Talk:Hilbert's problems

Second problem and consistency proof

The current table entry for the second problem is quite odd; it has several long footnotes that really belong as content in the article about Hilbert's second problem. But there isn't any article that describes the second problem, and the article that should (Hilbert's second problem) redirects to consistency proof, which is about consistency proofs in general but not about Hilbert's second problem.

Here is what I would like to do:

1. Change table to say:

2nd

Prove that the axioms of arithmetic are consistent.

Partially resolved: Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself.

2. Move the footnotes from this page to Hilbert's second problem (no longer redirected to consistency proof) and use them to write an article about the second problem.

Thoughts? CMummert 14:59, 5 January 2007 (UTC)

That would definitely be an improvement on the current situation. --Zundark 15:31, 5 January 2007 (UTC)

I was the guy who added all these footnotes. The reason is, from my reading -- that is, from the published stuff I've encountered -- I would say that the matter is actually "resolved" -- but some assert "in the affirmative", some assert "in the negative". Thus it has become a "matter of debate" -- at least in the literature. Kleene "resolves" it as a resounding NO ... given certain assumptions/qualifications. (And Kleene 1952 has a long discussion at the end about Gentzen's proof, so he isn't ignoring the issue). And the quote by Nagel and Newmann is particularly damning with respect to Gentzen's proof -- they do not accept it as a resolution: period. The problem is: this is what the literature is saying, not us. If it were left to me, I would word it as below. I agree that some of this could be moved to another article, but the summary here requires a more sharply-pointed "A matter of debate; some hold it as resolved in the affirmative, some in the negative" for the casual reader.

2nd

Prove that the axioms of arithmetic are consistent.

A matter of debate. Given the results of the subsequent 50 years, Hilbert's question is too vaguely worded to resolve the matter with a simple yes or no. Gentzen proved in 1936 that consistency of arithmetic follows from the well-foundedness of the ordinal $\epsilon _{0}$. Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself.

wvbaileyWvbailey 16:02, 5 January 2007 (UTC)

Yes, some people think that it "is" resolved (in each direction) and some think it is not; that is what I want to cover in depth in the new article on Hilbert's second problem. Since you have read up on it, you might be interested in helping with that article.

The wording you suggest overstates the extent of "debate". Everyone understands everyone else's position about the interpretation of Godel and Gentzen's proofs, and the "debate" boils down to which side you are sympathetic towards. How about this version, parallel to the first problem:

2nd

Prove that the axioms of arithmetic are consistent.

Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. There is not consensus on whether these results give a solution to the problem as stated by Hilbert.

It is true that we must reflect what the literature says, but this issue requires extra care because a good understanding of these issues took a while to form and so some literature from the early to mid 20th was written before the issues were as well understood as they are now. CMummert 16:31, 5 January 2007 (UTC)

The write-up of Gentzen's proof which I read in Mendelson also uses a kind of infinitary logic which I find more problematical than the well-foundedness of epsilon zero. A separate article would be good. I changed the "see also" section of "consistency proof" accordingly. JRSpriggs 05:46, 6 January 2007 (UTC)

I haven't read Mendelson's exposition. It is possible to recast the Gentzen's proof in a way that uses an ω-rule to deal with the induction axiom, but the proof can also be done without infinitary inference rules. CMummert 16:43, 6 January 2007 (UTC)

I would be satisfied if the entry were more like this (put the no? not?-consensus first and then in time order Godel followed by Gentzen):

2nd

Prove that the axioms of arithmetic are consistent.

There is not consensus on whether the results of Gödel 1931 and Gentzen 1936 give a solution to the problem as stated by Hilbert: Gödel's second incompleteness theorem shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0.

I expect the sub-article will result in quite a snarl (which should be quite interesting). Kleene's quote is in context of the intuitionists: the issues around what Kleene calls "the completed infinite" and Brouwer's fundamental objection to it, and around Church's thesis (cf p. 318, §62 Church's Thesis, Chapter XII Partial Recursive Functions). There is (at least one) very large gorilla in the room. wvbaileyWvbailey 16:21, 6 January 2007 (UTC)

That order isn't parallel to the box for the first problem. Do you favor reversing that one as well?

I believe they are fine as they appear now excepting a link to the new article should be added within the box (there are no footnotes now, so the reader needs a link).wvbaileyWvbailey 18:49, 7 January 2007 (UTC)

As with all the other problems, the problem number 2nd links to the article with details. CMummert 18:52, 7 January 2007 (UTC)

Yes you are correct; it links nicely. Thanks. wvbaileyWvbailey 19:26, 7 January 2007 (UTC)

As for the other article, it isn't a "subarticle" (same issue as with the algorithm articles), it's independent. The issue is not as contentious as you make it out to be; what is required to prove the consistency of PA is well understood. The situation is quite similar to the status of the continuum hypothesis. CMummert 16:43, 6 January 2007 (UTC)

Actually, I don't think the situation is very like that of CH, and for precisely the reason you state: The status of the 2nd problem is well understood, and is unlikely to change, barring some spectacular development (say, the success of Edward Nelson's project to prove that arithmetic is actually inconsistent). CH is not like that. It is reasonably possible that, thirty years from now, it will be widely agreed, based on Woodin's, work, that the continuum is $\aleph _{2}$. Oh, not a full consensus I think, but maybe the sort of weak "it's the natural thing to assume" sort of consensus that we see now for large cardinals. --Trovatore 07:37, 7 January 2007 (UTC)

Since there was no vocal opposition, I have split out the article Hilbert's second problem. The discussion should move to that article's talk page. CMummert 17:10, 6 January 2007 (UTC)

Unresolved or Open?

I see in the archived talk that there was a long discussion about whether problem 6 was mathematical. Now it's 'Unresolved' in the table but it is not clear if there is a difference between this status and the 'Open' of several other problems. --Angelastic 06:38, 1 May 2007 (UTC)

Marc Goossens had the last word on this, in a long piece on 18 December 2006, which concluded that more thought was needed. Here are my thoughts, at similar length but with a more definite suggestion.

The history of physics has demonstrated it to be open-ended thus far. Experiments continue to yield new facts not predicted by any extant theories of physics, and there is no end in sight to this. Conceivably a final theory may one day emerge that predicts the outcome of all future repeatable experiments, but how would we know this?

We might think it complete because it is so beautiful. In 1900 this was in fact how physicists viewed the situation. Gravitation, electrodynamics, and optics were described perfectly by Newton, Maxwell, Young, and their elaborators: Lagrange, Fresnel, etc. Those who accepted the law of large numbers as applicable to nature felt that this made the thermodynamics of Maxwell, Boltzmann, and Gibbs laws of physics, a status which, for those who had not fully grasped the logic of the situation, made it applicable at all scales including small numbers. Kelvin summarized the situation in his century, for which he gained great notoriety in the next, by suggesting that all that remained was to determine the constants of nature to ever greater precision, as though they were the digits of pi.

In 1900 small chinks in the armor of the understanding of physics were just starting to open up. Michelson and Morley had demonstrated to a precision of 1 in 40 that there was no sign of the Earth's movement through the putative ether as measured by the velocity of starlight (it should have fluctuated on an annual basis). And the Rayleigh-Jeans radiation law, first propounded in 1900, failed mysteriously (but fortunately, or we couldn't go outdoors for fear of skin cancer) in the ultraviolet.

But in 1900 hardly any responsible physicist would dare to say that either of these were supernatural phenomena, not explained by the current understanding of the laws of nature. Planck in 1900 and Einstein in 1905 stuck their necks out by proposing radical changes to the laws of nature. Einstein's was more radical, requiring the distortion of, and a leakage between, space and time, whereas Planck's merely required fudging the (then very new) Rayleigh-Jeans law to make it work, with no real insight into why the fudge worked, though he arrived at his eponymous constant to high precision in the process. Einstein stuck his neck out a second time (the one that eventually won him his only Nobel prize), also in 1905, to say that light was discrete, based on his observation of the photoelectric effect, a discreteness which Planck himself rejected!

Even if Hilbert had been so on top of the physics literature as to be immediately aware of Planck's amended radiation law, its formulation as a fudge rather than a fundamentally different basis for physics would not have been sufficient to persuade him that Kelvin was mistaken. Far more plausible is that a great mathematician like Hilbert would have responded to Kelvin's proposal that physics only needed greater precision of measurement with the counterproposal that greater precision of description was also needed, which mathematics had started to bring under control during the latter half of the 19th century with a vision of logic taking it far beyond Aristotle's syllogisms. One could then predict (i.e. calculate) the constants of physics instead of having to measure them to ever greater precision.

The general world view of physicists in 1900 makes Hilbert's 6th problem eminently reasonable.

Since we have a very different view today of the same situation, I would think it best to describe the status of Hilbert's 6th as "Unresolvable until we are sure, if ever, that all future phenomena and measurements will follow from the understanding of physics at that time," and pointing out that Lord Kelvin's view of physics as a subject now complete but for precision of measurement was the prevailing one when Hilbert posed Problem 6.

I suggest leaving this open for comments for a bit before doing anything about the table entry in question. If after a week or two no one has raised serious issues not yet resolved, I would consider the entry fair game for change. --Vaughan Pratt (talk) 21:26, 24 December 2008 (UTC)

Any objections to this change? If not I will make it reasonably soon. --Vaughan Pratt (talk) 11:02, 7 January 2009 (UTC)

What printed sources are there to which we might turn to help editors make a decision? Bill Wvbailey (talk) 14:51, 7 January 2009 (UTC)

My comment is years later than those above, but I believe the writers make a fundamental (and pretty obvious) blunder in assuming that Hilbert's use of the term 'physics' was (or should be) open-ended. What was he talking about? Either he was talking about the physics known to him OR he was talking about some nebulous universe of possible extensions of the understood physics of the day, with new (unknown with all sorts of possible structure) laws, relationships and structure. I submit that the latter possibility is very unlikely (but if it is, then clearly the conjecture is too vague to be resolved). I suggest that all uses of the word "physics" be replaced by the term "[Classical] physics" since its almost certain that is what he meant. Actually, I doubt the square editorial brackets are necessary, but for quotes should be inserted. Our purpose here is, after all, to convey meaning, not random sequences of characters on a page. Given that he meant classical physics, the 6th problem HAS been solved - it is neither open nor unresolved.173.189.79.42 (talk) 23:29, 15 May 2015 (UTC)

Dead link

The Mathematical Gazette, March 2000 (page 2-8) "100 Years On" seems to be a dead link Randomblue 07:43, 30 June 2007 (UTC) Randomblue

I've removed it. --Zundark 07:53, 30 June 2007 (UTC)

Stilted and pretentious language

Is no one else bothered by the language and tone used in this article, which is obfuscatory, stilted and jargony? Some examples:

"This might be put down to the eminence of the problems' author."

Why not just "due to Hilbert's eminence"?

"would go on to lead" Why all these conditionals? Hilbert DID go on to lead etc.

"On closer examination, matters are not so simple."

What?? Whose examination? What matters? Is the author trying to say that Hilbert's reputation as a mathematician is not the only reason the Hilbert problems are influential? Yes; probably the fact that most of the problems were deep and interesting was worth as much or more than Hilbert's reputation in the mathematical community. But I have read the paragraph several times and am still not sure what role this sentence is supposed to play.

"The mathematics of the time was still discursive."

Again, what?? I had no idea what this sentence was supposed to mean until I clicked on the footnote. Putting in footnotes to clarify the meaning of your sentence is classic bad writing. All of the footnotes should be assessed and either reincorporated into the text or removed.

Well, I'm sorry you don't like the writing, which is mine. By the way, you have just used a classic "passive construction". I looked at this last night and was surprised at how little has changed in this article, recently. By simply listing lots of objections, you are registering disapproval. That's about all. The mathematics of 1900 was "discursive". That's the precise word. I can amplify, for example with the way Poincaré wrote. Charles Matthews 08:46, 30 September 2007 (UTC)

Okay: I wish YOU would reassess all the footnotes and either reincorporate them into the text or remove them. I am sorry that you do not realize that explaining what you mean in little footnotes at the end is not a good writing style. (Nor are footnotes used for purposes of clarification in many other WP articles.)

I still don't know what the sentence involving discursive is supposed to mean. I looked up the word discursive, and got: "1. passing aimlessly from one subject to another; rambling. 2. proceeding from reason or argument rather than intuition." If you meant 1., much justification is needed (more than just Poincare; he is only one mathematician). From the context I doubt you meant 2. But my point is that I can't tell what you mean! I am a native speaker of the English language and a professional mathematician. If I can't understand what your sentences are supposed to mean, who are you writing for?

"In 1900 Hilbert could not appeal to....permanently change its field."

Yes, like most people living at time t, Hilbert was able to mention things that occurred at time t - x but not at time t + x, where x is a positive number. But what is the point??

"...nothing, as a naive assumption might have had it, about spectral theory."

Yes, it is truly the height of naivete to think even for one second that Hilbert might have propounded a problem on spectral theory --- again, what?? Where is all this coming from?

"In that sense the list was not predictive."

Another sentence that makes sense only by virtue of its footnote, which contains:

"as it did not roll with the way mathematical logic would pan out."

Is this intentionally horrible writing? Is it supposed to be amusing?

I think the author is trying to say that 20th century mathematics was marked by a greater abstraction than is evident in Hilbert's problems and that certain fields which became important soon afterwards are not emphasized. But I'm honestly not sure.

Again, the following paragraph (especially the first sentence, with the confusing use of "belie") makes the simple idea -- that some of Hilbert's problems are, by modern standards, not stated precisely enough for us to say with assurance whether or not they have been resolved -- sound quite complicated.

"With all qualifications...and personally acquainted)." Another awful sentence. If the smaller size of the mathematical community in 1900 is an important point in understanding something about Hilbert's problems (this is not made clear to me), it can be developed in a separate sentence.

Comparing Hilbert's problems to the Weil conjectures seems strange, because HP is a list of 23 problems in vastly different areas of mathematics, some but not all of which are given in the form of a precise conjecture; WC are three closely related, precisely formulated conjectures in arithmetic algebraic geometry. This distinction is much more important than what Weil was "perhaps temperamentally unlikely" to do.

No mention of the Clay Math Institute's millennium problems??

"It is quite clear that..."

Good writing refrains from saying things that are quite clear.

Footnote 9 makes no sense. Moreover, I'm willing to hear (Sir) Michael Atiyah's negative opinions about Klein and his programme, in part because they are probably relevant to the point he is trying to make. But an anonymous encyclopedist bashing Klein makes me want to roll my eyes.

Constance Reid's biography is mentioned casually, without proper reference.

"The theory of functions of a complex variable...quite neglected...[no] neat question, short of the Riemann hypothesis." Huh?? The Riemann hypothesis was, then and now, a much more important problem than the Bieberbach Conjecture (as even Louis de Branges would agree). Saying that complex variables is neglected makes no sense.

"One of Hilbert's strategic aims was to have commutative algebra and complex function theory on the same level."

I don't even clearly understand what this statement means (and not for a lack of knowledge about commutative algebra and complex function theory), but in that it seems to claim something about Hilbert, I would like to see a reference. What does it mean for complex analysis and comutative algebra "to change places"??

"[Hurwitz and Minkowski] were both close friends and intellectual equals."

It is not the job of encyclopedists to intellectually rank great mathematicians of 100 years ago. Anyway, how does this square with the claim that Poincare was Hilbert's only rival?

Oh well, if you know the Constance Reid biography, you'll know what this is about. Hilbert learned a great deal from those two in his early days. They talked to each other as equals. By the way, this whole introduction is context. It is supposed to make some sense of the problem list. Charles Matthews 08:50, 30 September 2007 (UTC)

If you wanted me to know about Reid's biography of Hilbert in order to understand the article, you should include it as a formal reference, which you have not done. Moreover an encyclopedia article does not include biographical information without reference -- it reads as though you were Dave Hilbert's close personal friend, and we should take your word for it.

Wouldn't it make more sense to list the problems and then talk about which of them have been resolved?

There is no reference to the results of Gentzen mentioned in regard to H2.

H3: Why does it not say it was resolved by Dehn (the inventor of the Dehn invariant)?

H7: Again, say it was resolved by Gelfond and Schneider (independently).

No references for the resolution of H22 or H23? Plclark 05:58, 30 September 2007 (UTC)Plclark

Please feel free to work on the article. Charles Matthews 08:50, 30 September 2007 (UTC)

Editing it would involve removing whole sentences and paragraphs that I detailed above. Is there no one here to explain what these passages are supposed to mean, or why they are supposed to be there? Plclark 00:01, 1 October 2007 (UTC)Plclark

I rewrote the introductory section completely. It is just a first attempt, but the idea is to make the discussion more specific and mathematical, and to eliminate biographical information / pseudo-historical judgments that are not referenced. Everything I wrote in that paragraph can be backed up with precise references (and should be). I deleted the section on Hilbert's manifesto entirely, because it was a sequence of claims, both vague and unsubstantiated, about Hilbert and not his problems. If anyone wishes to restore this text, please include proper documentation. Plclark 01:13, 1 October 2007 (UTC)Plclark

The Continuum Hypothesis as Hilbert´s first problem

The Continuum Hypothesis (CH) is often said to be Hilbert´s First Problem. I just want to point out that CH was just half of what Hilbert presented as the first problem, at least in the actual talk he gave. (No, I wasn't there, but I've read a transcript.)

The second half was about wether the continuum could be well ordered, and was emphasized just as much as CH. Quote: "The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element, i. e., whether the continuum cannot be considered as a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out."

That second half has been resolved completely, not by proving it, but by making it an axiom of Set Theory, i.e. The Axiom of Choice (AC) which is completely equivalent to saying that every set can be well ordered.YohanN7 (talk) 16:22, 27 November 2007 (UTC)

Hilbert apparently thought there might be an actual construction of such a well-order of the continuum, without assuming AC. As the latter is independent of ZF, it is still a valid question whether well-orderability of the continuum is a consequence of pure ZF. I don't know the status of this, but if this hasn't been settled, the second half of the 1st problem has not been resolved in a way that I find satisfactory.  --Lambiam 20:45, 27 November 2007 (UTC)

OK, first of all, let's keep in mind that when Hilbert proposed his list, there was no ZF. (Well, of course it existed Platonistically, the same way that Hamlet, prince of Denmark existed before Shakespeare, but Hilbert did not have it in mind.) And Zermelo's proof of the wellordering theorem had not been enunciated either. So "without assuming AC" is a non-sequitur here.

Just the same I'm happy to answer the technical question -- whether the continuum can be wellordered is definitely independent of ZF. In fact that's the natural way to prove that AC is independent of ZF, and I think it's how Cohen proved it, though I'm not 100% on that. --Trovatore (talk) 20:57, 27 November 2007 (UTC)

And, if I might add, as far as I understand AC and the Well Ordering Theorem (WO) are equivalent (in ZFC). Thus WO and ZF must be independent as ZF is just ZFC without AC (is it?). Thus WO cannot be proven in ZF. Please correct me if I am wrong. Still, that wouldn't mean that any particular set (like the continuum) could not be well ordered without AC or WO though I doubt that it could. —Preceding unsigned comment added by 217.208.31.144 (talk) 12:06, 28 November 2007 (UTC)

Yes, but it is conceivable that the continuum could be well-ordered while some other set could not. So AC, being equivalent to WO, implies the well-orderability of the continuum, but the converse might not hold in some model of ZF.  --Lambiam 15:06, 28 November 2007 (UTC)

Right, there are models of ZF in which the continuum has a wellordering but, say, the powerset of the continuum doesn't. I think if you start with L, add a generic subset of R using countably closed forcing, and then look at the L(P(R)) of the forcing extension, that should do it for you -- basically the same argument should go through as for the simpler case of ¬AC -- assume P(R) has a wellordering in that model, pick the least set of reals (in that wellordering) that's not in the ground model, and exploit homogeneity to show that you can already calculate it in the ground model. No warranty on that but I think it should probably go through with a bit of work. (Note that the R of the final model is just the R of L, and keeps the same wellordering that it had in L). --Trovatore (talk) 20:33, 28 November 2007 (UTC)

Yeah, thats what I meant too. By the way, I looked it up. There are quite a few things that are equivalent to AC, most notably Zorns Lemma (ZL) which is frequently used in "everyday mathematics". It struck me that the mathematician who on purpose works entirely within ZF must be very careful in order not to "accidentaly" use AC or results that rely on it. It might not be very apparant on the surface.217.208.31.144 (talk) 16:43, 28 November 2007 (UTC)

Not quite sure how to edit this Talk page, but I find it *highly* misleading to categorize this as "resolved." And, if it is "consensus" that its independence is a resolution of some sort, I would really really like to see some kind of sources to this effect. Put it this way: Hilbert posed the problem on the assumption that CH has a definite truth value. Relative to ZF it doesn't, but how is that a resolution of the problem as Hilbert posed it (granted it's an enormous step forward in attempts to achieve an answer)? — Preceding unsigned comment added by 171.66.221.130 (talk) 17:16, 13 August 2014 (UTC)

Nice catch. This was changed on 5 June 2013 by an anonymous editor who didn't say what he/she was doing in the edit summary. Slipped by me. I've put back the original language. --Trovatore (talk) 17:27, 13 August 2014 (UTC)

More specific criticisms would be helpful

The section on "Ignorabimus" is currently marked with flags for use of weasel words and lack of neutrality/verifiability. A little while ago, someone identified specifically objectionable sentences and removed them. I had no problem with this. Since I then did not know what, if anything, was still problematic, I removed the tags. They have now been unremoved.

I would be happy to provide any documentation necessary, but I need to know specifically which passages remain problematic. 68.223.61.59 (talk) 19:01, 6 June 2008 (UTC)Plclark

Here are a few statements from the section that I think ought to be attributed to a source:

1. The resolutions of some of the Hilbert problems would have been surprising/disturbing to Hilbert.
2. The significance of Gödel's work was dramatically illustrated by its applicability to one of Hilbert's problems.
3. Hilbert believed that we always can know what the solution is.
4. There is no mathematical consensus whether the results of Gödel and/or Cohen give definitive negative solutions.
5. The formalization of the problems to which these solutions apply is quite reasonable.
6. But it is not the only possible one.

The uses of "arguably", "presumably", and "(not) necessarily" are somewhat weaselly. The terminology "(in a certain sense) our own ability to discern whether a solution exists" is strange, as this is a limitation on formal systems independent of any person's abilities.  --Lambiam 01:39, 7 June 2008 (UTC)

Thanks, that is helpful. I will have to do some digging around in the library for references for these. Would it be sufficient, do you think, to cite some well-regarded biography of Hilbert's (e.g. the one by Constance Reid) or do you want citations from Hilbert's own writing? If the latter, I am not the person for the job, since I cannot read German.

On the other hand, identifying certain words and prhases as "weaselly" seems less useful to me. (Or rather -- and more amusingly -- SOMEWHAT weaselly.) If one can back these statements up with specific citations, then I don't think the phrasing itself is problematic. For instance, in the "presumably" I am indeed making a presumption -- I am extrapolating the very well-known and well-documented opinions of a certain figure to an event that took place after his death. It would be wrong to remove the "presumably", but as long as these opinions of Hilbert can be documented, I think the sentence is appropriate and useful. The sentiment could be rephrased so as to make it less personal, but in my opinion this would be to the article's detriment: one of the intriguing things about Hilbert's problems is that they were all made by a particular person, who was brilliant enough to select absolutely wonderful and influential problems for a century's worth of work and opinionated enough to have strong ideas about how these problems should work out. The fact that the solutions to some of these problems were not at all what he was expecting is, to me, an important and compelling part of the story, and the personal angle need not be deemphasized. Plclark (talk) 01:25, 9 June 2008 (UTC)Plclark

A comment about the "our own ability to discern whether a solution exists": I whole-heartedly disagree that the negative solution of H10 is "independent of any person's abilities". The statement of H10 is "To devise a process according to which it can be determined in a finite number of operations..." What does this mean? I think all can agree that (i) one meaningful way of construing this "process" is as the Church-Turing notion of an algorithm, but (ii) it is at least conceivable that there is some other, broader way of construing it. If you or I want to solve H10, the most clearcut thing we could do would be to write a computer program. Our skills and insights might differ, but the negative solution of H10 implies that that does not matter here: we will both fail. Again, I grant the possibility that there _may_ be some supra-algorithmic process that succeeds where an ordinary algorithm would fail (cf. Roger Penrose's "The Emperor's New Mind"), but of course there may not be either. Therefore it seems quite accurate to say that the negative solution of H10 shows that -- in a certain sense -- we are not able to systematically solve Diophantine equations. I honesly think this is all NPOV. It would be POV to assert that human beings cannot in any reasonable sense perform algorithmically impossible tasks, although with the single exception of Roger Penrose I have never met a mathematician who thinks otherwise. Plclark (talk) 01:50, 9 June 2008 (UTC)Plclark

For clarity, I am not the person asking for citations. I was merely trying to be helpful by pointing out where citations might reasonably be required in accordance with the Wikipedia policies and guidelines by editors not familiar with the subject matter. I think, though, that well-chosen attribution of the points I have identified will improve the quality of the article. The "somewhat" weaselly words do not bother me personally, but they are nevertheless somewhat weaselly – whose reservation is it that is thusly expressed?

I disagree, however, on the issue of H10. If you grant the Church–Turing thesis, we know that not only you and me will fail to devise a process that does the job, but also visitors from the Aldebaran system, as well as any collection of dactylographic monkeys, for the reason that no effective method for systematically solving Diophantine equations exists. If provably no one, however able, can do it, then the inability to do it is independent of any person's abilities; the task itself is intrinsically impossible. If, on the contrary, the statement depends on denying the Church–Turing thesis, it thereby represents a point of view that is definitely not generally accepted. I think you will need a solid citation to back your point of view.  --Lambiam 17:25, 9 June 2008 (UTC)

As I said, your comments certainly are helpful. (And just as certainly, putting a tag on an article without explaining why is not very helpful.) On H10...I agree about Aldebarans and dactylographic monkeys and all that. Moreover I find the Church-Turing thesis to be quite plausible. So we agree on almost everything, with the possible exception that I would have to say that denying C-T is a coherent stance. Maybe I should say it is a coherent philosophical stance: it would be another thing entirely to deny C-T in a way which has some mathematical content, but it is possible (the point being that anything which is sufficiently vaguely phrased is possible!). It seems like the sentence in question reads in a way different from what I intend, so it should be fixed, by me or someone else. Plclark (talk) 22:57, 9 June 2008 (UTC)Plclark

baseball statistics

"Baseball's Hilbert Problems: 23 Burning Questions"[1]. Heh. 66.127.55.192 (talk) 01:57, 19 February 2010 (UTC)

Summary

The summary section says 11 has an accepted answer, but the table says otherwise — Preceding unsigned comment added by 98.109.170.24 (talk) 22:20, 9 June 2011 (UTC)

Also, the 13th problem in the table says "Unresolved", but is still colored green. I couldn't figure out how to fix it and I'm not sure I'm not missing something anyway... — Preceding unsigned comment added by 124.148.188.82 (talk) 10:44, 17 July 2011 (UTC)

Ditto problem 22, whose color I changed to pink to match the "Unresolved" in the table. -- Elphion (talk) 01:59, 25 October 2018 (UTC)

Nonsensical sentence

Under the section Ignorabimus this sentence appears:

"In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible."

Unfortunately the phrase "the original problem is impossible" is a meaningless phrase. (A mathematical *problem* is not the kind of thing that is either possible or impossible.) The footnote ([6]) makes this nonsensical phrase no clearer.

Perhaps someone who knows what it is supposed to mean can express it clearly.Daqu (talk) 14:49, 5 December 2012 (UTC)

See Proof of impossibility: some problems cannot be solved, e.g. the Halting problem. Bill Wvbailey (talk) 01:49, 6 December 2012 (UTC)

Right, so the issue seems to be, is that the sort of problem Hilbert was talking about? If the question is "how do I do such-and-such", then the answer "there is no way to do such-and-such; it's impossible" is a perfectly legitimate answer. But if the problem is, "is proposition A true or false?" then the answer "the problem is impossible" doesn't really make sense, which is presumably what Daqu was getting at.

So the text as reported is clearly confusing and needs to be changed, but to change it, we need to understand what Hilbert was actually talking about, assuming he actually did say something along these lines. Let's start with the last point — has it been demonstrated that he did say something that could have led to the quoted sentence? If so, what exactly did he say? --Trovatore (talk) 01:55, 6 December 2012 (UTC)

Maybe this will be of use -- from Hilbert by Reid 1996. My reading of the following is that Hilbert is talking of the first way you stated it: "If the question is "how do I do such-and-such", then the answer "there is no way to do such-and-such; it's impossible" is a perfectly legitimate answer."

"[Paris 1900 speech were Hilbert presents his problems:] Hilbert had concluded the extensive introduction to his problems with a stirring reiteration of his conviction ("which every mathematician shares, but which no one has yet supported by a proof") that every definite mathematical problem must ncecessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. He had then taken the opportunity to deny publicly and emphatically the " Ignoramus et ignorabimus " -- we are ignorant and we shall remain ignorant -- which the writings of Emil duBois-Reymond had made popular during the century which was passing:

" 'We hear within us the perpetual call. There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.' " (Reid 1996:72)

"But it was also in the course of this talk [Zurich, 1917] that he brought up certain questions which revealed for the first time since 1904 in public utterance his continued interest in the subject of the foundations of his science:

"The problem of the solvability in principle of every mathematical question.

"The problem of finding a standard of simplicity for mathematical proof.

"The problem of the relation of content and formalism in mathematics.

"The problem of the decidability of a mathematical lquestion by a finite procedure." (p. 150]

"An example was Hilbert's attitude toward the question of the solvability of every definite mathematical problem. At Paris he had spoken in ringing tones of the axiom of the solvability of every problem, 'the conviction which every mathematician shares, although it has not yet been supported by proof.' He was convinced that in mathematics at least 'there is no ignorabimus. ' Yet, at Zurich [1917], he listed among the epistomological questions which he felt should be investigated the question of the solvability in prinicple of every mathematical question." (p. 174)

" 'The true reason, according to my thinking, why Comte cound not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem.'

"He denied again, at the end of his career, the "foolish ignorabimus of duBois-Reymond and his follwers. His last words into the microphone were firm and strong:

" 'Wir müssen wissen. Wir werden wissen.'

" We must know. We shall know. " (p. 196)

Bill Wvbailey (talk) 16:07, 6 December 2012 (UTC)

So there's nothing in any of that that, in my reading, supports the claim that "Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible". Quite the opposite. I'm thinking that the claim should probably be removed, unless someone can figure out where it came from. --Trovatore (talk) 17:20, 6 December 2012 (UTC)

Oh, I didn't see the bit about "the proof of the impossibility of its solution". This is very peculiar; it does not match at all with the rest of the text. I have no immediate opinion on what sort of example he might have had in mind. This is very problematic — it's hard to ignore it, but it's also hard to find a non-speculative way of presenting it. Maybe the best thing would be just to present the quotation somewhere and let the reader try to guess what it means, rather than substituting our own guesses. --Trovatore (talk) 18:50, 6 December 2012 (UTC)

I was going to suggest exactly the same, i.e. in the context of 1900 we might present the following as a quotation:

"his conviction . . . that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution . . ." (Reid 1996:72)

But, based on the context of "the decision problem" especially later in the 1920s after his thinking evolved, he believed no mathematical question would admit of the possible alternative: "undecidable”. In other words, he believed in a "universal decider" for any mathematical question. But this seems to imply notions of "true and false". From the article Decision problem:

The Entscheidungsproblem, German for "Decision-problem", is attributed to David Hilbert: "At [the] 1928 conference Hilbert made his questions quite precise. First, was mathematics complete. . . Second, was mathematics consistent. . . And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle be applied to any assertion, and which was guaranteed to produce a correct decision on whether that assertion was true" (Hodges, p. 91). Hilbert believed that "in mathematics there is no ignorabimus' (Hodges, p. 91ff) meaning 'there is no limit to what can be known'. See David Hilbert and Halting Problem for more.

But this seems to complicate matters. Maybe what happened here was iWvbailey)ndeed an evolution of Hilbert's thinking -- in 1900 the question he proposed was too vague, but by 1927 it had evolved into one quite precise and answerable (but wouldn't that vindicate him? The universal "decider" for mathematics is "impossible" and, and Church and Turing provided impossibility proofs to that effect). I am confused. Wvbailey (talk) 22:04, 6 December 2012 (UTC)Bill

Bill (user Wvbailey): Apparently you did not read carefully my comment that opened this section. Of course I know that some problems cannot be solved finitely (depending on what axiom system is being used).

But my comment is about clear wording. To repeat: "This problem is impossible" is a meaningless phrase. "It is impossible to solve this problem" expresses the impossibility of solution clearly. In this case it is not the problem that is impossible; it is its solution.Daqu (talk) 22:44, 13 December 2012 (UTC)

Yes, the issue exists. The question is what to do about it.

If we're going to paraphrase Hilbert's "proof of impossibility" comment in any intelligent way, beyond just quoting it, we need to understand what he meant. On its face, it seems to contradict the rest of what he said. --Trovatore (talk) 22:52, 13 December 2012 (UTC)

The sentence really is nonsensical; according to all of the historical information provided by Wvbailey, I think we should just state Hilbert's 1st idea, and hilbert's 2nd idea. Like this: T *In a Paris 1900 speech, Hilbert said that every mathematical problem must have a solution. For a problem like "Is this preposition true?", the answer "we cannot determine whether it is true or false, and here is why" is a valid solution, the same as "yes" or "no" could be valid solutions.* Qsimanelix (talk) 13:55, 15 July 2021 (UTC)

Hilbert's fourth problem

I think it is misleading simply to say, as the article does, that the problem is "too vague" and to leave it at that. Although the problem does have various interpretations, the primary one is surely the one given by Hilbert himself.

We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.[2]

In fact, in 1973 Pogorelov gave a complete characterization of such geometries in dimensions 2 and 3. Busemann confirmed that the method can be extended to all dimensions. Although the information that it is "too vague" is properly sourced to Gray's summary of the status of the Hilbert problems, my impression on reading [3] and [4] is that, in the primary sense intended by Hilbert, as understood by specialists, the problem has been solved to the extent that it can ever be. The idea that the problem is too vague is most likely related to the following situation, as described by Busemann:

The fourth problem concerns the geometries in which the ordinary lines, i.e. lines of an n-dimensional (real) projective space Pn or pieces of them, are the shortest curves or geodesics. Specifically, Hilbert asks for the construction of these metrics and the study of the individual geometries. It is clear from Hilbert’s comments that he was not aware of the immense number of these metrics, so that the second part of the problem is not at all well posed and has inevitably been replaced by the investigation of special, or special classes of, interesting geometries.[5]

In [6], Busemann characterizes the problem as "(1) Determine all geometries satisfying these conditions. (2) Study the individual ones." Then he writes: "The discovery of the great variety of solutions showed that part (2) of Problem 4 is not feasible. It is therefore no longer considered as part of the problem. But many interesting special cases have been studied since 1929." Papadopoulos agrees, writing "The second part of the problem is indeed very wide, and because of that it is likely to remain open forever." [7] In my view, this should take nothing away from the solution of part (1), and the discovery that there are simply too many solutions to "study the individual ones." Likewise, the fact that the problem can also be re-interpreted in other ways doesn't alter the fact that the formulation of it that Hilbert stated in the quote above has been solved.

I propose therefore that something be said in the table about Pogorelov's solution. 184.171.221.61 (talk) 08:37, 23 December 2013 (UTC)

I've just noticed that the Encyclopedia of Mathematics writes "Final solution by A.V. Pogorelov" in its entry on Hilbert's problems. [8] 184.171.221.61 (talk) 10:03, 23 December 2013 (UTC)

I suggest to first edit Hilbert's fourth problem to insert the history of the question that you have summarized (Pogorelov's solution included), and then correct the table as "partially resolved" or "resolved" (I have not a clear opinion), with an explaining note similar to that of fifth problem. D.Lazard (talk) 11:28, 23 December 2013 (UTC)

6th Problem - again

In the discussion of the 6th problem, no effort is made to distinguish the pre-quantum mechanical, pre-relativistic Physics that Hilbert must have been referring to in 1900 (since neither had been 'discovered' yet) from modern physics. I think the term 'physics' should be changed to 'Classical physics'. (I saw no mention of this in the archived (pre-2007) talk-page section on Problem 6.) It also seems to me, given this qualification, that Problem 6 HAS BEEN COMPLETED. (as stated Kolmogorov's axiomatics seems to complete the problem, but I am NOT an expert in this, so must defer to others.) Thus, the statement made in the article that the the axiomatics of [classical] physics is now more "remote" seems false, although it almost certainly is less important (since classical physics is no longer regarded as a fundamentally adequate basis for describing the physics of our Universe).

I contend:

1. 'physics' should be qualified as 'classical physics' or perhaps '[classical] physics if in a direct quote. (For the mathematical basis of the Standard Model, see the - yet to be claimed - Clay prize for Yang–Mills existence and mass gap problem) and

2. "both more remote and " should be deleted (its not only arguably wrong, but 'remote' is a weasle word)173.189.79.42 (talk) 22:17, 15 May 2015 (UTC)

Some issues

The following sentence:

There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.

has no source and I believe it is vague and/or plainly false (as some comments above point out, too). The problems are examined in details, for example, in the book edited by Browder (about 110 pages for Problem 6 !). As another source, the Encyclopedia of Mathematics mentions them, asserting 4 is solved and 6 is very far from solved in any way... (but) there is a great deal of interest and activity in (the axiomatic approach represented by) topological quantum field... (etc.)

In my opinion the sentence should be deleted.

Turning to an argument I know a bit better, it is uncontroversial that the second problem deals with a theory much stronger than Peano Arithmetic (I guess that the definition itself of PA has been given in subsequent years). (Now I see that this has been discussed clearly also in Talk:Hilbert's second problem.) The theory mentioned by Hilbert should deal at least with real numbers, but at the end Hilbert even mentions "Cantor’s higher classes of numbers and cardinal numbers". It is a quite common opinion that Gödel's incompleteness gives a negative solution to problem 2, though this is not accepted by some. On the other hand, Gentzen deals only with PA; only a much stronger result could be considered as a possible solution to the second problem. Gentzen-like results, so far, can be considered only as very partial solutions (see, e.g., Simpson, Subsystems of Second Order Arithmetic). As far as I know, a Gentzen-like result for second order arithmetic is presently beyond reach [9], let apart full set theory! [10]. I suggest to delete the reference to Gentzen as a possible solution. In view of the subsequent formulations of Hilbert program, it is debatable whether Hilbert could have considered Gentzen proof as "finitistic". However, the point is that Gentzen proof deals with a theory which is too weak.

By the way, parts of the section "Ignorabimus" are historically inaccurate: Hilbert program has been developed after the problem list. Paolo Lipparini (talk) 17:40, 11 October 2015 (UTC)

Table sorting glitch

Sorting the table of problems by problem number yields a very strange order, with 1st in the middle.Retardednamingpolicy (talk) 13:20, 21 February 2016 (UTC)

What do you mean by "1st in the middle"? The numbering and and the resulting sorting of the table are due to Hilbert himself. Most of the problems are referred to in the literature (and often in article titles) by their number. This cannot be changed by Wikipedia nor by anybody. D.Lazard (talk) 14:15, 21 February 2016 (UTC)

Huh? I literally mean 1st in the middle. 10-19, then 1, then 20-23, then 2-9. Just try it. Retardednamingpolicy (talk) 14:47, 21 February 2016 (UTC)

On my browser the problems are displayed in the normal ordering. It seems that you try to sort the table with an algorithm that treats the problem numbers as character chains, in which case the ordering is correct, as usually the alphabetic characters appear after the digits in the lexical ordering. Wikipedia tables are not designed for being sorted by external programs nor for being sorted in a different way. D.Lazard (talk) 15:38, 21 February 2016 (UTC)

Did you actually try sorting by problem number, as in click on Problem in the header? For me the table is displayed in the proper ordering as well when the page first loads, but if you wanna sort by something else and then revert to the original sorting, that's not possible. I tried on Chrome, Firefox and Internet Explorer, Windows 8.1, so clearly it's not just me.Retardednamingpolicy (talk) 16:58, 21 February 2016 (UTC)

Understood. The table was mistakenly marked as "sortable". I have edited it for fixing the problem. D.Lazard (talk) 17:19, 21 February 2016 (UTC)

Dubious

Is Hilbert's 16th problem really too vague to be resolved? The problem seems well-posed to me, and the article on the 16th problem also seems to treat it (and attempts to solve it) seriously. Same goes for the various journal articles dealing with the problem. I'm tagging the statement in the summary section as dubious, and reverting the status back to "unresolved" for now. Banedon (talk) 02:54, 8 August 2016 (UTC)

Removed references

The books contain information of some generality. For example, the 10th problem is directly connected to (interpretations) of the 2nd problem, e.g. "it concerned a decision problem for Diophantine equations that was related to unpublished results he [Goedel] had obtained years earlier as corollaries to his incompleteness theorems." (e.g. from Dawson "Logical Dilemmas" 1997:238). But if they are not there already these references probably could be moved to the relevant sub-articles (the Matijasevich reference is indeed in the 10th problem article). I'd prefer them to stay, but I'll leave your edit in place and won't dispute the matter further. Bill Wvbailey (talk) 17:38, 10 August 2016 (UTC)

I'm going to put them back. This article is a bit of an anomaly in that it gathers together quite disparate topics in a way that might ordinarily violate the principle that an article should be about a single topic, except for the very noteworthy connection of going through Hilbert's challenge. When a reader drills down into the article, he/she may very well want references on one of the individual problems, even if that reference is not talking about the subject of the article as a whole, meaning the problems considered together. --Trovatore (talk) 20:36, 10 August 2016 (UTC)

The problem with that is that there are 23 problems, and if we allow references to individual ones in this general article, then the number of possible references balloons quickly. We also have articles on each individual problem, which are the obvious places to look if one is interested in those. The current article even says "for details on the solutions and references, see the detailed articles that are linked to in the first column" just before the table. I think leaving the references out is preferable. Banedon (talk) 00:52, 11 August 2016 (UTC)

The references section is of quite manageable size; I don't think we have a problem with too many. --Trovatore (talk) 01:27, 11 August 2016 (UTC)

That's right now, but what about in the future? If we keep these references then adding references to problem 1, problem 2, etc would make sense and "improve" the article. The Riemann Hypothesis article, Hibert's problem 8, contains many such references. Is there a reason not to copy those references over? Banedon (talk) 06:04, 11 August 2016 (UTC)

Well, we could certainly copy one or two especially useful ones; I can't see any problem with that. Sometimes WP has a tendency to overdo references sections and they get hard to find anything in, but I don't think we're near that line in this particular article. --Trovatore (talk) 06:36, 11 August 2016 (UTC)

18th problem: 3 parts?

According to Hilbert's eighteenth problem, the 18th problem has three parts, the first of which was answered by Bieberbach. This page only lists two parts. Is there a reason for that? Should this page be updated? 178.11.76.129 (talk) 09:14, 31 July 2020 (UTC)

Reference 7 404s.

Reference 7 (http://www.networkworld.com/community/node/33361) is a dead link that just goes to a "Page not Found" page. I don't know what, if anything, it should be replaced by. — Preceding unsigned comment added by 82.40.177.40 (talk) 14:53, 8 February 2021 (UTC)

Create articles for each problem?

It seems this article struggles somewhat to talk about all the problems together in a properly explanatory manner. Hilbert was a massively influential mathematician and all of his 23 problems invoke incredibly complex topics. The relevance of these topics cannot meaningfully be explained by simply throwing in a links to Axiomatic set theory, for example. Multiple of the problems are both noteworthy enough and difficult enough to explain that they deserve their own articles. EditorPerson53 (talk) 12:19, 29 April 2023 (UTC)

Each of the Hilbert's problem has already its own article. They can be reach by clicking on the first column of the table. The relevance of these topics cannot meaningfully be explained by simply throwing in a links to Axiomatic set theory, for example: everybody agree with this assertion, and it is probably for this reason that axiomatic set theory is not linked to in this article. D.Lazard (talk) 12:36, 29 April 2023 (UTC)