The Problem of Parallels As a Protestant Issue in Late Eighteenth-Century Hungary

chapter 8 The Problem of Parallels as a Protestant Issue in Late Eighteenth-Century Hungary

János Tanács

Summary

The paper presents a mathematical problem that intersects with intertwined religious and political values and conflicts.* The problem of parallel lines, which led to the creation and formation of non-Euclidean geometries, is shown to suit the denomi- national-political expectations and values of the Protestant intellectual milieu at the end of the eighteenth century, while there was no place for the mathematical problem in the Catholic parts of the Hungarian Kingdom. Complex and conflicting social values played a formative role, determining indirectly but decisively which was the receptive environment for this major mathematical breakthrough. This paper suggests that the two value-systems set the boundaries for both the positive and the negative reception of the problem at that time. The disputed question of parallels conforms to complex social expectations and values, and these determined, indirectly but forma- tively, what was or was not a relevant and significant mathematical problem. Contrary to the received view, there was no general interest in the problem as such in the sec- ond half of the eighteenth century, nor did even the leading mathematical school of the period appreciate it. Only a specific segment of the striving Protestant intellectual milieu, mostly German-speaking mathematicians in a marginal position, considered it relevant. Consequently, this paper offers a radically new narrative, suggesting that the Problem of Parallels was not merely a mathematical question philosophically motivated by Kant’s ideas, but that it was deeply embedded in the German-speaking Protestant intellectual milieu.

* This work was supported by OTKA Grant K72598 and by TAMOP-4.2.2/B-10/1–2010–0009. I thank Tamás Demeter, Gábor Kutrovátz, Benedek Láng, Márta Fehér and Tihamér Margitay for fruitful discussions, and especially Gábor Zemplén for numerous comments on earlier drafts. I would also thank Karl Hall for proof-reading my paper as well as for his useful recommendations.

Introduction

In this paper, I examine indirect but determining religious, confessional, political, philosophical, and educational factors which played a role in mathematicians’ engagement with the Problem of Parallels at the end of the eighteenth century in Hungary. These will show the hidden but formative effect of intertwined religious and political values, which determined the relevance and the importance of the problem for János Bolyai, son of Farkas, whose contribution is central in the creation of non-Euclidean geometry. Farkas Bolyai was already engaged with the question, so I begin by discussing the factors which played a role in the family predilection for parallel lines.1 The long-lasting historical process which eventually resulted in the creation and formation of non-Euclidean geometry in the 1830s is usually referred to as mathematical work on the Problem of Parallels (or the Theory of Parallels). In this process, some attempts focused on the conceptual side of the question, trying to clarify the conception of parallelism: i.e. how the co-planar non- secant straight lines would continue if they were extended infinitely. Others concentrated on the demonstrational side of the difficulty. They aimed to clarify the exact status of the so-called Parallel Postulate of Euclid.2 This latter question focused on the identification of whether the Parallel Postulate (the Fifth Postulate) was an axiom similar to Euclid’s other axioms or if it was a theo- rem which could be deduced from the other axioms and postulates (called the residual system of axioms). In some cases, attempts were made to replace the Parallel Postulate by a simpler and more plausible mathematical proposi- tion than the original. In these cases, the simplicity and the plausibility of the proposed substitution, compared to the original, was disputed and finally cast in doubt. The problem was finally resolved by János Bolyai, and independently by Nikolai Ivanovich Lobachevskii. As the Problem of Parallels acquired relevance and significance for János Bolyai through his father Farkas, it is obviously interesting to look at the factors that played a role in Farkas Bolyai’s engagement with the question. The literature on the history of the Problem of Parallels supports the view that there was consistent general interest in the problem until the discovery of the Bolyai-Lobachevskii non-Euclidean geometries, and that Immanuel Kant’s

1 Boyer C.A., A History of Mathematics (New York: 1968) 585–588; Gray J., János Bolyai, Non- Euclidean Geometry, and the Nature of Space (Cambridge, MA: 2004) 43–49. 2 Gray, János Bolyai, Non-Euclidean Geometry, and the Nature of Space 17–32; Eves H., Foundations and Fundamental Concepts of Mathematics (Boston: 1997) 51–60; Kárteszi F., János Bolyai. Appendix: The Theory of Space (Budapest: 1987) 14–18.