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homepageIn this talk, I will be examining a few public statements made during and after World War I in which the nature of mathematics is put in relation to the role it played in the war effort. The actors' characterization of a French 'style' as both formal yet not overly abstract and useful for applications yet not geared towards them will be analyzed. A few suggestions regarding some further developments of mathematics in the interwar period will be presented. I will bring some excerpts to discuss.
will follow soon
My presentation will focus on two articles by Richard Courant.
The first one "Bernhard Riemann und die Mathematik der letzten hundert
Jahre" was published in the journal "Die Naturwissenschaften" in
September 1926, the second "Über die allgemeine Bedeutung des
mathematischen Denkens" was published in the same journal in February
1928.
In these articles Courant reflects on the peculiar position of
mathematics between the natural sciences and the humanities. He
describes how "conceptual diffusion" and "expansion of mathematical
contents", the two main currents he sees at work within mathematical
research, influence the development of the discipline. Courant also
tries to find patterns for the future progress of mathematics by
analysing the "major lines of its historical becoming".
Courant's considerations (may) lead to the following questions:
Keeping Courant's understanding of mathematics in mind, what are the
consequences for his attitude toward applications? Can a position such
as Courant's fruitfully be interpreted in terms of a
modern/counter-modern distinction? My answer to this second question
will be sceptical.
This study tackles the various dimensions of a complex phenomenon of universalization in the history of mathematics. Our methodology resorts to a statistical survey in order to bring out a corpus and a historical periodization (1920-1938) along the line of ?scale games? between the local and the global, the short-term and the long-term (1870-1939). The international ?theory of matrices? of the 1930?s highlights the interrelations and the structuring effects of local practices on the production of global ones. Because such practices were peculiar to individuals or networks and because they were resorting to some tacit knowledge, local ways of thinking, disciplinary ideals and internal philosophies, they point to some cultural and social issues related to disciplines and communities formations, evolutions and connections. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the "modernity" of linear algebra without focusing on issues related to the origins of theories or structures. The adoption of a universal terminology of ?matrices? sheds some light on the interrelation of the ?abstract? nature attributed to matrices in the theory of bilinear forms (most prominent at the beginning of the 20th century) and the various operatory processes resorting to pictorial representations which were often developed for some issues related to "applications". This talk especially raises some issues related to the collective phenonemons of appropriations and circulations of texts (especially between France and the US at the beginning of the 20th century) and point out the role of some pedagogical values of ?simplicity? and ?generality? taken up by new communities of teachers-cum-researchers in the constitution of unifying algebraic theories.
Mathematicians and historians of mathematics will mostly agree
in
acknowledging the period roughly delimited by 1890 and 1930 as a
special
period of deep change in the discipline in all respects: new
methodologies arose, new mathematical entities were investigated and
concomitantly new sub-disciplines arose, the relation between
mathematics and its neighboring disciplines was transformed, the
internal organization into sub-disciplines was completely reshaped,
important areas of research in the previous century receded into the
background or were essentially forgotten, new philosophical conceptions
were either implicitly espoused or explicitly discussed, etc.
"Modernism" is a concept that has helped understand many important, and
equally thoroughgoing processes that took place at roughly the same
time
in many other areas of Western culture, such as the art, music,
literature, etc. Is this concept equally useful in understanding this
important period in the history of mathematics?
In my talk I would like to discuss this latter question, while
indicating some possible limitations and expected problems inherent in
the idea pf "Modernist mathematics". At the same it I would like
to suggest some possibly illuminating ways in which the idea of
“Modernism” could be of help when studying the
history of mathematics .
In previous discussions of modernity or modernism in mathematics it has become almost commonplace to identify the 'autonomous', pure mathematics of the early 20th century with a 'modern(ist)' trend while the 'heteronomous', applied mathematics of the period is counted as 'traditional' (or even counter-modern). One of the texts which appears to serve as a basis for such an interpretation is Felix Hausdorff's fragment "Der Formalismus" (undated, ca. 1903-1904). The text introduces the distinction between 'autonomous' and 'heteronomous' mathematics and couples it with the distinction between 'pure' and 'applied' mathematics. The talk will discuss this text in some detail, with the intention of showing that even such a text does not rule out the possibility of a non-traditional, perhaps even 'modernist' understanding of applications in mathematics.
After a long period (20 years!) during which Dedekind’s innovative theory of ideals found no continuators, the years 1894 and 1895 saw a series of interrelated papers on the topic published in the Nachrichten der königliche Gesellschaft der Wissenschaften zu Göttingen by Hilbert, Hurwitz, and Dedekind himself in direct response to them. Polemics was served, for the young friends Hilbert and Hurwitz were favouring an algebraic approach that combined Kroneckerian elements with Dedekind’s basic notions of field and ideal, and this happened to contradict key ideas of Dedekind’s “arithmetisation” of algebra. The discussion among these authors allows for a very rich methodological analysis, and one can see their decisions as strategic moves in the attempt to delineate the architectural plan of development for modern algebra and number theory. The debate in the Göttingen Nachrichten was framed by the publication of Dedekind’s third and last version of ideal theory (Vorlesungen, 189 4) and Hilbert’s famous Zahlbericht (1897), which was highly successful in disseminating the new number-theoretic views. By way of appetizer, I offer here a sentence in which Hilbert reacts to Dedekind’s publication: “wie in Vielem so hat die neue Ausgabe von Dedekind mich auch darin enttäuscht, dass die Theorie der quadratischen Körper sich im ganzen Buch zerissen und zerstreut, ohne innere und äussere Einheit, und ohne den modernen Standpunkt der Idealtheorie behandelt findet” (Hilbert to Hurwitz, 17.III.1894). Our main question will be to understand what the “modernen Standpunkt” was or meant for the actors. (Above and beyond this, consideration of the views of key “modern” algebraists and number theorists in the 1930s makes it possible to deepen the analysis, but for reasons of time and focus we shall not enter into that in the present talk.)
Primary References: mainly four papers published in the
„Nachrichten der königl. Gesellschaft der
Wissenschaften zu Göttingen“, years 1894 &
1895:
- Hilbert 1894. Grundzüge einer Theorie des galoisschen
Zahlkörpers
- Dedekind 1894. Zur Theorie der Ideale
- Hurwitz 1894. Ueber die Theorie der Ideale
- Dedekind 1895. Ueber die Begründung der Idealtheorie
Studying the French Mathematical Society (SMF) from its creation in
1872 till the first world war, I had noticed that a significant part of
the papers written by its members has been published in popularisation
or educational journals, the two most important journals being the
annual proceedings of the French Association for the Advancement of
Science and the international educational journal, founded in 1899,
l’Enseignement mathématique, the foreign journal where the
members of the SMF published most frequently.
These decades were those of the beginnings of the French Third Republic
which promoted Science, scientific and technical progress linked to the
industrial development, one of its most crucial values. In this
context, the issues of teaching and popularisation of mathematics took
a particular importance and many mathematicians were engaged in them
arguing on modernity, applications and mathematics. Emile Borel, one of
the two most prominent “French modernists” quoting J. Gray,
has been one of the most prominent advocates, in the name of modernity,
of the use of applications and practical exercises in the mathematics
teaching for the elites.
Our reference here will be a pedagogical conference by Borel in
1904. Borel has also been an active actor of popularisation of science,
writing papers, books and creating a journal : I will give an overview
of the way he used his cultural journal, La revue du mois, to promote
applications of mathematics.
Considering Borel’s Notices sur les travaux scientifiques, I will
argue that the importance Borel gave to applications was not confined
to education and popularisation. Even if Borel was
“abstract”, quoting once more J. Gray, he did not
“let applications to others”; he was involved from the very
beginning and his doctoral thesis in 1895, in applications of function
theory in physics, molecular mechanics and statistic mechanics.
Mathematics was not reduced to modern analysis.
Lastly, considering two papers Borel wrote in 1925 and in 1931, after
the 1929 crisis, I will examine how Borel’s arguments on
modernity, pure science and applications took into account the new
ideological, social, economical and political contexts of the twenties.
This presentation is centred on two papers, a discussion, and a review in the history of applied mathematics: (1) Nicolas Rashevsky’s talk “Physico-mathematical aspects of cellular multiplication and development” presented at the Cold Spring Harbor Symposia on quantitative biology in 1934, and the discussion following the talk where a certain tension between Rashevsky and the biologists surfaced. (2) John von Neumann’s paper “Über ein ökonomische Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes” published in 1937 and a review published in The Review of Economic Studies in 1945. Based on analyses of these sources it will be discussed whether (shades of) modernism can be found in those papers and the question, whether modernism can be useful for understanding the migration of mathematics in these fields will be raised.
The ubiquitous modernisation of Scandinavian society in the first decades of the twentieth century relied and depended on technological and theoretical scientific advances and the professionalisation of the engineers. On the Continent, applied mathematics became an important part of the modernisation process, in particular in – but not limited to – the form of so-called “applied mechanics” such as hydro- and aerodynamics. During the first decades of the twentieth century, Scandinavia was in many was a culturally connected region, but the development of applied mathematics seems to have taken quite distinct routes in Denmark, Norway, and Sweden. It is the purpose of this paper to describe and analyse some of these differences, in particular as they are connected to issues of modernisation and internationalisation.
In Scandinavia, the processes of modernising society through applied mathematics were concurrent with efforts to establish Scandinavian research in pure mathematics. Beginning in the 1880s, Scandinavian mathematicians sought to position themselves on the international map of research in pure mathematics with initiatives such as the journal Acta Mathematica, participation in international congresses and the establishing of Scandinavian congresses of mathematics from 1909. As research papers in pure mathematics are often silent on the relation to applied mathematics and society at large, this paper exploits the more informal communications of congresses and celebratory speeches as focal insights which need to be explained against the background of regional developments in mathematics and transformations of society in Scandinavia.
By focusing on the Scandinavian discussions about applications of mathematics surrounding the Abel centennial in 1902, the Scandinavian congresses of mathematicians 1909-1925, and the international congresses of applied mechanics 1922-1930 this paper illustrates and analyses how the Norwegian and Swedish communities of applied mathematics developed much earlier than the Danish one and aims to explain some of the reasons for such differences.
An important source about circulation of scientific ideas can be found in texts written by scientists who were involved in politics. In my talk, I shall present a text by Volterra placed by Borel as opening of his journal "Revue du Mois". This text brings some information both about Volterra's action and Borel's interest towards applied mathematics.
We will present a few examples of conceivable claims to
modernity from
the applied mathematical statistics and probability theory from the
1920s on. The discussion will be based on the idea that modernity
needs an antagonistic other. Furthermore, whenever possible, we want
to test the hypothesis that elements of holistic thinking are an
indicator of a not so modern approach. We will thus propose to discuss
different well-known antagonisms of the time - which have been
studied before.
The overall message will be that matters are complicated - possibly
because we choose to look at complicated cases. One of them will be
Paul Lévy, in particular his article "Théorie des
erreurs. La loi de
Gauss et les lois exceptionnelles", our aim being to place it within
the larger debate around 1920 about versions of the Central Limit
Theorem.
Another example we will look at is Chapter V: Experimental
epidemiology, of Major Greenwood's 1935 book : Epidemics and Crowd-
diseases.
In my mind, the aim of the talk & discussion ought to be to
determine
whether it would make sense to include these texts in the data base of
the project.
In 1948 (occasion unknown) H. Weyl gave a talk on automorphisms ("similarities") in geometry and physics and their relation to congruences understood as material transformations of bodies. The talk seems to address a scientifically educated and philosophically interested audience not specialized in mathematics. It spans a bridge from the Leibniz/Clarke discussion to early gauge theoretic considerations in modern physics, with Klein's Erlanger program (+ a side remark to Helmholtz) and special relativity as two intermediate piers. In the typically Weylian style we do not find separation lines between considerations in ("pure") mathematics (simple and well explained ones only), in theoretical physics or in philosophy (epistemology). So it does not fit the criteria known to me from recent discussions on "modernism" in mathematics. But in my opinion it gives a striking introduction into basic questions of what mathematics and foundations of physics in the 20th century was about (and to a certain degree still is). In this sense, it goes deep into what modern (as a historical attribute -- in the sense of 19/20th century) mathematics and physics is about. A typoscript of the talk is preserved in the ETH manuscript collection (ETH Hs 91a:31). Because of the archive's restriction, no scan is made, but a Xerox will be available at the workshop.
In the first volume of his new journal ZAMM von Mises published his programmatic paper “On the tasks and goals of applied mathematics”. Here, von Mises tried to address on a general level several of the most fundamental philosophical and practical problems of the application of mathematics to reality. At the same time von Mises gives a general overview of the state of the art of engineering mathematics, for instance as far as the theory of plasticity and aerodynamics are concerned.
Von Mises makes principal remarks about the delimitation of “applied mathematics” from other parts of mathematics. He talks about its relation to analysis, in particular differential equations, and about the respective role of numerical, graphical and instrumental methods. He picks up the old discussion, dating back to Felix Klein, of an alleged divide between “Approximations- and Präzisionsmathematik.” The importance of rigor even in applied mathematics von Mises explains among other things by the fact that engineers can only afford to spend a small part of their time on mathematics and are therefore in need of the idealizations and simplifications which come with the introduction of rigor.
Von Mises’ decision to prioritize engineering mathematics in the new journal was partly based on a pragmatic point of view, namely the recognition that the mathematical problems of technology “were not represented at any other place.” Thus, together with mathematical statistics, nearly all major applications of the theory of probability were missing from von Mises’ programmatic paper. For the same partly pragmatic reason even physics, the traditional field of application of mathematics, was not prioritized in ZAMM. The sidestepping of probability and mathematical physics made ZAMM from the outset a venue of a somewhat different strand of the modernization of mathematics and science, compared to the notion of modernity usually conveyed in historical accounts.
This, however, is only half of the picture when looking on von Mises himself. His parallel work in probability and mathematical physics and another paper by him of the same year 1921 (also originally published in ZAMM), entitled “On the present crisis of mechanics” revealed that to von Mises the notion of modernity in science was very much based on the interaction and interpenetration between classical, rational mechanics and contemporary fields of pure mathematics, then exemplified by tentative approaches to the difficult and even today (2009) problematic notion of turbulence. Von Mises proposed the abandonment of the “naive concept of causality” and to extend it according to the needs of physics. Von Mises was leaning - already in classical statistical mechanics - toward an “epistemological” indeterminism (which in his case was very much connected to the use of mathematics as an “idealizing” method) as opposed to an “ontological” indeterminism. To quote, as P. Forman does, von Mises as one of the major examples for the alleged “adaptation by German physicists and mathematicians to a hostile intellectual environment,” in particular towards acausality in physics, is neither compatible with von Mises’ adherence to the philosopher Ernst Mach nor with his perspective on the peculiar role of mathematics, in particular probability, within physics.
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