https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/

NUMBER THEORY
Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

By ERICA KLARREICH

September 20, 2018

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

dv said:

https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/

NUMBER THEORY
Titans of Mathematics Clash Over Epic Proof of ABC Conjecture

By ERICA KLARREICH

September 20, 2018

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years.

“ABC conjecture” sounds vaguely familiar from somewhere.

> In a report posted online today, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a “serious, unfixable gap” within a mammoth series of papers by Shinichi Mochizuki.

> Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct.

> A line of reasoning near the end of the proof of “Corollary 3.12” in Mochizuki’s third of four papers is fundamentally flawed.

Huh? “A line of reasoning”?

> Mochizuki couldn’t convince Scholze and Stix that his argument was sound, but they couldn’t convince him that it was unsound. Mochizuki has now posted Scholze’s and Stix’s report on his website, along with several reports of his own in rebuttal.

LOL. If the greatest pure mathematicians of today spend all their time debating whether a proof is correct, it just goes to prove that pure mathematics has been heading in the wrong direction since Hilbert defeated Brouwer. Hilbert himself bolstered Brouwer’s position enormously by publishing conjectures.

mollwollfumble said:

LOL. If the greatest pure mathematicians of today spend all their time debating whether a proof is correct, it just goes to prove that pure mathematics has been heading in the wrong direction since Hilbert defeated Brouwer. Hilbert himself bolstered Brouwer’s position enormously by publishing conjectures.

This is a very unusual case, perhaps unique, due to the author’s obscurantism. He appears to have buried a dodgy logical leap under 1000 pages of packing foam.

I’m going to expand on the Hilbert vs Brouwer debate, as I see it (which is probably totally different to the way that Brouwer saw it), for helping me to sort things out in my own head.

mollwollfumble’s (mis-)interpretation of Brouwer.

• A conjecture is more valuable than a proof.
• If no counterexamples of a conjecture can be found in about a fortnight then that conjecture can be accepted as true until a counterexample is found.
• Even a conjecture known to be false is valuable.

To illustrate the third point, consider the two statements:
Statement A: pi = 3.14159
Statement B: pi is irrational
Statement A is provably false and statement B is provably true. But from a computational point of view, the false statement “pi = 3.14159” is more valuable than that true statement “pi is irrational”.

To take another illustration.

mollwollfumble said:

I’m going to expand on the Hilbert vs Brouwer debate, as I see it (which is probably totally different to the way that Brouwer saw it), for helping me to sort things out in my own head.

mollwollfumble’s (mis-)interpretation of Brouwer.

• A conjecture is more valuable than a proof.
• If no counterexamples of a conjecture can be found in about a fortnight then that conjecture can be accepted as true until a counterexample is found.
• Even a conjecture known to be false is valuable.

To illustrate the third point, consider the two statements:
Statement A: pi = 3.14159
Statement B: pi is irrational
Statement A is provably false and statement B is provably true. But from a computational point of view, the false statement “pi = 3.14159” is more valuable than that true statement “pi is irrational”.

To take another illustration.

Okay but in this case what we are talking about is an element of a proof.
Quite probably, the ABC conjecture is correct. But that doesn’t mean that Mochizuki’s proof is correct.

mollwollfumble said:

I’m going to expand on the Hilbert vs Brouwer debate, as I see it (which is probably totally different to the way that Brouwer saw it), for helping me to sort things out in my own head.

mollwollfumble’s (mis-)interpretation of Brouwer.

• A conjecture is more valuable than a proof.
• If no counterexamples of a conjecture can be found in about a fortnight then that conjecture can be accepted as true until a counterexample is found.
• Even a conjecture known to be false is valuable.

To illustrate the third point, consider the two statements:
Statement A: pi = 3.14159
Statement B: pi is irrational
Statement A is provably false and statement B is provably true. But from a computational point of view, the false statement “pi = 3.14159” is more valuable than that true statement “pi is irrational”.

To take another illustration.

I’m going to have to do some background research on this, but I disagree that the conjecture pi = 3.14159 is valuable.

The approximation pi = 3.14159 is very valuable, but it isn’t a conjecture.

What does “coprime” mean?

The Rev Dodgson said:

What does “coprime” mean?

Two numbers are coprime if they have no common factors (other than 1).

8 and 21 are coprime, eg.

The linked article is worth reading in its entirety.

That was an interesting article. i vaguely remember some hoohaa when Mochizuki released his notes and hadn’t heard anything about it since.

Anyway, pretty soon I’m sure we’ll have a computer program that can analyse any series of logical statements and say whether they are valid or not.

Then we’ll only need to prove that the logic of the program is correct.

The Rev Dodgson said:

Anyway, pretty soon I’m sure we’ll have a computer program that can analyse any series of logical statements and say whether they are valid or not.

Then we’ll only need to prove that the logic of the program is correct.

Of course, I have written an instructive little story along these lines before:

What the Tortoise Said to Achilles

>No expert who claims to understand the arguments has succeeded in explaining them to any of the (very many) experts who remain mystified.

Sounds like they need to give the hand-waving a rest and just use a calculator.

The Rev Dodgson said:

Anyway, pretty soon I’m sure we’ll have a computer program that can analyse any series of logical statements and say whether they are valid or not.

Then we’ll only need to prove that the logic of the program is correct.

just run it through the program.

I have a question. But first some definitions.

• Kepler’s discovery (1609 to 1619) that planets have elliptical orbits is applied mathematics, not a conjecture in pure mathematics or a proof in pure mathematics.
• Claims by several people (late 1660s) that the inverse square law of gravity gave Kepler’s elliptical orbits is a conjecture in pure mathematics.
• Newton’s Principia (1686) included a proof in pure mathematics that the inverse square law of gravity gives Kepler’s elliptical orbits.

With that in mind, my question is:

When was the last year in which a proof in pure mathematics helped engineering?

mollwollfumble said:

I have a question. But first some definitions.

• Kepler’s discovery (1609 to 1619) that planets have elliptical orbits is applied mathematics, not a conjecture in pure mathematics or a proof in pure mathematics.
• Claims by several people (late 1660s) that the inverse square law of gravity gave Kepler’s elliptical orbits is a conjecture in pure mathematics.
• Newton’s Principia (1686) included a proof in pure mathematics that the inverse square law of gravity gives Kepler’s elliptical orbits.

With that in mind, my question is:

When was the last year in which a proof in pure mathematics helped engineering?

FFT algorithm done in the mid 60s?

dv said:

mollwollfumble said:

LOL. If the greatest pure mathematicians of today spend all their time debating whether a proof is correct, it just goes to prove that pure mathematics has been heading in the wrong direction since Hilbert defeated Brouwer. Hilbert himself bolstered Brouwer’s position enormously by publishing conjectures.

This is a very unusual case, perhaps unique, due to the author’s obscurantism. He appears to have buried a dodgy logical leap under 1000 pages of packing foam.

Don’t we all. I know my PhD thesis did.

sibeen said:

mollwollfumble said:

I have a question. But first some definitions.

• Kepler’s discovery (1609 to 1619) that planets have elliptical orbits is applied mathematics, not a conjecture in pure mathematics or a proof in pure mathematics.
• Claims by several people (late 1660s) that the inverse square law of gravity gave Kepler’s elliptical orbits is a conjecture in pure mathematics.
• Newton’s Principia (1686) included a proof in pure mathematics that the inverse square law of gravity gives Kepler’s elliptical orbits.

With that in mind, my question is:

When was the last year in which a proof in pure mathematics helped engineering?

FFT algorithm done in the mid 60s?

I surprised FFT was so recent.

There must be some recent pure mathematics in Finite Element Analysis and fast solution of large sets of sparse simultaneous equations.

What about Monte Carlo methods and the like? Must be some recent stuff there.

And probabilistic stuff in general.

But for my line of work I think the most recent conjecture would be:

As hangs the flexible line,
so but inverted will stand the rigid arch.

R. Hooke 1705.

The Rev Dodgson said:

sibeen said:

mollwollfumble said:

I have a question. But first some definitions.

• Kepler’s discovery (1609 to 1619) that planets have elliptical orbits is applied mathematics, not a conjecture in pure mathematics or a proof in pure mathematics.
• Claims by several people (late 1660s) that the inverse square law of gravity gave Kepler’s elliptical orbits is a conjecture in pure mathematics.
• Newton’s Principia (1686) included a proof in pure mathematics that the inverse square law of gravity gives Kepler’s elliptical orbits.

With that in mind, my question is:

When was the last year in which a proof in pure mathematics helped engineering?

FFT algorithm done in the mid 60s?

I surprised FFT was so recent.

There must be some recent pure mathematics in Finite Element Analysis and fast solution of large sets of sparse simultaneous equations.

What about Monte Carlo methods and the like? Must be some recent stuff there.

And probabilistic stuff in general.

But for my line of work I think the most recent conjecture would be:

As hangs the flexible line,
so but inverted will stand the rigid arch.

R. Hooke 1705.

I’m sure there were FFT algorithms well before the 60s, but I do know that the algorithm that is widely used now days was developed back in the 60s. If the blokes who had come up with it had been able to patent it they would have ended up very, very rich. :)

The Rev Dodgson said:

But for my line of work I think the most recent conjecture would be:

As hangs the flexible line,
so but inverted will stand the rigid arch.

R. Hooke 1705.

Pfft, bit of hyperbole there.

sibeen said:

The Rev Dodgson said:

But for my line of work I think the most recent conjecture would be:

As hangs the flexible line,
so but inverted will stand the rigid arch.

R. Hooke 1705.

Pfft, bit of hyperbole there.

More catenary than hyperbole, I think.

The Rev Dodgson said:

sibeen said:

The Rev Dodgson said:

But for my line of work I think the most recent conjecture would be:

As hangs the flexible line,
so but inverted will stand the rigid arch.

R. Hooke 1705.

Pfft, bit of hyperbole there.

More catenary than hyperbole, I think.

Yeah, I know that but was using poetic license – ie, lying.

oh my cosh, surely with all those brains you 2 could find some way to tie them together