Workshop: Adolf Hurwitz and David Hilbert. Two universal ...oswald/hilbert.pdf · Würzburg N. Oswald Workshop: Adolf Hurwitz and David Hilbert. Two universal mathematicians. Nicola

Würzburg

N. Oswald

Workshop: Adolf Hurwitz and David

Hilbert.

Two universal mathematicians.

Nicola Oswald

July 22, 2014

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WorkshopA definition...

a place where things are made or repaired

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WorkshopA definition...

a place where things are made or repaired

a class or series of classes in which a small group of peoplelearn the methods and skills used in doing something

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Research questionTeacher-Student-Relation.

Question

Adolf Hurwitz and David Hilbert:

When was the turning point in their relationship? Who

benefited from whom?

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1857 - 1919German mathematics.

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1859Adolf Hurwitz, born in...

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Adolf Hurwitz.A zealous character.

* 1859 in Hildesheim

”He was of enormous reliability, loyalty and love

of justice.”

Ida Samuel-Hurwitz, Biographical Dossier, Archive ETH Zurich.

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Student of H.C.H. Schubert


of justice.”


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1881 Doctorate with advisor FelixKlein


of justice.”


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1882 Habilitation in Göttingen


of justice.”


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1884 Professorship in Königsberg


of justice.”


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1892 Professorship at Polytechnic(ETH) Zurich


of justice.”


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1892 Professorship at Polytechnic(ETH) Zurich

| 1919 in Zurich


of justice.”


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Adolf Hurwitz.A mathematical talent with prominent contacts.

Additive Geometry: Theorem of Chasles (1876)

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Modular Forms (1881), Riemann-Hurwitz-Formula (1893)

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Theorem about zeros of sequences of functions (1889)

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Results on continued fraction expansions (since 1882)

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Results on continued fraction expansions (since 1882)

Approximation Theorem (1891)

Since 1888 regular exchange with

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David Hilbert.As well a zealous character.

* 1862 in Königsberg

” [...] Then Hilbert’s greatness is based on an

overwhelming sensibility.”

Otto Blumenthal, Lebensgeschichte, Collected Papers of David Hilbert, 1932

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Student of Adolf Hurwitz




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1885 Doctorate supervized byFerdinand Lindemann




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1885/6 at the University ofLeipzig (Felix Klein), Stay in Paris




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1886 Habilitation in Königsberg




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1895 Professorship in Göttingen




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1895 Professorship in Göttingen

| 1943 in Göttingen




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Königberg 1884 - 1892Triumvirat.

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Königsberg 1884 - 1892Hilbert.

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Königsberg 1884 - 1892Hilbert.

”At that time still a student, Hurwitz soon encouraged me for ascientific exchange and I was lucky, that by being together withhim, in the easiest and most interesting way, I got to know thedirections of thoughts of the at that time opposite, howevereach other superbly complementing shools, the geometricalschool of Klein and the algebraic-analytical school of Berlin.[...]”’Adolf Hurwitz’, David Hilbert, 1921

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Königsberg 1884 - 1892Hilbert as a student.

”New ideas were stimulated by the Mathematical Colloquium[...], however, in particular by the walks with Hurwitz ”preciselyat 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 histeacher Hurwitz. Their roles were still clear defined.

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The corpus of investigation:Adolf Hurwitz’s estate in the ETH Zurich.

In the directory HS 582 and 583 of the archive in Zurichreferences to Hilbert can be found:

Greeting cards from conferences

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Lectures notes of Hilbert, edited by Julius Hurwitz

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A biographical dossier of Ida Samuel-Hurwitz

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A letter of condolence to Ida Samuel-Hurwitz by DavidHilbert

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A letter of condolence to Ida Samuel-Hurwitz by DavidHilbert

Remarks about Hilbert in the register of ...

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Hurwitz’s mathematical diaries.An overview.

”Since his habilitation in 1882, Hurwitz took notes of everythinghe spent time on with uninterrupted regularity and in this wayhe left a series of 31 diaries, which give a true [...] .”’Adolf Hurwitz’, David Hilbert, 1921

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The corpus of investigation:Hurwitz’s mathematical diaries.

Diary No. 4, 1885

Diary No. 5, 1886 - 1888

30 diaries,ca. 6000 pages

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Diary No. 4, 1885

Diary No. 5, 1886 - 1888


from March 1882until September 1919

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Diary No. 4, 1885

Diary No. 5, 1886 - 1888



reviewed and registered byGeorg Pólya

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Diary No. 4, 1885

Diary No. 5, 1886 - 1888




Contents include NumberTheory, Geometry,Complex Analysis as wellas Algebra

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Diary No. 4, 1885

Diary No. 5, 1886 - 1888




Contents include NumberTheory, Geometry,Complex Analysis as wellas Algebra

and at least 15 directreferences about Hilbert.

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Hilbert.Publications - fields of interest.

� 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

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� 1895 - 1903 ”Grundlagen der Geometrie” includingComplements (GG)

� 1895 ”Über die gerade Linie als kürzeste Verbindung zweierPunkte” (Complement I in GG)

� 1900 ”Über den Zahlbegriff” (Complement VI in GG):Axiomatization of Arithmetic

� 1900 Hilberts 23 Mathematical Problems, InternationalCongress of Mathematics in Paris (T at ICM)

� 1902 - 1910 Complex Analysis

� 1904 - 1910 Linear Algebra, Functional Analysis:”Grundzüge einer allgemeinen Theorie der linearenIntegralgleichungen” with Supplements (GZ with six Sup.)

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� 1907 (published 1910) Analysis meets Geometry:Analytical refounding of Minkowski’s theory of volumes andsurfaces of convex bodies (Sixth Sup. of GZ)

� 1907 Analysis meets Number Theory: ”Beweis für dieDarstellbarkeit der ganzen Zahlen durch eine feste Anzahln-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 dieGrundlagen der Logik und der Arithmetik”(T in Heidelberg in Sup. VII, GG)

� 1922 - 1934 Hilbert Programme on consistency,Formalism, Proof Theory, ....

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Hilbert in the diaries...... first references.

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!

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According to Blumenthal... prime ideals.

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 secondwas Hurwitz’s published proof of the same theorem one yearlater, which Hilbert gave preference in his ’Zahlbericht’.”Lebensgeschichte, Otto Blumenthal, 1932

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Theory of Invariants.No. 8, p. 207; No. 14, p. 204; No. 25, p. 77

No. 14: 1896 I.1. - 1897 II.1., p. 204 ”Hilberts 2nd Theorem”

”This is probably the easiest way to regard Hilbert’s proof ofTheorem II (Ann 36. p. 485).”

”Hilbert’s Theorem is also valid for forms, which coefficients areintegers of a finite number field.”

Hurwitz is still on top: He completed and generalized Hilbert’stheorem.

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Zahlbericht.No. 15: 1897 II.1. - 1898 III.19., p. 175 ’Concerning Hilbert’s

”Report on Number Fields” ’

”Concerning Chapter V of Hilbert’s report we remark thefollowing.”

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 νk1νk2

= νk12”. First Hurwitz verified this

consequence of Hilbert’s formula νk1νk2

= νk12, then continued

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”I assume that the theorem holds: If K is a composition ofk1, k2, furthermore k12 is the greatest common divisor of k1 andk2, we have ννk12

= νk1νk2

, with basic ν, ν12, ν1, ν2 ideals of thenumber fields K , k12, k1, k2.”

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Finally, Hurwitz concluded a new generalized theorm:

”[...] consequently the generalized theorem holds: ν = ν1·ν2ν

,where ν

1 is a common divisor of ν1 and ν2. Or also: In theequation ν1ν2 = νν12 · j is ν12 · j a common divisor of ν1 andν2.”

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Zahlbericht.No. 16: 1898 III.20. - 1899 II.23., p. 129 ”Concerning

Hilbert’s report pag. 287”

Hurwitz refered to the later called ’Hilbert’s Symbol’...

.... and in particular to the equation:

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Zahlbericht.No. 16: 1898 III.20. - 1899 II.23., p. 129 ”Concerning

Hilbert’s report pag. 287”

Within one page Hurwitz gave the proof:

On the one hand, Hurwitz used Hilbert’s ’Zahlbericht’ astextbook, on the other hand, he improved and complemented italso.

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Hilbert’s Axiomatization.No. 19: 1901 XI.1. - 1904 III.16., p. 29 ”Hilberts axiomatische

Größenlehre” (Theory of quantities)

What makes Hurwitz’s entry so interesting is the exact copyingof

”II. Axioms of Calculation. [...]” Hilbert’s axioms aboutoperations, calculation, order and continuity...

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Hilbert’s Axiomatization.No. 19: 1901 XI.1. - 1904 III.16., p. 29 ”Hilberts axiomatische

Größenlehre”

... as well as Hilbert’s conclusions:

”Some notes about the dependence of the axioms were addedby Hilbert: [...]”and Hilbert’s new terminology:

”What follows is the existence of the ”Verdichtungsstelle” (asHilbert expresses himself.)”

It seems that the Axiomatization is a completely new topic forHurwitz. In any case, he benefited from his progressive formerstudent Hilbert.

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Integral Equation V.No. 21: 1906 II.1. - 1906 XII.8., p. 166 ”Hilbert’s Vth

supplement on integral equations”

In the fifth supplement on page 459 Hilbert wrote: ” The valuesare [...] significantly determined by the kernel(s,t); I called themEigenvalues resp. Eigenfunctions [...].”

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Integral Equation V.No. 21: 1906 II.1. - 1906 XII.8., p. 166 ”Hilbert’s Vth

supplement on integral equations”

Hurwitz familiarized himself with the terminology as well as itsapplication:

Obviously Hurwitz did neither know about Hilbert’s methodsconcerning integral equations before, nor his newly introducedterminology.

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Minkowski’s theory of convex bodies.No. 18: 1900 XII. - 1901 X.

In 1907 David Hilbert succeeded in developing the analyticalfoundation of Minkowki’s theory of volumes and surfaces ofconvex bodies in his sixth supplement.There is a diary entry about a colloquium talk of Hurwitz from21.01.1901:

”Minkowski’s theroems on convex bodies.” Hurwitz was notsatisfied:

”It remains doubtful if simple results can be discovered.”31 / 40

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Minkowski’s theory of convex bodies.A consequence of No. 18?

One year later, in 1902, Hurwitz published the article ’Surquelques applications geometriques des séries de Fourier’, inwhich he tried a respective new foundation of Minkowski’stheory ”using his theory of spherical functions [...], however, heonly had a partial success. Hilbert, with his powerful tool onintegral equations, replaces the spherical function by moregeneralized ones and passes through.”(’Lebensgeschichte’, Otto Blumenthal, 1932)

Hilbert is on the fast lane.

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Waring’s Problem.No diary entry...

On November 20, 1907 Hurwitz published a note on

Waring’s Problem

For each positive integer k does there exist a positive integern(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” Is the nth power of x2

1+ x2

2+ x2

3+ x2

4equal to a sum of 2nth

powers of a linear rational form of x1, x2, x3, x4, and does theWaring Conjecture hold, it is also valid for 2n.”

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Waring’s Problem.... instead a nice quotation.

Hilbert deduced ”from a general principle, which was used byHurwitz 1897 for the theory of invariants, a formula [...]” withwhich he finally solved Waring’s problem.

Therewith, Hilbert gave a wonderful proof of his emancipationfrom his former teacher Hurwitz:

”Because he fought together with a master of Hurwitz’s highlevel and won with the weapons from Hurwitz’s armor chamberon a point, when [Hurwitz] had no prospect of success”(’Lebensgeschichte’, Otto Blumenthal, 1932)

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An answer...... to the question of research.

The turning point

At the latest in 1907 there was a significant turning point in thestudent-teacher-relation of Hurwitz and Hilbert!

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The personal relation.Livelong...

Greeting cards from the ’Dirichletkommers am 13. Februar1905’ and the ’Landau-Kommers 18. Jan. 1913’, signed byHilbert (and other mathematicians):

”Sending warm greetings, wishing good recovery and hoping fora soon reunion longer than the last time. Hilbert”.

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The personal relation.... Hilbert directly in the mathematical diaries...

In the register made by Georg Pólya, the ’32nd diary’, thefollowing notes can be found:”the first nine volumes and table of contents are for the purposeof 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 progresssivedevelopment and at the same time they are a rich treasure trovefor interesting and for further examination appropriate thoughtsand problems.”(’Adolf Hurwitz’, David Hilbert, 1921)

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The personal relation..... and even longer.

In a letter of condolence to Ida Samuel-Hurwitz with dateDecember 15, 1919 Hilbert wrote that for Pólya and him the”matter of publishing the Hurwitz’s treatises [is] of utmostconcern”.He offers, ”The negotiations could be done verbally withSpringer 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.

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Thank you.

Thank you for your attention!

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Announcement.... evening programme!

This evening, we will meet in the Biergarten ”Am alten Kranen”!

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Documents

Workshop: Adolf Hurwitz and David Hilbert. Two universal ...oswald/hilbert.pdf · Würzburg N. Oswald Workshop: Adolf Hurwitz and David Hilbert. Two universal mathematicians. Nicola