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August 9th, 2018, 07:01 AM   #21
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Originally Posted by zylo View Post
Objectivity doesn't matter, Impression does.

The NYT and Fox announce a remarkable new thesis, and Romsek proves absolutely and irrevocably that it is wrong, but nobody sees his proof, other than a few colleaqjes. Is the thesis correct?

Denis, thanks for the publicity.
Ah. This explains zylo's posts here. He believes that "true" or "false", what is fact and what is not, is a matter of majority opinion.
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August 9th, 2018, 03:22 PM   #22
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Originally Posted by zylo View Post

The NYT and Fox announce a remarkable new thesis, and Romsek proves absolutely and irrevocably that it is wrong, but nobody sees his proof, other than a few colleagues. Is the thesis correct?
if a tree falls down in a forest...
Thanks from Denis

Last edited by skipjack; August 9th, 2018 at 07:18 PM.
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August 9th, 2018, 04:34 PM   #23
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Originally Posted by Country Boy View Post
Ah. This explains zylo's posts here. He believes that "true" or "false", what is fact and what is not, is a matter of majority opinion.
But it is!! The concept of what constitutes mathematical proof is historically contingent. Gauss gets credit for the first complete proof of the Fundamental Theorem of Algebra, but his proof would not be regarded as sufficiently rigorous by today's standards.

A mathematical proof is whatever a preponderance of working professional mathematicians say it is.

Case in point, Mochizuki's claimed proof of the abc conjecture. Briefly, a known genius mathematician has been working for a decade on his own area of research that nobody else understands. One day he drops 500 new pages in four related papers, and claims to have proved the abc conjecture, a famous and highly difficult conjecture in number theory. To give an example, if abc is true then Fermat's Last Theorem is a trivial consequence.

This happened several years ago. Since then, nobody understands the proof. People have been working on it. People have been complaining that Mochizuki hasn't been forthcoming enough in explaining himself. Mochizuki expresses disappointment that everyone is so lazy and dim. This has been going on for years. A handful of specialists are slowly making progress understanding his ideas.

Is this a proof or not?

Moral of the story: Proof is a social construct. Something is proved when mathematicians say it is. There's no objective standard.

For anyone who hasn't heard about this amazing (and ongoing) story, someone wrote up a very nice popular article about it in 2013. http://projectwordsworth.com/the-paradox-of-the-proof/

There are more technical summaries of the situation as it evolves that can be found via Google.

Mochizuki, by the way, graduated from Princeton at age 16 and got his doctorate under one of the world's leading number theorists. That's why he's taken seriously and not labelled a crank. Again, social factors. Nobody can evaluate his math, not even specialists.

Last edited by Maschke; August 9th, 2018 at 04:37 PM.
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August 10th, 2018, 01:17 AM   #24
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Originally Posted by Maschke View Post
But it is!! The concept of what constitutes mathematical proof is historically contingent. Gauss gets credit for the first complete proof of the Fundamental Theorem of Algebra, but his proof would not be regarded as sufficiently rigorous by today's standards.

A mathematical proof is whatever a preponderance of working professional mathematicians say it is.

Case in point, Mochizuki's claimed proof of the abc conjecture. Briefly, a known genius mathematician has been working for a decade on his own area of research that nobody else understands. One day he drops 500 new pages in four related papers, and claims to have proved the abc conjecture, a famous and highly difficult conjecture in number theory. To give an example, if abc is true then Fermat's Last Theorem is a trivial consequence.

This happened several years ago. Since then, nobody understands the proof. People have been working on it. People have been complaining that Mochizuki hasn't been forthcoming enough in explaining himself. Mochizuki expresses disappointment that everyone is so lazy and dim. This has been going on for years. A handful of specialists are slowly making progress understanding his ideas.

Is this a proof or not?

Moral of the story: Proof is a social construct. Something is proved when mathematicians say it is. There's no objective standard.

For anyone who hasn't heard about this amazing (and ongoing) story, someone wrote up a very nice popular article about it in 2013. The Paradox of the Proof | Project Wordsworth

There are more technical summaries of the situation as it evolves that can be found via Google.

Mochizuki, by the way, graduated from Princeton at age 16 and got his doctorate under one of the world's leading number theorists. That's why he's taken seriously and not labelled a crank. Again, social factors. Nobody can evaluate his math, not even specialists.
I agree 100% with your post. But it needs to be said that nowadays there is a very good consensus on what a proof is.
Let me clarify, a proof is a certain collection of statements, each of which are connected by inference rules. Any computer or human familiar with the limited set of axioms and inference rules, can verify whether a proof is correct. So in that sense, math is not subjective.

Of course, nobody ever does it that way. People are merely content with showing that it can be done. And this is where the subjective elements come in, and where a majority opinion can agree a proof is correct, even though it is not correctly spelled out.

But in principle, it can be done. If I ever publish a proof, and nobody agrees with me, I can always write it down in basic chunks of elementary statements and inference rules. I can then let a computer check it, and everybody will HAVE to agree. It will take a god awful long time though, but in principle it can be done.

And it HAS been done before:

Quote:
In 1611 Johannes Kepler had conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014 Hales announced the completion of a formal proof using automated proof checking, removing any doubt.
from wikipedia: https://en.wikipedia.org/wiki/Sphere_packing

So if people disagree with zylo, or with the abc-formula proof, they can always put in a formal proof and let the computer check it. Awful work, but it is a certain one to make sure everybody accepts you as correct. It's the only proofs in math that are not subjective, but objective.

Also see, metamath: Home Page - Metamath
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August 23rd, 2018, 01:36 AM   #25
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You should be more specific, this is an interesting post but the question is too general. Do you mean Asian students in Asia or Asian students who go abroad to study? Or second generation Asian abroad? Depending on the group the answer can be quite different.
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