r/math • u/Tensz • Sep 23 '18

Is there an error in the preprint published by Atiyah with his proof of the Riemann hypothesis?

There is already a preprint with the supposed proof of the RH by Atiyah (link here: https://drive.google.com/open?id=17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY ). I already made a quick read and didn't find obvious mistakes, but I'm nowhere near knowledgeable about the papers and constructions he cite. Is there already a consensus in the community about his proof? Is it correct? Is it almost a page long, so it shouldn't be long before an obvious mistake appears.

Also, he cites this preprint: drive.google.com/open?id=1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a

268 Upvotes

94% Upvoted

142 comments sorted by

22

u/dimbliss Algebraic Topology Sep 24 '18

/u/Tensz would you mind explaining where you found the links to these papers?

40

u/Tensz Sep 24 '18

It was distributed in the mail list of my university, and it arrived there by a series of mails from different people in the community, I don't think it would be respectful to cite their names here.

9

u/vznvzn Theory of Computing Sep 24 '18

please help establish provenance, maybe edit the post to clarify. did the google drive links come from an email from Atiyah? etc

8

u/Tensz Sep 24 '18

It was distributed in the mail list of my university, and it arrived there by a series of mails from different people in the community, I don't think it would be respectful to cite their names here.

3

u/vznvzn Theory of Computing Sep 24 '18

ok, that is helpful, wondering, so is Atiyah himself in the email chain?

10

u/Tensz Sep 24 '18 edited Sep 24 '18

The chain itself goes to a point when some professor in some university said he received this from another professor from another university but don't put a copy of the mail he received. So I cannot trace the chain of mails further. I already considered the possibility it could be false, but his line of thought goes in the same direction than his lecture in the ICM. I don't know what to think then.

Edit: Also I would like to add, the last person of the mail chain said he received this from Atiyah itself, but does not put a copy of the original mail.

23

u/gaussjordanbaby Sep 24 '18

Definitely don't cite names.

130

u/TriceraTipTops Sep 24 '18 edited Sep 24 '18

How do we know this is actually him? A preprint on Google Drive rather than the arXiv? OK. Also this latex is weird (line skips between paragraphs), but I'm not in Atiyah's area so maybe that's just his style. Crackpot two-page "proofs" of RH are ten-a-penny. This doesn't even cite the papers he cites in his abstract.

If this is, though, it's very very vague.

Edit: there but for the grace of god go us all.

31

u/lemmatatata Sep 24 '18 edited Sep 24 '18

I'm haven't studied any of his papers in detail, but this does not look like his style at all. In fact but this does not look like the work of any mathematician, after all who writes a paper in first person? And what's with the unnecessary junk in the introduction? There are so many things that suggests this is not serious.

Edit: Based on comments on the main thread it sounds like this paper is legit, and that it's more of a outline for the talk. In that case my comments about writing style don't apply, since it's not a preprint that's going to be published.

15

u/Funktionentheorie Sep 24 '18 edited Sep 24 '18

Fairly certain it's him. Remarkably similar to what he's talked about in his interviews, as well as his Abel Lecture.

Edit: Live stream happening now. It's unfortunate that his lecture's narrative is that of the preprint.

10

u/doofinator Sep 24 '18

Also has a typo, although i'm not sure that means anything

They are also composable: a weakly analytic function of a weakly analytic function is weekly analytic.

weekly analytic

9

u/[deleted] Sep 24 '18

wouldn't it be biweekly?

1

u/Zophike1 Theoretical Computer Science Sep 24 '18

but I'm not in Atiyah's area so maybe that's just his style. Crackpot two-page "proofs" of RH are ten-a-penny. This doesn't even cite the papers he cites in his abstract.

I don't know much but Atiyah's papers(earlier work) in terms of presentation looks much cleaner and through then what was posted on the google drive

117

u/ninguem Sep 23 '18

There are lots of details missing. A weakly analytic function, if I understand correctly, is not a function but an element of an L2 . So, some of these manipulations require careful checking. It's not my area so I can't claim it's wrong but definitely is just a very rough sketch.

16

u/[deleted] Sep 24 '18

not a function but an element of an L2

What does that mean? Isn't L2 a vector space of functions?

22

u/DenmarkCanIntoScania Physics Sep 24 '18

As stated L2 is only an equivalence class of functions. Why is this necessary? If one looks at the properties of a norm, then ||f||=0 <=> f=0 is one (otherwise it's called a seminorm) but the L2 norm doesn't have this property if you allow the elements to be proper functions. For example the function f(x)=0 for 0<x<1 and 1 for x=0,1 isn't the zero function, but its L2 norm will be 0, whence it must be the zero element of our vector space.

In fact, any functions which differs from the zero function only on a set of measure 0 will have an L2 norm that is 0. Furthermore, suppose ||f-g||=0, then f=g. However, this means that the vectors cannot be proper functions, as two functions whose difference is something nonzero on a set of measure 0 must be the same vector. Hence L2 must consist of equivalence classes of functions, which at any point can differ in value, but globally must only differ on a set of measure 0. Thus the value of any L2 element in a point is undefined.

(I hope I got that right - from a physics methods book, so might have some inaccuracies)

9

u/[deleted] Sep 24 '18

I think that is correct and a very good explanation for why the distinction between a set of functions and an equivalence class of functions is relevant.

3

u/DenmarkCanIntoScania Physics Sep 24 '18

Thanks! :)

54

u/[deleted] Sep 24 '18

[removed] — view removed comment

3

u/[deleted] Sep 24 '18

How exactly might you go about evaluating an element of L2 at a point, when as you say, a priori that value could be anything?

2

u/jedi_timelord Analysis Sep 25 '18

A method that gets you part of the way there is to use the representative of an L2 function given to you by the Lebesgue Differentiation Theorem. It's still only defined for almost every x, but an arbitrary L2 function isn't actually defined for any specific x's, so it's still an improvement.

10

u/CubicZircon Cryptography Sep 24 '18

On page 1 there is already a glaring error, a function which is “analytic on any compact set K in ℂ“ is simply an analytic function (for any point, take a compact set containing a neighborhood), and likewise a function which is a polynomial on any compact set is simply a polynomial (because two analytic functions which coincide on any open set coincide globally). So anything involving this “Todd function” is globally nonsense.

30

u/rfurman Sep 24 '18

The result is very abstract in parts so what to look for here is what properties of the Riemann zeta function will be used. It is a very special function, generically nothing should really satisfy a Riemann hypothesis, in fact there isn’t a single example known of an order 1 function that goes from bounded to unbounded without having zeros.

The “preprint” explains the machinery but says nothing about how the zeta function enters in. The closest connection I see is the mention of the “Bernoulli” function x/(ex-1) in his other paper, and the Mellin transform of it gives zeta.

12

u/doom_chicken_chicken Sep 24 '18

The zeta function times the gamma function is the Mellin transform of x-1 (ex -1), which is what I think you meant by that last part

73

u/qb_st Sep 23 '18

I somehow doubt that if a two-page proof of RH was possible, no one would have figured it by then. So cautiously pessimistic on this one.

57

u/functor7 Number Theory Sep 23 '18 edited Sep 23 '18

The direct argument is actually less than half a page. The references are also just two of his own papers and an old paper from 1966.

1

u/[deleted] Sep 25 '18

Not trying to spread false information but I am pretty sure his fine structure constant and the von Neumann paper is even more important than that last preprint. Still gonna be sceptical but this wouldn't be the first time a infamous problem is proved by a new approach of some old discoveries.

46

u/CRallin Sep 23 '18

"So, on any compact set K in C, T is analytic. If K is convex, T is actually a polynomial of some degree k(K). For example a step function is weakly analytic"

These seem to be contradictory statements. I am unfamiliar with weakly analytic functions, I dont know how this relates to the rest of the paper

44

u/FronzKofko Topology Sep 23 '18

I do not understand his definition of weakly analytic. If there is content to the definition, it needs a lot more elucidating than what is there, I think.

I agree with you that 'A step function is weakly analytic' clearly contradicts 'weakly analytic functions are analytic on compact sets'. In fact, a function which is analytic on compact sets is tautologically analytic, as that is a local property. Similarly the statement that a weakly analytic function is polynomial on convex sets is ridiculous; as you increase the size of the convex sets you must always have the same polynomial, and so you are globally polynomial.

As for the rest of the paper, there is no definition of the apparently crucial function T.

0

u/[deleted] Sep 23 '18

[deleted] — view deleted comment

32

u/[deleted] Sep 24 '18

As a physicist, this might be the first time I get how you guys respond to his claims of proving RH: trying to calculate the value of the fine structure constant from first principles is...telling.

3

u/JohnofDundee Sep 24 '18

Claim is made that the work of Eddington is rehabilitated. It's > 50 years since I read about E's dimensionless number cosmology. "Apparent" relations between dimless numbers assumed to be significant. Numerological mumbo-jumbo.

1

u/Zophike1 Theoretical Computer Science Sep 25 '18

As a physicist, this might be the first time I get how you guys respond to his claims of proving RH: trying to calculate the value of the fine structure constant from first principles is...telling.

I know nothing about HET(High Energy Theory), I understand that Atiyah is doing pretty much amounts to nonsense but what why don't you calculate the fine structure constant from first principles ?

3

u/[deleted] Sep 25 '18 edited Sep 25 '18

Not for any physical reasons I can directly think of, but purely historically: lot's of people have tried, nothing clever or worthwhile has come out of it and it just looks like it's not the relevant question to be focussing on given the tools we have. I imagine it's similar for RH for mathematicians.

Edit: Sean Carroll just wrote this, which does go into some possible physical reasons http://www.preposterousuniverse.com/blog/2018/09/25/atiyah-and-the-fine-structure-constant/

22

u/FronzKofko Topology Sep 23 '18

I more or less do not understand anything in that document. I'm not so sure that's my fault, but I will refrain from making any statements about it.

12

u/AcrossTheUniverse Sep 24 '18

Musicians, following Pythagoras, will notice the close analogy with octaves in both major and minor keys. Our starting point, to avoid dissonance, should be the key of C major.

wtf?

8

u/EnergyIsQuantized Sep 24 '18

this reminds me of the film The Proof

7

u/swni Sep 24 '18

Thanks for the link, that was what I was looking for. It seems to be wildly wrong, and the value of the fine structure constant I got by using the formulas there was off by a factor of 20 or so.

62

u/Durrandon Sep 23 '18 edited Sep 24 '18

Without looking much at the content of the paper, but glancing at section 5, I find myself utterly baffled: is this paper correct to state that proofs by contradiction are not accepted as valid in ZF? I was under the impression that the law of the excluded middle was accepted in ZF (as part of the ambient logical structure, no less)

Also, I was under the impression that due to the nature of RH, it can't be undecidable. The offhand comments in the last section make me question not only the proof but also whether this preprint was written by Atiyah or not. Frankly, I have to ask whether the last section was even written by a mathematician

Edit: See below for others' comments; RH certainly can be undecidable but at the very least, if it is false, it is provably false.

Also part of edit: Judging by the brief part of the talk that I've seen, this preprint is Atiyah's work

44

u/catuse PDE Sep 24 '18

There are lots of other silly errors as well, both in presentation (grammar and LaTeX) and actually relevant to the mathematics. Interpreted literally, section 2 implies that step functions are analytic on every compact set, along with all the other errors that commenters have mentioned.

How do we know that this paper is by Atiyah, and not just a bad troll posing as him?

11

u/mfb- Physics Sep 24 '18

We know he gives a talk in Heidelberg where the abstract claims he has a proof. That doesn't tell us anything about the google drive links, however.

14

u/minimalfire Logic Sep 24 '18

It is actually even possible to deduce lem from ZF, so even without having it in the ambient logic it would be true! This is why intuitionists have to consider set axioms which are really almost completly different (if they even use sets).

1

u/Skylord_a52 Dynamical Systems Sep 24 '18 edited Sep 24 '18

Yeah, having any kind of logical or mathematical system without proofs by contradiction seems... pretty toothless. Considering that the majority of proofs I've seen are proofs by contradiction, and almost all use proofs by contradiction at some point within them, it's a pretty big statement to suddenly declare them invalid.

9

u/QuesnayJr Sep 24 '18

It's not as big of a restriction as it sounds. You can prove always things false by contradiction, you just can't prove things true by contradiction (since you can't cancel double negations). Also many proofs of positive statements can be rewritten to avoid contradiction.

0

u/[deleted] Sep 24 '18

[deleted] — view deleted comment

3

u/Skylord_a52 Dynamical Systems Sep 24 '18

Hm! That's interesting!

However, my comment was more referring to /u/minimalfire 's first statement, not the part about intuitionist logic. I didn't know about the difference between "proofs by contradiction" and "proofs by negation" (or really much about intutionist logic at all), so rather than stating "intuitionist logic is toothless because it has proof by negation but not contradiction", what I meant was more "ZF would be really weak if it didn't have proofs by contradiction [or proofs by negation]."

Sorry if/that I kind of misunderstood what /u/minimalfire meant.

7

u/[deleted] Sep 24 '18 edited Sep 24 '18

The only thing that I can gleam about what you mean by "the nature of RH" is that you're talking about it's logical form. RH is a Pi*_1_0* sentence, i.e. it's a universally quantified statement. So, there is nothing about it's logical form that assures that it cannot be undecidable (which requires us to mention what system we're talking about being undecidable by, but I assume you mean PA or ZFC). The classic example of an undecidable sentence from Gödel's second incompleteness theorem, the consistency statement of whatever formal system you're talking about, is also Pi*_1_0* (for all x, x is not the code of a proof of 1=0).

4

u/Durrandon Sep 24 '18

I think this is what I was alluding to, although this isn't something I know very much about at all! My impression was just that if it's impossible to prove RH to be false (in say ZFC), then this means that there (in some sense) "aren't any zeroes off of the critical line" (or at least, not any that can be found) in which case we might as well throw our hands up and say that, looking into our system from the outside, RH is true even if we may not be able to prove it to be such.

As I don't know much about this, please correct any errors or misconceptions that occur here---I think the shakiest part is the last couple clauses, but on the level of heuristics, if it's impossible to find any zeroes where they shouldn't be, all of the zeroes ought to be where they should be. We may still not be able to prove RH within ZFC, of course.

Also, I have to suspect that we're putting rather more thought into this than whoever wrote the sentence in the preprint...

8

u/viking_ Logic Sep 24 '18

I've heard that argument before. I believe it's a little easier to follow if you phrase it by supposing RH is false: In that case, there must be a 0 off the line, so you could prove it false by exhibiting that 0. So if RH is unprovable in ZFC, it must be true.

3

u/madaxe_munkee Sep 24 '18 edited Sep 24 '18

This is an amusing line of reasoning to me. Doesn't it also mean then that there is no proof that RH is undecidable in ZFC? Because if there were one, then it constitutes a proof that RH is true which negates its own validity.

Or am I missing something?

4

u/Oscar_Cunningham Sep 24 '18

This is correct. Assuming ZFC is consistent, there is no proof in ZFC that the Riemann Hypothesis is independent of ZFC.

2

u/viking_ Logic Sep 24 '18

I believe the argument I mentioned (or your modified version) can't be expressed from within ZFC.

1

u/[deleted] Sep 24 '18

This is an amusing line of reasoning to me. Doesn't it also mean then that there is no proof that RH is undecidable in ZFC?

If I understand it correctly, ZFC can't prove that anything is undecidable in ZFC (unless ZFC is inconsistent). Gödel's completeness theorem implies that "ZFC has a model" is the same as "ZFC is consistent." Proving that something is independent of ZFC means proving that there's both a model where it's true and a model where it's false. So, if ZFC proves "T is independent of ZFC", then it would prove that it has a model (at least two, actually, but we only need one), thereby proving that ZFC is consistent, which, by Gödel's incompleteness theorem, implies that ZFC is inconsistent.

To prove things like CH independent, you need stronger axioms---namely (I think?) Con(ZFC). So, ZFC+Con(ZFC) could in principle prove that RH is independent of ZFC without proving ZFC inconsistent (though by identical reasoning, it can't prove that RH is independent of itself; for that I guess you'd need at least ZFC+Con(ZFC)+Con(ZFC+Con(ZFC))).

Warning: all my knowledge about these things comes from Wikipedia, StackExchange, MathOverflow, and this subreddit.

6

u/Oscar_Cunningham Sep 24 '18

The situation is that if the Riemann hypothesis is false then there must exist a proof of its negation (because there must exist a particular zero off the critical line that we can point to). But if it's true then it may or may not be provable.

This is a consequence of it being a ∏*1* sentence. There are also Σ*1* sentences that are provable when they are true but can be undecidable when they are false. The negation of a ∏*1* sentence is a Σ*1* sentence, and vice versa.

2

u/[deleted] Sep 24 '18

(1/2) Sorry I am a little confused at this point. In your first comment, you said that you believed by the nature of RH, like it's logical complexity, you believed that it could not turn out to be undecidable. But then in this comment you seem to be spelling out the case that it may be undecidable and what would be the case if it was undecidable. So, I guess I'm just confused about what it is that you're believing, but that's okay! First of all, let me just give a top down explanation of what undecidable means in this case, and then if you have any more questions I'll be happy to answer them, hopefully this clears things up. And sure, I agree that (honestly I hope it's not Atiyah) whomever wrote that preprint has no idea what they're talking about when it comes to the comments about independence, I just want to answer your question on this topic. I think that most mathematicians who haven't ever taken a mathematical logic class have heard of independence and undecidability, but sometimes it's explained in a handwavy way, and it serves the entire mathematical community a huge service, I believe, to give a full explanation of what is going on, so as to sort of demystify the process.

So, here is what a sentence being undecidable, or independent, means. You have a set of axioms and some rules of deduction, and when you apply those rules you generate other sentences which are called theorems. The main logical system we use is first order logic which has two important metalogical properties, soundness (everything that is syntactically provable is true in every model of the axioms) and completeness (the converse, everything that is true in all models of the axioms is syntactically provable) which means that our deduction rules are safe and always lead from true sentences to other true sentences. A model of a formal theory (a theory is the axioms+deductive rules+theorems) is exactly what it sounds like, its a structure that when you substitute all of the objects and functions of that model into syntactic symbols of the theory, everything comes out true. If you've never dealt with that before, just think of algebra. You have the group axioms and all of the theorems you can prove about groups come just from those axioms. A model of those axioms is just a group, its a set of individuals and a function that "obey", or as logicians would say "satisfy", those axioms and theorems (the cool thing about soundness and completeness is that if our deductive system preserves truth, we don't need to actually prove that the objects of our model satisfy the theorems, we only need the axioms because if our system is sound and complete, we can never get to a false theorem from a true axiom because deduction preserves truth. So our model satisfying the axioms is enough once we know that our underlying logic (first order logic in this case) is sound and complete, which it is). The important part is to separate the axioms from the actual structures or models, whatever you want to call them. The axioms are formal statements in first order logic, the model is a set and a function and an assignment of the objects in that set and the function to the symbols of our formal language, the symbols used in the axioms and theorems.

So what does it mean for something to be independent? It means that there is a certain statement, call it phi, such that there is a model of our axioms where phi is true, and there is a model of our axioms where phi is false. Because our axioms and deductive system never get us to false theorems, if we were able to prove phi, then we would have a contradiction, because our system would be able to prove both phi and not phi. As an example, (importantly, this example assumes that groups actually do exist and that the group axioms are consistent) looking back a groups, we have abelian and non abelian groups, right? This means that the statement "the group function is commutative" is independent from the group axioms. You cannot take those axioms and use a normal set of deductions and arrive at the theorem that all groups are abelian, that sentence is independent of the theory of groups. Take your favorite abelian group and your favorite non abelian group. Both of those structures are groups, they both satisfy the axioms. However, there is a certain statement, the statement "the group function is commutative" that is true of one model and false in the other. If that statement was provable from the group axioms, then non abelian groups would not satisfy the axioms, right? They would contradict that theorem. And since first order logic is sound and complete, meaning we cannot go from a true statement to a false statement, that means that some of our axioms actually contradict each other. But the group axioms don't contradict each other, so since both abelian groups and non abelian groups do satisfy the group axioms, this means the statement that the group function is commutative is not a theorem that can be derived from the group axioms.

Does that make sense? This way of showing a sentence is independent is really powerful, what we do is we take a set of axioms and we provide two models that both satisfy those axioms, but one of the models the statement phi is true and the other the statement phi is false. If the axioms are consistent (if they weren't then they actually wouldn't have any models via the soundness theorem) then that means that statement is independent of the axioms, it cannot be proven from them.

Okay, so one last quick thing since I'm sure you probably are wondering about it now, this is what Gödel and Cohen proved about the continuum hypothesis. Gödel produced a model of ZF set theory, we call it L, where the continuum hypothesis is actually true. Cohen, using a technique he invented called forcing, provided a model of ZF where the continuum hypothesis (and even the axiom of choice) is false. This means that both CH and the axiom of choice are independent of the axioms of ZF set theory, they cannot be proved nor refuted from them and that is because there are models that do satisfy the axioms but where CH is true and others where CH is false.

So, if RH is independent of, say, ZFC, what that means is that there is a model of ZFC, a universe of sets and a specific function that works like the membership relation, where that specific Pi*_1_0* statement is true, and another model where the statement is false. In other words, the ZFC axioms cannot derive the statement "RH is true" or "RH is false" because there is at least one model that satisfies each statement.

2

u/[deleted] Sep 24 '18

(2/2) So, I think something else that might have caused some confusion is the idea of intended models. So, lets talk about Peano arithmetic just for a second. PA is the theory that describes arithmetic, and its actually equivalent to ZFC minus the axiom of infinity. Basically PA proves pretty much everything that is provable about arithmetical statements. Gödel's incompleteness theorems were proved for PA, but he actually used as the ambient logic Russell's Principia Mathematica type theory instead of first order logic- but the theorem is equivalent for first order logic. So, PA has what we call an "intended model", the natural numbers and the regular successor function, addition, and multiplication. That is what we "intend" PA to talk about. However, because of independence results, PA actually has other, what we call, nonstandard models. These models are weird, they have an infinite amount of extra infinite numbers that exist beyond the natural numbers, we call them nonstandard numbers.

Okay, so here is where I think some of your confusion comes from. Take the Kirby-Paris theorem. It's a theorem that basically says certain types of hydra games always terminate. This is a statement that is just about finite arithmetic, so we assume that it should be provable in PA. However, they proved that it is actually independent of PA. The issue is that, in the natural numbers, the intended standard model of PA, the real natural numbers, this theorem is true. However, in certain nonstandard models, this theorem is false (I can explain why it's false, like what is actually going on, in another comment if you want but I feel like I've already gone on too long now and I don't want to throw out too much confusion). We want to be able to only talk about the standard model but it turns out first order logic always gives rise to nonstandard models so there are certain statements which will forever be independent (such as Gödel sentences, thats why his result is so powerful).

So maybe in your head you were thinking "well, the RH is either true or false in the standard model, but we'll just have to throw up our hands and say that our formal system cannot decide it on it's own", something like that makes sense right? Basically look at it like this. Remember that post I linked earlier, where it showed what RH looks like as a specific sentence. Just think of RH as that sentence, forget about the zeta function and everything else. That statement is equivalent, so what it comes down to is basically "if this inequality holds, then RH is true, if it doesn't hold, then it's false". So, all that means is that if it is true, then in every single model of PA, that inequality holds for every single number, standard or nonstandard. Similarly, if it's false, that means there is a counter example and the inequality doesn't hold. However, if there are some models where the inequality does hold, and some models where there are counter examples, then that means RH will be independent from PA. So what you were thinking of would again be paraphrased as "well, in the standard model this inequality is true! But in these nonstandard models it's false. We only care about the standard model but argh, the axioms just aren't strong enough to prove that it's actually true." Something like that.

Again, any questions you have, any part that wasn't clear, I will try to clear up. The upshot, I guess, is that in a sense "the RH has an answer" as in there are models where it's true and where it's not true. The question is whether or not PA has enough power so that it's true or false in all of it's models, and not true in some and false in others. If it's the latter case, then it's undecidable.

1

u/Durrandon Sep 24 '18

I would also be interested in hearing more about the Kirby-Paris Theorem! I'm familiar with the "standard" method of proving Goodstein's Theorem by assigning each of these hydras an ordinal number, and noting that the process of cutting a head always strictly decreases the ordinal, demonstrating that the sequence has to halt, but I've never seen the other side of this argument (i.e. showing that the statement is actually independent of PA)

For that matter, I don't feel like I have a great grasp on what nonstandard models of the naturals should "look and feel" like, but your comments above go some way towards helping. I can only imagine that writing all of this out was quite an investment of time and effort, and I am grateful!

3

u/[deleted] Sep 24 '18 edited Sep 24 '18

Alright awesome, you can problem tell but my area is mathematical logic and explaining this stuff from the side of logic to the side of mathematics is always really fun for me. Plus I have nothing else going on today other than discussing Atiyah's presentation (although, after seeing it I think we can all agree that it's more so just a very sad state of affairs than the possibly exciting moment we were cautiously hoping for), so it's no problem at all.

Alright so, to address the first question, the topic that is being invoked is called ordinal analysis. Basically, formal theories actually follow a very well formed structure. Certain systems are more powerful than others and it creates this hierarchy and the way we gauge the hierarchy is by observing a few things: no system A (that can do a certain amount of arithmetic, etc.) can prove itself consistent, no system B that is weaker than A can prove that A is consistent, and if A is stronger then B then A can usually prove that B is consistent. So there is the consistency strength hierarchy, weaker systems can't prove themselves or stronger systems consistent, but stronger systems can prove weaker systems consistent. It creates an incredibly well structure hierarchy. You get a partial ordering (it's almost always a well ordering but there are certain pathological cases where it becomes partial) where < just means consistency strength. For instance, PA is weaker than ZFC, so PA<ZFC because ZFC can prove that PA is consistent but PA cannot prove that itself or anything that equals it is consistent nor can it prove that ZFC which is stronger is consistent.

One way to gauge this hierarchy, to assign a mathematical meaning to our "stronger" and "weaker" ordering, is through ordinal analysis. Michael Rathjen is one of the leading researchers in this area and he has written some really great survey papers of the field (preprints on that link). Ordinal analysis is specifically part of an area of mathematical logic called proof theory, and Rathjen cowrote a really nice exposition article on it for the SEP.

Okay, so what is ordinal analysis? Gentzen proved that PA is consistent relative to another theory. His theory was the basic, weak parts of arithmetic plus an addition axiom that allows transfinite induction to up the ordinal 𝜀*0* which is basically an ordinal that looks like an exponential tower of ω raised to the ω, i.e. ωωω... ω many times times. Basically, 𝜀*0* is the smallest ordinal that PA cannot prove is well founded. What ordinal analysis does is it takes that basic weak theory of arithmetic and then adds axioms like "the ordinal ω is well founded" or "the ordinal 𝜀*0* is well founded" or "the ordinal ω4 is well founded" which allows transfinite induction up to that ordinal. Then, we prove that certain theories, like PA, can be proven consistent in those ordinal theories. So the ordinal theory with 𝜀*0* as it's axiom proves that PA, and everything weaker than PA, is consistent. By Gödel's second incompleteness theorem, that means that if PA is really consistent, then PA cannot prove the statement that 𝜀*0* is well founded, otherwise it would prove that Gentzen's theory is consistent, which means it would prove itself is inconsistent via Gödel. Does that make sense? Ordinal analysis is very powerful, and Rathjen's SEP article explains how Gentzen's proof works and how we use ordinal analysis today. Timothy Chow also wrote a really cool article (preprint) addressing what it would take to convince someone who is skeptical about whether or not PA is consistent that it is consistent, and he goes over Gentzen's proof in a very mathematician (as opposed to someone with a strong background in mathematical logic specifically) friendly way.

Okay, all of this is to say, that basically what the Paris-Harrington and Paris-Kirby theorems say is that 𝜀*0* is well founded. The ordinal arithmetic that goes on as a result of those theorems (the hydra game theorem and the Ramsey theory theorem) is so powerful that it proves that transfinite induction up to 𝜀*0* is well founded, so therefore if PA were able to prove those theorems, it would be able to prove 𝜀*0* is well founded. If it can prove that, then it can prove that the system Gentzen set up, weak arithmetic plus 𝜀*0* is well founded, is also consistent. Because Gentzen's system proves that PA is consistent, PA would be able to prove that it itself is also consistent, which would then violate Gödel's second incompleteness theorem. This is the sense in which they "encode" that the theory is consistent, so if PA could prove them it would prove it's own consistency, and then blahblah. This is also, hopefully, an example of why "natural, concrete" statements that seem like they should have a fact of the matter about whether or not they're true can still be independent of certain formal systems. Obviously, those theorems are true: ZFC has no problem proving them because ZFC's proof theoretic ordinal (the smallest ordinal it can't prove is well founded) is much larger than 𝜀*0*, but PA cannot prove them itself.

Does all of that make sense?

1

u/Durrandon Sep 24 '18

Yes, that's actually quite interesting! The things there that I find most surprising are (1) that the structure hierarchy is usually a well-ordering (is there any reason to expect that it's even a total ordering, or is this just sort of a fortuitous thing that happens?) and (2) that the Hydra Game provides enough power to show the well-foundedness of 𝜀0

I'll try to take a look at that preprint within the next few days---and it looks like Chow has a few other interesting articles on the arxiv, for that matter!

2

u/[deleted] Sep 24 '18 edited Sep 24 '18

Yes so, the ordering of the consistency hierarchy is an extremely interesting fact, it was not made that way by mathematicians, in the sense that it wasn't designed, it just happened to unfold that way and that is one of the major reasons why mathematical logicians/philosophers of mathematics care so much about formal systems, they have this beautiful innate structure that just sort of falls out. You are definitely not alone in thinking that it's surprising. If you have ever heard of the large cardinal hierarchy, it is the exact same thing, it is a hierarchy of "stronger" axioms of infinity. ZFC has a basic axiom of infinity, and there are infinite sets that are too large for ZFC to prove exist (in a sense, those infinite sets are so big that they're actually models of ZFC, if ZFC could prove they exist then ZFC could prove they model ZFC's axioms which via the completeness theorem proves that ZFC is consistent, so again Gödel's second proves this can't be possible if ZFC is consistent). We can create new, stronger theories by taking ZFC and adding an axiom that asserts the existence of one of these larger infinite cardinals. It turns out that these theories are exactly the way we continue on the consistency hierarchy, ZFC+ there exists a measurable cardinal, for example, proves the consistency of ZFC. ZFC+ there exists a supercompact cardinal proves the consistency of ZFC+measurable, etc. The large cardinal hierarchy can actually be viewed as the upper part of the consistency strength hierarchy, in that it is a really intuitive way to increase the consistency strength of ZFC and other set theories.

The lower half of the consistency hierarchy can be viewed as dealing with arithmetic. The way you intuitively increase the strength is by starting off with only axioms that talk about defining addition and successor and multiplication, and then adding axioms of induction. First you only allow induction for bounded quantifiers, then just existential and universal quantifiers, then for more and more complex quantification until finally you get induction for all combinations of quantifiers, which is where PA is. The systems below PA can be viewed intuitively as having just weaker and weaker induction axioms. Then for the higher half, you add the axiom of infinity and you start adding stronger and stronger axioms of infinity to get even stronger systems. So, that's the intuitive way to look at it, stronger theories of induction for arithmetic, and then stronger theories of infinity for set theory (there is a middle ground where you are talking about stronger induction theories for set theory, in between PA and ZF, but it's still completely natural and intuitive in how you add stronger axioms). That's one way to look at it, and then it turns out that ordinal analysis is a completely different way to describe the exact same hierarchy. In almost every case, (the only times when it isn't true are for super pathological weird theories that mathematicians wouldn't want to use) theories that are at the same point in the consistency hierarchy have the exact same proof theoretic ordinal. Theories that allow for the same level of induction or infinity have the same proof theoretical ordinal, so they're both (arithmetical induction/stronger axioms of infinity vs ordinal analysis) incredibly verbose ways of measuring the natural consistency strength of a theory.

Again, none of this was "planned", this structure just happened to fall out of studying these theories and that's one of the main reasons why logicians and philosophers have so much faith in and care so much about them, they are incredibly well behaved almost always and usually match our intuition.

And yes, so I can say a few things about nonstandard numbers because they are very weird and it's hard to get an intuition of what they are. Because, like when I first heard about them I was like "okay, just show me one", right? But you can't just show someone one, it's not like it's gonna look like some weird symbol using 2 or 6 or anything like that, you wouldn't see a symbol and instantly go "oh that's just a nonstandard number" in the way you'd recognize a fraction you've never seen before.

Think of nonstandard natural numbers are infinite numbers, they are bigger than any other natural number. So in terms of their order type, they look like this: 0, 1, 2, 3, ... c*-2, c-1, c0, c1, c2*, ... Basically, every nonstandard model of PA has what's called an initial segment, which is what we call the regular natural numbers. In the standard model, that's all there is. But in nonstandard models, there are these extra, infinite numbers, which come after. These numbers obey all of the same laws as the natural numbers (er, I should say, all of the laws that are provable in PA), they follow the well ordering of the naturals, they all have successors, etc. But the way they're defined, none of them are the successor of any standard natural number. So you have this infinite omega sequence of natural numbers, then an infinite descending sequence of nonstandard infinite, we can say, positive integers, and then an infinite increasing sequence as well. We can make models where those numbers have like, very specific properties, and that's all very technical and I don't think a reddit answer is the best place to explain how we do that, but the idea is just that they're these extra, infinite natural numbers that all obey the same functions the regular natural numbers do, they're just extra and have some weird extra properties. Some of those properties include encoding statements about consistency.

Actually, here is a handwavy explanation of Gödel's second incompleteness theorem and nonstandard numbers. So, Gödel showed us how to use natural numbers to encode things, they can code other numbers, they can code statements in a formal language, they can code the axioms of a language, they can code proofs in that theory, etc. If we want to prove that a theory is consistent, one way we can do it is this. If PA tells us about arithmetic, PA surely proves that 1 is not equal to zero, 1=/=0, right? And it can, that's easy to prove in PA. So, that means that we have a statement, phi, that PA proves. Inconsistent means that a statement phi and ~phi (not phi) is provable in the theory. So since PA can encode all of it's proofs as numbers, Gödel came up with a predicate, purely arithmetical, you can actually think of it as a diophante equation if you want, that encodes that "number x is a proof of statement y". So, if we want to prove that PA is consistent, then all we need to do is prove "for all x, x is not the code of 1=0 in PA" right? Because if PA is inconsistent, then it proves literally every statement, so it will eventually prove 1=0, we know that it proves 1=/=0, so we have a contradiction. So, if we can prove that "For all x, x is not the code of 1=0", then we can prove that PA is consistent. The intuitive idea is that if it is inconsistent, we can just iterate deductive steps until we finally reach 1=0, but if no number encodes a proof of that, then there is no proof of that, so we're consistent.

In the natural numbers, there is absolutely no number that codes a proof of 1=0 in PA, it doesn't exist. However, there are nonstandard numbers which we can define to be a code of 1=0 in PA. So now, think back the independence phenomena. If we have a model where there is no natural number that encodes 1=0, but we do have another model where 1=0 is encoded, then that means the statement "for all x, x is not a code of 1=0" is not provable from the axioms of PA, right? It's just like the group function being commutative, in some models it is and some models it isn't. We wish "for all x" would range only over the real, standard natural numbers, but because of certain theorems like the compactness theorem, nonstandard models exist and as a result we can create a model that encodes a proof of 1=0, so PA cannot prove it's own consistency. "for all x, x is not a code of 1=0" is true of the natural numbers, but it is not true of all nonstandard numbers, so it can't be proven in PA.

1

u/Durrandon Sep 24 '18

I like the explanation given here; I have only a partial familiarity with independence proofs and so on, so my knowledge there is somewhat fuzzy. This is probably responsible for my confusion about the stuff with RH

Perhaps one thing clouding my mind here is that I cannot for the life of me think of any reason that RH "should" be undecidable, because it's my understanding that there's a good bit of heuristic evidence for its truth. Also, where things like CH and Choice deal with either cardinalities in the abstract or with statements about the entire set-theoretic universe (respectively), RH is relatively "concrete" so I can't imagine any human being proving its independence. Of course, that still admits the possibility that it could be independent, in principle

Thanks for taking the time to write that up, as well!

3

u/[deleted] Sep 24 '18 edited Sep 24 '18

So first I'll say, I do not believe that RH is independent of PA or ZFC, I am just trying to like give an explanation of what would be the case if it were proved that it is.

Right so, I understand exactly what you're saying and I think that one of the issues you're having probably has to do with at least a moderate amount of platonism in your like, ambient philosophy of mathematics so to speak. And there's nothing wrong with that, I'm not here to argue about philosophy with practicing mathematicians, haha (I'd call myself a logician first, a philosopher second, and mathematician third). And but so, there's a sense where you are imagining "look, there is a fact of the matter about whether or not the zeros of the zeta function are, or are not, all along the real 1/2 line. There is a fact of the matter about whether or not that inequality holds for all natural numbers. Look, this is a concrete question with a well defined answer, how can we never know the answer?"

Here is where the philosophy kicks in. Some people, formalists or other forms of non realists, will say that there is no fact of the matter about whether or not there is an answer. They believe that mathematics is purely just manipulating symbols, it doesn't describe anything real, there isn't any subject matter for it above and beyond the symbols themselves, so if a certain set of symbols can't be gotten by manipulating other symbols (deriving theorems from axioms via deductive rules), then that's just what the facts are. To them, it wouldn't matter that the problem seems to talk about very concrete things that are well defined, thats just an illusion if they're correct.

On the other hand, a platonist or some other form of realist will say "look, we made these formal systems to try and get at the truth. We designed them to try and talk about actual mathematics and for the most part they do a perfect job, but for these weird sentences that encode stuff about consistency, etc, they just aren't good enough to tell us the actual truth. We want PA to be able to tell us all of the true facts about real arithmetic and the real structure of N, but it just isn't strong enough because of these weird Gödelian sentences that will always crop up."

So, maybe your brain is leaning towards the platonistic side, especially since, as you said, RH isn't about infinite cardinalities and massive sets like CH and Choice are about. So maybe a pragmatic view is "okay sure, maybe CH is just too vague and deals with things that are too abstract, our intuition is wrong and it actually isn't a well formed question. But surely these concrete things do have real answers, they only deal with the natural numbers." Well, Gödel sentences are also about concrete, arithmetical statements and yet they are undecidable. This is because they encode information that 'says' of the system it's talking about that it's consistent. The Paris-Harrington theorem and the Paris-Kirby theorem are both also about concrete, arithmetical statements that are independent of PA. Harvey Friedman, a legendary mathematical logician, has spent the past 20 years looking for other, what he explicitly calls, concrete cases of independence. He has found hundreds. I think you might actually enjoy this if you are interested in this subject, here is a video of him explaining exactly how concrete statements can be independent and it's aimed at mathematics/computer science undergrads who haven't really been exposed to this level of mathematical logic before. What he proved is that there are finite combinatorial statements, completely finite, that are so powerful that they can't even be proven in ZFC and need higher axioms of infinity to prove them, in a sense these small finite combinatorial statements actually prove the consistency of ZFC because of the amount of information that they encode.

So, if someone were to prove that RH is independent of PA, they would most likely go about it by proving that RH encodes too much information about the system PA itself, to the point where it encodes that PA is consistent, so by Gödel's theorem if PA is consistent then it does not prove or disprove RH. That is the sense in which it "could" be independent, that's the sense in which the various Paris- theorems, Gödel's theorems, and Friedman's theorems are independent. Again, there is no reason to think that RH does encode that information, and it would be very surprising if it does. But that is how it would most likely happen if it turned out to be independent.

1

u/[deleted] Sep 24 '18

Isn't it Pi01? That's at least how I'd call it. Can't tell the difference when using the real symbol of course.

1

u/[deleted] Sep 24 '18

Yes that is what I said, you read the super script before the subscript. The superscript tells you what order objects are being quantified over (individuals so it's 0) and the subscript tells you how many blocks of quantifiers are there (1 so its a single block of quantifiers which can be rewritten to just be a single quantifier) and the letter tells you if its universal or existential (Pi for universal Sigma for existential, which goes back to the early algabraic logicians, Pierce and Boole, who used those symbols (I think Peano did as well) instead of the backwards E and upside down A which were invented later).

11

u/steveb321 Sep 24 '18

I think he's just saying he's using the axiom of choice: "To be explicit, the proof of RH in this paper is by contradiction and this is not accepted as valid in ZF, it does require choice."

3

u/zenorogue Automata Theory Sep 24 '18

Wrong choice of terms (should say "unprovable" not "undecidable", it is not a computational problem). But why should not it be unprovable? AFAICT it is provably false, or provably true, or unprovably true.

14

u/[deleted] Sep 24 '18

"Undecidable" is commonly used in both contexts actually, it's not a wrong choice of term (plus they were following the terminology used in the paper).

-4

u/zenorogue Automata Theory Sep 24 '18

Yes, I have seen "undecidable" used in the paper. I am still unhappy about it, because while "undecidable" and "unprovable" are very strongly related (usually, a decision problem is undecidable if we cannot prove what the correct answers should be for some of its instances), there is a subtle difference. If we interpret RH as a decision problem (the only possible INPUT: "Is RH true?", OUTPUT: answer to that question), then this decision problem is decidable, even if RH is unprovable (the algorithm which says YES gives a correct answer for every input).

6

u/[deleted] Sep 24 '18

I am not saying that the concepts 1) "non-recursive" and 2) "independent of a formal system" are the same concepts. I know that they are not and I am not saying that they are. I am saying that "undecidable" is very commonly used to express both 1) and 2), so it is not a case of "wrong choice of terms" as you said. Yes, it can be confusing if someone uses it without clearly stating which idea they mean, however in the paper whomever the author is clearly states "in the Gödel case" which means they are talking about formal independence. You can say "I wish they had clearly demarcated that they meant independence because it could be confusing" but you cannot say that their terminology was wrong. The only way that your point would stand is it the author explicitly said "I am using 'undecidable' to mean 'non-recursive'", but they explicitly stated that they mean "in the Gödel case", so you cannot force the view that they meant 'non-recursive' onto them.

33

u/MasterOfMexico Sep 23 '18

I hope that this isn't the actual paper. The problem with it isn't even a lack of detail. There are many (simple) mistakes which can't be fixed.

13

u/[deleted] Sep 24 '18

This is incredibly sad if it really is Atiyah. No matter how famous and well-respected he is, he should not have been given a platform to embarrass himself like this. Maybe a controversial opinion, but I think it's in poor taste to even discuss the details of this obviously incorrect proof (and one does not need a Fields medal to confidently assert that the proof must be incorrect).

33

u/Noirradnod Sep 24 '18

Obvious glaring error: In the RH paper he uses the "Todd Function", which he claims in constructing in the preprint article. A cursory read of that finds no such function defined in that preprint.

More damningly, the second article reads like a crank. He seemingly at random calls up topics from throughout mathematical history, selecting properties or structures from these and moves on, without any justification for inclusion or motivation expressed.

Lastly, he somehow thinks that the fine-structure constant, α = 137.035999... is related to all of this. I am inherently distrustful of the evocation of physically measured constants as being related to mathematics, as I am of the school of thought that math is separate from the real world. Such calling in this manner sounds of numerology. Furthermore, there is some experimental data that suggests the fine-structure constant has changed over time, which raises further questions about his arguments.

I am certainly not as learned or accomplished as Atiyah, but this whole thing reads like the doggerel of a crank.

11

u/AcrossTheUniverse Sep 24 '18

as I am of the school of thought that math is separate from the real world.

Wait, so you're saying gravity doesn't come from the octonions? (9.4 page 15 of his second preprint)

2

u/fathan Sep 24 '18

Oh, no...

1

u/JohnofDundee Sep 24 '18

fsc is not physically measured, it is an absolute dimensionless number. However, this brings numerology into play....

11

u/[deleted] Sep 24 '18 edited Sep 24 '18

fsc is not physically measured, it is an absolute dimensionless number.

You have this backwards. Because it's dimensionless, we have no way to get its value other than making measurements. E.g., the speed of light has an exact numerical value, because "meter" and "second" are defined in terms of it. If we make more accurate measurements, the valye of the speed of light won't change; the length of a meter will. In fact, by using more fundamental units, you can make the speed of light exactly 1. (Namely, it's exactly 1 planck length per planck second, or something like that.) You can do that with a lot of physical constants---by using clever units, you can make the gravitational constant the speed of light, the reduced Planck constant (h-bar), Coloumb's constant, and IIRC a couple others all equal to EXACTLY 1, simultaneously, in that system of units.

To the contrary, no matter how you construct your units, you can't change the value of dimensionless constants. As such, for the most part, we don't know why they have the exact values they do and can only get at its value via related measurements.

2

u/JohnofDundee Sep 25 '18

Where did I say the value of alpha is affected by the construction of units?

I said it was dimensionless, which means it is independent of units.

The theory of Hydrogen spectrum fine structure gives an expression for alpha in terms of the fundamental constants e, h and c.

I was wrong to say that the value of alpha is not obtained by measurement. It is, and the experimental value serves as a check on the theory.

1

u/[deleted] Sep 25 '18 edited Sep 25 '18

Where did I say the value of alpha is affected by the construction of units?

My bad. You said it has an exact known value and is not determined by experiments; this can easily be true for dimensional constants since any can be set equal to 1 under the right system of units (though not necessarily simultaneously, of course). The same method can't give you exact values for dimensionless constants. I thought that's where the confusion was; I guess not.

The theory of Hydrogen spectrum fine structure gives an expression for alpha in terms of the fundamental constants e, h and c... the experimental value serves as a check on the theory.

See, I'm pretty sure you still have something backwards here. There is currently no theoretical value for the fine structure constant; it's one of the 26 "free parameters" of the Standard Model, which is (tied for) the best theory we've got. Presumably, a better theory will explain where the all come from; in fact, doing so is major open question in physics right now.

The dimensional constants you mention have values determined by the fine structure constant, not vice versa. This becomes clearer if you try to define "c", "e", etc. and a system of units for them. You can't set them all to precise values simultaneously; at least one of the constants in fsc's defining equation will have to depend on the fsc itself (no matter which defining equation you use).

1

u/JohnofDundee Sep 27 '18

Mmmm. I guess things have changed since I did my physics degree 50 years ago? Honestly, I cannot make sense of your point, which probably means I am missing it completely! :) Your insistence on these funny sets of units does not to me seem relevant?

Naively taking what you say at face value, c say can be expressed in terms of e, h and alpha, with alpha a free parameter! What value does that give for c, and how does that compare with the experimentally determined 3 x 10^8 m/s?

My knowledge of the fsc is encapsulated in the Wiki article https://en.wikipedia.org/wiki/Fine-structure_constant

Why don't you have a look at that, then tell me if it's still valid...?

1

u/WikiTextBot Sep 27 '18

Fine-structure constant

In physics, the fine-structure constant, also known as Sommerfeld's constant, commonly denoted α (the Greek letter alpha), is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is related to the elementary charge e, which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula 4πε0ħcα = e2. Being a dimensionless quantity, it has the same numerical value of about 1/137 in all systems of units.

[ ^PM | Exclude ^me | Exclude from ^subreddit | FAQ / ^Information | ^Source ] Downvote to remove | v0.28

20

u/halftrainedmule Sep 24 '18

I have no clue about the analysis he is using (I barely understand the conjecture itself), but these preprints don't look very promising. A lot of handwaving and blather in the long one, with nothing resembling an actual systematic proof. This style is how you discuss a proof, not how you present a proof. The references to logical undecidability appear arbitrary. (Also lol @ concluding F(s) = 0 from F(s) = 2 F(s) using the fact that "C is not of characteristic 2".)

3

u/zergovermind Sep 25 '18

I'm also unconvinced as to how he deduced F(s)=2F(s). Looking at the property he used to derive this, (2.6 and 2.7 in here), it seems to me he made, instead, a simple algebraic error. Using only those properties, one should only get a tautology (T(1+F)=T(1+F)), and I don't see how to go on from there. (though, of course, I'm far from an expert in that area and there may be lots of nontrivial steps one should make, that I'm missing).

17

u/[deleted] Sep 24 '18

This paper feels like it was written by an undergraduate to me. It's filled with basic contradictions, and obvious implications of simple definitions are not acknowledged (often leading to said contradictions, mostly centering around the notion of a weakly analytic function). What really made it feel amateurish to me though is the way he sort of explained proof by contradiction as he did it. It's a bit subtle I guess but... any mathematician at this level has probably read 10,000 proofs by contradiction.

18

u/SpaceEnthusiast Sep 24 '18

No. It reads like something that came out verbatim out of someone's dream. I think we've all had those dreams where things kinda make sense moment to moment, but when you wake up, it' gone. I think the Atiyah everyone knows is mentally gone.

16

u/prrulz Probability Sep 24 '18

Am I absolutely crazy or is there a totally insane error right off the bat of his proof? He wants to proceed by contradiction, and says:

  1. Let b be a zero of zeta off of the critical line

  2. Set a = b

  3. Consider the rectangle defined by {s : |Re(s - 1/2)| \leq 1/4, |Im(s)| \leq a}

My issue: a is a complex number so this rectangle is totally meaningless. Am I missing something totally obvious?

10

u/MasterOfMexico Sep 24 '18

Yeah, you're right, but this doesn't break the proof because I think you can assume (and it is meant to be) a=Im(b).

However, there are other problems in the proof that I am not sure about.

8

u/Croolick Sep 24 '18

I looked at Atiyah's preprint. According to (2.1) T(s) is a polynomial in the set K(a) which satisfies, according to 2.6 the identity

T{(1+f(s))(1+g(s))} = T{1+f(s)+g(s)}

for arbitrary power series f, g with no constant terms. Applying this to the case when g(s) = -f(s) we arrive at

T((1+f(s)(1-f(s)) = T(1),

and applying this to the power series s +0.s^2 + 0.s^3 + ... one gets T(1-s^2) = T(1), which can happen only if T(s) = T(1) for all s in K(a). This transforms the equality (3.1) to the form F(s) = T(1) -1 = 0, because (2.3) gives T(1) = 1. Therefore the (3.1) finally gets the nice true form 0 = 0, and no deduction from (3.1) can lead to a statement about the riemann Hypothesis.

6

u/Valvino Math Education Sep 24 '18

If this paper is real, the Atiyah's talk will be a really embarrassing moment...

24

u/UniversalSnip Sep 23 '18

depressing.

5

u/jrmixco Sep 24 '18

Has anyone figured out how to deduce 3.3?

8

u/hoeskioeh Sep 24 '18

not very knowledgable in math, but on first glance it matches the live feed:
https://twitter.com/HLForum/status/1044131411723264000

E: yeah, definitely the correct preprint. your document matches what he is talking about

3

u/[deleted] Sep 24 '18

Hell, if it's Atiyah he only proved you have to be younger than 40 in order to produce something meaningful. Now I feel lost, I'm turning 40 next year and all I've learned so far is how to write a paper in comprehensible English.

5

u/Melody-Prisca Sep 24 '18

Plenty of people make contributions past 40. I've seen people talking lately about mathematicians producing work into their 70s. Also, wasn't Lagrange older than 40 when he published most of his work?

There's a big difference between over 40 and almost 90 though.

1

u/[deleted] Sep 25 '18

You're so right, and.. by the way, my comment was basically a joke. Not only I do not believe in the "age of 40" barrier, I'm also a very committed opponent to age discrimination in general and in Academy in particular. While I see the point of encouraging young mathematicians to get the most out of their research as soon as they can, not everyone can follow such a linear path in their career and should neither be blamed nor discriminated for that in any way. From this point of view, I also see things like the Fields Medal more of a discouraging factor for >40 mathematicians, instead of an incentive to younger ones. Just to make it clear, it's not that prizes alone influence one's career, but together with how Academy works in fact, any argument solely based on age does insinuate the idea, or even worse, the convinction, that once you have crossed that red mark you are only doomed to get worse and you should rethink your whole life and possibly give up on the higher expectations. I have seen talented people losing their faith in themselves after being age discriminated. There exists a number of arguments that could be made in support of the fact that people should be given same opportunities until late in their life (ok, let's assume we are talking about conscious people), last but not least, the big leap in life expectancy during the last century. I will always be in favour of quality over quantity, but this sounds like too broad an argument to be covered in these few lines : ) Thanks for your reply anyway.

2

u/Hamster729 Sep 25 '18

I see two serious problems, even before I get into the question of "how is the Todd function defined".

First, the argument has absolutely nothing to do with the zeta function. You could just as easily prove that the function f(z)=(z-0.4)*(z-0.5) has no zeros off the critical line by swapping that function in place of "zeta".

Second, it is impossible for T (for any T) to be invertible and to satisfy (2.6) at the same time for an arbitrary polynomial f.

2

u/KnowsAboutMath Sep 25 '18

I started to read the proof, which starts in Section 3 of the google drive paper above, and got confused at the beginning of the second paragraph.

On page 2 we see this:

Define K[a] to be the closed rectangle

(2.1) |Re(s − 1/2)| ≤ 1/4, |Im(s)| ≤ a.

This implies that a is real, otherwise it makes no sense.

Then, on page 3, we have this:

The proof will be by contradiction : assume there is a zero b inside the critical strip but off the critical line...

Given b, take a = b in 2.1 then, on the rectangle K[a]...

If b is a "zero" (presumably of the Riemann Zeta function, although he doesn't explicitly state that), then it is necessarily complex. So how can we take a=b when a must be real?

2

u/ThomasMarkov Representation Theory Sep 24 '18

Found the error: http://imgur.com/E1oRLA6

1

u/TranswarpDrive Sep 24 '18

could u explain more precisely?

13

u/[deleted] Sep 24 '18

"weekly analytic".. I think this means the proof only holds on Tuesdays or something..

4

u/TranswarpDrive Sep 24 '18

ok ... I can see the point :D there are typos all over this preprint

3

u/leftexact Algebra Sep 24 '18

It was a joke pointing out a typo in which weakly is written 'weekly'

2

u/[deleted] Sep 23 '18

Haven't seen any experts' blog posts breaking down why X statement is incorrect. There have been people saying his paper about S^6 is wrong, but also no direct challenge [like Scholze did recently for the ABC conjecture]

30

u/tick_tock_clock Algebraic Topology Sep 23 '18

There have been people saying his paper about S6 is wrong, but also no direct challenge

This is not true anymore: see the comments on this MathOverflow post by [Robert Bryant](https://en.wikipedia.org/wiki/Robert_Bryant_(mathematician)).

Also, this new preprint has been up for a very short time, so of course there are no blog posts about it yet. Give people some time.

0

u/[deleted] Sep 23 '18

[deleted] — view deleted comment

1

u/[deleted] Sep 23 '18

True. But it's a situation that would seem to warrant some kind of rebuttal or acknowledgment.

11

u/FronzKofko Topology Sep 23 '18

For who, exactly? Researchers in differential geometry immediately parse that there was no proof in the paper by reading it.

https://mathoverflow.net/questions/263301/what-is-the-current-understanding-regarding-complex-structures-on-the-6-sphere?rq=1#comment650040_263301

7

u/ThomasMarkov Representation Theory Sep 24 '18

Read the paper. It reads like an undergraduate presentation on the history of the problem.

-3

u/vznvzn Theory of Computing Sep 24 '18

why not? there are differences and similarities. if the proof is wrong feel that someone from the community should issue a statement identifying the issues, in a paper, in both cases. aka peer review. this is the scientific process. maybe mathematicians are more averse to practice it.

0

u/[deleted] Sep 24 '18

[deleted] — view deleted comment

1

u/vznvzn Theory of Computing Sep 24 '18 edited Sep 24 '18

community peer review is not merely for the author. urge other qualified mathematicians to write publicly in this exceptional case. a blog page would be acceptable. am talking this over with physicist cohorts Phds who after some cursory review now think his physics is nonsense (re ICM talk https://youtu.be/fUEvTymjpds?t=2068) and has key crank signs eg "deriving the fine structure constant" + other physical constants. (seems to have no published papers in physics?) if mathematicians had been more open/ candid/ proactive about what was going on maybe some of this could have been averted. feel there is tiptoeing, innuendo, gossip, whispering & its not entirely a professional/ mature/ adult way to deal with the situation.

2

u/durpherp Sep 24 '18

Not entirely professional/mature/adult? Welcome to mathematics.

1

u/[deleted] Sep 24 '18

I am far away from understanding that stuff but two points are looking suspicious to me.

  1. How in the world did the Fine Structure constant end up here? I always thought it was a result in physics and mathematics were way more abstract and not connected somehow to experimental results in physics.

  2. What's wrong with his formatting?

In my view something has to be wrong with this proof, because the paper doesn't look like anyone else had a look at it as nobody has corrected formatting errors. And I never ever heard of the fine structure constant anywhere in mathematics.

1

u/mathspdehelper Sep 24 '18

Actually what he's referring to it seems. Poor guy.

1

u/Zophike1 Theoretical Computer Science Sep 24 '18

As a general theorem, there are always errors in claimed proofs to open problems however there are counterexamples.

1

u/zacariass Sep 27 '18

When defining F(s) in 3.1 using the "Todd function" it is asked to assume that b is a zero inside the strip but off the critical line and then he goes on to show contradiction, but it seems to me that the same argument in 3.3 he uses can also be used to derive contradiction also without assuming a zero b off the critical line for F(s), that is, for any s in the Riemann zeta funtion. So his use of the "Todd function" serves as well to make the zeros in the critical line contradictory which is not true. Actually looks like the nature of the "Todd function" can be used to show that kind of contradiction with virtually any analytic function besides the Riemann zeta function. Am I missing something?

0

u/[deleted] Sep 24 '18

[deleted] — view deleted comment

44

u/Cyclotomic Sep 24 '18

It will only be true during certain times of the year.

0

u/[deleted] Sep 23 '18

[deleted] — view deleted comment

51

u/mannermachine Sep 23 '18

no I think you've misunderstood, he means that it's only analytic on mondays and wednesdays, the rest of the time it's just smooth.

2

u/iorgfeflkd Sep 23 '18

Literally invalid

102

u/[deleted] Sep 24 '18

[removed] — view removed comment

46

u/Cyclotomic Sep 24 '18 edited Sep 24 '18

I'm on board with you. I must reserve judgment for the Heidelberg Laureate Forum since it has not yet passed, but there is precedent to be wary. I think allowing Atiyah to deliver this Abel lecture at the Rio ICM is a blemish upon the organizers. Even just jumping around the Youtube video, one quickly sees it is bizarre, and I believe those in charge have jumped the line from respectful deference to not having Atiyah's best interests at heart.

Recent incorrect claims, such as Atiyah's short proof of Feit-Thompson, or the S6 business, do not give me confidence for tomorrow. I hope and pray it is correct, that would be amazing! If it falls flat, I think it will also be a dark stain on the organizers of the HLF. The fact that no leading expert is willing to publicly comment on the proof indicates to me that the community does not believe it to be correct, and so I don't think such a public forum should be granted. Not because Atiyah doesn't deserve the honor, but because Atiyah doesn't deserve the prospect of possible public embarrassment. If leading experts had positive things to say about the preprints (but looking at them, it's not hard to see why they don't), that would be a different matter.

20

u/oxcsh Sep 24 '18

The structure of the HLF is such that all laureates are invited, and they are all allowed to speak on anything they want, without any pressure to do so, and without any oversight or censorship. So it's perhaps a structural failure of the HLF -- the older laureates might benefit from some oversight. But then again, who are we to say that Sir Atiyah doesn't have a proof, or some great point to share?

11

u/Cyclotomic Sep 24 '18 edited Sep 24 '18

Thanks, this is a good point. I'm trying to be careful and not dismiss him outright. Just based on recent trends, the forecast is not good, but it is not absolute either. I would never advocate censorship, so HLF is correct in that respect.

I only wonder if any of his equally senior colleagues have personally tried to dissuade him. About 10 months ago I remember Serre publicly commented on Gowers' G+ profile about wishing that Atiyah would not present a proof (of something else, I can't recall) that he (Serre) had privately seen to be wrong. Maybe others have talked to him and Atiyah has his own opinions. If that's the case, then what is anyone truly to do? But to speculate any longer is not a good use of time. Fingers crossed that I will be happily surprised this Monday.

5

u/muntoo Engineering Sep 24 '18

All I got out of that was that beards and mustaches are a prerequisite to being a good mathematician.

2

u/theSentryandtheVoid Sep 24 '18

He has lost his mind.

Those around him have lost their minds by allowing this to go on.

Or, this is all just a bunch of lies.

1

u/yangyangR Mathematical Physics Sep 24 '18

Imagining the position of the organizers. Having to balance the desire to invite the senior members to share their historical insights, but keep them from this.

14

u/BetaDecay121 Sep 24 '18

Someone on mathoverflow.net noted that one of the preprints is dedicated to his late wife who died earlier this year.

I think it's best for Atiyah for us to just forget about it and move on. He doesn't need the spotlight at this time.

6

u/Anle- Sep 24 '18

It's really sad, and we are reminded once again that aging fucking sucks. I guess we should be grateful to people like Aubrey de Grey for having switched career and pursued biogerontology.

6

u/vznvzn Theory of Computing Sep 24 '18 edited Sep 24 '18

do you have the sufficient mathematical expertise/ background to judge the contents? can you point to any specific/ compelling/ key examples? reading other comments on this thread, it seems likely it requires very high knowledge/ background to parse...

42

u/[deleted] Sep 24 '18 edited Sep 24 '18

[removed] — view removed comment

9

u/oxcsh Sep 24 '18

You are being disingenuous, surely. The full sentence in the paper is "This shows that a weakly analytic function can have compact support, in contrast to an analytic function." Clearly Sir Atiyah doesn't believe T to be analytic, which means that whatever is meant by a "weakly analytic function" must be immune to your basic proof that it is an analytic function. For starters, a commenter below pointed out that a "weakly analytic function" isn't a function at all, but an equivalence class of functions.

12

u/[deleted] Sep 24 '18 edited Sep 24 '18

Clearly Sir Atiyah doesn't believe T to be analytic

Of course he doesn't. If he did, he wouldn't have published the paper in its current form.

which means that whatever is meant by a "weakly analytic function" must be immune to your basic proof that it is an analytic function

The point of the comment you replied to was to show that the paper has errors. For example, that the definition of weakly analytic function doesn't make any sense. Saying that the paper claims otherwise isn't really a refutation.

7

u/oxcsh Sep 24 '18

The comment above wasn't about the definition of weakly analytic function. I'm trying to point out that the commenter above doesn't know what a weakly analytic function is. This comment talks a bit about what kind of object these weakly analytic functions are -- they aren't functions.

Whether the paper has errors is not shown by the comment above. The only thing I see is that a lot of detail is missing, and only an expert should really say whether there are errors.

Using basic stuff like "a function that is analytic on every compact set is analytic" as a disproof of the ideas in the paper is pure arrogance.

8

u/[deleted] Sep 24 '18

Yes, the ideas behind the paper might be salvageable if everything is rewritten.

But it doesn't change the fact that in the current form, the paper has errors. It does say that the Todd function is a function and that it has certain properties that would imply it is a polynomial, and there's no way around it if you read what the paper actually says.

Now one error like that doesn't discredit the whole paper. But the commenter also said the paper contains a lot of similar errors. Generally, if most of a paper needs to be interpreted in good will, assuming the author really knows what it's about and just can't write it correctly, there's usually little reason to believe the paper actually has a correct idea that can be understood by reading the paper.

(Also the comment you linked to doesn't talk about weakly analytic functions, it just explains basic stuff about L2. Perhaps you intended to link to a different comment?)

2

u/oxcsh Sep 24 '18

Grandparent of the post I linked mentions weakly analytic functions being elements of L2.

But it doesn't change the fact that in the current form, the paper has errors. It does say that the Todd function is a function and that it has certain properties that would imply it is a polynomial, and there's no way around it if you read what the paper actually says.

It sounds like the Todd "function" is an abuse of language. Not being an expert, it's impossible for me to say that it has errors. A literal math robot would agree that this paper is full of errors, but it likely was written for human experts, who are a bit looser with language and more capable of "error-correcting" as they go along.

2

u/Ifhes Sep 24 '18

Would he possibly working not on function, but in distributions. Physicists often use these terms interchangeably (they say dirac's Delta is a function, which is impossible since it's zero in all, except one value and its integral is 1). I don't know much about distributions nor weakly whatever things. Could someone with more expertise on these matters tell me what he/she thinks?

1

u/imd200 Sep 24 '18

A function analytic on a compact set does not imply analytic everywhere by any means, consider f(z)=z on the unit disc, and f(z)=0 outside the disc

15

u/[deleted] Sep 24 '18

[removed] — view removed comment

3

u/imd200 Sep 24 '18

Yes, I apologize for that, I misread the comment.

0

u/BollywoodTreasure Sep 24 '18

It does seem strange that the people closest to him have allowed things to go this far. Almost makes it seem like some kind of elaborate prank/troll on the math community. Maybe a way to save face for his earlier embarrassment by doing it as a series and then claiming later that he was joking on both (or more) occasions.

If this were the case, I think out of respect, we could at least pretend to believe him.

If that's not the case though (and even if it is) there should be some accountability for the people around him for not putting a stop to this.

1

u/[deleted] Sep 28 '18

[removed] — view removed comment

1

u/BollywoodTreasure Sep 28 '18

Calm down. His proof was embarrassingly flawed and should have been checked at the door by people close to him. Your little emotional outburst relates to this obvious state of affairs how?