Talk:Hilbert's problems

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Archive: before 2007

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[edit] 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 ε0. 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)

[edit] 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)

[edit] 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)

[edit] 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

[edit] 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)

[edit] 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