- #1

alexmahone

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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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- MHB
- Thread starter alexmahone
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- #1

alexmahone

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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?

- #2

tkhunny

- 256

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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

alexmahone

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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

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- 810

Except that the sum rule \(\displaystyle S = \frac{a_0}{1 - r}\) for an infinite geometric seriesThe proof has been checked only by finite number of people, so we can't be 99.999...% sure it's right.

-Dan

- #5

MountEvariste

- 87

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

-Dan

- #6

S.G. Janssens

Science Advisor

Education Advisor

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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.

- #7

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

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

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

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

MountEvariste

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- #9

tkhunny

- 256

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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".

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