**Carl Johannes Thomae**was born in Laucha (Unstrut), which is about 35 km south west of Halle. His father was Karl-August Thomae, the rector of a school in Laucha, and his mother was Emilie Gutsmuths. Carl Johannes was the eldest of his parents two children and was not expected to survive to adulthood. For the first four years of his life he was extremely frail but eventually he gained in strength and by the age of five he was able to enjoy a normal childhood.

When he was five years old Thomae entered the primary school in Laucha. His father, as rector of a school, was an important figure in the town. When Thomae was nine years old his father became his teacher, giving him an excellent grounding to enter the gymnasium. He then entered the cathedral gymnasium in Naumburg (Saale) which is about 10 km south of Laucha. His teacher at the gymnasium, Moritz Hülsen, saw what an outstanding pupil he had in Thomae and coached him with additional work beyond the usual course of study. Thomae's work in mathematics was quite outstanding and it won for him a scholarship which he was awarded from October 1860.

Thomae entered the University of Halle in 1861, which was essentially his local university. There he attended lectures by Carl Neumann, who was a privatdozent at the time, on applied mathematics and also by Eduard Heine who was an ordinary professor. It was Heine who had the greatest influence on Thomae, giving him a love for function theory which was to set the course of his research for the whole of his career. In 1862 Thomae, following the usual tradition of German students of his time, moved to study at another university. This time he chose to study at Göttingen beginning in the autumn. He would have taken courses by Riemann, but sadly Riemann caught a heavy cold which turned to tuberculosis. His lectures were taken over by Schering and Thomae went on to undertake his doctoral studies supervised by Schering.

After being awarded his doctorate in 1864, Thomae went to Berlin where he studied elliptic functions with Weierstrass for two terms. After this he returned to his home town of Laucha and there worked on a number of mathematical papers. In 1866 he submitted work *On the introduction of ideal numbers* to the University of Göttingen as his habilitation and began lecturing there.

In June 1866 a war broke out between Austria and Prussia. It was a war which was sought by Bismarck who used as a pretext a dispute over the administration of Schleswig and Holstein. Austria and Prussia had captured this from Denmark in 1864 and jointly administered it for two years. Military preparations began fairly early in 1866 and hostilities broke out in the middle of June. Thomae took part in the campaign in Bohemia, where the main Prussian armies met the main Austrian forces. Thomae took part in three battles in this short campaign, the most decisive being the Battle of Königgrätz on 3 July. The war ended on 23 August with the signing of the Treaty of Prague. The Prussian victory resulted in Austria's exclusion from Germany as Bismarck had intended. After the Seven Weeks' War (as this short war is called) Thomae returned to Göttingen and gave a lectures on determinants and on the differential and integral calculus.

In 1867 Thomae was appointed as a privatdozent at Halle where he became a colleague of Heine and Cantor. He submitted another habilitation thesis, this time containing two works: *De propositione quadam Riemanniana in analysi* Ⓣ and *Über die Differentialgleichungen für die Module der Abelschen Integrale* Ⓣ. While he worked as a privatdozent at Halle, major changes were taking place in Prussia. Prussia led the German states to victory over France in the Franco-Prussian War of 1870-71 and in 1871 the German Reich (German Empire), with William I of Prussia as its emperor, came into existence.

Thomae was promoted to extraordinary professor at Halle in 1872 and two years later he moved to the University of Freiburg where he was appointed as ordinary professor. The vacancy arose since Paul du Bois-Reymond, who held the chair at Freiburg from 1870 to 1874, had moved to the chair at the University of Tübingen where he succeeded Hankel. Also in 1874 Thomae married Anna Uhde in Balgstädt near his home town of Laucha; their son Walter was born on 5 November of the following year. At this point a great tragedy struck the family, for Anna died five days after Walter was born.

After spending five years at Freiburg, Thomae moved to Jena in 1879 when he accepted the chair there. At Jena he became a colleague of Gottlob Frege who had been appointed as a privatdozent at Freiburg four years earlier. Both Thomae and Frege spent the rest of their careers at Freiburg. The relationship between Thomae and Frege is an interesting one. Scientifically they were vehemently opposed; their dispute was a public one carried out in the pages of the *Jahresberichte der Deuschen Mathematiker-Vereinigung*. We will look at the issues involved in this dispute below. On the personal side Dathe [2] claims their relations were quite friendly. Gabriel's evidence [4], however, may suggest that their scientific dispute spilled over into their personal relations.

Thomae was Dean of the Philosophy Faculty in 1884, 1891, 1898, and for a fourth time in 1905. He was also elected Rector of the university in 1901. Thomae married for a second time in 1892. His marriage in Jena to Sophie Pröpper gave him a second child, this time a daughter Susanne born in 1893. Susanne went on to become a singing teacher, while his son Walter studied the theory of art. Thomae retired in 1914 but continued to publish papers up to 1919. He died in 1921 after a short illness.

The approach of both Riemann and Weierstrass strongly influenced Thomae's study of function theory. One of Thomae's first papers is *Die allgemeine Transformation der Thetafunktionen mit beliebig vielen Variablen* Ⓣ (1864). He also published the text *Theorie der ultraelliptischen Funktionen und Integrale erster und zweiter Ordnung* Ⓣ (1865). An important formula, which is still often used today, is Thomae's formula which expresses branch points of hyperelliptic curves in terms of hyperelliptic theta constants. This first appeared in *Bestimmung von d lg* *θ*(0, 0, ... , 0) *durch die Classenmoduln* Ⓣ (1866) and is also studied in his important paper *Beitrag zur Bestimmung von* *θ*(0, 0, ... , 0) *durch die Klassenmoduln algebraischer Funktionen* Ⓣ (1870). Both of these two papers were published in *Crelle's Journal*. In the second of the papers Thomae also showed that the roots of a polynomial can be expressed in terms of hyperelliptic theta functions. Also in 1870 he produced the first examples which show that joint continuity of a function *f* : **R**^{n} → **R** does not follow from separate continuity. He also discovered methods of solving difference equations giving solutions in the form of definite integrals. Thomae was the first to attempt to introduce "trans-Archimedean numbers" but Cantor argued that these were unworthy of the name of magnitude or quantity. He is also famed for the introduction of the Thomae *θ*-gamma function.

Cantor had discovered that the points in *n*-dimensional space could be put in 1-1 correspondence with the line. In a letter of 1877 to Dedekind he said:-

*I see it but I don't believe it*.

Thomae's textbook *Elementare Theorie der analytischen Funktionen einer komplexen Veränderlichen* Ⓣ published in 1880 contained an introduction to arithmetic that fitted together and extended many earlier ideas. He begins his textbook with the statement that:-

He then went to construct the rational numbers using Weierstrass's approach, then continued with a construction of the real numbers using the Cauchy sequence type of definition already published by Cantor and Heine. Only the positive integers had a concrete existence, while all other numbers were interpreted as signs. Following Hankel's ideas, Thomae wrote in his book that these numbers should be:-... the whole of pure mathematics is concerned with relations between numbers.

Frege, Thomae's colleague, strongly opposed these remarks. He held the opposite view to Thomae, and tried to set up arithmetic on a purely logical basis. In the second edition of his textbook published in 1890, Thomae tried to address the concern's of his colleague while still believing in his type of approach. He greatly extended the chat at the beginning of the book presenting the "formal arithmetic" approach which he summarised as follows:-... viewed as pure schemes without content[whose]right to exist[depended on the fact]that the rules of combination abstracted from calculations with integers may be applied to them without contradiction.

Frege did not calm down, however, and became even more vehemently opposed to Thomae's point of view. Some of Frege's sarcastic and biting criticisms of Thomae's approach are quoted in [9], see also [4].The formal conception of numbers requires of itself more modest limitations than does the logical conception. It does not ask, what are and what shall the numbers be, but it asks, what does one require of numbers in arithmetic. For the formal conception, arithmetic is a game with signs which one may call empty; by this one wants to say that(in the game of calculation)they have no other content than that which has been attributed to them concerning their behaviour with respect to certain rules of combination(rules of the game). Similarly a chess player uses his pieces, he attributes to them certain properties which condition their behaviour in the game, and the pieces themselves are only external signs for this behaviour. To be sure, there is an important difference between the game of chess and arithmetic. The rules of chess are arbitrary; the system of rules for arithmetic is such that by means of simple axioms the numbers can be related to intuitive manifolds, so that they are of essential service in the knowledge of nature. - The formal standpoint relieves us of all metaphysical difficulties, that is the benefit it offers to us.

Thomae's final four papers are *Die Liebmannsche Formel für das Ponceletsche Dreieck* Ⓣ (1918), *Über die harmonischen Kovarianten zweier Kegelschnitte* Ⓣ (1918), *Die harmonische Kovariante zweiter Art für zwei Kegelschnitte mit vier reellen Schnittpunkten* Ⓣ (1919), and *Über die Cassinischen Kurven* Ⓣ (1919).

**Article by:** *J J O'Connor* and *E F Robertson*

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