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

In summary: Conjecture" - Will its name change when it is proven? As a conjecture, it is, by definition, unproven.0.99999... = 1The proof has been checked only by finite number of people, so we can't be 99.999...% sure it's right.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 S = \frac{a_0}{1 - r} for an infinite geometric series is a proven fact. So 0.99999... = 1 is a certainty.
  • #1
alexmahone
304
0
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?
 
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  • #2
Alexmahone said:
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.
 
  • #3
tkhunny said:
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.
 
  • #4
Alexmahone said:
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
 
  • #5
topsquark said:
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.
 
  • #6
Alexmahone said:
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?
June29 said:
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.
 
  • #7
June29 said:
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
 
  • #8
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)
 
  • #9
June29 said:
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".
 

1. What is the Poincare conjecture?

The Poincare conjecture is a mathematical problem in topology, proposed by Henri Poincare in 1904. It states that any closed 3-dimensional manifold is topologically equivalent to a 3-sphere.

2. What is the significance of the Poincare conjecture in mathematics?

The Poincare conjecture is considered one of the most important and challenging problems in mathematics. It has implications in various fields such as topology, geometry, and physics. Its proof would also lead to a better understanding of the fundamental structure of space.

3. What is the current status of the Poincare conjecture?

The Poincare conjecture was one of the seven Millennium Prize Problems, announced by the Clay Mathematics Institute in 2000. In 2006, Grigori Perelman presented a proof of the conjecture, which was later verified by other mathematicians. In 2010, Perelman declined the Fields Medal and the $1 million prize for solving the conjecture.

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

It is possible, as with any mathematical proof, that there could be an error in the proof of the Poincare conjecture. However, Perelman's proof has been extensively reviewed and verified by other mathematicians, and no significant errors have been found so far.

5. How does the Poincare conjecture relate to the more general problem of the Geometrization conjecture?

The Poincare conjecture is a special case of the more general Geometrization conjecture, which states that every closed 3-dimensional manifold can be decomposed into pieces that are geometrically similar to one of eight standard geometries. The Poincare conjecture is the special case where the manifold is topologically equivalent to a 3-sphere. Perelman's proof also resolved the Geometrization conjecture.

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