Hilbertovi problemi
Hilbertovi problemi, to su 23 problema, od kojih je trinaest postavio matematičar David Hilbert da bi na Drugom međunarodnom kongresu matematičara u Parizu, 8. avgusta 1900. godine bilo dodato još deset, ovde broj 1, 2, 6, 7, 8, 13, 16, 19, 21, i 22. Neki od ovih problema su zapravo područja za istraživanje, a zajedno sa ostalima bili su primer narastanja čitavih disciplina, vremenom, iz malih „problema“. Potpuna lista od 23 problema objavljena je kasnije, posebno u prevodu na engleski 1902. od strane Mary Frances Winston NewsonMeri Frensis Vinston Njuson u Bulletin of the American Mathematical Society.[1]
Ignorabimus uredi
Prateći Gotloba Fregea i Bertranda Rasela, Hilbert je pokušao da definiše matematiku logički koristeći metod formalnih sistema, tj. finitističke dokaze iz dogovorenog skupa aksioma.[2] Jedan od glavnih ciljeva Hilbertovog programa bio je finitistički dokaz konzistentnosti aksioma aritmetike: to je njegov drugi problem.[a]
Međutim, Gedelova druga teorema o nepotpunosti daje precizan smisao u kome je takav finitistički dokaz konzistentnosti aritmetike dokazivo nemoguć. Hilbert je živeo 12 godina nakon što je Kurt Gedel objavio svoju teoremu, ali izgleda da nije napisao nikakav formalni odgovor na Gedelov rad.[b][v]
Hilbertov deseti problem ne postavlja pitanje da li postoji algoritam za odlučivanje o rešivosti Diofantskih jednačina, već traži konstrukciju takvog algoritma: „da se osmisli proces prema kojem se u konačnom broju operacija može odrediti da li je jednačina rešiva u racionalnim celim brojevima“. To što je ovaj problem rešen pokazivanjem da ne može postojati takav algoritam protivreči Hilbertovoj filozofiji matematike.
Raspravljajući o svom mišljenju da svaki matematički problem treba da ima rešenje, Hilbert dopušta mogućnost da bi rešenje moglo biti dokaz da je originalni problem nemoguć.[g] On je naveo da je poenta znati na ovaj ili onaj način šta je to rešenje, i verovao je da se to uvek može znati, da u matematici ne postoji „ignorabimus“ (tvrdnja čija se istina nikada ne može saznati).[d] Ostaje nejasno da li bi on smatrao rešenje desetog problema kao primer ignorabimusa: ono što je dokazano da ne postoji nije celobrojno rešenje, već (u izvesnom smislu) sposobnost da se na specifičan način razazna da li rešenje postoji.
S druge strane, status prvog i drugog zadatka je još komplikovaniji: ne postoji jasan matematički konsenzus o tome da li su rezultati Gedela (u slučaju drugog zadatka), ili Gedelovog i Kohenovog (u slučaju prvog problema) daju definitivna negativna rešenja ili ne, jer se ta rešenja odnose na izvesnu formalizaciju problema, koja nije nužno jedina moguća.[đ]
Dvadeset četvrti problem uredi
Hilbert je prvobitno uključio 24 problema na svoju listu, ali je odlučio da ne uključi jedan od njih na objavljenu listu. „24. problem“ (u teoriji dokaza, o kriterijumu jednostavnosti i opštim metodama) ponovo je otkrio nemački istoričar Ridiger Tile 2000. godine u Hilbertovim originalnim beleškama u rukopisu.[5]
Problemi uredi
- Kantorov problem kardinalnog broja kontinuuma.[6]
- Konzistentnost aksioma aritmetike.
- Jednakost zapremina dva tetraedra jednakih baza i visina.
- Problem prave linije kao najkraćeg rastojanja između dve tačke.
- Koncept Lijevih grupa neprekidnih transformacija, bez pretpostavke diferencijabilnosti.
- Matematički tretman aksioma fizike. Može li se fizika aksiomatizovati?[7][8]
- Iracionalnost i transcendentnost izvesnih brojeva, oblika , npr. ,; .
- Problem prostih brojeva, Rimanova hipoteza.
- Opšti dokaz teorema recipročnosti teorije brojeva.
- Opšte rešenje Diofantove jednačine.
- Kvadratna forma proizvoljnog celobrojnog algebarskog polja.[9]
- Kronekerova teorema, konstrukcija holomorfne funkcije.[10]
- Nemogućnost rešenja opšte jednačine 7-og stepena funkcijama sa samo dva argumenta.[11][12][13][14]
- Problem konačnosti izvesnih funkcija.
- Strogo zasnivanje Šubertovog neprebrojivog računa (Schubert).
- Problem topologije algebarskih krivih i površi.
- Reprezentacija končane forme kvadrata.
- Izgradnja prostora iz kongruentnog poliedra.
- Jesu li rešenja problema varijacija uvek analitička?
- Opšti problem granične vrednosti.
- Dokaz egzistencije rešenja linearne diferencijalne jednačine za monodromsku grupu.
- Uniformizacija analitičkih relacija pomoću automorfnih funkcija.
- Dalji razvoj metoda računa varijacija.
Napomene uredi
- ^ See Nagel and Newman revised by Hofstadter (2001, p. 107),[3] footnote 37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page 109[3] and was not modified there by Hofstadter (p. 108).[3]
- ^ Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p. 198–199).[4] Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p. 199).[4]
- ^ Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p. 217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p. vii).[4]
- ^ This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the Law of Excluded Middle be employed in proofs. See much more at Brouwer–Hilbert controversy.
- ^ "This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. 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." (Hilbert, 1902, p. 445.)
- ^ Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of being mirrored inside Principia Mathematica (footnote 39, page 109). The authors conclude that the prospect "is most unlikely."[3]
Reference uredi
- ^ Hilbert, David (1902). „Mathematical Problems”. Bulletin of the American Mathematical Society. 8 (10): 437—479. doi:10.1090/S0002-9904-1902-00923-3
. Earlier publications (in the original German) appeared in Hilbert, David (1900). „Mathematische Probleme”. Göttinger Nachrichten: 253—297. and Hilbert, David (1901). „[no title cited]”. Archiv der Mathematik und Physik. 3. 1: 44—63, 213—237.
- ^ van Heijenoort, Jean, ur. (1976) [1966]. From Frege to Gödel: A source book in mathematical logic, 1879–1931 ((pbk.) izd.). Cambridge MA: Harvard University Press. str. 464ff. ISBN 978-0-674-32449-7.
A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational "crisis" that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927). - ^ a b v g Nagel, Ernest; Newman, James R. (2001). Hofstadter, Douglas R., ur. Gödel's Proof. New York, NY: New York University Press. ISBN 978-0-8147-5816-8.
- ^ a b v Reid, Constance (1996). Hilbert
. New York, NY: Springer-Verlag. ISBN 978-0387946740.
- ^ Thiele, Rüdiger (januar 2003). „Hilbert's twenty-fourth problem” (PDF). American Mathematical Monthly. 110: 1—24. S2CID 123061382. doi:10.1080/00029890.2003.11919933.
- ^ Cantor, Georg (1878). „Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Mathematik. 1878 (84): 242—258. doi:10.1515/crll.1878.84.242.
- ^ Corry, L. (1997). „David Hilbert and the axiomatization of physics (1894–1905)”. Arch. Hist. Exact Sci. 51 (2): 83—198. S2CID 122709777. doi:10.1007/BF00375141.
- ^ Gorban, A.N.; Karlin, I. (2014). „Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations”. Bulletin of the American Mathematical Society. 51 (2): 186—246. arXiv:1310.0406 . doi:10.1090/S0273-0979-2013-01439-3 .
- ^ Hazewinkel, Michiel (2009). Handbook of Algebra. 6. Elsevier. str. 69. ISBN 978-0080932811.
- ^ Houston-Edwards, Kelsey. „Mathematicians Find Long-Sought Building Blocks for Special Polynomials”.
- ^ Abhyankar, Shreeram S. „Hilbert's Thirteenth Problem” (PDF).
- ^ Vitushkin, A.G. „On Hilbert's thirteenth problem and related questions” (PDF).
- ^ Chebotarev, N.G., On certain questions of the problem of resolvents
- ^ Hilbert, David (1927). „Über die Gleichung neunten Grades”. Math. Ann. 97: 243—250. S2CID 179178089. doi:10.1007/BF01447867.
Literatura uredi
- Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford, UK: Oxford University Press. ISBN 978-0-19-850651-5.
- Yandell, Benjamin H. (2002). The Honors Class: Hilbert's problems and their solvers . Wellesley, MA: A.K. Peters. ISBN 978-1-56881-141-3.
- Thiele, Rüdiger (2005). „On Hilbert and his twenty-four problems”. Ur.: Van Brummelen, Glen. Mathematics and the Historian's Craft: The Kenneth O. May lectures. CMS Books in Mathematics / Ouvrages de Mathématiques de la SMC. 21. str. 243—295. ISBN 978-0-387-25284-1.
- Dawson, John W. Jr. (1997). Logical Dilemmas: The life and work of Kurt Gödel. A.K. Peters.
A wealth of information relevant to Hilbert's "program" and Gödel's impact on the Second Question, the impact of Arend Heyting's and Brouwer's Intuitionism on Hilbert's philosophy. - Browder, Felix E., ur. (1976). „Mathematical Developments Arising from Hilbert Problems”. Proceedings of Symposia in Pure Mathematics XXVIII. American Mathematical Society.
A collection of survey essays by experts devoted to each of the 23 problems emphasizing current developments. - Matiyasevich, Yuri (1993). Hilbert's Tenth Problem. Cambridge, MA: MIT Press. ISBN 978-0262132954.
An account at the undergraduate level by the mathematician who completed the solution of the problem. - Cohen, Paul Joseph (2008). Set theory and the continuum hypothesis. Mineola, New York City: Dover Publications. ISBN 978-0-486-46921-8.
- Dales, H.G.; Woodin, W.H. (1987). An Introduction to Independence for Analysts. Cambridge.
- Enderton, Herbert (1977). Elements of Set Theory. Academic Press.
- Gödel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
- Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in Mathematical Developments Arising from Hilbert's Problems, Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1
- McGough, Nancy. „The Continuum Hypothesis”.
- Wolchover, Natalie (15. 7. 2021). „How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer”.
Spoljašnje veze uredi
- Hazewinkel Michiel, ur. (2001). „Hilbert problems”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104.
- „Original text of Hilbert's talk, in German.”. Arhivirano iz originala 2012-02-05. g. Pristupljeno 2005-02-05.
- „David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900.” (PDF).
Mathematical Problems public domain audiobook at LibriVox
