## Hilbert’s twenty-fourth problem (2001)

Venue: | American Mathematical Monthly |

Citations: | 9 - 4 self |

### BibTeX

@ARTICLE{Thiele01hilbert’stwenty-fourth,

author = {Rüdiger Thiele},

title = {Hilbert’s twenty-fourth problem},

journal = {American Mathematical Monthly},

year = {2001},

volume = {110},

pages = {2003}

}

### OpenURL

### Abstract

1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

### Citations

393 | Modular elliptic curves and Fermat’s last theorem
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Citation Context ...treated successfully since the 1920s [8:1, pp. 241–268], [115, pp. 171–202]. On the other hand, he was quite right in his prognosis that Fermat’s Last Theorem would be solved in the twentieth century =-=[112]-=-. In 1970, the year Yurij Matiyasevich showed that Hilbert’s tenth problem is unsolvable [70] (seealso[115, pp. 85–114]), J. Fang wrote [18, p. 123]: “The tenth problem, like the eighth, is not likely... |

387 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte der
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Citation Context ...istic reasoning cannot prove the consistency of an axiomatic system, because the necessary formulas for proving consistency are not derivable within this system (Gödel’s second incompleteness theorem =-=[26]-=-). In a different sense, in his metamathematics, Hilbert intended to establish such concepts as consistency by intuitive justifications, i.e., by appealing to self-evident principles of reasoning and ... |

277 |
Every planar map is four colorable
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(Show Context)
Citation Context ... is estimated to require at least two thousand pages. Nevertheless, such proofs, including those of Fermat’s Last Theorem by Andrew Wiles [97], [112] or of the “Four-color Problem” by Appel and Haken =-=[3]-=-, are examples of the kind of finite mathematics that Hilbert hoped to establish by his program (see [101]). Wiles’s proof can be carried out by hand; by contrast, no proof unaided by technology is ye... |

240 | Ring-theoretic properties of certain Hecke algebras
- Taylor, Wiles
- 1995
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Citation Context ...usand pages, while a complete proof of Ramanujan’s conjectures is estimated to require at least two thousand pages. Nevertheless, such proofs, including those of Fermat’s Last Theorem by Andrew Wiles =-=[97]-=-, [112] or of the “Four-color Problem” by Appel and Haken [3], are examples of the kind of finite mathematics that Hilbert hoped to establish by his program (see [101]). Wiles’s proof can be carried o... |

100 |
A note on the entscheidungsproblem
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- 1936
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Citation Context ..., the proofs are carried out completely within the framework of number theory. (Noteworthy in this connection is the fact established by Church that Gödel’s results cannot be obtained by finite means =-=[11]-=-.) In the axiomatic approach, the “tree” of all possible mathematical formulas growing from axioms is not only extremely expansive but, as Gödel pointed out, even disconnected. However, from a finitar... |

76 | Grundzüge der theoretischen Logik - Hilbert, Ackermann - 1928 |

69 | A Mathematician’s Apology - Hardy - 1940 |

51 | The lengths of proofs
- Pudlak
- 1998
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Citation Context ...increasing field of research has developed natural links with the search for the most effective and shortest algorithms, an endeavor with clear ties to Hilbert’s vision of proof simplicity (see [66], =-=[83]-=-, [116]). One might ask: How many proofs do mathematicians publish each year? A backof-the-envelope calculation yields a rough approximation: multiplying the number of journals by the number of yearly... |

49 | Foundations of Mathematical Logic - Curry - 1963 |

45 | Dreams of a Final Theory - Weinberg - 1992 |

38 | Partial realizations of Hilbert’s program
- Simpson
- 1988
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Citation Context ...t these foundational objectives were attainable in a modified sense. From our viewpoint, however, a distinction is to be made between the program Hilbert intended to carry out and its realizable part =-=[94]-=-. We extract another quotation from Hilbert’s Paris lecture [37, p. 444]: “Occasionally it happens that we seek the solution under insufficient presuppositions or in an 18 c○ THE MATHEMATICAL ASSOCIAT... |

36 | The Classical Decision Problems - Gurevich - 1997 |

36 |
Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 37
- Gödel
- 1930
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Citation Context ...3]). Hans Hahn (1879–1934) communicated Hilbert’s extended program to the Vienna Circle, and in 1930 a young Austrian mathematician named Kurt Gödel (1906–1978) demonstrated in his Ph.D. dissertation =-=[24]-=- thatfirstorder predicate logic is complete, i.e., every true statement in it can be derived from its axioms (see [93]). Even the things one most confidently expects do not always come to pass. In one... |

32 |
Algére locale
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Citation Context ...gers) is finitely generated, a fact now known as Hilbert’s Basis Theorem [1], [42, lecture 35], [58:2, no. 16, pp. 199–270] (for generalizations, see [88], [8:2, pp. 431–444], [61, pp. 387, 391], and =-=[92]-=-). Accordingly, this special part of algebra can be placed in the kind of finitary framework we have been discussing. As to the reduction of proofs to an algebraic calculus, it may well be that what H... |

27 |
Zur Widerspruchsfreiheit der Zahlentheorie
- Ackermann
- 1940
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Citation Context ... was due to Gerhard Gentzen, who in the 1934 paper “Die Widerspruchsfreiheit der Zahlentheorie” [21] established the consistency of number theory (see [72]), which Ackermann did independently in 1940 =-=[2]-=-. In accordance with Gödel’s results, such investigations must resort to principles that lie outside pure number theory. For example, transfinite induction is used, but January 2003] HILBERT’S TWENTY-... |

20 | Randbemerkungen zu Hauptproblemen der Mathematik, II, Fundamentalsatz der Algebra and Grundlagen der - Weyl |

17 | Modern Algebra and the Rise of Mathematical Structures - Corry - 1996 |

16 | From Kant To Hilbert: A Source Book - Ewald - 1996 |

16 | Mathematische Grundlagenforschung. Intuitionismus - Heyting - 1934 |

12 | editor): Mathematical Developments Arising from Hilbert Problems - Browder - 1976 |

11 | Zur Hilbertschen Beweistheorie - Neumann - 1927 |

10 | Hilbertprogramm und kritische Philosophie, Göttingen: Vandenhoeck - Peckhaus - 1991 |

9 |
The Hilbert Challenge
- Gray
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Citation Context ...ly. Hermann Weyl (1885–1955) once remarked that “We mathematicians have often measured our progress by checking which of Hilbert’s questions had been settled in the meantime” [110, p. 525]. (See also =-=[31]-=-and[115].) Hilbert and his twenty-three problems have become proverbial. As a matter of fact, however, because of time constraints Hilbert presented only ten of the problems at the Congress. Charlotte... |

8 | Automating the search for elegant proofs
- Wos
- 1998
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Citation Context ...cellation in 1901. 5. SIMPLICITY AND RIGOR. It is widely believed among mathematicians that simplicity is a reliable guideline for judging the beauty (see [84, sect. 4–5], [106, chap. 6]) or elegance =-=[113]-=- of proofs, but like all aesthetic principles, such a criterion is highly subjective. Can one really say that certain mathematical proofs are simpler than others? In other words, the question of what ... |

7 | 2000), ‘A sideways look at Hilbert’s twenty-three problems of 1900
- Grattan-Guinness
(Show Context)
Citation Context ...ttan-Guinness presented an interesting overview of Hilbert’s problems in the Notices of the American Mathematical Society, discussing the form in which each of the twenty-three problems was published =-=[30]-=-. Yet, in dealing with the celebrated problems from this viewpoint, he failed to mention the most interesting problem of Hilbert’s collection: the canceled twenty-fourth. Hilbert included it neither i... |

7 |
The Honors Class. Hilbert’s Problems and their Solvers, A K Peters, 2002. 38 U. Zannier, Lecture Notes on Diophantine Analysis (with an appendix by Francesco Amoroso), Edizioni Della Normale
- Yandell
- 2009
(Show Context)
Citation Context ...mann Weyl (1885–1955) once remarked that “We mathematicians have often measured our progress by checking which of Hilbert’s questions had been settled in the meantime” [110, p. 525]. (See also [31]and=-=[115]-=-.) Hilbert and his twenty-three problems have become proverbial. As a matter of fact, however, because of time constraints Hilbert presented only ten of the problems at the Congress. Charlotte Angas S... |

6 |
Hilbert’s Invariant Theory Papers
- Ackerman, Hermann
- 1978
(Show Context)
Citation Context ...ofs to an algebraic calculus by means of formal logic would allow one to decide which of two given proofs is simpler merely by comparing the number of operations involved in each. In invariant theory =-=[1]-=-, Hilbert had already touched upon the question of how to express relations in terms of a finite basis (see also his 1897 lecture [42]). From such a viewpoint, it would seem possible to arrange mathem... |

6 | The Foundations of Mathematics for the Working Mathematician - Bourbaki - 1949 |

5 |
The one-hundred anniversary of the death of invariant theory
- PARSHALL
- 1990
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Citation Context ... the invariants in question are invariants of tensors or, to employ the terminology of Hilbert’s time, invariants of an n-ary form of degree m under linear transformations ([39], [42]; see also [74], =-=[79]-=-, [89]). Old papers in invariant theory typically consisted of masses of endless algorithmic computations, whereas by viewing invariants in a broader framework Hilbert proved his theorems in a few pag... |

4 | Über Hilberts Gedanken zur Grundlegung der Arithmetik. Jahresbericht der Deutschen - Bernays - 1922 |

4 | A shorter proof of the Ramanujan congruence mod 5, This MONTHLY 104 - Drost - 1997 |

4 | Elements de Logique Mathematique. Theorie des modeles. Societe Mathematique de - Kreisel, Krivine - 1966 |

4 | How reliable is a computer-based proof - Lam - 1990 |

4 |
The International Congress of Mathematicians in
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- 1900
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Citation Context ...ed only ten of the problems at the Congress. Charlotte Angas Scott (1858–1931) reported on the Congress and Hilbert’s presentation of ten problems in the Bulletin of the American Mathematical Society =-=[91]-=-. The complete list of twenty-three problems only appeared in the journal Göttinger Nachrichten in the fall of 1900 [37], and Mary Winston Newson (1869–1959) translated the paper into English for the ... |

4 | Hilbert and his mathematical work - David - 1944 |

3 | What does logic have to tell us about mathematical proofs - Feferman - 1979 |

3 |
Grundlegung der elementaren Zahlenlehre
- Die
- 1930
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Citation Context ...ld necessarily be successful, especially if pursued by means of the reduction of proofs to an algebraic calculus or “logical arithmetic” [44], [45]. That was still his belief in the 1920s ([48], [51]–=-=[56]-=-; see also [66]). Hilbert wanted to achieve his objective of establishing the consistency of axiomatized mathematics by proceeding from a purely finitary standpoint, i.e., he accepted only those facts... |

3 | Recursive Functions and Metamathematics - Murawski - 1999 |

3 | Hilbert’s new problem - Wos, Thiele - 2001 |

2 | Towards a Philosophy of Modern Mathematics - Fang - 1970 |

2 |
Grundlagen der Mathematik, 2 vols
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Citation Context ...d to it, delivering a course “Principien der Mathematik” [Principles of mathematics], which was worked out by Paul Bernays (1888–1977) [47] and which finally led to the book Grundlagen der Mathematik =-=[60]-=-. The basic idea of Hilbert’s proof theory is to ensure the validity of arguments by reducing mathematics to a finite number of rules of inference that govern the manipulation of formulas. The set of ... |

2 |
Two turning points in invariant theory
- ROTA
- 1999
(Show Context)
Citation Context ...ed the finiteness question only in principle, 12 c○ THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 110swithout any indication that we can actually calculate certain numbers; [42, lecture 37], [74], =-=[89]-=-.) The solvability of this problem in finitely many operations is evident, but from an epistemological viewpoint such computing tasks, which have been undertaken by Yasumasa Kanada, Takahashi, Fabrice... |

1 | Redefining Geometrical Exactness - Bos - 2001 |

1 | The beginning of research in the calculus of variations. Osiris - Caratheodory - 1937 |

1 | Zermelo and the Skolem paradox - Dalen, Ebbinghaus |

1 | in Biographical Dictionary of Mathematicians - Freudenthal, Hilbert - 1991 |

1 |
The Collected Papers of Gentzen
- Gentzen
- 1969
(Show Context)
Citation Context ...helm Ackermann (1896–1962) and Paul Bernays (1888– 1977). The first major variation of Hilbert’s program was due to Gerhard Gentzen, who in the 1934 paper “Die Widerspruchsfreiheit der Zahlentheorie” =-=[21]-=- established the consistency of number theory (see [72]), which Ackermann did independently in 1940 [2]. In accordance with Gödel’s results, such investigations must resort to principles that lie outs... |

1 | physikalische Grundlegungsversuch in der Biologie und das psychophysische Problem, Ratio 1 - Gierer, Der - 1970 |

1 |
zu der Diskussion zur Grundlegung der Mathematik am Sonntag, dem 7
- Nachtrag
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(Show Context)
Citation Context ...lace on September 7 (one day before Hilbert’s famous speech) that Gödel made an offhand remark on a work in progress, which is now known as Gödel’s incompleteness theorem ([33, pp. 147–148], see also =-=[25]-=-). Already in November 1930 the Leipzig journal Monatshefte für Mathematik und Physik received the epoch-making, 25-page article “Über formal unentscheidbare Sätze” [On formally undecidable propositio... |

1 |
die Länge von Beweisen, Erg
- Über
- 1934
(Show Context)
Citation Context ...TY. It may be very hard to define the simplicity of proofs, but we nevertheless recognize “simplicity” in proofs when we see it. Different criteria for assessing simplicity (for example, proof length =-=[27]-=-, total number of symbols in the proof, the absence of certain terms, the number of basic operations) may become relevant as the context changes so that different facets of simplicity may be emphasize... |