Widely accepted mathematical results that were later shown to be wrong? (2010) | Hacker News Reader

Widely accepted mathematical results that were later shown to be wrong? (2010)

mathoverflow.net

112 pointsbelter47 comments

It's kind of the opposite scenario (a mathematical result that people are widely skeptical of, but that could be correct), but this reminded me, I wonder what's going on with Mochizuki's proof of the ABC conjecture.

The proof situation is pretty much unchanged. The proof uses a very obscure branch of mathematics, so that few people are capable of a real critique. Those few have mostly said that it's not complete, but Mochizuki and a few others disagree.

It's kind of a philosophical question at this point about what actually comprises a valid proof, if other mathematicians can't make use of it.

So the conjecture is still mostly considered unproven. That could change at any point but there have been no real advances in some time.

> very obscure branch of mathematics

This is an understatement. He developed a completely different branch of mathematics through papers that spanned hundreds of pages with very dense and very different ideas from mainstream mathematics.

If it was just slightly obscure, maybe all but half a dozen mathematicians in the world might be able to understand his work. As it stands, at it's best, his proof is so opaque that even the brightest (Scholze) can only kind of understand it. At it's worst, they've rightfully identified it as flawed and that even Mochizuki himself doesn't understand that it's wrong.

Make proof using computer system, obviously.

Mochizuki is unenthusiastic.

[1] Section 1.12 of https://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logic....

everything relies on consensus anyway

> what's going on with Mochizuki's proof of the ABC conjecture.

The assumption at this point is that it's broken.

The onus is on Mochizuki to either clarify the proof or use the tools of the proof on some other already established problem to show their validity.

Hm one answer mentions some of Euclid’s proofs, but I think you could add that there were also false/widely accepted proofs that his Fifth Postulate could be deduced from the others and was therefore redundant. IIRC, such proofs made subtle use of the geometry being flat, which you (of course) need the Fifth to pin down.

Edit: Wikipedia mentions some such false proofs but doesn’t say how widely accepted they were:

https://en.wikipedia.org/wiki/Parallel_postulate#History

I studied maths at uni 30 years ago and have forgotten it all and I love how completely uncomprehensible all of these examples are if you're not in the field:

The Busemann-Petty problem (posed in 1956) has an interesting history. It asks the following question: if K and L are two origin-symmetric convex bodies in Rn such that the volume of each central hyperplane section of K is less than the volume of the corresponding section of L, does it follow that the volume of K is less than the volume of L: Voln(K)≤Voln(L)?

Its such a dense avalanche of concepts. If you coded up a few examples using actual values to run the calculations in the much-less-dense format of code (say js or python or fortran) how many lines of code would it be? Fifty? Hundreds?

That depends very much on how K and L are defined.

The code to calculate whether any possible cross-section of K is greater than the corresponding cross-section of L is trivial if they are both (hyper)spheres. If they are polyhedra (polytopes), then it would be a lot more involved. And if their boundaries are defined by even more complicated surfaces, it could be fiendishly difficult.

The relationship between declarative statements about infinite families of continuous objects, and imperative code that can be implemented on a discrete computer in finite time, tends to be very very non-trivial.

Aren't you comparing the code for calculations with the jargon of the question? I feel like the proofs would be quite long.

Surely the example which disproves the hypothesis would be smaller than the proof that it is valid for all values?

Could be, but sometimes the description of an object can be quite involved if it's constructed in some bizarre form (e.g., the intersections of some already complicated objects, for instance)

If you read the actual text of that mathoverflow answer you'll see that it was first disproved for dimensions >=12 and that all the rest of the work was in dimensions 3 thru 11.

I absolutely love this question. It cannot be answered!

Just trying to think of what the equivalent in code would be, and so calculating actual values seems like the closest analogue, but I know its not really comparable. I'm grasping for ideas of how can we compare the concept-density of mathematical notation with the concept-density of code?

Unless you are programming in Python, it should be possible in one line.

I also wonder what widely accepted mathematical conjectures were later proven to be wrong (for example, a hypothetical answer would be if it was proved that P = NP, since most computer scientists today believe that P =/= NP).

Analytic number theory has seen a fair number of such conjectures. The first that comes to mind is the Pólya conjecture [0]. The conjecture stated that for any positive integer N > 2, there are at least as many positive integers less than N with an odd number of prime factors as there are with an even number. The smallest counterexample is N = 906,150,258. [0] https://en.m.wikipedia.org/wiki/P%C3%B3lya_conjecture

I have a half answer: not a result, but a proof that was found to be invalid.

And it's a famous proof too: Euclid's very first proposition: "On a given straight line to construct an equilateral triangle."

Euclid's proof assumes that two circles intersect, but there is no axiom to ensure this. There is no Principle of Continuity.

I think this is holding Euclid's work to a higher standard that didn't exist at that time. I believe you're referring to "Proof-checking Euclid"(2019)[0], in which the authors used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I.

Euclid Book I was written 2,300 years ago. I think it's reasonable that some "additional" axioms were occasionally implied. As [0] states, "[that] gap is filled by adding a 'circle–circle' axiom, according to which if circle C has a point inside circle K, and also a point outside circle K, then there is a point lying on both C and K." I'm not sure, but I feel like that might be reasonable to do for a reader of Euclid Book I in 300 B.C.

So is the proof "invalid"? Yeah maybe, according to modern definitions. But I don't think the logic of that part of the proof was actually flawed, just under-presented.

0: https://link.springer.com/content/pdf/10.1007/s10472-018-960...

> There is no Principle of Continuity.

This goes back to the question of what are the axioms and what is a proof.

I guess Euclid would just disagree.

A few years ago I read an article about how solar panels were about to gain efficiency after a widely-accepted "fact" in physics textbooks was shown to be wrong.

If I recall correctly, it was some characteristic of electrons in the atomic structure.

At the moment I can't find a reference. Anybody remember this?

I always found it fascinating for that newton's law of gravity was found false from Einsteins equations or that Einstein equations simplified to newton's for specific conditions.

Safe to say Fermat was wrong about his most famous result, given that he couldn't possibly have had the tools to pull it off.

He was wrong about the proof, not the result.

No, something made him think he had obtained that particular result legitimately, but he was clearly wrong about that. His "result" was just a lucky guess.

The 4-color conjecture. An erroneous proof was believed to be correct for about a decade. (mentioned in the link but is an example that came to mind)

Having only taken math through advanced calculus and differential equations, none of the examples make much sense to me.

But there is the "Monty Hall problem", where many mathematicians believe(d?) the incorrect result.

https://en.wikipedia.org/wiki/Monty_Hall_problem

So - my great beef with the “Monty Hall Problem” is that it is often poorly, comically so, articulated such that one is led to draw the 50-50 conclusion because of the wording.

Then, someone springs the probability for the problem they poorly described and feels proud they fooled us all!

I _hate_ the Monty Hall problem for this reason. It’s not a function of statistics but rather a function of an author’s technical writing ability.

The problem certainly becomes clearer if you state that Monty never opens the door with the prize. But I have to admit that even when presented with the Monty Hall problem extension, 100 doors, with Monty offering 99 chances to change, it was still not "intuitive" to me that one should change.

I don't think it is just poor wording.

And the idea that probabilities can shift based on Monty's knowing where the prize is certainly not intuitive, in part because many of us do not intuit accurately about probability.

What's the correct wording?

I honestly don't see much of an issue with the standard description straight from Wikipedia:

> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I would just point out the obvious that the game show host will never pick the door with the car.

Not quite good enough! You have to make it clear that the host will always "open another door" no matter what, and that the contestant knows that fact. Otherwise, the contestant might reasonably think that he might only be offered the chance to change when the host knows that a change will be the wrong move.

You could say that the wording given kind of implies the always, but it's not completely clear. So, the claim that it's almost always worded poorly is reasonable.

> You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Left unspecified: does the host _always_ follow this exact procedure unconditionally? Does the player know this to be a fact?

Is the hosts algorithm exacly:

1) let the player choose a door 2) open _a_ door that was not the players choice that the host knows contains a goat 3) let the player change choice if they'd like

If it's not, then depending on what the host's algorithm is, then it's not even a question of probability or the odds might differ. Also depending on what the player knows or doesn't it may stop being a question of probability.

saying "Is it to your advantage to switch your choice?" is actually hinting at the correct answer; should be worded more neutrally, simply giving the choice to stay or switch.

most statisticians knew the correct answer, I presume

The difficulty of the Monty Hall Problem is not due to vague language (although some statements of it do suffer from vagueness). It is a genuinely astonishing result that many professional mathematicians, including statisticians, have tripped over. You should read about the story of Marilyn vos Savant and her columns discussing the Monty Hall Problem in Parade magazine, and the associated controversy. E.g. <https://chance.amstat.org/2022/11/monty-hall/>.

"vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"

I’ve been convinced since ‘91 that they Monty Hall problem will eventually get added to the list.

Monty hall is really easy to work out if you consider the 100 door version.

It isn’t trivial if the host has to open 98 or still 1 door though.

I mean the one where you're left with 2 doors.

Yes, but my intuition doesn’t find obvious the jump from open-one to leave-two. These options feel at least equally noteworthy and nothing hints which one you should use.

Well you can also look at opening 95 of the 100.

The important takeaway is that the probability is not 1 divided by the number of doors left. And that is true no matter how many doors he opens, if it's a fixed nonzero number.

If you ask yourself "at what point did the odds stop being 1 in 100", the answer is obviously not "when he opened the last door out of many" or "midway through opening the many doors", so it has to be when he opened the very first door. And if opening the first door out of 90+ changes the odds, it's hard to see a reason why opening the first door out of 1 wouldn't change the odds.

List of common misconceptions > Science, technology, and mathematics > Mathematics: https://en.wikipedia.org/wiki/List_of_common_misconceptions#...

Is the limit of 1/x equal to the limit of 2/x; is ±infinity_1 equal to ±infinity_2?

I'm not sure what you're implying about the limits, but the limit is exactly zero isn't it?

Not sure what you imply the misconception and the correct answer to be in this case.

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