Talk:Hilbert's problems/Archive 1

message: I'm not sure I understand, or agree with, the status given for the 6th problem: "Non-mathematical". As I understand it, Hilbert was intending the same thing for physics which had already occured for Euclidean geometry. Quoting Hilbert:
:Geometry is a science which essentially has developed to such a state that all its facts may be derived by logical deduction from previous ones. Comptely different from e.g. electricity theory or optics where even today new facts are still being discovered."
and
:Now also all other sciences are to be treated following the model of geometry, first of all mechanics, but then also optics and electricity theory.
How is such a program "non-mathematical"? (See Sauer ).
Paul August ☎ 21:09, 10 March 2006 (UTC)
replies: text: It requires empirical observations.  General relativity could be axiomatized, for instance, but it's not a full and correct theory of physics.  We don't have one.  Thus, the problem is basically one of a lack of empirical observations, insofar as the problem is to axiomatize the correct laws of physics. —Simetrical (talk • contribs) 02:16, 23 May 2006 (UTC)

text: Isn't it impossibly naive to expect solving all physical problems in the near future(even more so back then). I mean... most of the mathematical problems are proving one theorem or the like whereas this is discovering all the hidden laws of physics and essentially completing all of science since all knowledge about all systems could then be deduced straightforwardly. Elentirmo 15:52, 24 August 2006 (UTC)
thread_title: 6th problem, "non-mathematical"?

message: 3rd problem summary seems really wrong to me, almost reverse of the actual problem... However I'd let someone more competent than me on the subject matter sort that out... JidGom 00:36, 8 Dec 2003 (UTC)
replies: text: I agree. I have removed them all (copied below for reference). --Zundark 10:35, 30 Dec 2003 (UTC)
thread_title: 3rd problem

message: I'd say the way most set theorists see the independence of CH from ZFC is that we need better axioms for set theory, namely set theory needs compelling large cardinal axioms capable of settling this question.  And there, my impression is that most set theorosts lean towards thinking CH is false: good large cardinal axioms will probably reveal structure of intermediate cardinality.
:Well, large cardinals of the sort we know today can't settle CH, because they're preserved under small forcing. Steel suggests that there might be things we would still recognize as large cardinal axioms--"natural markers of consistency strength"--that could settle it. --Trovatore 08:22, 20 July 2005 (UTC)
replies: text: Damn straight! --Trovatore 08:22, 20 July 2005 (UTC)
thread_title: Problem #1

message: To summarise Hilbert's 2nd problem "Are the axioms of arithmetic consistent?" and then claim it is 'solved' by Gödel's incompleteness theorem is silly (if you think arithmetic might be consistent) and shows a fundamental failure to understand what is at stake.  Goedel's result only shows that the kind of finitary proof Hilbert sought was impossible; in fact combinatorial principles that go beyond standard arithmetic axiomatisations such as first-order Peano arithmetic can be used to prove consistency of arithmetic, such as Gerhard Gentzen achieved with his consistency proof.
Both of these links need to be changed, and the claim that the first is settled should be changed, possibly also the second ---- Charles Stewart 15:37, 21 Oct 2004 (UTC)
thread_title: Problem #2

message: So I followed the link to Hilbert's speech, and it appears to me that (at least in the speech; I haven't found the subsequent written list), Hilbert's question was not "is CH true?" but rather "is CH provable?".  (He doesn't seem to have explicitly considered that it might be refutable.)  Provable from what exactly, he doesn't specify (note that ZFC had not been formulated at the time). --Trovatore 03:43, 27 July 2005 (UTC)
thread_title: What is the 1st problem, exactly?

message: The table states that the problem was solved, yet the page Hilbert's twelfth problem states that this is an open problem. So which one is correct? Samohyl Jan 20:44, 9 August 2005 (UTC)
replies: text: The specific case of complex quadratic fields is solved; the general, broad question is wide open. Charles Matthews 08:25, 3 November 2005 (UTC)
thread_title:  12th problem 

message: There is a discussion on german wikipedia if hilbert did present 10 or 23 problems at first. In a german translation of a speach given by him in Paris 1900 he talks about all 23 problems. If someone has any clues to where the 10 comes from, he may share it on our discussion there. You're welcome to participate in english. Maybe someone even has the original french paper. Chrislb --84.163.201.189 18:33, 11 August 2005 (UTC)
replies: text: An as I may cite from David Hilbert: He put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. So his speach was actually the first publication - the original paper should then point out the correct number. Chrislb --84.163.201.189 18:43, 11 August 2005 (UTC)
thread_title:  10 vs. 23 problems 

message: There are too many sentences in parentheses. Things should be reworded to not need this. --MarSch 12:11, 3 November 2005 (UTC)
thread_title:  parentheses 

message: Charles Matthews suggested, over at Wikipedia talk:WikiProject Mathematics that redirects like Hilbert's first problem should be filled out in some sort of "buffer page". Whether this is necessary or not in the general case, I certainly agree in the specific case of the first problem.
If you look at Hilbert's speech (there's a link on the main page), he raises two points in the first problem: the continuum hypothesis (or possibly, the provability of the continuum hypothesis), and the issue of the existence of a wellordering of the reals. The latter question is not currently touched here. I see a wide range of possible positions as to its status:
# Solved affirmatively by Zermelo's proof, using the axiom of choice.
# Solved negatively by Cohen's proof of independence from ZF
# Open, because Hilbert wanted to "find" a wellordering, which implies definability, and the existence of a definable wellordering of R is independent of ZFC
# Solved as neither "yes" nor "no", for exactly the reason stated above
# Solved negatively, because no definable wellordering can be proved to be a wellordering, using just ZFC
# Solved negatively, because Hilbert would have wanted something much better than just "definable"; he would have expected something definable in some natural combinatorial way, and if we specify try to pin down that notion according to descriptive set theory, we can prove that no such simple wellordering exists.
The page could also discuss what's consistent with large cardinals--it can be shown that large cardinals of the sort understood today cannot exclude definable wellorders, though they can push the lower bound on their complexity up pretty high, but not higher than Δ22, by a result of Saharon Shelah. --Trovatore 17:41, 3 November 2005 (UTC)
replies: text: Thanks. It's pretty convincing on the basic point, which is that the unmediated redirect is untenable. Charles Matthews 17:56, 3 November 2005 (UTC)

text: Both this page and the second problem would benefit from this, although another fix would be to have a section in the existing target page that explicitly deals with these problems of interpretation.  For the 2nd problem, I already plan on writing a "2nd problem" section, though it's on the sometime/maybe section of my tasks list at the moment. --- Charles Stewart 18:57, 3 November 2005 (UTC)
replies: text: There is a question here as regards the speech: The section is apparently entitled "Cantors Problem von der Mächtigkeit des Continuums", which doesn't mention wellordering the reals. That's from the text here. Question: is this section title from Hilbert's notes, or from some later compilation or something? Hilbert definitely mentions the wellordering problem in the section, towards the end. --Trovatore 19:57, 3 November 2005 (UTC)
replies: text: IIRC Hilbert published his address, so the section headings should be from that publication. --- Charles Stewart 21:01, 3 November 2005 (UTC)
replies: text: Then perhaps among the ambiguities to be discussed in Hilbert's first problem is whether or not the question of wellordering the reals is in fact part of the problem. --Trovatore 21:15, 3 November 2005 (UTC)
thread_title: Hilbert's first problem page

message: Look here for original German versions of the published paper. Charles Matthews 22:39, 3 November 2005 (UTC)
thread_title: German texts

message: Has a superscript 6, but the footnote refers to problem 16 as well.  Is this a typo or should there by another footnote? njh 04:05, 5 January 2006 (UTC)
replies: text: I've cut out the comment about the 16th, since it contains nothing crucial. The dedicated page is a better place for updates. Charles Matthews 09:43, 5 January 2006 (UTC)
thread_title:  Problem 18 

message: Would anyone object if I changed the formatting of the footnotes. I can't stand those "up arrow thingies".  I think this looks better:
===Tabulated information===
Hilbert's twenty-three problems are:
Problem	Brief explanation                                                                                                                                                                                                   	Status
1st    	The continuum hypothesis (that is, there is no set whose size is strictly between that of the integers and that of the real numbers).                                                                               	No consensus.
2nd    	Prove that the axioms of arithmetic are consistent (that is, that arithmetic is a formal system that does not prove a contradiction).                                                                               	Partially resolved: Some hold it has been shown impossible to establish in a consistent, finitistic axiomatic system - However, Gentzen proved in 1936 that consistency of arithmetic followed from the well-foundedness of the ordinal \epsilon_0, a fact amenable to combinatorial intuition.
3rd    	Can two tetrahedra be proved to have equal volume (under certain assumptions)?                                                                                                                                      	Resolved - no, using Dehn invariants.
4th    	Construct all metrics where lines are geodesics.                                                                                                                                                                    	Too vague.
5th    	Are continuous groups automatically differential groups?                                                                                                                                                            	Resolved.
6th    	Axiomatize all of physics.                                                                                                                                                                                          	Non-mathematical.
7th    	Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?                                                                                                                                    	Resolved - Yes - Gelfond's theorem or the Gelfond–Schneider theorem).
8th    	The Riemann hypothesis (the real part of any non-trivial zero of the Riemann zeta function is ½) and Goldbach's conjecture (every even number greater than 2 can be written as the sum of two prime numbers).	Open.
9th    	Find most general law of the reciprocity theorem in any algebraic number field.                                                                                                                                     	Partially resolved.
10th   	Determination of the solvability of a Diophantine equation.                                                                                                                                                         	Resolved - Matiyasevich's theorem implies that this is impossible.
11th   	Solving quadratic forms with algebraic numerical coefficients.                                                                                                                                                      	Resolved.
12th   	Extend Kronecker's theorem on abelian extensions of the rational numbers to any base number field.                                                                                                                  	Open.
13th   	Solve all 7-th degree equations using functions of two parameters.                                                                                                                                                  	Resolved.
14th   	Proof of the finiteness of certain complete systems of functions.                                                                                                                                                   	Resolved
15th   	Rigorous foundation of Schubert's enumerative calculus.                                                                                                                                                             	Resolved.
16th   	Topology of algebraic curves and surfaces.                                                                                                                                                                          	Open.
17th   	Expression of definite rational function as quotient of sums of squares.                                                                                                                                            	Resolved.
18th   	Is there a non-regular, space-filling polyhedron? What is the densest sphere packing?                                                                                                                               	Resolved.
19th   	Are the solutions of Lagrangians always analytic?                                                                                                                                                                   	Resolved.
20th   	Do all variational problems with certain boundary conditions have solutions?                                                                                                                                        	Resolved.
21st   	Proof of the existence of linear differential equations having a prescribed monodromic group                                                                                                                        	Resolved.
22nd   	Uniformization of analytic relations by means of automorphic functions.                                                                                                                                             	Resolved.
23rd   	Further development of the calculus of variations.                                                                                                                                                                  	Resolved.
=== Notes ===
: Cohen's independence result, showing the continuum hypothesis to be independent of ZFC (Zermelo-Frankel set theory, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved. One contemporary view is that it may be the case that set theory should have additional axioms, capable of settling the problem.
: A matter of opinion, not shared by all. Gentzen's result shows rather precisely how much needs to be assumed to prove that Peano arithmetic is consistent. It is widely held that Gödel's second incompleteness theorem shows that there is no finitistic proof that PA is consistent (though Gödel himself disclaimed this inference [this needs a reference-- wvb]).  The following is from Nagel and Newman, pp. 96 and 97: "This imposing result of Godel's analysis should not be misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic. What it excludes is a proof of consistency that can be mirrored by the formal deductions of arithmetic- Footnote 29.[Footnote 29 gives an example of the trisection of an angle-- it is possible, although not with straight-edge and compass]. Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then-footnote 30" [Footnote 30: Describes Gentzen's proof, which uses transfinite induction; "30: "Gentzen's proof depends on arranging all the domonstrations of arithmetic in a linear order according to their degree of 'simplicity'...but Gentzen's argument cannot be mapped onto the formalism of artimetic. Moreover, although most students do not question the cogency of the proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency."[italics added]...."But these [meta-mathematical] proofs cannot be irepresented within the arithmetical calculus; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program."
: According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
: Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
: Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
: Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).
----
Paul August ☎ 14:09, 9 February 2006 (UTC)
replies: text: Hmm … Trov, last I recall the problem occurred only if we had "use MathML if possible (experimental)" selected in "Math" preferences, and then only for certain files containing TeX and certain combinations of the "ent" templates. We had created a test page which produced the problem. However when I now look at that test page I no longer see a problem. Do you? Paul August ☎ 20:18, 9 February 2006 (UTC)
replies: text: No. But it seems to be an intermittent bug. Sometimes I see the problem, then I don't see it on the same page minutes later. --Trovatore 23:57, 9 February 2006 (UTC)
thread_title:  Proposed format change for footnotes 

message: : Cohen's independence result, showing the continuum hypothesis to be independent of ZFC (Zermelo-Frankel set theory, extended to include the axiom of choice) is often cited to justify the assertion that the first problem has been solved. One contemporary view is that it may be the case that set theory should have additional axioms, capable of settling the problem.
: A matter of opinion, not shared by all. Gentzen's result shows rather precisely how much needs to be assumed to prove that Peano arithmetic is consistent. It is widely held that Gödel's second incompleteness theorem shows that there is no finitistic proof that PA is consistent (though Gödel himself disclaimed this inference [this needs a reference-- wvb]).  The following is from Nagel and Newman, pp. 96 and 97: "This imposing result of Godel's analysis should not be misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic. What it excludes is a proof of consistency that can be mirrored by the formal deductions of arithmetic- Footnote 29.[Footnote 29 gives an example of the trisection of an angle-- it is possible, although not with straight-edge and compass]. Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then-footnote 30" [Footnote 30: Describes Gentzen's proof, which uses transfinite induction; "30: "Gentzen's proof depends on arranging all the domonstrations of arithmetic in a linear order according to their degree of 'simplicity'...but Gentzen's argument cannot be mapped onto the formalism of artimetic. Moreover, although most students do not question the cogency of the proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency."[italics added]...."But these [meta-mathematical] proofs cannot be irepresented within the arithmetical calculus; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program."
: According to Rowe & Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved"; Rowe & Gray lists the fourth problem as too vague to say whether it has been solved.
: Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
: Problem 9 has been solved in the abelian case, by the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
: Rowe & Gray also list the 18th problem as "open" in their 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).
----
Paul August ☎ 14:09, 9 February 2006 (UTC)
thread_title:  Notes 

message: *I prefer this to the up arrows, too. --- Charles Stewart(talk) 16:00, 9 February 2006 (UTC)
replies: text: Hmm … Trov, last I recall the problem occurred only if we had "use MathML if possible (experimental)" selected in "Math" preferences, and then only for certain files containing TeX and certain combinations of the "ent" templates. We had created a test page which produced the problem. However when I now look at that test page I no longer see a problem. Do you? Paul August ☎ 20:18, 9 February 2006 (UTC)
replies: text: No. But it seems to be an intermittent bug. Sometimes I see the problem, then I don't see it on the same page minutes later. --Trovatore 23:57, 9 February 2006 (UTC)
thread_title: Comments

message: This article claims that 8 and 12 are unsolved.  The Millenium Prize Problems page says that only one of the problems is unsolved...Surely one of them should be changed... —The preceding unsigned comment was added by 128.135.226.69 (talk • contribs) 12:11, 27 March 2006  (UTC)
:I'm guessing this page is correct and changing Clay Mathematics Institute. —Simetrical (talk • contribs) 02:17, 23 May 2006 (UTC)
thread_title:  inconsistency with millenium prize problems page 

message: The "summary" section says that problem 23 is too vague to be considered solved.  The table lists it as solved.  This is inconsistent.  Also, the summary lists problems 5, 15, 21, and 22 as having solutions with partial acceptance, but the table just lists them as "resolvd" with no qualifications. Ken Arromdee 19:07, 31 March 2006 (UTC)
thread_title:  Summary inconsistency 

message: Whoever you are, you just did a disservice to the readers. I'm sure you are aware of this, but just forgot: This form of bibliographic reference is called an annotated bibliography. And, you erased an annotation by none other than Martin Davis. Good show, dude.wvbaileyWvbailey 13:38, 12 September 2006 (UTC)
replies: text: I removed some POV words and phrases from references added by User:Martin Davis and others. Whoever User:Martin Davis is in real life, he seems to be a new contributor to Wikipedia (judging from his contributions list), and so may not be aware of Wikipedia:Neutral point of view. Subjective terms such as "excellent", "wonderful" and "foolish" are unencyclopaedic, and are as out of place in references as they would be in the body of a Wikipedia article. To quote from Wikipedia:What Wikipedia is not, "Wikipedia is not a place to publish your own thoughts and analyses". Gandalf61 15:23, 12 September 2006 (UTC)
thread_title:  Removed "P.O.V." from the references