# Oskar Perron

### Quick Info

Born
7 May 1880
Frankenthal, Pfalz, Germany
Died
22 February 1975
Munich, Germany

Summary
Oskar Perron was a German mathematician best known for the Perron paradox:
Suppose the largest natural number is N. Then if N > 1 we have N2 > N contradicting the definition. Hence (!) N = 1.

### Biography

Oskar Perron's father was the merchant and banker Jakob Heinrich August Perron (1850-1925), born on 30 May 1850 in Frankenthal. Heinrich, the son of leather trader Valentin Perron (1807-1888) and Katharina Schaaff (1818-1865), married Auguste Leinenweber (1857-1924) from Pirmasens on 30 May 1876. Auguste was the daughter of tanner Ludwig Leinenweber (1826-1870) and Katharina Von Gerickten (1835-1878).

Oskar began his schooling at the Volksschule in 1886 before moving to the Latin School in the autumn of 1889. The Latin school should have provided five years of schooling but in 1893 it was extended to six years of study. He then spent two and a half years studying mainly classics at the Humanist Gymnasium in Worms and, despite his father wishing him to continue in the family business, he studied mathematics in his spare time. In 1898 Perron was awarded his Abitur and later that year he began his studies of mathematics and physics at the University of Munich. In keeping with the custom of the time to spend semesters at different universities, he also studied at the University of Berlin. He undertook research for his doctorate at Munich advised by Ferdinand von Lindemann and in 1902 he was awarded the degree by the Ludwig-Maximilian University of Munich for his 43-page thesis Über die Drehung eines starren Körpers um seinen Schwerpunkt bei Wirkung äusserer Kräfte .

After the award of his doctorate, Perron studied at Tübingen University and Göttingen University, where he worked with David Hilbert. He published Note über die Konvergenz von Kettenbrüchen mit positiven Gliedern in 1905. It had been lectures at Munich by Alfred Pringsheim on continued fractions that had been a major influence on Perron and this 1905 paper on continued fractions continued Pringsheim's work on the topic. In 1898 Pringsheim had introduced the term 'unconditional convergence' of a continued fraction and had also given what is now known as the Pringsheim criterion which insures the convergence of a continued fraction. Perron also published Über eine Anwendung der Idealtheorie auf die Frage nach der Irreduzibilität algebraischer Gleichungen in 1905. Perron went on to complete his habilitation at Munich and was appointed a lecturer there in 1906. On 28 July 1906, he married Hermione Perron, who was related to him via a number of different routes. From this marriage there were three daughters, Hertha, Erika and Hedwig.

Perron's work on continued fractions led to him publishing the book Die Lehre von den Kettenbrüchen in 1913. For extracts of reviews of this work see THIS LINK.

In 1910 Perron accepted the offer of a post as extraordinary professor at Tübingen and then, on 13 December 1913, he became an ordinary professor at Heidelberg, taking up the appointment in 1914. However World War I disrupted his career and, in 1915, he undertook war work which was to earn him the Iron Cross. His first military service was in the Landsturm, a third-line reserve force consisting of older men. They saw active service on the eastern front. Later he served as a lieutenant in a survey unit until 1918. At the end of the war he returned to Heidelberg where he taught until 30 September 1922 when he was appointed to a chair at Ludwig-Maximilian-University of Munich. This chair had become vacant due to Alfred Pringsheim retiring. He became a colleague of Constantin Carathéodory and Heinrich Tietze and the three professors became known as the "Munich mathematical triumvirate".

In [5] 218 publications by Perron are listed in a bibliography which he compiled himself. These publications cover a wide range of mathematical topics. His work in analysis is certainly remembered through the Perron integral. However he also worked on differential equations, matrices and other topics in algebra, continued fractions, geometry and number theory. Perron published a number of important texts. In addition to the work on continued fractions mentioned above, which in fact ran to three editions the last being a two volume version in 1954/57, he published an important text on irrational numbers in 1921. This text was designed to require only school level mathematics as a prerequisite and the topic was skilfully developed in a beautiful self-contained way. Again this was a text which ran to several editions and Perron revised the text in 1960 when he was aged 80.

For extracts from reviews of various editions of Perron's Irrationalzahlen see THIS LINK.

One of the things he is best-known for is the Perron paradox which highlights the danger of assuming that a solution to a problem exists. He introduced this as part of the discussion of Steiner's attempted proof of an isoperimetric problem. The paradox runs as follows:
Suppose the largest natural number is $N$. Then if $N > 1$ we have $N^{2} > N$ contradicting the definition. Hence (!) $N = 1$.
A two volume work on algebra first appeared in 1927 with the third revised edition being published in 1951. For extracts from reviews of various editions of Perron's Algebra see THIS LINK.

Hitler's Nazi party came to power in early 1933. Perron later wrote:-
When [the Nazis] were not at the helm and with the Communists were developing the roughness, I thought it was actually the same whoever won the next election. For both would endeavour to pretend to the world that they were the true bearers of culture, and consequently science would not go badly. But I deceived myself. The Nazis were serious about their reforms and began the destruction of the universities on the first day.
On 1 April 1933 there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter the university. On 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired but there were exemptions for those who had fought for Germany in World War I. Perron was invited to lecture at Göttingen University in the summer of 1933. He later wrote:-
In the summer of 1933 I was invited to a lecture in Göttingen, the stronghold of mathematics. The professors were already packing their suitcases, and a student mob dominated the field. And then came the most painful disappointment of my life. What I had imagined was quite impossible, happened. A whole series of outstanding scientists, not just opportunists who thought they could get rid of their competition, rushed to be Hitler's mouthpiece, did not hesitate to speak of the good Aryan and the bad "foreign" Jewish physics and mathematics, and thus ridicule their German fatherland to all foreign countries. I had a lot of fighting with them, and my wife was often afraid that with my mouth I would destroy myself.
The first "fighting" came in 1934 when Perron was president of the German Mathematical Society. In that year Ludwig Bieberbach gave a lecture on racial mathematics. Bieberbach also published an open letter in the journal of the German Mathematical Society attacking Harald Bohr. Perron, as president of the German Mathematical Society, wrote to Harald Bohr in May 1934 (see [1]):-
I completely understand your excitement over Bieberbach's talk. I am myself horrified, must however, for today, confine myself to telling you that in any case, the German Mathematical Society, which has at all times reckoned the large number of foreign members to its honour, stands completely apart from these views.
Certainly at this time Perron seemed unable to understand that Bieberbach could really believe the things he was saying about race.

Perron, together with Helmut Hasse and Konrad Knopp, decided to make a public declaration that Bieberbach's open letter had been published in the Society's journal against the wishes of the other editors. Fearing that if they published this in the Society's journal, Bieberbach as an editor might be able to delete it, they decided to produce a separate page, printed at their own cost, and circulate this to members of the Society. However, without the agreement of the Society's Board (which Bieberbach was on), the publisher refused to circulate the declaration. The September 1934 meeting of the German Mathematical Society at Bad Pyrmont was a difficult one for many of the members but particularly for Perron as President. Bieberbach, as the Society's secretary, invited a number of students to the meeting. Perron, as president, was in the chair and, following the constitution, asked non-members to leave. Bieberbach's secretarial report introduced the topic of his open letter to Harald Bohr. Perron wanted to keep the discussion to the fact that Bieberbach had published the letter in the Society's Journal without discussing it with the other editors. However, he failed to keep the discussion on this point and the discussion became political. It was a very difficult meeting for Perron and, as a consequence, there was a move by the Reich Ministry to remove Perron from his post, but it seems to have been dropped. Perron later wrote:-
But I made it, and never came into conflict with the Gestapo, except once with a miserable block guard, who wanted to ban the Frankfurter Zeitung, which I of course let go. The local colleges were naturally on my side in the fight against the Nazis. There was no DC switch. Commands were bypassed or ignored. Never before were so many Jewish authors recommended in the lecture as after the ban.
Holidays in Switzerland gave Perron the opportunity to see what the rest of the world was reporting about the Nazis. He later wrote:-
In the 1930s, I regularly visited Switzerland, partly to stroll around on the four-thousand metre mountains, but to a large extent also to read foreign newspapers and talk with colleagues about Nazi atrocities. But also the Swiss too, if they wanted to speak openly, were just as anxious as we were. People were afraid of Nazis and there were abductions.
To see how much he risked in attacking the Nazi racial views we need only look at the Preface he wrote for the second edition of Irrationalzahlen which was published in 1939. Olive Hazlett writes in [6]:-
The preface of this new edition seemed in some ways the most interesting part, on account of the reason that the author gives for presenting Dedekind's theory of irrational numbers rather than the theory of Cantor and Méray. In 1921, he did not seem to think it necessary to give any reason for basing his treatment on Dedekind's work; but in 1939, he devotes most of the preface to justify his giving Dedekind's rather than Cantor's theory. He refers to an article by Bieberbach and the famous one by Hardy in "Nature" on the J-type and S-type of mathematicians. One could easily wonder just what lies behind these careful justifications. However that may be, we wish him well, for Perron has done yeoman service in writing textbooks for universities and technische Hochschulen.
Also in 1939 he wrote a letter to the rector of Munich University, Philipp Broemser (1886-1940), which made absolutely clear his views on the Reich:-
Magnificence!
I cannot participate in the celebration of the Lecturers' Union Academy organized by the Reich Leader of University Teachers, Professor Dr Walter Schultze.
Reason:
Since I am neither a member of the Lecturer's Academy nor a lecturer's union at all, my participation can only be thought of as a scientific honour. Now, however, I am a member of a number of German scientific academies, and, against these institutions and their members, the Reich Leader of University Teachers has expressed his contempt by stating that the German academies have not done anything scientifically since Leibniz and are now only societies of calcified scientific veterans.
Two things are possible. Either the Reich Leader of University Teachers is right with this low opinion or is not right. In the first case, it cannot be a pleasure for the Reich Leader of University Teachers to see among his honourable guests such inferior scientific personalities; I would at least spare him this sight as far as my person is concerned. In the second case, however, I cannot be expected to be honoured by a man who has wrongfully denigrated the academies and their members, and probably I would have to listen defencelessly, if the honourable guests are condemned in the same way.
Heil Hitler!
O Perron
Let us note that Walter Schultze (1894-1979) was a physician and World War I aviator who had joined the Nazi party when it was founded in 1919 and became Reich Leader of University Teachers in 1935. In that role he implemented Nazi racial policies in the German universities. We also note that Perron had been honoured with election to the Heidelberg Academy of Sciences in 1917, the German Academy of Scientists Leopoldina in 1919, the Bavarian Academy of Sciences in 1924, and the Göttingen Academy of Sciences in 1928.

By 1941 there was a report on Perron by the Nazi party in response to a request he had made which stated (see [11] or [12]):-
Although a legal attitude is to be expected from him, it can be said on the other hand that Perron is not a very faithful follower of the movement.
The Dean of Science at Munich during the years of World War II was Wilhelm Carl Gottlieb Müller (1880-1968) who had studied physics, mathematics and philosophy. Müller had been appointed in 1939 to succeed Arnold Sommerfeld as Professor of Theoretical Physics at Munich but many felt it was for political rather than scientific reasons. He argued in favour of "German physics" and had many disagreements with Perron. Much later Perron wrote:-
... after the final victory by the Americans, mathematics here has remained completely intact both at the university and at the Technische Hochschule, while, for example, a representative of theoretical physics [Wilhelm Müller] who had pushed towards the Ministry, who often tried to frighten us with his good relations with the Führer headquarters, had to disappear immediately.
Indeed Müller was dismissed in 1945.

Although Perron formally retired in 1951, he continued to teach certain courses at Munich until 1960. However even when he ended his teaching at the age of 80 he was still able to continue with a vigorous research program, publishing 18 papers between 1964 and 1973. Perhaps most remarkable of all Perron's books was his text on non-euclidean geometry which he published at the age of 82. Frank, in [5], writes:-
This work won the approval of the entire mathematical world due to its great worth and masterful presentation. It is of interest not only to students of mathematics and physics but also especially to teachers of mathematics.
For an extract from a review by Hans Schwerdtfeger see THIS LINK.

Despite the large amount of mathematics which Perron produced over a long career, he also had other interests. These are described in [5] and included:-
... his love of the mountains of his surroundings. No vacation would have been complete without the mountains. As well as higher mountains, he climbed the 2200 meter Totenkirchl in the Wilder Kaiser more than twenty times, the last time when he was 74.
As well as the honours Perron received through election to the academies mentioned above, he was also awarded an Honorary doctorate from the University of Tübingen in 1956, an Honorary doctorate from the University of Mainz in 1960 and the Bavarian Order of Merit in 1959.

### References (show)

1. S L Segal, Mathematicians under the Nazis (Princeton University Press, Princeton, 2003).
2. T A A Broadbent, Review: Die Lehre von den Kettenbrüchen (Second edition, 1950) by Oskar Perron, The Mathematical Gazette 36 (315) (1952), 66.
3. J L Burchnall, Review: Irrationalzahlen (Second edition, 1939) by Oskar Perron, The Mathematical Gazette 24 (259) (1940), 137-138.
4. A B Coble, Review: Algebra (2 volumes) Vol 1 Die Grundlagen, Vol 2 Theorie der algebraischen Gleichungen (1927), by Oskar Perron, Amer. Math. Monthly 36 (4) (1929), 224-225.
5. E Frank, In memoriam Oskar Perron, Journal of Number Theory 14 (1982), 281-291.
6. O C Hazlett, Review: Irrationalzahlen (Second edition, 1939) by Oskar Perron, Bull. Amer. Math. Soc. 46 (1) (1940), 15.
7. O C Hazlett, Review: Algebra (2 volumes) Vol 1 Die Grundlagen, Vol 2 Theorie der algebraischen Gleichungen (1927), by Oskar Perron, Bull. Amer. Math. Soc. 34 (1) (1928), 115-116.
8. J Heinhold, Oskar Perron, Annual Report of the DMV 90 (4) (1988), 184-199.
9. K Huther, Der Mathematiker Oskar Perron, Frankenthal einst und jetzt 1 (1966), 14-18.
10. A J Kempner, Review: Irrationalzahlen (1921), by Oskar Perron, Bull. Amer. Math. Soc. 29 (1) (1923) (1) (1923), 34-36.
11. F Litten, Oskar Perron - Ein Beispiel für Zivilcourage im Dritten Reich, Mitteilungen der Deutschen Mathematiker-Vereinigung 3 (1994), 11-12.
12. F Litten, Oskar Perron - Ein Beispiel für Zivilcourage im Dritten Reich, Frankenthal einst und jetzt (1/2) (1995), 26-28.
13. F Litten, Oskar Perron, Neue Deutsche Biographie 20 (Duncker & Humblot, Berlin, 2001), 196-197.
14. J L Nayler, Review: Die Lehre von den Kettenbrüchen (1913) by Oskar Perron, The Mathematical Gazette 7 (106) (1913), 159-160.
15. M S Rees, Review: Algebra Second edition (volume 1, 1932, volume 2 1933) by Oskar Perron, Amer. Math. Monthly 40 (8) (1933), 484-485.
16. L A Rubel, Review: Irrationalzahlen (Second edition reprint, 1951) by Oskar Perron, American Scientist 45 (4) (1957), 298A, 300A.
17. H Schmidt, Oskar Perron, Bavarian Academy of Sciences Yearbook 1976 (1976), 217-227.
18. W T Scott, Review: Die Lehre von den Kettenbrüchen (Third edition). Vol. I: Elementare Kettenbrüche (1954) by Oskar Perron, Bull. Amer. Math. Soc. 61 (6) (1955), 594.
19. W T Scott, Review: Die Lehre von den Kettenbrüchen (Third edition). Vol. II: Analytisch-funktionentheoretische Kettenbruche. (1957) by Oskar Perron, Bull. Amer. Math. Soc. 64 (5) (1958), 299-300.
20. G L Watson, Review: Irrationalzahlen (Second edition reprint, 1951) by Oskar Perron, Biometrika 44 (1/2) (1957), 299.
21. G N Watson, Review: Die Lehre von den Kettenbrüchen (Third edition). Vol. I: Elementare Kettenbrüche (1954) by Oskar Perron, The Mathematical Gazette 41 (338) (1957), 309-310.