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Gyula Kőnig

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Born
16 December 1849
Győr., Hungary
Died
8 April 1913
Budapest, Hungary
Summary
Julius König was a Hungarian mathematician who worked on algebra, number theory, geometry, set theory and analysis. He is remembered today for König's paradox.
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Biography

Julius König (or Gyula Kőnig) was the son of Zsigmond Kőnig (1824-1907) and Wilhelmine Freyer (1828-1883). Zsigmond König was a wealthy wholesale spice merchant who was born on 17 January 1824 into one of the oldest Jewish families in Győrsziget, a district of Győr, Hungary. He was a member of the committee set up by the Győr Jewish community in 1861 to build a synagogue. He married Wilhelmine Freyer in Győrsziget on 9 July 1848. She was the daughter of Ábrahám Freyer, who was a teacher in the Jewish school in Bratislava, and is described as "gentle and cultured". Her brother, Salamon Freyer was the rabbi of Győrsziget from 1839 to 1860. Gyula Kőnig was the only child of Zsigmond and Mina Kőnig.

He went to both primary and secondary school in Győr. Recognising his talent, and expecting him to follow a scientific career, in 1859 his parents enrolled him in the Benedictine Gymnasium, a school with a high reputation in the lively trading city of Győr. He was an excellent student throughout. At first he excelled in literature and it looked at one stage as though this would be the topic that he would study at university. He had developed fine linguistic abilities, speaking German at home while the Gymnasium gave him perfect Hungarian and a Hungarian identity. After studying for eight grades at the Győr Benedictine Gymnasium he graduated in 1867.

After leaving the Benedictine Gymnasium in Győr when he was sixteen years old, he went to study medicine in Vienna, matriculating in February 1868. One has to understand the position of Jews in Hungary at this time where the medical profession was considered suitable for Jews while other careers were not. In addition to the course that Kőnig followed in the Medical School at the University of Vienna he also made a study of mathematics and the natural sciences. He attended mathematical lectures as a first-year medical student on differential and integral calculus and also Franz Moth's lecture on differential equations. Franz Xaver Moth (1802-1879) had studied at the University of Prague and was a professor at Salzburg and Linz before being appointed Professor of Mathematics at the University of Vienna in 1849. König obviously believed that mathematics is an important tool for exact medical research.

He went to Berlin spending one semester there in 1868-69. He was greatly influenced by the three mathematicians, Ernst Kummer, Leopold Kronecker, and Karl Weierstrass. Feeling now that he wanted to follow a course in the natural sciences, although still retaining his medical interests, he next went to Heidelberg which had an excellent reputation for science with famous men such as Robert Bunsen, Gustav Kirchhoff and Hermann von Helmholtz on the teaching staff. Kőnig's interests were broad at this stage in his career and he attended lectures by the physiologist Julius Bernsten, the pathologist Wilhelm Heinrich Erb, the philosophers Eduard Gottlob Zeller and Karl Alexander von Reichlin-Meldegg.

Encouraged by Helmholtz, he conducted physiological research that required both laboratory experiments and theoretical considerations. His first research paper was written on a topic suggested by Helmholtz and considered a medical topic. The paper, Beiträge zur Theorie der elektrischen Nervenreizungen Ⓣ, was published by the Imperial Academy of Sciences in Vienna (now called the Austrian Academy of Sciences) in 1870. It is an early work on the physical and mathematical foundations of electrical nerve impulse transmission, highlighting its focus on the theory of nerve transmission.

Perhaps the most significant event which turned Kőnig's interests firmly towards mathematics happened in 1869 when Leo Königsberger left Greifswald to take up the appointment of a chair of mathematics at Heidelberg. Königsberger had been greatly influenced by Weierstrass's lectures on elliptic functions and this was the topic which interested him at this time, so he in turn influenced Kőnig to also undertake research on elliptic functions. Kőnig worked on his doctoral dissertation under Königsberger's supervision and submitted his 24-page thesis Zur Theorie der Modulargleichungen der elliptischen Functionen Ⓣ to Heidelberg passing his doctoral examination summa cum laude in July 1870; his thesis was published in the following year. It contains six chapters: I. The modular equations; II. Series expansion of the roots; III. The dimension of the modular equation; IV. The discriminant; V. Determination of the numerical functions f(n)f(n) and f′(n)f'(n); and VI. The polynomial g(u)g(u).

After being awarded a doctorate from Heidelberg, Kőnig went to Berlin where he spent six months attending lectures by Weierstrass and Kronecker. He then returned to Budapest where he was appointed as a docent at the University in 1872. He became a professor at the Teacher's College in Budapest in August 1873 and, in the following year, was appointed to the Technical University of Budapest. This was a period when improvements in Hungary's economic position led to a need for education and consequently a need to train more school teachers. The mathematical level began to rise steadily and the Technical University of Budapest had been established in 1871 with the right of issuing diplomas. Kőnig joined the University at a time when there were other talented and enthusiastic mathematicians on the staff. In November 1874 he was promoted to full professor at the Technical University of Budapest.

Kőnig took a lead in a new venture which these mathematicians embarked on, namely the founding of a new mathematical journal Müegyetemi Lapok Ⓣ. The first issue appeared in 1876 and a part was produced each month. Sadly only 3 volumes and 30 issues appeared before it was forced to close. The editors wrote in the final issue:-
With this issue the Technical University Journal has come to the end of its career. It seems that here in Hungary no mathematical journal can exist without financial support. Of course, if there were only half as many readers of mathematics as there are teachers, things would be quite different.
It was with great sadness that Kőnig was forced to admit that the venture which he had thought so highly of could not continue after so short a time.

Perhaps it was to teaching, rather than to research, that Kőnig made the most important contributions. He raised the level of mathematics teaching at the Technical University to a high standard, in fact it was a level which the engineers must have found very demanding. Gusztáv Rados writes [20]:-
Kőnig's curriculum made it possible for the audience to become acquainted with all the modern disciplines of mathematics. He himself changed the subject of his lectures as often as possible and it would be almost difficult to list the many lectures he gave during his nearly 40-year teaching career. Initially, in his lectures for the students of the Technical University entitled "Algebraic Analysis" and "Theory of Higher Degree Equations", later in his lectures on Analysis I and II, he summarised on a scientific basis everything that he found important and indispensable for technical practice. However, the focus of his lectures and scientific work was on special lectures. His versatility and work force were amazing, as evidenced by the 20 different special lectures he gave. The subjects of these were: determinants and systems of linear equations; the algebra of linear transformations; algebraic equations from the functional point of view; Galois theory; elimination theory; the theory of algebraic curves; elementary number theory; the theory of algebraic integers; the arithmetic theory of algebraic quantities; functions of complex variables and elliptic functions; functions of real variables; definite integrals, Fourier series and spherical functions; systems of total differential equations and the theory of integration of first-order partial differential equations; calculus of variations; probability calculus; political arithmetic; set theory. Most of his lectures were accompanied by one or more dissertations presenting new research results, with which he filled in the gaps he noticed in the theory he had elaborated on in his lectures, or provided answers to new questions that had arisen there.
Szénássy writes [21]:-
Kőnig was extremely conscientious and he expected the same of his students. He went into details whatever the subject be and analysed everything thoroughly even for students of the Technical University. He emphasised the importance of the quality rather than the quantity of knowledge, and he tried to get his students to acquire an exact mathematical thinking.
A fuller description is gave by Rados [20]:-
Kőnig delivered his lectures with oratory skill and great eloquence, in which he was able to captivate the attention of his listeners and arouse their interest in new questions with his brilliant entertaining and illumination comments, the constant highlighting of the guiding ideas, the raising of questions that were more original than those studied, with his crystal-clear logic and sharp criticism. In his lectures, his main concern was not easy comprehensibility, but the pursuit of absolute truth, the scientific rigour that was always and everywhere valid. Therefore, those of weaker talent could only find it more difficult to follow the explanations of his lectures, in which he only outlined his subject in broad strokes and deliberately avoided giving the impression that the discipline discussed in the lecture was already completed and a finished thing; on the contrary, he boldly pointed out open questions, their unsolved nature and the new open questions that would arise after their solution.
He contributed to the Technical University in other roles too. He was on three occasions elected Dean of the Engineering Faculty 1886-1890 and also on three occasions he was Rector of the University 1891-1893. In his inaugural speeches as rector he described his educational philosophy saying that he considered a university as a place which reconciles the work of nature and of society in the interest of the culture and the enrichment of the nation. He proposed the introduction of art history and literature lectures in order to make engineering education more versatile. He also emphasised the importance of teaching economics and law.

Several attempts were made in the 1870s to have Kőnig admitted to the Hungarian Academy of Sciences. The first was on 31 March 1873 when Jenő Hunyady, a corresponding member of the Academy, recommended Kőnig as a corresponding member citing his eight papers and his leading role in organising the manuscripts of Farkas Bolyai and János Bolyai. It failed to gain enough support but on 31 March 1874 Loránd Eötvös repeated the recommendation for Kőnig. It again failed, as did another attempt in 1876 but on 21 March 1880 he was elected as a member of the Hungarian Academy of Sciences after the physicist Kálmán Szily, a leading member of the Academy, made his recommendation. In it he criticised the Academy for repeatedly failing to accept him and queried how they could have failed to elect "such an outstanding scientist as Gyula Kőnig who is known to the entire mathematical world." Elected a corresponding member in 1880 he was elected a full member in 1889. In 1894 he became secretary of the Department of Mathematics and Natural Sciences of the Hungarian Academy of Sciences, a position he held for the rest of his life.

On 22 June 1882 Kőnig married Erzsébet Oppenheim (1862-1916) (known as Elise). She was born on 17 July 1862 in Pest, the daughter of Sámuel Oppenheim, a wine merchant in Pest, and Hermina Stern. Gyula and Elise König had two sons György Dezső König (1883-1944) and Dénes Kőnig (1884-1944), a well-known mathematician with a biography in this archive. György Kőnig studied law at university, had a political career as an education official, but was also a literary historian and translated French novels into Hungarian. He lost his eyesight at an early age, so he retired in 1921. Even after this, however, some of his novels were published in French. Both Kőnig and his wife were of the Jewish faith when the boys were born. As a result of the anti-Semitic movements that arose in Hungary in the early 1880s, they decided to have their children baptised as Christians. György Kőnig was baptised into the Christian faith in June 1883 and Dénes Kőnig in November 1884. Kőnig and his wife were baptised on 12 June 1889.

Gyula Kőnig worked on a wide range of topics in algebra, number theory, geometry, set theory, and analysis. One of his early ideas was a paper of 1872 which looked at intuitive ways to prove the consistency of non-Euclidean geometries. He published many research papers in analysis, but his greatest significance in this area comes from the excellent textbooks which he wrote on the topic. His most important work written in 1903 is based on a fundamental study by Kronecker published in 1892. Kőnig developed Kronecker's polynomial ideals and presented many results on discriminants of forms, elimination theory and Diophantine problems. Kőnig's work on polynomial ideals influenced Hilbert, Lasker, Macaulay, Emmy Noether, van der Waerden and Gröbner but they simplified his ideas so Kőnig's work is now only of historical interest.

In the last eight years of his life Kőnig's interests turned towards set theory and he contributed to the continuum hypothesis. In August 1904, at the International Congress of Mathematicians at Heidelberg, he announced that the continuum hypothesis was false [21]:-
... what a sensation the announcement of the title of Kőnig's lecture .. stirred among the participants of the Congress. All section meetings were cancelled so that everyone could hear his contribution.
Kőnig's proof contains an error in that he applied a theorem due to Felix Bernstein in a case where it does not hold. It was a little while before Zermelo found the error in the proof but it appears that Felix Hausdorff was probably the first to realise the error. In the paper by Kőnig in the Proceedings of the Congress, he states:-
... if Bernstein's theorem were to hold in general, then the continuum could not be well-ordered.
In 1905 Felix Bernstein published a short note correcting his theorem. Kőnig continued to work on set theory and in particular published Über die Grundlagen der Mengenlehre und das Potenz des Kontinuums Ⓣ (1905). Miriam Franchella writes [5]:-
Kőnig presented what is known as his 'paradox' in 1905: there is a number that at the same time is and is not finitely definable. It was used by him as a step for (trying to) prove that the continuum was not well-ordered, and only later he shifted the focus to the contradiction itself and tried to find solutions. ... Kőnig stressed in a footnote at the end of the 1905 paper that his difference between classes and sets solved the problem of those paradoxes of the theory of ordinal numbers which Burali-Forti drew to the attention of the mathematicians.
Poincaré criticised Kőnig's 1905 paper, so Kőnig reacted by working even harder on set theory. In 1906 he gave a two-pages proof of the equivalence theorem that Poincaré presented to the Académie des Sciences as Sur la théorie des ensembles Ⓣ. In the same year Kőnig also published Über die Grundlagen der Mengenlehre und das Kontinu-umproblem Ⓣ which examined his 'paradox'. Eventually he came to realise that he was confusing two separate things, namely writing the definition of a number and writing the number itself. Miriam Franchella writes [5]:-
Kőnig became conscious of the necessity of keeping the two things distinguished and used this distinction to let the paradox disappear. That was the end of the tortuous life of Kőnig's paradox, which has a singular story with respect to the other paradoxes: it was considered a tool for proving the falsity of the continuum hypothesis; it was individuated as a problem in itself, and a very strange solution, with further peculiar consequences, was proposed; and finally it seemed to be not problematic at all.
Kőnig retired from his post at the Technical University on 31 October 1905 but he continued to lecture there particularly on topics that he was interested in. Clearly retirement had been undertaken so that he could spend more time on things which he wanted to do - is this not the reason why many academics take early retirement? He spent the last part of his life working on his own approach to set theory, logic and arithmetic, which was published in 1914, the year after his death, as Neue Grundlagen der Logik, Arithmetik, und Mengenlehre Ⓣ. He had been working on the final chapter at the time of his death. The publication was completed by König's son Dénes Kőnig and the book had a Preface by Gyula Kőnig and a Preface by his son Dénes Kőnig. We give an English translation of both Prefaces at THIS LINK.

We should mention Kőnig's contribution to mathematics teaching at secondary schools in Hungary. He worked out the algebra part of the syllabus and wrote some fine textbooks to support the teaching. He also served on the Board of the Hungarian Academy of Sciences for 19 years. As a member from 1898 of the Board of Directors of the Franklin Society Literary Publishing House and Institute, the largest publishing house in Hungary, he had a wealth of expertise and experience to share with his colleagues [20]:-
... from 1900 he was the controller of the entire management of the publishing house. No important matter was handled without him. He recognised that the publishing house could become a monopolistic force in the Hungarian book market if it acquired the competing Wodianer Publishing House. He successfully led the complicated business negotiations, and in 1904 the publishing house was purchased together with its publications and contracts. Subsequently, at the meeting of March 1904, the board of directors of the association offered him the position of CEO, on the recommendation of Zsigmond Kornfeld, director of the Hungarian General Credit Bank. ... He developed the publishing house into a modern large-scale enterprise, the most prestigious factor in Hungarian book production. The aim of his management was to serve Hungarian literature and science in a modern way.
We mentioned at the beginning of this article that in his young days Kőnig had a love of literature, so it is easy to see why he was so involved in the Literary Publishing House and also why one of his sons followed a literary career. Kőnig retained his love of literature and reading was throughout his life his favourite hobby.

Let us end with a quote from Gusztáv Rados [20]:-
Among the greats was Gyula Kőnig, who since the Bolyai family was the most talented researcher in the field of mathematics in our country (Hungary), the richest in merits and successes, and whom death snatched from the ranks of the living on 8 April 1913, in the prime of his life and labour, almost while working. This was a sad day for Hungarian scientific and public life, of which he was a significant factor, even sadder for his numerous students who adhered to him with never-ending gratitude, who respected him as their benevolent friend and brilliant teacher.
Other Mathematicians born in Hungary
A Poster of Gyula Kőnig

References (show)

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Written by J J O'Connor and E F Robertson
Last Update March 2026