Elli Heesch, Heinrich Heesch and Hilbert’s eighteenth problem: collaborative research between philosophy, mathematics and application

Abstract

This paper examines the hitherto unknown scientific collaboration between the siblings Elli Heesch (1904–1993) and Heinrich Heesch (1906–1995). Heinrich Heesch, a well-known mathematician, was spearheading the early development of the computer-aided proof of the four-colour theorem. Much less is known about his sister Elli Heesch, a philosopher and logician. Together with her brother she investigated tiling problems and worked out a solution of Hilbert’s 18th problem. In 1944, Elli and Heinrich Heesch wrote a joint treatise on the industrial application of the tessellation method, which was of great interest to the German war and armaments industry. The collaboration of the Heesch siblings illustrates individual, disciplinary, cultural, and political aspects of knowledge production. The common interplay of close family relations and socio-political conditions that we find here underlines the fact that women’s contributions to solving mathematical problems often remained invisible.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Heinrich Heesch did pioneering work in developing methods for a computer-aided proof of the four colour theorem which turned out to be fundamental for the computer-aided proof by Kenneth Appel and Wolfgang Haken (1977). The four color theorem states that any map – a division of the plane into any number of regions – can be colored using no more than four colors in such a way that no two adjacent regions share the same color. Heinrich Heesch was the first to investigate the notion of ‘discharging’ for probing the theorem. Between 1967 and 1971, Heesch made several visits to the United States, where bigger and faster computers were available, working with Hermann Haken at University of Illinois at Urbana-Champaign and with Karl Durre and Yoshio Shimamoto at Brookhaven National Laboratory. During the crucial phase of his project, the German national research fund DFG cancelled financial support. Appel and Haken announced (Appel and Haken 1977) that they had proven the theorem. They were assisted in some algorithmic work by John A. Koch (Appel et al. 1977). Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. The unusual nature of the proof – it was the first major theorem to be proved with extensive computer assistance – and the complexity of the human-verifiable portion aroused considerable controversy (Wilson 2014).

2 For the following presentation of Elli und Heinrich Heesch’s life and carrier I consulted, among other sources: (i) Münster University Archive. Doctoral file Elli Heesch. Faculty of Philosophy 65/2737 (Universitätsarchviv Münster. Promotionsakte Elli Heesch. Philosophische Fakultät. Bestand 65, Nummer 2737). (ii) University and State Library Münster. Heinrich Scholz collection. Sign: N. Scholz (Universitäts- und Landesbibliothek Münster. Nachlass Heinrich Scholz. Sign.: N. Scholz). (iii) Göttingen State and University Library. Heinrich Heesch collection (including Elli Heesch collection). Sign.: Cod. Ms. H. Heesch (Niedersächsische Staats- und Universitätsbibliothek Göttingen. Nachlass Heinrich Heesch. Teilnachlass Elli Heesch. Sign.: Cod. Ms. H. Heesch). The Heinrich and Elli Heesch collection in the Göttingen State and University Library consists of 11 boxes, 16 folders, 7 mpn, 2 rolls (Cod. Ms. H. Heesch; Inventory number: Acc. Mss. 1995.30). Göttingen received the collection as a gift from Professor Hans-Günther Bigalke in 1995. My own archival research on Elli Heesch would not have been possible without the information that Adelheid-Hamacher-Hermes gathered on Elli Heesch. Hamacher-Hermes’ article (2008) is the only one that has been published on Elli Heesch up to the present day.

3 Heinrich Heesch’s doctoral thesis is composed of two articles published in 1929 in the Zeitschrift für Kristallographie (Heesch 1929a, 1929b). A third paper followed in 1930 in the same journal (H. Heesch 1930). In these papers, Heesch introduced the so-called black-white groups (a subclass of magnetic space groups). Specifically, Heesch introduced an anti-symmetry operation to the 32 crystallographic point groups which gives a total of 122 magnetic point groups. An example of an anti-symmetry operation is magnetic spin so that each atom side can have one of two possible values: spin up or spin down if the magnetic values are not randomly arranged, but aligned. A further example is the case where one atom is colored in one colour, say black, and the other identical atom symmetrically related in position is coloured in a different colour, say white. The concept was more fully explored by Alexei Vasilievich Shubnikov in terms of ‘colour symmetry’ (Shubnikov and Belov 1964).

4 For these and further information concerning Elli Heesch’s curriculum see the following archival documents: (i) Münster University Archive. Doctoral file Elli Heesch. Faculty of Philosophy 65/2737 (Universitätsarchviv Münster. Promotionsakte Elli Heesch. Philosophische Fakultät. Bestand 65, Nummer 2737); (ii) North Rhine-Westphalian State Archives Münster. Staff records A Nr. Schulkollegium H-158: Heesch, Elli, 1 p. handwritten, undated (Nordrhein-Westfälisches Staatsarchiv Münster. Personalakten A Nr. Schulkollegium H – 158: Heesch, Elli, Lebenslauf, 1 S. handschriftlich, undatiert).

5 Both Elli Heesch’s doctoral thesis and her article published two years later are relatively short. In his expert review of the doctoral thesis, Heinrich Scholz expressly emphasized the clarity and brevity of the work: ‘this brevity is the expression of an effortful and responsible concentration, the stages of which I have followed with interest and the pursuit of which I have emphatically favoured.’ Quoted from Scholz’ expert opinion on Elli Heesch’s doctoral, 2 p. handwritten, dated 25 January 1932. 25 January 1932. Münster University Archive. Doctoral file Elli Heesch. Faculty of Philosophy 65/2737 (Universitätsarchiv Munster. Promotionsakte der Elli Heesch Nr. 2737, Philosophische Fakultät). The translation is my own.

6 Noteworthy, even before Elli Heesch’s doctoral thesis went to the publisher, it was mentioned by Emerich Franzis (1933). Scholz himself quotes Elli Heesch’s work in his article ‘Die Wissenschaftslehre Bolzanos. Eine Jahrhundertbetrachtung’ (Scholz 1937). Concerning the reception of Elli Heesch’s work on Bolzano see Beth 1944; Bochenski, 1951/1978; Buhl 1961; Berg 1962; Morscher 1973.

7 See Heinrich Heesch’s letter to David Hilbert Göttingen, 14 November 1933. Göttingen State and University Library. David Hilbert collection. Sign.: Cod. Ms. D. Hilbert 145 A (Niedersächsische Staats- und Universitätsbibliothek Göttingen Nachlass David Hilbert. Cod. Ms. D. Hilbert 145 A).

8 In 1934, at the time when Elli was already working in Tübingen and Innsbruck, her paper ‘Psychische Wellen’ (Heesch 1934) appeared in the journal of the former Eucken-Bund Die Tatwelt. Zeitschrift für Erneuerung des Geisteslebens.

9 In her letters to her brother from this period, Elli tells of her life and work in Prague (see Göttingen State and University Library, Cod. Ms. H. Heesch 241.). Elli’s contacts with members of the Viennese Circle do not come as a surprise. At that time Heinrich Scholz was in close contact with Rudolf Carnap and Otto Neurath, among others. Some letters between Elli Heesch and Otto Neurath are in The Vienna Circle Archive, Otto Neurath Collection, Inv. nr. 243, Noord-Hollands Archief, Haarlem, The Netherlands. In the Brentano-Archive in Prague Elli Heesch is not listed as a proper member of the Prague Brentano Society. However, she is listed as a visiting scholar in the society’s annual report (see Binder 2019, 272). (Also listed is the Polish philosopher and logician Janina Hosiasson-Lindenbaum, a student of Władysław Tatarkiewicz.)

10 There are five letters from Elli Heesch to Otto Neurath and six letters from Otto Neurath to Elli Heesch in the Vienna Circle Archive, Otto Neurath Collection, Inv. nr. 243, Noord-Hollands Archief, Haarlem, The Netherlands.

11 The mathematician Arnold Scholz wrote to Olga Taussky in a postcard Kiel, 4 July 1935: ‘Dear Oli! Thank you very much for your last two cards from Bryn Mawr and Oxford! I hope you are feeling quite well in England! The last time Foradori from Innsbruck, now Berlin, was here, whom you perhaps still know slightly, a very nice, modest person. Miss Heesch also attended his lecture. She and her brother, who live currently in Kiel, have nothing to live on.' Quoted from Lemmermeyer and Roquette 2016, 425). The translation is my own.

12 Steinitz’ Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie, which Elli Heesch attended in the winter semesters 1921/22 and 1923/24, were published posthumously in 1934, as volume XLI of the Grundlehren der mathematischen Wissenschaften, edited by Hans Rademacher (Steinitz 1934). Steinitz showed ‘how one can formulate the criteria that are necessary and sufficient for the existence of a convex geometric polyhedron that is combinatorially given, and established that all such convex realizations are determined up to isomorphism of convex polyhedral’ (Grünbaum 2007, 446). The characterization of combinatorially defined polyhedra, which admit a realization as convex polyhedra in space, is presented today as one of the main results of polyhedron theory and was known to Elli Heesch from Steinitz’s lectures she attended. Steinitz’s fundamental work on group theory, especially on polyhedra, had a lasting influence on Heinrich Heesch’s research, too. During his time as Hermann Weyl’s assistant in Göttingen, Heinrich Heesch turned his attention to tiling problems.

13 An obituary appeared in the Hamburger Abendblatt, 21 September 1993 (Staatsarchiv Hamburg, Zeitungsausschnitt-Sammlung A 758).

14 Originally published as ‘Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris 1900,‘ Gött. Nachr. 1900, pp. 253–297, Vandenhoeck & Ruprecht, Göttingen. Translated for Bulletin of the American Mathematical Society 8 (July 1902), pp. 437–479, with the author’s permission, by Dr. Mary Winston Newson, 1902.

15 BArch R 3/653. German Federal Archives: Bundesarchiv. Bestandssignatur: R 3 (Reichsministerium für Rüstung und Kriegsproduktion), Bestandsart/-typ : Schriftgut, Staatliche Unterlagen (1936–1946). https://invenio.bundesarchiv.de/invenio/direktlink/d0623c75-6b94-4b5c-8865-e95399671b80/ (accessed 27 April 2023).

16 The coding scheme also includes subscripts, as in the rather complicated type CG1CG2G1G2. However, it is outside the scope of this paper to explain these in detail.

17 The term ‘Matilda effect’ was coined in 1993 by science historian Margaret W. Rossiter in honour of the suffragist Matilda Joslyn Gage (1826–1898), who fought for women’s rights and for the recognition of women’s scientific achievements (Rossiter 1993).