Würzburg
N. Oswald
Workshop: Adolf Hurwitz and David Hilbert. Two universal mathematicians.
Nicola Oswald
July 22, 2014
1 / 40 Workshop A definition...
Würzburg
N. Oswald
a place where things are made or repaired
2 / 40 Workshop A definition...
Würzburg
N. Oswald
a place where things are made or repaired a class or series of classes in which a small group of people learn the methods and skills used in doing something
2 / 40 Research question Teacher-Student-Relation.
Würzburg
N. Oswald
Question Adolf Hurwitz and David Hilbert: When was the turning point in their relationship? Who benefited from whom?
3 / 40 1857 - 1919 German mathematics.
Würzburg
N. Oswald
4 / 40 1859 Adolf Hurwitz, born in...
Würzburg
N. Oswald
5 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert 1881 Doctorate with advisor Felix Klein
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert 1881 Doctorate with advisor Felix Klein 1882 Habilitation in Göttingen
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert 1881 Doctorate with advisor Felix Klein 1882 Habilitation in Göttingen 1884 Professorship in Königsberg
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert 1881 Doctorate with advisor Felix Klein 1882 Habilitation in Göttingen 1884 Professorship in Königsberg 1892 Professorship at Polytechnic (ETH) Zurich
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A zealous character.
Würzburg
N. Oswald * 1859 in Hildesheim Student of H.C.H. Schubert 1881 Doctorate with advisor Felix Klein 1882 Habilitation in Göttingen 1884 Professorship in Königsberg 1892 Professorship at Polytechnic (ETH) Zurich | 1919 in Zurich
”He was of enormous reliability, loyalty and love of justice.”
Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.
6 / 40 Adolf Hurwitz. A mathematical talent with prominent contacts.
Würzburg
N. Oswald Additive Geometry: Theorem of Chasles (1876)
7 / 40 Adolf Hurwitz. A mathematical talent with prominent contacts.
Würzburg
N. Oswald Additive Geometry: Theorem of Chasles (1876)
Modular Forms (1881), Riemann-Hurwitz-Formula (1893)
7 / 40 Adolf Hurwitz. A mathematical talent with prominent contacts.
Würzburg
N. Oswald Additive Geometry: Theorem of Chasles (1876)
Modular Forms (1881), Riemann-Hurwitz-Formula (1893)
Theorem about zeros of sequences of functions (1889)
7 / 40 Adolf Hurwitz. A mathematical talent with prominent contacts.
Würzburg
N. Oswald Additive Geometry: Theorem of Chasles (1876)
Modular Forms (1881), Riemann-Hurwitz-Formula (1893)
Theorem about zeros of sequences of functions (1889)
Results on continued fraction expansions (since 1882)
7 / 40 Adolf Hurwitz. A mathematical talent with prominent contacts.
Würzburg
N. Oswald Additive Geometry: Theorem of Chasles (1876)
Modular Forms (1881), Riemann-Hurwitz-Formula (1893)
Theorem about zeros of sequences of functions (1889)
Results on continued fraction expansions (since 1882)
Approximation Theorem (1891)
Since 1888 regular exchange with
7 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann 1885/6 at the University of Leipzig (Felix Klein), Stay in Paris
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann 1885/6 at the University of Leipzig (Felix Klein), Stay in Paris 1886 Habilitation in Königsberg
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann 1885/6 at the University of Leipzig (Felix Klein), Stay in Paris 1886 Habilitation in Königsberg 1892 Professorship in Königsberg
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann 1885/6 at the University of Leipzig (Felix Klein), Stay in Paris 1886 Habilitation in Königsberg 1892 Professorship in Königsberg 1895 Professorship in Göttingen
” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 David Hilbert. As well a zealous character.
Würzburg * 1862 in Königsberg N. Oswald Student of Adolf Hurwitz 1885 Doctorate supervized by Ferdinand Lindemann 1885/6 at the University of Leipzig (Felix Klein), Stay in Paris 1886 Habilitation in Königsberg 1892 Professorship in Königsberg 1895 Professorship in Göttingen | 1943 in Göttingen ” [...] Then Hilbert’s greatness is based on an overwhelming sensibility.”
Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932 8 / 40 Königberg 1884 - 1892 Triumvirat.
Würzburg
N. Oswald
9 / 40 Königsberg 1884 - 1892 Hilbert.
Würzburg
N. Oswald
10 / 40 Königsberg 1884 - 1892 Hilbert.
Würzburg
N. Oswald
”At that time still a student, Hurwitz soon encouraged me for a scientific exchange and I was lucky, that by being together with him, in the easiest and most interesting way, I got to know the directions of thoughts of the at that time opposite, however each other superbly complementing shools, the geometrical school of Klein and the algebraic-analytical school of Berlin. [...]”
’Adolf Hurwitz’, David Hilbert, 1921
11 / 40 Königsberg 1884 - 1892 Hilbert as a student.
Würzburg
N. Oswald
”New ideas were stimulated by the Mathematical Colloquium [...], however, in particular by the walks with Hurwitz ”precisely at 5 o’clock in the afternoon next to the apple tree”. ”
Lebensgeschichte, Otto Blumenthal, 1932 Let’s start the analysis... In the beginning naturally it was Hilbert, who benefited of his teacher Hurwitz. Their roles were still clear defined.
12 / 40 The corpus of investigation: Adolf Hurwitz’s estate in the ETH Zurich.
Würzburg
N. Oswald In the directory HS 582 and 583 of the archive in Zurich references to Hilbert can be found: Greeting cards from conferences
13 / 40 The corpus of investigation: Adolf Hurwitz’s estate in the ETH Zurich.
Würzburg
N. Oswald In the directory HS 582 and 583 of the archive in Zurich references to Hilbert can be found: Greeting cards from conferences Lectures notes of Hilbert, edited by Julius Hurwitz
13 / 40 The corpus of investigation: Adolf Hurwitz’s estate in the ETH Zurich.
Würzburg
N. Oswald In the directory HS 582 and 583 of the archive in Zurich references to Hilbert can be found: Greeting cards from conferences Lectures notes of Hilbert, edited by Julius Hurwitz A biographical dossier of Ida Samuel-Hurwitz
13 / 40 The corpus of investigation: Adolf Hurwitz’s estate in the ETH Zurich.
Würzburg
N. Oswald In the directory HS 582 and 583 of the archive in Zurich references to Hilbert can be found: Greeting cards from conferences Lectures notes of Hilbert, edited by Julius Hurwitz A biographical dossier of Ida Samuel-Hurwitz A letter of condolence to Ida Samuel-Hurwitz by David Hilbert
13 / 40 The corpus of investigation: Adolf Hurwitz’s estate in the ETH Zurich.
Würzburg
N. Oswald In the directory HS 582 and 583 of the archive in Zurich references to Hilbert can be found: Greeting cards from conferences Lectures notes of Hilbert, edited by Julius Hurwitz A biographical dossier of Ida Samuel-Hurwitz A letter of condolence to Ida Samuel-Hurwitz by David Hilbert Remarks about Hilbert in the register of ...
13 / 40 Hurwitz’s mathematical diaries. An overview.
Würzburg
N. Oswald
”Since his habilitation in 1882, Hurwitz took notes of everything he spent time on with uninterrupted regularity and in this way he left a series of 31 diaries, which give a true [...] .” ’Adolf Hurwitz’, David Hilbert, 1921
14 / 40 The corpus of investigation: Hurwitz’s mathematical diaries.
Würzburg
N. Oswald 30 diaries, ca. 6000 pages
Diary No. 4, 1885
Diary No. 5, 1886 - 1888
15 / 40 The corpus of investigation: Hurwitz’s mathematical diaries.
Würzburg
N. Oswald 30 diaries, ca. 6000 pages from March 1882 until September 1919
Diary No. 4, 1885
Diary No. 5, 1886 - 1888
15 / 40 The corpus of investigation: Hurwitz’s mathematical diaries.
Würzburg
N. Oswald 30 diaries, ca. 6000 pages from March 1882 until September 1919
Diary No. 4, 1885 reviewed and registered by Georg Pólya
Diary No. 5, 1886 - 1888
15 / 40 The corpus of investigation: Hurwitz’s mathematical diaries.
Würzburg
N. Oswald 30 diaries, ca. 6000 pages from March 1882 until September 1919
Diary No. 4, 1885 reviewed and registered by Georg Pólya Contents include Number Theory, Geometry, Complex Analysis as well as Algebra
Diary No. 5, 1886 - 1888
15 / 40 The corpus of investigation: Hurwitz’s mathematical diaries.
Würzburg
N. Oswald 30 diaries, ca. 6000 pages from March 1882 until September 1919
Diary No. 4, 1885 reviewed and registered by Georg Pólya Contents include Number Theory, Geometry, Complex Analysis as well as Algebra and at least 15 direct
Diary No. 5, 1886 - 1888 references about Hilbert.
15 / 40 Hilbert. Publications - fields of interest.
Würzburg
N. Oswald 1885 - 1892 Algebra: Theory of Invariants 1890 ”Ueber die Theorie der algebraischen Formen” (P in GA Bd. 2) 1892 Theorem on irreducibility 1892 - 1899 Number Theory: Theory of Number Fields 1893 Simplification of the Hermite-Lindemann proof of the transcendence of e and π (P in GA Bd. 1) 1894 ”Zwei neue Beweise für die Zerlegbarkeit der Zahlen eines Körpers in Primideale” (L in Munich DMV + P) 1896 ”Die Theorie der algebraischen Zahlkörper”, often called ’Zahlbericht’ (P on demand of DMV, in GA Bd. 1) 1891 - 1902 Geometry: Axiomatization of Geometry
16 / 40 Hilbert. Publications - fields of interest.
Würzburg
N. Oswald 1895 - 1903 ”Grundlagen der Geometrie” including Complements (GG) 1895 ”Über die gerade Linie als kürzeste Verbindung zweier Punkte” (Complement I in GG) 1900 ”Über den Zahlbegriff” (Complement VI in GG): Axiomatization of Arithmetic 1900 Hilberts 23 Mathematical Problems, International Congress of Mathematics in Paris (T at ICM) 1902 - 1910 Complex Analysis 1904 - 1910 Linear Algebra, Functional Analysis: ”Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen” with Supplements (GZ with six Sup.)
17 / 40 Hilbert. Publications - fields of interest.
Würzburg 1907 (published 1910) Analysis meets Geometry: N. Oswald Analytical refounding of Minkowski’s theory of volumes and surfaces of convex bodies (Sixth Sup. of GZ) 1907 Analysis meets Number Theory: ”Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)” (P in GA Bd.1) 1902 - 1918 Axiomatization of Physics and Mechanics: Theory of Relativity 1904 - 1934 Mathematical Foundation 1904 Axiomatization of theory of numbers: ”Über die Grundlagen der Logik und der Arithmetik” (T in Heidelberg in Sup. VII, GG) 1922 - 1934 Hilbert Programme on consistency, Formalism, Proof Theory, ....
18 / 40 Hilbert in the diaries...... first references.
Würzburg
N. Oswald No. 6: 1888 IV. 1889 XI. p. 44 ”On Noether’s Theorem (concerning a message of Hilbert)” p. 45 ”Hilbert’s Fundamental Theorem” p. 93 Studies on convergent series, ”Hilbert proved the mentioned theorems as follows” No. 7: 1890 IV.9. - 1891 XI. p. 94 ”[...] the figures of Hilbert”
Hurwitz noticed Hilbert as student: Their exchange begins!
19 / 40 According to Blumenthal ... prime ideals.
Würzburg
N. Oswald
Talk of Hilbert in Munich (1893), annual meeting DMV: ”Two new proofs of the decomposability of numbers of a field into prime ideals”
”It was the first result from the walks with Hurwitz. The second was Hurwitz’s published proof of the same theorem one year later, which Hilbert gave preference in his ’Zahlbericht’.”
Lebensgeschichte, Otto Blumenthal, 1932
20 / 40 Theory of Invariants. No. 8, p. 207; No. 14, p. 204; No. 25, p. 77
Würzburg
N. Oswald No. 14: 1896 I.1. - 1897 II.1., p. 204 ”Hilberts 2nd Theorem”
”This is probably the easiest way to regard Hilbert’s proof of Theorem II (Ann 36. p. 485).”
”Hilbert’s Theorem is also valid for forms, which coefficients are integers of a finite number field.”
Hurwitz is still on top: He completed and generalized Hilbert’s theorem.
21 / 40 Zahlbericht. No. 15: 1897 II.1. - 1898 III.19., p. 175 ’Concerning Hilbert’s ”Report on Number Fields” ’
Würzburg ”Concerning Chapter V of Hilbert’s report we remark the N. Oswald following.”
Here, we consider a number field K and subfields ki resp. concerning ’Grundideale’ ν and νi :
”According to Hilbert p. 209 [...] the equation νν12 = ν1ν2
would lead to νk1 νk2 = νk12 ”. First Hurwitz verified this
consequence of Hilbert’s formula νk1 νk2 = νk12 , then continued to compare it with his assumption: 22 / 40 Zahlbericht. No. 15: 1897 II.1. - 1898 III.19., p. 175 ’Concerning Hilbert’s ”Report on Number Fields” ’
Würzburg
N. Oswald
”I assume that the theorem holds: If K is a composition of k1, k2, furthermore k12 is the greatest common divisor of k1 and
k2, we have ννk12 = νk1 νk2 , with basic ν,ν12,ν1,ν2 ideals of the number fields K, k12, k1, k2.”
23 / 40 Zahlbericht. No. 15: 1897 II.1. - 1898 III.19., p. 175 ’Concerning Hilbert’s ”Report on Number Fields” ’
Würzburg
N. Oswald Finally, Hurwitz concluded a new generalized theorm:
ν1·ν2 ”[...] consequently the generalized theorem holds: ν = ν , 1 where ν is a common divisor of ν1 and ν2. Or also: In the equation ν1ν2 = νν12 · j is ν12 · j a common divisor of ν1 and ν2.”
1 probably ν12 and ν were mixed up 24 / 40 Zahlbericht. No. 16: 1898 III.20. - 1899 II.23., p. 129 ”Concerning Hilbert’s report pag. 287”
Würzburg
N. Oswald Hurwitz refered to the later called ’Hilbert’s Symbol’...
.... and in particular to the equation:
25 / 40 Zahlbericht. No. 16: 1898 III.20. - 1899 II.23., p. 129 ”Concerning Hilbert’s report pag. 287”
Würzburg
N. Oswald
Within one page Hurwitz gave the proof:
On the one hand, Hurwitz used Hilbert’s ’Zahlbericht’ as textbook, on the other hand, he improved and complemented it also.
26 / 40 Hilbert’s Axiomatization. No. 19: 1901 XI.1. - 1904 III.16., p. 29 ”Hilberts axiomatische Größenlehre” (Theory of quantities)
Würzburg
N. Oswald
What makes Hurwitz’s entry so interesting is the exact copying of
”II. Axioms of Calculation. [...]” Hilbert’s axioms about operations, calculation, order and continuity... 27 / 40 Hilbert’s Axiomatization. No. 19: 1901 XI.1. - 1904 III.16., p. 29 ”Hilberts axiomatische Größenlehre”
Würzburg ... as well as Hilbert’s conclusions:
N. Oswald
”Some notes about the dependence of the axioms were added by Hilbert: [...]” and Hilbert’s new terminology:
”What follows is the existence of the ”Verdichtungsstelle” (as Hilbert expresses himself.)”
It seems that the Axiomatization is a completely new topic for Hurwitz. In any case, he benefited from his progressive former student Hilbert.
28 / 40 Integral Equation V. No. 21: 1906 II.1. - 1906 XII.8., p. 166 ”Hilbert’s Vth supplement on integral equations”
Würzburg In the fifth supplement on page 459 Hilbert wrote: ” The values N. Oswald are [...] significantly determined by the kernel(s,t); I called them Eigenvalues resp. Eigenfunctions [...].”
29 / 40 Integral Equation V. No. 21: 1906 II.1. - 1906 XII.8., p. 166 ”Hilbert’s Vth supplement on integral equations”
Würzburg
N. Oswald Hurwitz familiarized himself with the terminology as well as its application:
Obviously Hurwitz did neither know about Hilbert’s methods concerning integral equations before, nor his newly introduced terminology.
30 / 40 Minkowski’s theory of convex bodies. No. 18: 1900 XII. - 1901 X.
Würzburg In 1907 David Hilbert succeeded in developing the analytical N. Oswald foundation of Minkowki’s theory of volumes and surfaces of convex bodies in his sixth supplement. There is a diary entry about a colloquium talk of Hurwitz from 21.01.1901:
”Minkowski’s theroems on convex bodies.” Hurwitz was not satisfied:
”It remains doubtful if simple results can be discovered.” 31 / 40 Minkowski’s theory of convex bodies. A consequence of No. 18?
Würzburg
N. Oswald One year later, in 1902, Hurwitz published the article ’Sur quelques applications geometriques des séries de Fourier’, in which he tried a respective new foundation of Minkowski’s theory ”using his theory of spherical functions [...], however, he only had a partial success. Hilbert, with his powerful tool on integral equations, replaces the spherical function by more generalized ones and passes through.”
(’Lebensgeschichte’, Otto Blumenthal, 1932)
Hilbert is on the fast lane.
32 / 40 Waring’s Problem. No diary entry...
Würzburg
N. Oswald On November 20, 1907 Hurwitz published a note on Waring’s Problem For each positive integer k does there exist a positive integer n(k) such that every natural number is the sum of at most n(k) kth powers of natural numbers?
Again Hurwitz succeeded again only partially. He proved 2 2 2 2 ” Is the nth power of x1 + x2 + x3 + x4 equal to a sum of 2nth powers of a linear rational form of x1, x2, x3, x4, and does the Waring Conjecture hold, it is also valid for 2n.”
33 / 40 Waring’s Problem. ... instead a nice quotation.
Würzburg
N. Oswald Hilbert deduced ”from a general principle, which was used by Hurwitz 1897 for the theory of invariants, a formula [...]” with which he finally solved Waring’s problem.
Therewith, Hilbert gave a wonderful proof of his emancipation from his former teacher Hurwitz: ”Because he fought together with a master of Hurwitz’s high level and won with the weapons from Hurwitz’s armor chamber on a point, when [Hurwitz] had no prospect of success”
(’Lebensgeschichte’, Otto Blumenthal, 1932)
34 / 40 An answer...... to the question of research.
Würzburg
N. Oswald
The turning point At the latest in 1907 there was a significant turning point in the student-teacher-relation of Hurwitz and Hilbert!
35 / 40 The personal relation. Livelong...
Würzburg
N. Oswald Greeting cards from the ’Dirichletkommers am 13. Februar 1905’ and the ’Landau-Kommers 18. Jan. 1913’, signed by Hilbert (and other mathematicians):
”Sending warm greetings, wishing good recovery and hoping for a soon reunion longer than the last time. Hilbert”.
36 / 40 The personal relation. ... Hilbert directly in the mathematical diaries...
Würzburg
N. Oswald In the register made by Georg Pólya, the ’32nd diary’, the following notes can be found: ”the first nine volumes and table of contents are for the purpose of editing temporarily at Prof. Hilbert in Göttingen” and
”22. for editing temporarily at Prof. Hilbert in Göttingen”. The above mentioned quotation of Hilbert is to be continued ”[...] which give a true view of his constantly progresssive development and at the same time they are a rich treasure trove for interesting and for further examination appropriate thoughts and problems.”
(’Adolf Hurwitz’, David Hilbert, 1921)
37 / 40 The personal relation. .... and even longer.
Würzburg
N. Oswald
In a letter of condolence to Ida Samuel-Hurwitz with date December 15, 1919 Hilbert wrote that for Pólya and him the ”matter of publishing the Hurwitz’s treatises [is] of utmost concern”. He offers, ”The negotiations could be done verbally with Springer by a local, very skillful, math. colleague.” Some years later his efforts turn out to be successful: Hurwitz’s ’Mathematischen Werke’ were published in 1932.
38 / 40 Thank you.
Würzburg
N. Oswald
Thank you for your attention!
39 / 40 Announcement. ... evening programme!
Würzburg
N. Oswald This evening, we will meet in the Biergarten ”Am alten Kranen”!
40 / 40