Could there be an error in the proof of the Poincare conjecture?

[alexmahone]

When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it contains no mistakes increases. But does it ever reach 100%? After all, the referees and checkers are human as well and they could theoretically have overlooked some subtle flaw in the proof.

So, I claim that we can be almost certain that the Poincare conjecture has been proved, but in theory we can never be 100% sure.

Comments?

 

[tkhunny]

When Grisha Perelman submitted his proof of the Poincare conjecture, he may have been reasonably sure that it contained no mistakes. But he could not have been 100% sure as he is, after all, human. Each time it was checked, say by the referee of an academic journal, the probability that it contains no mistakes increases. But does it ever reach 100%? After all, the referees and checkers are human as well and they could theoretically have overlooked some subtle flaw in the proof.

So, I claim that we can be almost certain that the Poincare conjecture has been proved, but in theory we can never be 100% sure.

Comments?

1) 0.99999... = 1

2) "Conjecture" - Will its name change when it is proven? As a conjecture, it is, by definition, unproven.

 

[alexmahone]

1) 0.99999... = 1

The proof has been checked only by finite number of people, so we can't be 99.999...% sure it's right.

 

[topsquark]

The proof has been checked only by finite number of people, so we can't be 99.999...% sure it's right.

Except that the sum rule \(\displaystyle S = \frac{a_0}{1 - r}\) for an infinite geometric series is a proven fact. So 0.99999... = 1 is a certainty.

-Dan

 

[MountEvariste]

Except that the sum rule \(\displaystyle S = \frac{a_0}{1 - r}\) for an infinite geometric series is a proven fact. So 0.99999... = 1 is a certainty.

-Dan

I'm confused now.

 

[S.G. Janssens]

So, I claim that we can be almost certain that the Poincare conjecture has been proved, but in theory we can never be 100% sure.
Comments?

This is a difficult claim to substantiate mathematically. How would you model this probabilistically (sample space, event space, probability measure) such that terms such as "almost certain" and the "law of large number of reviewers" become precisely defined and formulated?

I'm confused now.

An infinite (i.e. limiting) decimal expansion is well-defined in terms of a geometric series.

On a more "soft" level:

Mathematics is a human activity, and so is peer review. However, the bigger the claim, the more numerous and scrutinous the reviewers will be. In general, errors will always remain possible, also because reviewing is a "ungrateful" task: If a reviewer checks a tedious argument, he may spend (possibly a lot of) time on it, while the author(s) reap the benefits - no matter the outcome of the check.

 

[topsquark]

I'm confused now.

Here's a derivation. Please let us know if you have any difficulties with it.

\(\displaystyle 0.99999... = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \text{ ...}\)

\(\displaystyle = \left ( \frac{9}{10} \right ) \left ( 1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \text{ ...} \right )\)

This is a geometric series with \(\displaystyle a_0 = \frac{9}{10}\) and \(\displaystyle r = \frac{1}{10}\)

So
\(\displaystyle S =\frac{9}{10} ~ \sum_{n = 0}^{\infty} \left ( \frac{1}{10} \right ) ^n = \frac{\frac{9}{10}}{1 - \frac{1}{10}}\)

\(\displaystyle = \frac{\frac{9}{10}}{\frac{9}{10}} = 1\)

-Dan

 

[MountEvariste]

I know that $0.{\overline 9} =1.$ I was confused by how the conversation got to whether or not $0.{\overline 9} =1$ is proven/certainty! (Rofl)

 

[tkhunny]

I know that $0.{\overline 9} =1.$ I was confused by how the conversation got to whether or not $0.{\overline 9} =1$ is proven/certainty! (Rofl)

1) Set the level certainty that satisfies you.
2) Find enough reviewers.

"Certainty" in this mortal world is only an intellectual construct. Search more for "sufficiency".