Otto Toeplitz was born in Breslau, Germany (now Wrocław, Poland) in 1881 and died in Jerusalem in 1940. He received his doctoral degree from the University of Breslau in 1905 under Jacob Rosanes and Friedrich Otto Rudolf Sturm with a dissertation entitled Über Systeme von Formen, deren Funktionaldeterminante identisch verschwindet (On Systems of Forms whose Functional Determinant Vanishes Identically). The following year, Toeplitz left Breslau and went to Göttingen. In his tribute to Toeplitz, Heinrich Behnke describes that move as one from “a quiet provincial town to a gleaming metropolis” (Behnke Reference Behnke1963, 2), – an apt expression, for Göttingen at that time was blessed with the luminary presence of Felix Klein, David Hilbert, and Hermann Minkowski. And it was not long, Behnke tells us, before Toeplitz was included among the students of Hilbert's inner circle. Toeplitz's doctoral dissertation had already touched on topics related to systems of bilinear and quadratic forms, but with Hilbert Toeplitz's interests in this direction crystallized and became the work on infinite linear, bilinear, and quadratic forms and infinite matrices for which he is known. His research in these areas and, generally, on linear operators on infinite dimensional spaces, continued after he left Göttingen and became a professor at the University of Kiel in 1913 and of Bonn in 1928.

But Toeplitz's mathematical accomplishments are not our main concern here. What interests us is his involvement in the history of mathematics. Still, it is important to know that he was a mathematician of the first rank, for his identity as a proponent of the history of mathematics was bound together with his identity as a mathematician (and, as we shall see, as a teacher of mathematics). Abraham Robinson underlines this fact in his biography of Toeplitz for the Dictionary of Scientific Biography (DSB) (Robinson Reference Robinson and Gillispie1970, 428) in recalling that Toeplitz “held that only a mathematician of stature is qualified to be a historian of mathematics.”Footnote ¹ That said, it cannot be claimed that his work in the history of mathematics as such was remarkable, at least relative to his deep mathematical work; however, he had, by all accounts, a profound interest in the history of mathematics, and he made efforts to promote its study. Together with the historian of ancient science and former assistant of Richard Courant, Otto Neugebauer, and the Plato specialist Julius Stenzel, Toeplitz established Quellen und Studien zur Geschichte der Mathematik (Sources and Studies in the History of Mathematics) in 1929, and it was Toeplitz who first received for the journal what later became Jacob Klein's famous Greek Mathematical Thought and Origins of Algebra.Footnote ² Toeplitz's enthusiasm for those papers mirrored his particular interest in the relationship between Greek thought and Greek mathematics: “He was a classical scholar able to read Greek texts and he knew his Plato just as well as his Gauss and Weierstrass” (Born Reference Born1940, 617). The breadth of Toeplitz’ scientific interests is also reflected in the good relations he cultivated with the philosopher and psychiatrist Karl Jaspers. Indeed, Jaspers dedicated his 1923 book Die Idee der Universität (The Idea of the University) to Toeplitz.Footnote ³

That a book about education was dedicated to Toeplitz is worth noting in connection to our own story. For where Toeplitz's interest in the history of mathematics was decisive was in his role as an educator. Toeplitz's father was a school teacher, and teaching was of great importance to Toeplitz. It is not by chance then that Behnke entitled his tribute to Toeplitz mentioned above, “Der Mensch und der Lehrer” (The Man and Teacher). In the early 1930s Toeplitz and Behnke initiated yearly “meetings for cultivating the relations between university and high school” (“Tagungen zur Pflege des Zusammenhangs von Universität und höherer Schule”) (Schubring Reference Schubring2008; Hartmann Reference Hartmann2009, 186–195) and they jointly founded the education oriented journal Mathematische und physikalische Semesterberichte (Semester Reports on Mathematics and Physics) in 1932 (Hartmann Reference Hartmann2009, 199–209). In nearly every issue of this small journal Toeplitz published articles on mathematics education. A study of these interesting papers is still lacking and would be a rewarding task.Footnote ⁴

These three streams in Toeplitz's intellectual life – mathematics, history of mathematics, and teaching – came together in what he called the “genetic method” for teaching. Toeplitz's lectures on the calculus using the “genetic method” were edited by his former student Gottfried Köthe and published posthumously in 1949 as Die Entwicklung der Infinitesimalrechnung: Eine Einleitung in die Infinitesimalrechnung nach der genetischen Methode, vol.1 (another volume was intended) and later republished in English as The Calculus: A Genetic Approach in 1963 (Toeplitz Reference Toeplitz and Köthe1949 and Reference Toeplitz1963). Although the book was never published in Toeplitz's lifetime, the idea of the genetic method, its motivation, and, essentially, the plan of the future book were set out in the paper by Toeplitz that we have translated for this issue of Science in Context.

The paper, “Das Problem der Universitätsvorlesungen über Infinitesimalrechnung und ihrer Abgrenzung gegenüber der Infinitesimalrechnung an den höheren Schulen” (Toeplitz Reference Toeplitz1927), was the published version of an address Toeplitz delivered at a meeting of the German Mathematical Society held in Düsseldorf in 1926. The primacy of Toeplitz's educational focus is immediately evident in his title, “The Problem of University Courses on Infinitesimal Calculus and Their Demarcation from Infinitesimal Calculus in High Schools”: no mention whatsoever of history is to be found, nor of the genetic approach which is based on history and which is the centerpiece of the address. Toeplitz wants to be understood as solving a specifically educational problem, namely, that of finding how to introduce the calculus to beginning university students.

The difficulty, as he saw it, was defined by three basic dilemmas, or “moments.” The first concerns two existing schools of thought as to what the guiding principle should be for such a course. One school of thought maintains that beginning university students should be given an introduction to the calculus that is exact and rigorous, the other, that the course should be intuitive and approachable. Neither “path” (Richtung) alone is completely satisfying, but nor is a hybrid that takes a little from each; they are, Toeplitz said, unbridgeable (unüberbrückbar).

The second moment is closely related to the first. In some ways, it is a version of the first. It is the tension between two aims of an introductory course in calculus, one being that students acquire the necessary tools and concepts to ground further work in mathematics and science, and the other, that students acquire a taste for the subject. The latter is particularly important to Toeplitz. He wants students to appreciate how mathematics can be exciting and beautiful. Yet, particularly in the case of calculus, it is all too easy, he believes, to destroy any pleasure of the subject by a tendency to teach mere rules and formulas. He also stresses in this connection that he wants to reach students who are capable of studying mathematics seriously, but who are not necessarily mathematically minded persons.

This concern for the personalities of the students is at the heart of the third moment. The difficulty here is that there are two different groups of students who are likely to take a first course in calculus. One group comes from the science-oriented Oberrealschule and the other from the humanist Gymnasia, and each is problematic in its own way. One can easily guess the advantages of the Oberrealschule students and the disadvantages of the Gymnasium students, but Toeplitz is also astute enough to recognize that the latter have something to offer and not all is well with the former. As we remarked above, Toeplitz himself had a firm humanist training, so he could well appreciate the value of an education obtained at the Gymnasia. The problem with the Oberrealschule students is precisely what others might see as an advantage, namely, their greater exposure to the technical side of calculus. Toeplitz points out that this too often and too easily prevents them from seeing that there is more to know, that knowing some techniques of calculus is not the same as knowing calculus. This is the greater challenge for Toeplitz.

The third moment refers to students’ previous high school education in a more concrete and pointed way than do the first and second moments. This was not by chance. Toeplitz had very much in mind a set of reforms in German high school education involving the incorporation of the infinitesimal calculus into the mathematical syllabus of the Gymnasia. Bringing the infinitesimal calculus into the Gymnasium was the most important project among Felix Klein's educational initiatives, and in 1925 infinitesimal analysis finally became part of the official syllabus at Prussian Gynmasia.Footnote ⁵ This was just one year before Toeplitz's address, which is why Toeplitz, unable to hide his objections, refers to the teaching of calculus to high school students with a discernable edge, calling it a fait accompli. But so it was, and, therefore, it became part of the overall difficulty in setting up a first year calculus course at the university level.

Toeplitz's suggestion is that all three moments and the difficulties they identify can be addressed if one gives proper attention to the history of mathematics: if one takes the development of mathematical ideas as a guide for teaching, he claims, not only will the drama of that development be revealed to students but so will the logic and interconnection of mathematical ideas themselves. This is the core of what Toeplitz called the “genetic method.” But Toeplitz goes on to say that there are, in practice, two ways in which the “genetic method” can be taken up. These he calls the “direct genetic method” and the “indirect genetic method.”

In the first case, the “direct method,” the historical development of mathematical ideas guides the presentation of ideas in the classroom – it is a way of teaching. It answers the first moment, for example, not by bridging the divide between rigor and intuition but by providing a third alternative: the student arrives at mathematical ideas by following the same slow and gentle ascent by which the ideas themselves were arrived at historically. In other words, he does not want to present rigorous ideas in a softened intuitive fashion, or to use intuitive notions as a springboard for jumping to rigorous formulations, but, rather, to allow the rigorous ideas to unfold for the students according to the very same gradual process that they themselves unfolded and, by doing so, not only bring the rigorous ideas into the classroom but also to show in a natural way – and that is key – why those ideas slowly became clear and necessary.

The “indirect method” is not so much a way of teaching as it is a way to analyze problems of teaching and difficulties in learning: it is, as Toeplitz put it, “the elucidation of didactic difficulties, I should say, didactic diagnosis and therapy on the basis of historical analyses, where these historical analyses serve only to turn one's attention in the right direction.” The qualification at the end is crucial: the application of the “indirect method” does not necessarily mean that history itself is brought into the classroom, and generally it is not. In the guise of the “indirect method,” history takes on a role similar to the psychology of learning: teachers use the psychology of learning to guide their teaching strategies and decision-making, but it is not what they teach their classes. Thus, Toeplitz argued, from the historical claim that knowledge of the definite integral was possessed by the Greeks or, at any rate, predated other ideas from the calculus, the study of the calculus for modern students ought to begin with the definite integral.

Toeplitz's program was far more detailed and systematic than any other comparable effort; however, he stressed that his “genetic method” was not itself original.Footnote ⁶ Felix Klein, he said, adopted a similar principle in his teaching in 1911. In fact, this direction in Klein's thinking was evident more than a decade and a half earlier. In 1895, Klein delivered a talk at a public session of the Königliche Gesellschaft der Wissenschaften (Royal Society of Sciences) in Göttingen on the “Arithmetization of Mathematics” (Klein [1895] 1896). At the end of his talk, Klein made a short remark about the teaching of mathematics. He said that, in his view, a paradoxical situation exists whereby teachers at gymnasia tend to stress Anschauung too much while professors at universities do so too little. The university professors more than underemphasize Anschauung: they put it aside completely whenever possible. Klein argued that at least the elementary courses meant to introduce the beginner to higher mathematics should take Anschauung as a starting point, “since on a small scale the learner always passes naturally through the same development which the science has passed through on a large scale.”Footnote ⁷ Though he did not explicitly say so it was perfectly apparent to Alfred Pringsheim (1850–1941) that Klein was alluding here to the “biogenetic law” of the German zoologist Ernst von Häckel (1834–1919). Thus in a talk at the meeting of the German Mathematical Society in 1897, Pringsheim asked whether it is “suitable to transfer Häckel's principle of the concurrence between phylogenesis and ontogenesis in such an unrestricted way to teaching” (Pringsheim Reference Pringsheim1897, 74) as Klein had done in his talk of 1895. After a lengthy discussion involving mostly non-mathematical examples, Pringsheim concluded that “the principle referred to by Herr Klein as a principle of teaching appears to be anything but conclusive and at least needs an examination from case to case” (ibid., 75). With that, Pringsheim stated his own principle, namely “Every individual passes essentially through the same development as science itself as long as he is not shown a better way” (ibid.). It was to continue these discussions that Klein and Pringsheim were invited to give lectures on the issue of the university courses for beginners at the meeting of the following year, 1898. Significantly, Toeplitz refers precisely to these lectures right at the start of the 1927 paper.Footnote ⁸

Klein himself referred to the biogenetic law as a source of a teaching principle explicitly before 1911. He did so, for example, in the appendix to his Elementary Mathematics from an Advanced Standpoint (Klein [1908] Reference Klein, Hedrick and Noble1939), a book meant for mathematics teachers. Concluding his discussion of set theory, Klein states the law and its implications in a way that shows why Toeplitz should find it so enticing as a principle for his own pedagogical strategy:

The biogenetic law, as Toeplitz surely knew, had a long history.Footnote ⁹ In biology itself and as a law of cultural evolution, it was popularized by Ernst von Haeckel and was generally identified with him, as can be seen in the quotation above from Pringsheim's 1897 talk. It is fairly clear, moreover, that it was Haeckel's formulation of the rule, “ontogeny recapitulates phylogeny,” that the development of the individual organism follows the development of the species, which Klein and Toeplitz referred to as the biogenetic law. Haeckel of course did not invent the law; complete and consistent statements of it can be found early in the nineteenth century (see Gould Reference Gould1977 and Mayr Reference Mayr1994 for critical accounts). And there were variations of the law, ranging from more respectable forms in which the individual development merely paralleled species development to less respectable forms in which the latter actually caused the former. But what is centrally important is that the biogenetic law was, even when rejected, viewed as a scientific matter and a serious scientific hypothesis.

The educational implication of the “scientific” biogenetic law, namely, that children should be educated according to the principle that their own intellectual development follows the development of civilization, was grasped by educational thinkers almost from the moment the law began to make its appearance. Thus, for example, in Froebel's 1826 The Education of Man we read, “Every human being who is attentive to his own development may thus recognize and study in himself the history of the development of the race to the point it may have reached, or to any fixed point” (Froebel Reference Froebel and Hailmann2005, 40). This view was held also by the followers of the influential Friedrich Herbart (1776–1841) such as Tuiskon Ziller (1817–1883) who called it the Kulturstufentheorie, or the cultural epoch theory (see Gould Reference Gould1977, 149ff). More importantly for us, Florian Cajori (1859–1930), the Swiss-American educator and historian of mathematics, referred again to the law as it comes down through educational thinkers in his History of Elementary Mathematics with Hints on Methods of Teaching (Cajori Reference Cajori1896). Cajori opened the preface of that work with a quotation from Herbert Spencer in which the genetic principle is stated:

Cajori then used this to justify his own use of history of mathematics for mathematics teaching:

Curiously enough, Toeplitz never referred to the “cultural epoch theory” or any other of these educational versions of the “biogenetic law.” One might speculate that, unlike the “biogenetic law,” Toeplitz might have found these other theories were somehow unscientific. Whether or not Toeplitz accepted the “biogenetic law” in the literal way Haeckel framed it, the scientific status of the “biogenetic law” would certainly provide his own genetic approach with a firm basis as Felix Klein seemed to think regarding his own pedagogic method – for Klein, recall, referred to his approach, in the passage quoted above, “a natural and truly scientific method of instruction.” This “scientific” or “natural” rationale for the genetic approach, this historical oriented method of teaching mathematics, would also imply a view of history itself as something natural, like biological evolution. Seeing history in this way would, implicitly, allow him to approach turns in history as developments that could be rationally reconstructed on solid ground without resorting to a kind of logical axiomatic structure.

Yet, what is fascinating about Toeplitz's text is the tension between this implicit view of history and an explicit denial that what he is doing has anything to do with history. He says that a historian must write down everything that happens, good or bad, while he is interested in only what has successfully entered into mathematics. He tells us that his is not a course in history: “Nothing could be further from me than to lecture about the history of infinitesimal calculus: I myself ran away from such a course when I was a student. It is not about history, but about the genesis of problems, facts, and proofs, about the decisive turning points within that genesis” (Toeplitz Reference Toeplitz1927, 94; emphasis in the original).

On the other hand, the very reason why taking that genesis into account should help students is that it is natural, fitting to the students’ own ways of learning. Is it not for this reason that Toeplitz used the medical language of “diagnosis and therapy” in describing how historical analysis is supposed to benefit teaching? But this is only possible if historical ideas are themselves somehow natural, just as diagnosis and therapy presuppose certain biological facts. The distinction Toeplitz makes between history of mathematics and the genesis of mathematical ideas and techniques is ultimately, therefore, an artificial one. Indeed, Toeplitz did not hesitate to make historical pronouncements as if they were indisputable facts – that “The Greeks discovered the definite integral” (ibid., 96), that “the Dedekind cut is essentially in Euclid's fifth book” (ibid., 97), that “Barrow possessed differential and integral calculus in its entirety” (ibid., 98).

More significantly, Toeplitz makes what are really historical claims on the basis of his genetic approach, even while he denies it. Thus, having argued that the relationship between the definite integral on the one side and the differential calculus and indefinite integral on the other is what needs to be highlighted in the calculus course, he says, “You see clearly here the difference between genesis and history. Historians place the bitter priority quarrel between Newton and Leibniz in the foreground of the historical development of the differential calculus; from the genetic perspective, completely different moments form the central focus” (ibid., 97). But since these judgments of what should be in the foreground and what should not are based on history via the “indirect genetic approach,” they must have some status as history. One cannot just dismiss the issue by saying one judgment is for history and the other for education: parallelism is a symmetric relation.

By considering what he takes to be purely educational questions, Toeplitz finds himself, then, unavoidably even if unwittingly, adopting historical and historiographical positions – positions that, reflexively, also are perspectives on the historical character of mathematics itself. How such tensions and assumptions about mathematics, history of mathematics, and historiography are revealed in the alembic of educational thinking is what makes Toeplitz's text interesting in the study of historiography of mathematics. In general, the ways historiography of mathematics and teaching of mathematics, even without an immediate concern for history, may be deeply entangled should, in our view, be given much greater attention both in historical and educational research. For the latter, the matter is particularly important since the introduction of history of mathematics into mathematics teaching is taken up all too often in a purely instrumental fashion with little cognizance given to what it means to look at mathematics historically in the first place:Footnote ¹⁰ typically, it is not asked, for example, whether the ends mathematics education aims towards are necessarily in harmony with those pursued by the history of mathematics. Of course if the genetic principle, as Toeplitz understood it, were unproblematic then such questions would lose their force; but, if not – and if one is guided by the needs of teaching modern mathematics – then one would have to confront the difficulties of anachronism and its inevitable distortions of history. On the other side of the coin, the assumptions one makes concerning the efficacy of a historical approach to mathematics teaching are assumptions about what one actually learns from history: the educational question, therefore, becomes a question of historical knowledge.

In producing the translation of Toeplitz's 1926 address, we have tried to stay as close as possible to the German text, and we included the original page numbers in the 1927 published version. This was so that the reader may use our translation as a guide to the German text, rather than a stand-alone English version of Toeplitz's work. The disadvantage of this of course is that the English is sometimes slightly stilted. But here we must also point out that Toeplitz's own German style in this formal address was itself academic and not always as smooth and clear as it could be. We believe the reader will get a feeling not only of what Toeplitz was saying but how he said it.