TEORÍAS SOBRE EL IMPACTO DE LA EDUCACIÓN EN LA SOCIEDAD.

During the 19th _{century, science became increasingly important and popular.}

The industrial revolution, and towards the end of the century inventions such as steam power and telegraphs, raised society’s awareness of science and technology; and, as economic prosperity increased, higher education became more important. However, teaching was no longer considered the main activity of a university professor; research started to play an equally crucial part in the job description. This resulted in an ever-growing number of scientists slowly starting to collaborate internationally [55, p. 1]. In mathematics, this happened rather late in comparison to sciences such as astronomy, geology, or cartography.

One of the results of this new significance of science was the foundation of scientific societies; the first mathematical society was founded in Moscow in 1864. Other societies followed in most European countries and in North America. As more and more research was done, the number of mathematical journals and books published each year increased ‘at a rapid pace’ [55, p. 2]. One of these journals was the Jahrbuch über die Fortschritte der Mathematik1_{, first}

published in 1871. The Jahrbuch was the first of many bibliographical catalogues that provided mathematicians across the world with an overview of current research and developments in their respective fields. Soon it ‘became indispensable for mathematical research’ [ibid.].

However, all the editors and the reviewers were German. The editors appealed for international cooperation and indeed, from the second volume on, some of the reviewers were non-German. The number of different countries represented increased with each new volume. Interestingly, none of the reviewers were French. The French first published Répértoire bibliographique des sciences mathématiques in 1885. It was a catalogue of all mathematical publications, divided into various sections and subsections. The French editors were soon joined by international colleagues, too.

For quite a while, French and German mathematicians did not have any means of communicating mathematical ideas apart from exchanging letters directly. The Swedish mathematician Gösta Mittag-Leffler published papers by both French and German mathematicians in his journal Acta Mathematica (founded in 1882), thus providing ‘a privileged place for communication between German and French researchers where the patriotic sensibilities of the various protagonists would not be offended’ [28].

Comparatively late, in 1894, a joint international bibliographical project was launched, the Enzyklopädie der mathematischen Wissenschaften2_{, with the first}

issue being published in November 1898. Though of German origin, the emphasis of the project was on German-French cooperation. In fact, German and French mathematicians worked together to produce a French edition of                                                                                                                

1

“Yearbook on the Progress of Mathematics”

2

the encyclopaedia, which was ‘not merely a translation, but an adaptation’ as Dyck puts it in the preface to the first volume [60, p. XVIII]. The initiators of the project were Felix Klein in Göttingen, Heinrich Weber in Strasbourg and Franz Meyer, at the time professor at the Mining Academy in Clausthal. In contrast to the earlier bibliographical publications, the encyclopaedia included papers in a range of mathematical fields and also catered for physics, mechanics, and astronomy.

Unsurprisingly, the relations between France and Germany had greatly suffered due to the Franco-Prussian War (1870-1871). The French saw the reason for Prussia’s victory in its scientific superiority and therefore wanted to catch up with the scientific developments in Germany. Germany on the other hand was a young, strong empire with all the German states united, and it had imperialistic aspirations, wanting “a place in the sun”. A lot of effort was put into expanding and improving the Empire’s naval fleet, which was paid for with French reparations. Science was considered to be key to military development and as a result, the German government supported scientific research.

Whilst scientific progress was furthered in the name of patriotism in France and Germany, the respective governments did little to support international cooperation. In fact, political tension grew across Europe, yet the end of the nineteenth century also saw a drastic increase in scientific exchange and cooperation. A lot of this change was brought about by individuals or scientific societies rather than by governmental bodies.

In mathematics, most of the early international collaborations concerned bibliographical projects. Georg Cantor in Halle was one of the first to express the necessity of international collaboration beyond the bibliographical level. He was a fervent advocate of the idea of a mathematical society in Germany and proposed in 1888 that ‘German and French mathematicians should meet at a neutral site’ [55, p. 3], e.g. in Belgium, Switzerland or the Netherlands. Leaving international cooperation aside for a moment and looking at the state of mathematical cooperation in Germany itself, it is clear that Cantor’s

ideas were in accordance with the spirit of the time. Until the 1890s, there were hardly any opportunities for German-speaking mathematicians to cultivate friendships amongst each other. A few meetings of societies such as the Gesellschaft deutscher Naturforscher und Ärzte3 _{included mathematical}

sections where they could present their work. However, in Jena in 1890, a number of mathematics and science teachers founded the Verein zur Förderung des Unterrichts in der Mathematik und in den Naturwissenschaften4_{. This had}

become necessary because of attempts to reform higher education at the time. Furthermore, mathematics and science teachers wanted to stand their ground against the interests of the arts teachers [10, p. 257].

In the same year the German Mathematical Society was founded and Cantor became its first president. At the time he already had the idea of an international congress of mathematicians. At first, he was not taken very seriously by some of his colleagues, as is shown by a letter that Walther von Dyck wrote to Felix Klein in August 1890:

Recently G. Cantor wrote me about very high-flying plans regarding international congresses of mathematicians. I really do not know whether that is a real need.

[55, p. 3]

From 1894-1896, Cantor was in correspondence on the subject of international congresses with a number of mathematicians, including Aleksander Vasilyev, Charles Hermite, Camille Jordan, Henri Poincaré, Charles-Ange Laisant, Émile Lemoine, Klein and von Dyck. Cantor argued that a congress would serve as a much-needed international forum where the ever-growing mathematical community could present and discuss their work without prejudice. He himself needed such a forum to present his work, as not all his German colleagues approved of his new and radical ideas in set theory. The fact that he began to stress his non-German origin – his father came from

                                                                                                                3

“Society for German Natural Scientists and Physicians”

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Denmark, and Cantor himself was born in St Petersburg – made him fall out of favour with the German mathematicians even more.

One of the German mathematicians who did not see eye to eye with Cantor was Klein. However, he recognised the need for international cooperation when he attended the Congress of Mathematicians and Astronomers in Chicago in August 1893. It was one of the satellite conferences held on the occasion of the Chicago World’s Columbian Exposition, organised in order to celebrate the 400th

anniversary of Columbus’s discovery of America. The mathematical congress had 45 participants, four of whom came from countries other than the United States. These four international mathematicians were all Europeans. In fact, the centres of mathematics were all European at this time, ‘yet a mathematical conference as early as 1893 with participants from two continents was a historical event’ [55, p. 5].

Klein went to Chicago in his capacity as Imperial Commissioner of the German Emperor Wilhelm II. He took with him papers of several of his colleagues and also gave an opening address, The Present State of Mathematics. In his speech he pointed out the threat to mathematics of being split into different branches, the necessity of international collaboration and the benefits that mathematical societies brought to mathematics. He said that mathematicians ‘must form international unions, and I trust that this present World Congress […] will be a step in that direction’ [49, p. 135].

Klein and Heinrich Weber became the leading figures in organising an international congress on the German side. They got much more support from their peers than Cantor had received a few years previously, as German mathematicians expressed the wish for an international congress of mathematicians to be organised, particularly ‘in view of the successes achieved by international communication in other areas of science’ [10, p. 258]. However, nothing was done about organising such a congress: In 1895, the German Mathematical Society claimed to support the idea of an international congress in principle after French mathematicians had presented the idea to their German colleagues at the society’s annual meeting the year before, but

they refused to organise it [55, p. 7]. As for Cantor, he eventually abandoned the project, probably due to the fact that his ideas had met with so much resistance. He did attend the first congress though.

Two of Cantor’s most supportive correspondents on the subject of international congresses were the French mathematicians Charles-Ange Laisant and Émile Lemoine. They presented the idea of an international congress in the first volume of their journal L’Intermédiaire des mathématiciens and explained that it came from both French and foreign mathematicians. Besides Laisant and Lemoine, Cantor could also claim Poincaré’s support [28]. The idea that an international congress should be organised began to spread across Europe and beyond from 1894 onwards. The French and the American mathematical societies backed the idea of an international congress, but neither offered to organise it. It was agreed, however, that the congress should be permanent, be held at regular intervals of three to five years and follow a number of rules that were to be established. The French Mathematical Society at least declared that it would support a trial congress.