[The summaries here of Hilbert's problems are necessarily brief and sometimes a bit wide of the mark; see some corrections below -- djr] ============================================================================== From: Aleph Software Consulting Subject: Problem for the 21'st century Date: 3 Oct 1999 17:15:35 GMT Newsgroups: sci.math Keywords: Hilbert's problems. In 1900 Hilbert gave 23 problems before the 1990 International Congress of Mathematics at Paris. We are now only a few months from 2000. Here are the 23 problems ennumerated by Hilbert. What problems will be important for the next 100 years? 1. Cantor's Problem of the Cardinal Number of the Continuum 2. The Compatibility of the Arithmetical Axioms 3. The Equality of the Volume of Two Tetrahedra of Equal Bases and Equal Altitudes. 4. Problem of the Straight Line as the Shortest Distance Between Two Points. 5. Lie's Concept of a Continuous Group of Transformations without the Assumption of the Differentiability of the Functions Defining the Group. 6. Mathematical Treatment of the Axioms of Physics. 7. Irrationality and Transcendence of Certain Numbers. 8. Problems of Prime Numbers 9. Proff of the Most General Law Reciprocity in Any Number Field 10. Determination of the Solvability of a Diophantine Equation. 11. Quadratic Forms with Any Algebraic Numerical Coefficients. 12. Extensions of Kronecker's Theorem on Abelian Fields to Any Algebraic Realm of Rationality. 13. Impossibility of the Solution of the General Equation of the 7th Degree by Means of Functions of Only Two Arguments. 14. Proff of the Finiteness of Certain Complete Systems of Functions. 15. Rigorous Foundation of Schubert's Enumerative Calculus. 16. Problem of the Topology of Algebraic Curves and Surfaces. 17. Expression of Definite Forms by Squares. 18. Building Up of Space from Congruent Polyhedra. 19. Are the Solutions of Regular Problems in the Calculus of Variations Always Necessarily Analytic? 20. The General Problem of Boundary values. 21. Proof of the Existance of Linear Differential Equations Having a Prescribed Monodromic Group. 22. Uniformization of Analytic Relations by Means of Automorphic Functions. 23. Further Development of the Methods of the Calculus of Variantions. ---------------------------- I think the final paragraph of Hilbert's lecture is most appropriate: The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That is may completely fufil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples. ======================================================= For the next century what problems will be important? ============================================================================== From: m.j.vasko <76370.326@CompuServe.COM> Newsgroups: sci.math Subject: Re: Hilbert's problems Date: 7 Jan 1995 06:18:33 GMT Here is a brief list of 22 of David Hilbert's 23 problems, apparently first published in 1901 (sorry for the omission, but my source apparently has a rather large typo--one problem is flat out missing). The basic list was extracted from "The Harper Collins Dictionary of Mathematics", and I've added details and explanations where possible based on reference materials at hand. Some of the items contain comments regarding the problems status, and I please understand that based on my sources, that such such comments are up to date only as of about 1987-1989 or so. Note also that I use terms like "apparently shows" when probably all modern experts would use the term "proves", mostly because I am sometimes paraphrasing and sometimes abstracting information from more than one source. 1. Prove or disprove the Continuum Hypothesis. In 1938, Kurt Godel proved that the Zermelo-Frankel set theory axioms are not sufficient to disprove the Continuum Hypothesis, but it has apparently not been proved that the ZF axioms are self-consistent (maybe that's the rough equivalent of proving the parallel postulate). In 1963, Cohen apparently showed that the negation of the continuum hypthoses cannot be disproved, but my source adds no more detail. The continuum hypothesis states that the cardinality of the continuum is the smallest of all non-denumerable cardinals. As I translate that, it means that the infinity that represents the number of real numbers is the smallest possible infinity there is. Since both this hypothesis and its negation have been shown to be consistent with the standard axioms of set theory, it is considered to be undecidable. 2. Prove or disprove the consistency of the axioms of arithmetic. Godel's famous theorem apparently shows that it is impossible to prove the consistency of arithmetic using only methods available within arithmetic, and that adding rules to the system adds just enough to the system to show that it is still not possible to prove consistency. This is usually interpreted to mean that it is impossible to prove all true theorems in arithmetic (or disprove all false ones, or both--in any case, both Russell and Whitehead are surely disappointed). Godel's proof relied on an idea that might be stated something like (don't take me to task for this paraphrase): code the rules of arithmetic so that they are expressible as numbers, then show that it is possible to create a a formula that refers to itself, and that says that itself is not provable, but that appears to be consistent with all the rules of set theory/arithmetic...which is a contradiction [see Hofstadter's "Godel, Escher, Bach" for a fairly understandable, and somewhat entertaining ramp-up to the proof] 3. Prove whether or not all tetrahedra with the same base and altitude have the same volume. In 1900, Max Dehn proved that all such tetrahedra do NOT have the same volume. Here's a problem that sounds accessible to us amateurs. (note that Dehn's solution was apparently obtained before Hilbert published his list) 4. Construct all metrics in which straight lines are geodesics. 5. Determine to what extent we can approach Lie's idea of continuous groups of transformations without assuming that the transformations are differentiable. Gleason (1952) and Montgomery-Zippen (1955) are said to have solved this problem by proving that every locally Euclidean group is a Lie Group. Lie groups are groups in topology that can be assigned a structure such that both the group operation and inversion are analytic. 6. Axiomatize mathematical physics. My sources provide no further information or information on progress except to say that some has been made. 7. Determine whether a^b is transcendental if a is algebraic and b is irrational. My source says this has not been solved. Perhaps this means a general solution is not known, since I thought I had recently seen proofs that some example of such a number had been proved transcendental (pi^e? e^pi?). Gelfond, Schneider, and Baker are said to have made progress, and Alexander Gelfond was apparently the key worker on this problem. 8. Prove or disprove the Riemann Hypothesis. Said to be "notoriously" unsolved. This is the zeta hypothesis, which says that the zeta function ( zeta(s) = SUMOF 1/n^s [1...infinity] ) has all of its non-trivial zeroes on the line re(z) = 1/2. (all of the trivial zeroes occur at even negative integers). My source says that the zeta hypothesis is known to be true for the first "many million values" of s. 9. Find the most general law of reciprocity (quadratic reciprocity?) in an algebraic number field. Emil Artin is credited with a 1927 proof valid only for Abelian extensions to the number field and the non-Abelian case is still said to be open, with no mention of progress or lack thereof. Unfortunately, my source does not write about this problem in language that allows me to clarify it any further. 10. Find or prove the nonexistence of a method to show whether any given Diophantine equation is soluble. Matijasevich is credited with a 1970 proof that shows it is impossible to develop a method that works for all Diophantines, and I believe that I read somewhere that he did so using a method similar in spirit to Godel's methods for problem 1. 11. The study of Quadratic Forms with algebraic coefficients. This problem is listed only as "incomplete". Does anyone know how closely this is related to problem #9? 12. The study of extensions (required extensions to? completeness of?) to algebraic number field(s). Also listed as "incomplete". 13. Show that the general equation of the 7th degree cannot be solved by means of functions with only two arguments. Listed as partly solved. 14. Sorry. I don't understand group/set theory well enough to be sure of translating this correctly, and the ASCII set doesn't allow me to type it as is. Here's the best I can do, and Nagata is credited with a 1959 proof that shows the assertion is false. "The ring K "cap" k[x.1,...,x.n] is finitely generated over K, where K is a field, k[x.1,...,x.n] is a polynomial ring, and k is a subset of K is a subset of k(x.1,...,x.n)." (this is as close a quote as I can make in ASCII...the "cap" appears to be the intersection sign, basically an upside down "U"). 15. Provide a rigorous foundation for "Schubert's Enumerative Calculus". No comments given. No more detailed definitions supplied in the body of the text under any of the words in the problem statement. 16. The investigation of the topology of algebraic surfaces. Nothing more supplied. 17. Express a (any?) definite rational function as a quotient of sums of squares. Emil Artin is credited with a 1927 solution which shows that "a positive definite rational function is a sum of squares". I am not sure if this is a solution, an even better solution, or not quite a solution. A rational function is defined as a polynomial or a ratio of polynomials. The following definition is supplied for "positive definite": - adj. (of a matrix or a self-adjoint operator on Hilbert space) having > 0 for all x<>0. [is it possible that this solution is directly related to Artin's presumably partial solution to problem #9?] 18. Do space-filling polyhedra exist that are non-regular? No progress noted. Sounds like a fun one, though. 19. Are the solutions to Langrangians always Analytic? No progress noted. Based on the definitions of these terms in the body of the text, it is unclear if the Langrangian referred to is that commonly noted in physics (the blah) or a more abstract concept, but it seems likely to me that Hilbert is referring to the first definition (based on the earlier problem #6). In either case, it seems fairly likely that the intepretation to be placed on analytic is that the solutions should locally correspond to (agree with) their Taylor series expansions. This also seems to imply that the function possesses derivatives of all orders (though I may have the cart before the horse on the "derivative/taylor" interpretation). 20. Can every variational problem be solved, provided suitable boundary conditions can be set. No progress noted. I assume that "variational problem" refers to problems in the calculus of variations based on their highlighting of the phrase, which leads back to the calculus of variations only. 21. Oddly enough, this problem is missing. If anyone can supply its definition, please do. 22. Show that there always(?) exists a linear differential equation of the Fuchsian class with given singular points and monodromic group. This is noted as solved by Deligne in 1970, so I am assuming that it was proved true by Deligne, not false. From the way it reads, I guess it means that one supplies any singular point(s) and a(?) monodromic group, and that no matter what you specify for those that a linear differential equation can be constructed to fit the specifications, but that's just my interpretation. The dictionary doesn't define Fuchsian class. Singular points are defined as a point on a curve where the curve does not have a smooth tangent (such as when the curve crosses itself or is discontinuous). Definitions are supplied for the "monodromy theorem" and the "monodromy theorem of Darboux", but I can't be sure these apply here. Both of them seem to imply that certain uniformities exist over the entire range of a (complex) function when it is known to be uniform in certain ways with certain subsets of the range. 23. Develop the Calculus of Variations. No progress noted. Elsewhere, the source says that CoV was first developed by Euler in 1744, that both Newton and Bernoulli had solved problems using variational methods and that CoV is currently considered to be a major branch of analysis. Other references point to control theory, Euler-Lagrange equations, optimization theory, and the brachistochrone problem. ============================================================================== From: kevin2003@delphi.com (Kevin Brown) Newsgroups: sci.math Subject: Re: Hilbert's problems Date: 7 Jan 1995 20:49:19 GMT MV = M.J.Vasko MV> Here is a brief list of 22 of David Hilbert's 23 problems,... MV> The basic list was extracted from "The Harper Collins Dictionary MV> of Mathematics", ... MV> [1-20 deleted] MV> 21. Oddly enough, this problem is missing. If anyone can supply its MV> definition, please do. According to the "Encyclopedic Dictionary of Mathematics" (ed by Kiyosi Ito) the Hilbert's 21st problem was "To show that there always exists a linear differential equation of the Fuchsian class with given singular points and monodromic group. Solved by H. Rohrl and others (1957)." ============================================================================== Date: Mon, 07 Jun 1999 18:01:46 -0500 From: Tamara MIller To: feedback@math-atlas.org Subject: Hilbert's First Problem The description of Hilbert's First Problem contains some serious errors. First of all, proving the ZF axioms self-consistent is impossible (within ZF) by Godel's "second" incompleteness theorem. Second, the continuum hypothesis should be translated as meaning that the infinity that represents the number of real numbers is the smallest possible *uncountable* infinity there is, not the smallest possible infinity, since of course, the infinity that represents the number of natural numbers is the smallest possible infinity. Cantor proved the number of real numbers is strictly greater than the number of natural numbers, and the continuum hypothesis states that there is no other size of infinity in between the one representing the number of natural numbers and th one representing the number of real numbers. [...] ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Problem for the 21'st century Date: 3 Oct 1999 20:52:07 GMT Newsgroups: sci.math In article <37f7b499.757217109@news.erols.com>, wrote: >>In 1900 Hilbert gave 23 problems before the 1990 International Congress of >>Mathematics at Paris. A slightly expanded discussion is in the sci.math FAQ; see also http://www.math-atlas.org/95/hilb.list Proceedings of a conference reviewing progress on the problems are in "Mathematical Developments Arising from the Hilbert Problems", Proc Symp Pure Math AMS, vol 28 (1976) Some are solved, some proved unsolveable, some open. But I would like to comment that Hilbert's list was directed to mathematicians, not to young students; in particular, they are not interpreted as narrow, "prove-this" questions. Rather, they are open-ended directives drawing attention to questions of the form, "What can one say about..." >5. Is this list truely the Holy Grail of mathematics? Well, some of the questions are particularly central to mathematics of the 20th century. For example, there are many results which state, "If the Riemann Hypothesis is true, then...". But some of the questions are now seen as comparatively minor. >6. Similarily, Is this list relevant to you or has late 20th cent. math > gone in other directions? Where? (Okay, that was 2 questions). To a large extent Hilbert's questions _caused_ 20th-century mathematics to be as it became. The developments of mathematical logic and algebraic geometry in particular strike me as having been engendered by Hilbert. Hilbert couldn't have foreseen the impact of computers and technology on mathematics and science, which have indirectly nutured many now-large areas of mathematics, which were mere backwaters in 1900: combinatorics, numerical analysis, operations research, ... dave