whom and my number theory textbook is not at hand right now] that each even integer is the sum of two numbers x and y where x is prime and y has at most two non-trivial factors. [My textbook didn't include the proof as it is rather heavy weight stuff].

 

[Skeptoid]

... Otherwise you have two nasty counterexamples.

I can think of three.

 

[LW]

I can think of three.

Umm... Are you sure that you are using the conventional definitions of prime numbers and modular arithmetic?

 

[Skeptoid]

Sorry, not sure what I was thinking. Time to go to bed.

 

[Peter Soderqvist]

That Goldbach conjecture is unsolved or still an open question can be found here!

From Wikipedia, the free encyclopedia. Goldbach's conjecture
Goldbach's Conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:
Every even number greater than 2 can be written as the sum of two primes. http://en2.wikipedia.org/wiki/Goldbach's_conjecture

School of Mathematics and Statistics University of St Andrews, Scotland
Goldbach did important work in number theory, much of it in correspondence with Euler. He is best remembered for his conjecture, made in 1742 in a letter to Euler and still an open question, that every even integer greater than 2 can be represented as the sum of two primes. Goldbach also conjectured that every odd number is the sum of three primes. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Goldbach.html

 

[xouper]

LW: Better make that all sufficiently large primes... Otherwise you have two nasty counterexamples.

GAACK. There must be a gremlin in the system that edited my post while I wasn't looking and removed my usual qualifier for primes greater than 3.

 

[DanishDynamite]

GAACK. There must be a gremlin in the system that edited my post while I wasn't looking and removed my usual qualifier for primes greater than 3.

Yaadah, yaadah, yaadah. You are clearly a troll trying to evade a clear ◊◊◊◊ up.

 

[xouper]

Peter Soderqvist: That Goldbach conjecture is unsolved or still an open question can be found here!

Yes, of course. We already knew that part of it. However, the word "unsolved" does not mean the same as "unsolvable". The phrase "not yet proven" does not mean the same as "unprovable". Perhaps we are having a difficulty with the English language here, which would be quite forgivable. Good thing we aren't having this conversation in Swedish, since your English is far better than my Swedish.

 

[xouper]

DanishDynamite: Yaadah, yaadah, yaadah. You are clearly a troll trying to evade a clear ◊◊◊◊ up.

Some people just cannot be fooled.

I have demonstrated that trivial proof (primes > 3 in the form 6N+-1) so many times, I get sloppy and accidentally omit certain relevant details. I really should be more careful when I make mathematical comments. This board is full of pedants, and I'm one of them.