Ladies and Gentlemen!

When the Association of German Scientists and Physicians last met in Düsseldorf exactly twenty-eight years ago on September 24, a debate took place following lectures by Felix Klein and Alfred Pringsheim on roughly the same topic to which I would like to direct your attention today. The printed report of the Düsseldorf debate only remarked that, “It is not possible to go into details here,” so one can only guess how two of the most powerful teacher personalities among German mathematicians of that time had confronted one another with their diametrically opposed views on this topic and how they did so with their characteristically lively spirit.Footnote ²

The problems touched upon then are equally relevant today. A lifetime has passed; one generation of mathematicians who viewed set theory as an aberrant oddity has been replaced by another fed on set theory with their mothers’ milk; but when it comes to introductory courses on infinitesimal calculus the same polar opposition between the two standpoints exists now as it did then. The actual state of these courses at German universities today still shows the same colorful diversity of approaches. At one extreme, there is the strict observance that begins with a six-week “Dedekind treatment” (Dedekindkur)Footnote ³ and then goes on to derive concrete rules (p. 89) of differentiation and integration from general concepts of number and function as if they were necessary and natural consequences. At the other extreme, there is the intuitive path in which the magic of differentials is made to rule, and in which even by the last hour of a two semester course the fog which arises from indivisibles remains undispelled by the sunshine of a clear limit concept.Footnote ⁴ And between these two orthogonal directions of thought may be fitted diagonals of hundreds of shades.

It is precisely this variety of modes with which one tries to deal with the problem of teaching calculus that situates the magnitude and difficulty of the problem in the brightest light. The difficulty is felt by all of us, I believe, whether we take a particular interest in this kind of course, or whether, through an unconscious sense of how tricky the problem is, we entrust it to younger colleagues, in accord with the maxim, “Beginners’ courses for beginners.” It is solely out of the consciousness of this problem's difficulty that – just as in that debate twenty-eight years ago – I confine myself to the university course alone and set aside the problems of the infinitesimal calculus at institutes of technology (Technische Hochschulen), which are quite as complex, but also completely different.

At the foundation of our problem's difficulty there are, as I see it, three moments. The first moment is the opposition already mentioned. No compromise can resolve this opposition; there is an unbridgeable gulf in the very nature of the matter. The path of exactitude (die exakte Richtung) seeks to establish, from the first beginnings of mathematical studies, the same degree of rigor which, since Weierstrass, is no longer the exclusive province of leading mathematicians but belongs to the common sense of mathematics (allgemeinen guten mathematischen Ton); guided strictly by this idea, this path begins with establishing in all necessary detail a number concept which today forms the viable basis for actual research; in this way, it attains a certain uniformity of style and a certain aesthetic effect for those students who are ripe for such a course. But sitting among those, the greater number of students do not yet possess the same ability for abstract thinking in their first hour of university lectures, but have, rather, a hunger for intuitive and productive notions. The intuitive path aims to satisfy that hunger. Kiepert-Stegemann, in its earliest editions, is a perfect example of this trend carried out in a pure way; this work must contain a spark of real didactic genius from which it derives its success, even in its current versions.Footnote ⁵ (p. 90) A comparison with the later editions shows in this textbook, as well as in some other well-known textbooks on our subject, with particular clarity what matters to us most of all here, namely, how little any attempt to strike a compromise between the two trends improves things. In such later editions, for example, one or a few paragraphs with subtleties connected with the concept of convergence are inserted after the “intuitive” definition of series convergence, that is to say, with the growing number of editions, theorems converge to serious statements asymptotically. This was the result of sharp reproaches coming over and over again from the “exact” party. It is particularly clear here how far these attempts at improvement missed the heart of the matter and how little the charges hit the nail on the head. For even if a reader did not skip over those added paragraphs, on the basis of the intuitive definition he would still not have the methodical preparation needed to understand any of the subtleties concerned. The more exact the wordings the more incomprehensible they must remain to him.

It is now especially important to free oneself from thinking that the contentious opposition treated in this first moment is the main difficulty dominating our entire problem. It is precisely insight into the other inherent difficulties and confrontation with them that will lead us on the way and allow us afterwards to overcome the seemingly unassailable first difficulty.

I discern a second moment in the twofold aim set for this introductory course and the tension between those two aims. The immediate aim of this course is to provide the basis for later courses, for the more compact mathematical theories such as function theory, differential geometry, etc.Footnote ⁶ To begin with, it must therefore introduce mathematical tools, so to speak, techniques of differentiation and integration, then their application to geometrical and mechanical problems (the initial steps toward solving these (der Ansatz davon) must be learned), and finally the technique of the tidy limit concept, the so-called “Epsilontik.”Footnote ⁷ But beside this immediate aim, another arises by itself: the young student who has turned towards mathematics and is considering whether it is ultimately what he wants to do will want to know to what extent mathematics is exciting, to what extent it is beautiful, whether it is worth dedicating his life to it. We are not speaking here about those chosen few who know this by nature, as it were, who could not be (p. 91) anything else but mathematicians, those born as hard-core mathematicians (Stockmathematiker).Footnote ⁸ These 5 percent of the listeners in our introductory course – I am giving a liberal estimate here – learn techniques easily; they hardly need the lecture; we would not need to discuss the entire problem on their account. But it is precisely from these students that we later recruit university professors who design instruction unwittingly with the memory of their own training too much in mind. It is true: if we adapt instruction to the level of these 5 percent then certainly a few others will be carried along and drawn to a height they never would have reached in a course of a lower level. But these students who have been thus drawn upward – the facts show – are by no means all the students and, what is more important, by no means all those who are well-talented. Just as a piano player can be robbed of all joy by excessive finger exercises at the beginning of his studies, differential calculus that exercises dexterity by apodictically set rules can spoil the subject even for the able student – and it is not the worst students (Naturen) among our beginners who want to know also why things happen. And we should be clear right from the start: our considerations are directed not towards the 5 percent hard-core mathematicians, those who find every integral marvelous per se, nor the 50 percent worst students, whom our classrooms would be better off without, but towards the 45 percent decent and enjoyable students.

So if the course in infinitesimal calculus wants to reveal something of the true nature of mathematics to this circle of students we have just delineated, it is confronted with an undeniable obstacle, for in such a course one does not find the tight line running through the theory of functions, for example, nor the verve of Galois theory, nor the quiet peaks of ideal theory. Missing are the exciting events (spannende Ereignisse), the conclusive facts, the rousing problems. In this regard, the philologist, the historian is in a better position than the beginning calculus lecturer; there, starting with the first semester, the student attends all courses;Footnote ⁹ and even if he lacks phased instruction advancing systematically, as we provide in lecture courses, he is soon provided with an overview of his field (Wissenschaft). Even analytic geometry, the other beginning course in mathematics, is in a slightly better position; though not quite as dramatic, it easily allows for certain aesthetic accents to be made, whether one turns to the elegance of homogenous coordinates and forms, or to considerations of projective symmetry. I believe it is good at one point or another not to mince words about the weaknesses (p. 92) of the infinitesimal calculus in this regard, so that the problem at hand will be clearly understood.

The third moment is of a more recent date. The introduction of infinitesimal calculus into high schools, which brings such welcome relief to colleagues at institutes of technology, is not a cause for complete untroubled happiness among us university teachers. First of all, the convenience of a considerable number of our students’ already knowing techniques of differentiation and integration is vitiated by the fact that among them also sit students from humanistic secondary schools (gymnasiale Studenten) who offer instead what in principle are more important qualities. But more crucial is this: Oberrealschule studentsFootnote ¹⁰ persuade themselves all too easily that everything we have to teach them they have already learned at school, and that indeed they have learned it in simpler and better way, so that they go away feeling bored. On this basis, they are subsequently incapable of comprehending a serious treatment of function theory, and they bring down the level of the course. Schools are not able to make this problem easier for us, and it is surely not their fault. For if schools teach differential calculus at all – and, for the moment, I am not speaking about the question whether they should take this up in the first place—they have to accommodate the majority of their pupils who will not later become mathematicians, and it is certainly correct that, for the majority, no treatment of the differential calculus providing an adequate basis for university mathematics can be taken into account. For a high school teacher to present such differential calculus to people who will gain none of the fulfilment from that knowledge which our students relish in their advanced courses later, would be a downright crime. But be that as it may, for us the problem we have been speaking about exists and requires a solution.

The proposal presented here for contending with all these difficulties has been gradually developed and tested from my own lecture practice for the last nineteen years, and I hope someday to be able to present it in the form of a textbook; I should like to call it the genetic method. I started from the second of the three moments just described and said to myself: all these subjects of infinitesimal calculus studied today as canonized requisites, the mean-value theorem, Taylor series, the concept of convergence, the definite integral, and above all the differential quotient itself, and about which questions as to “Why is that so?” “How were these arrived at?” are never asked – all these requisites, nevertheless, must have been at one time objects of absorbing investigation, exciting activity, (p. 93) namely, at the moment of their creation. If one were to go back to the roots of the concepts, the dust of time and the abrasions of long use would fall away from them, and they would once more arise before us as beings full of life. And from this point onward a twofold path into practice would present itself: either one could present the discovery of the concept with all its drama directly to the students and let the problems, concepts, and facts emerge before them – this I should call the direct genetic method – or one [i.e., the teacher] could learn from this kind of historical analysis for oneself, what the essential meaning, the actual core of each concept is, and then, from that, draw conclusions for teaching the concept, which as such would no longer have anything to do with history – the indirect genetic method.

I would like first to speak about the direct genetic method. In order to obviate misunderstanding, I must begin with two preliminary remarks. One is that there is no intention at all that this idea should present something new. Felix Klein refers to the biogenetic law explicitly in the Düsseldorf talk mentioned above, and Pringsheim caricatures it in his reply; furthermore, you all know the book by Gerhard Kowalewski about the classical problems of the analysis of the infinite.Footnote ¹¹ Klein based his 1911 differential calculus course on the idea at least behind such thinking; and from conversations I had with him and, on his advice, with Heinrich Burkhardt,Footnote ¹² I learned something essential about it, after I had already tried out the whole idea earlier with other subjects in the years following 1907. But priorities are certainly not interesting in this context where we are not dealing with mathematical research. The lecture courses actually given today are generally not guided by such ideas – in part, they stand completely opposed to them; this is the circumstance that prompts me to speak about these things here. But above all, my talk concerns the way in which these ideas might be deepened and systematically developed so that they will be sufficient to attain completely the aims described above.

The other preliminary remark is meant to prevent the misunderstanding that we are dealing here with a “historical method.” Not without reason, this catchphrase is unpopular; the historical brings to mind the idea, which we, on the contrary, would particularly like to eliminate, of the old and antiquated, of the roundabout paths often followed by research, of the subjective and haphazard nature of scientific discoveries. It is especially important to me to draw a dividing line in this direction. (p. 94) The historian, including the historian of mathematics, has the task of recording all that has been, whether good or bad. But I want to select only the motives for those things in history which afterwards proved successful, and I want to make use of them directly or indirectly. Nothing could be further from my intention than to lecture about the history of infinitesimal calculus: I myself ran away from such a course when I was a student.Footnote ¹³ My motive is not history, but the genesis of problems, of facts and proofs, about the decisive turning points within that genesis.

Now, with this idea at hand, it is easy to show how I plan to achieve the goals I have set out. The second of the three moments mentioned above, the challenge of making the subjects come alive, was my starting point. It is clear that the direct genetic method sheds as much light on the subject as the subject has ever comprised. To mention just one example, when we are dealing with the concept of the sum of an infinite series, I start with the beautiful and fruitful discoveries in this area, beginning with those of Nicolaus Mercator and Newton, that bring about such a surprising improvement in the calculation of logarithms and other concrete (handgreifliche) problems – not forgetting to stress, naturally, the boldness of using infinite series this way; I then give an account of the wealth of Euler's discoveries and the awful mess Euler and the Bernoullis got themselves into by their use of divergent series; I describe how, in the end, they did not know whether it was that which was behind the inconsistencies arising, or whether it was the simultaneous use of imaginary quantities, still mysterious at the time. At that point, the student is ripe for the concept of series convergence, for the concept of a general power series and its interval of convergence; he sees the necessity of these concepts, and how by means of a few strokes of the pen all the wonderful discoveries described above become neatly proved facts – for the restoration of the original methods, and not of those methods based on Taylor's theorem that have unfortunately become habitual, is in fact very easy.

More important will be for me to describe how our method can confront the sharp opposition of the first moment. I consider it an inviolable axiom that by the end of a two-semester course, a beginner should have obtained a complete understanding and complete mastery of the technique of “epsilontic” operations, and that he did not bring such techniques with him from high school. The way this has been formulated already suggests the solution. Instead of launching (p. 95) the “epsilontic” methodology right away at the beginning with all the stops out, as one says of the organ, one should lead the student gradually up a gentle ascent to the peak of this technique, just as the organist uses one register after the other in a well-composed piece of organ music – and in this way not one of the 45 percent spoken of above will be left out. One should not organize the course according to subject matter, but according to this methodological viewpoint of ascending; the risks of non-commuting double limits should be kept methodologically as a highlight for the end, and not used unsparingly wherever they happen to occur in the subject.

But what does this have to do with the genetic method, you might ask? The genetic method provides the safest guide for carrying out this gentle ascent, which is not always easily found, to say the least. For if one follows the great progressive path of genetic development, along which all mathematical humanity has gone, one will observe that in general it has taken the gentle ascent from the easier to the more difficult, and, as a rule, one may take the individual, often explosive major developments as an indication of a progress in method. Didactic methodology can, in this way, endlessly learn from history.

Regarding the third moment, it is completely obvious that students for whom the historical side of things is likely to be entirely new will go through things which, in principle, they have already learned before with much greater pleasure when they are dressed up in this way; and by this mixture with history they will be brought imperceptibly to those points that are situated methodologically beyond their high school course. I believe I can say this has been wholly affirmed with my own students in recent years when it became relevant, owing to the rapid introduction of the differential calculus into high schools.

It cannot be the intention of this report to treat the question of the infinitesimal calculus in high school as such; I only wish to clearly indicate its demarcation. Allow me, at this point, to say a word about that demarcation. I completely leave aside the question as to whether or not it is good that schools incorporated differential calculus. It is not relevant now for it seems to me we are confronted with a fait accompli, an irreversible process which has already progressed too far to turn back even if one wanted to.Footnote ¹⁴ Schools have found in the wealth of problems coming from differentiation, integration, and their applications a domain (p. 96) richer and more multifaceted than the triangle constructions which in the past was the air they breathed. One can ask whether it is not a pity that synthetic geometry has not been cultivated instead, and whether even now something of it could be introduced into schools. The differential calculus is here – that is to say, the formal side, the technique of operation. As for the genetic method in this context, the direct genetic method in the schools could only gain limited relevance. I see the chief domain of school mathematics to be those developments which can, in principle, be resolved into several series of methodically ascending problems. Next to this, dramatic accounts will always play a secondary role. Even at the universities, by the way, they [i.e. dramatic accounts] will only be justified and successful when they run alongside intensive supplementary exercises. However, the more thoroughly we finally elaborate the system of exercises here, the more we will have the possibility of shaping the lecture course in a purely interpretive fashion and showing in it the larger [historical] backgrounds [of the subjects of the lectures]. – What is well worth considering for schools, however, is the indirect genetic method. To this I must now turn.

I enter straightaway in medias res; in this fashion, though, it will become clear at once how I understand the concept of the indirect genetic method.

That the definite integral should be treated before differential and integral calculus proper is the first thesis suggested by the indirect genetic method. This is what Burkhardt attempted 20 years ago, here and there; this is also what Dirichlet was said to have already done, according to a remark Kronecker once made in a course of lectures, even though it is surely not common practice (geltendes Recht). A glance at the historical development makes this thesis obvious. The Greeks discovered the definite integral; Archimedes elevated it to the height of a mature theory: only the Leibnizian symbol was lacking and, likewise, the concept of an arbitrary curve, the concept of a function. The modern age (Cavalieri, Fermat) introduced the function concept into the theory even before the rest of the infinitesimal calculus was discovered.Footnote ¹⁵ The doctrine of the definite integral is primary.

The second thesis is directly related to the first and properly exploits the possibilities provided by it: the relationship between the definite integral on the one side and the differential calculus and the theory of the indefinite integral on the other side is the central point of the infinitesimal calculus, and, as such, (p. 97) it is that which should be emphasized most vividly instead of smuggling it in along with differentials, as banally unnoticed as possible. This is the point that F. Klein worked out with great clarity in his lecture of 1911. There is a juncture in the history of the infinitesimal calculus which makes us conscious of the missing evidence of this connection in a particularly striking way: Fermat greatly advanced the theory of the definite integral; he had at his disposal, on the other hand – again in the terminology of the Greeks, naturally – the complete differential calculus, and he solved maximum problems as well as any pupil (Primaner Footnote ¹⁶) today; yet – in 1660 or thereabouts – he did not suspect that there could be any connection between these two separate spheres, as Barrow was ready to provide eight years later. You see clearly here the difference between genesis and history. Historians place the unfortunate priority quarrel between Newton and Leibniz in the foreground of the historical emergence of the differential calculus; from the genetic perspective, completely different moments move into the center of interest.

The third thesis: the foundational questions (introduction of the number concept) belong to the end of the two-semester course, not the beginning. That mathematics perfected this foundation a posteriori, and not a priori, is clear. Nevertheless, a word of explanation is in order since in this case there is also a historical complication, and the uncovering of it yields right away something new concerning the didactical side of the matter. In fact, what we call today a Dedekind cut is essentially in Euclid's fifth book,Footnote ¹⁷ and research – correct or not – takes Eudoxus as its author; in any case, it was at the very latest worked out by Eudoxus, and thus by about 360 B.C. But why, you might ask, does the whole matter not belong then to the beginning of the beginners’ course after all? And what, I have further to ask, is the real achievement of Dedekind? He himself has said this plainly, not in the paper on the cut, but later in the preface to Was sind und was sollen die Zahlen? There, he says explicitly that the cut as such is already in Euclid; however, before him, no one ever proved that $\sqrt 2 \cdot \sqrt 3 = \sqrt 6 $.Footnote ¹⁸ Thus, he himself sees neither the cut nor its definition as his achievement, but rather the establishment of the rules of calculation. It is here then that the entire difference between the Greek and the modern concept of number lies – and we learn from this that that difference contains the actual didactic difficulty, namely, while the Dedekind cut itself (p. 98) is a showpiece whose elegance the beginner very easily grasps, the ascent to the tedious rules of calculation involves the moment that leaves the beginner exhausted. And here lies the weakness of Dedekind's achievement as well. For his paper actually passes over this irksome side of things with hints and constructions that, in principle, are nothing else but a reversion to Cantor's definition of irrational numbers. Hence, from this historical analysis we infer a corollary to the third thesis: Cantor's definition of irrational numbers is to be preferred over Dedekind's – for which, incidentally, other reasons as well can be adduced.

I wish to single out just one more thesis, a fourth thesis, from the wealth of historical consequences furnished by the indirect genetic method: If one insists on using the concept of the differential in lectures, one must do so in terms of computational functions, that is, functions expressible in terms of calculation. To elucidate, let the following historical remarks be added. The stir Fourier caused and the difficult battle he had brought to bear against the authorities in Paris concerning the concept of an “arbitrary” function, as opposed to the computational function, which had dominated the entire eighteenth century,Footnote ¹⁹ is well known. It has been far less observed, however, that in the seventeenth century these two varieties of the function concept had originally developed along quite separate paths: the first, the concept of an “arbitrary” function, arose out of Cavalieri through Pascal to Barrow; and, then, completely independently, Newton and Leibniz made a transition to the idea of a computational function, coming from the Vieta-Descartes line (aus der Linie Vieta-Descartes herkommend). Again, there is a place in the historical development where the difference is seen in a striking way, namely, with Barrow. Barrow possessed differential and integral calculus in its entirety, including the relationship to the definite integral, all in the language of the Greeks, though formulated for arbitrary functions. Why then is Barrow not the discoverer of the infinitesimal calculus? Or to put the question more correctly – to what extent is he not its discoverer? For, in fact, to a large degree he is. What is essentially new in what Newton, and even more systematically, what Leibniz brings to the scene is really only the turn to the concept of a computational function and the use of the rich possibilities for application that emerge from this turn. It was not merely a great moral gesture that Barrow gave his teaching post over to his student Newton so that he himself could, with trouble, resume the theological career he had begun previously. Unconsciously, he seemed to be (p. 99) possessed by a feeling that a new epoch had commenced, one that set calculation in opposition to the language of the Greeks,Footnote ²⁰ and the feeling of not belonging made him turn away [from creative mathematical work] and become an administrator, editing Euclid and Archimedes in his spare time. Historians of mathematics, completely absorbed in the Newton-Leibniz controversy, have failed to clarify the much more important relationship between Newton and Barrow, and Newton's papers in this connection are said to lie somewhere neglected, and in a bad state.Footnote ²¹ Whatever the case may be, enough of this is clear to be able to derive the above thesis from it.

By way of such examples, I believe, it becomes clear what I mean by the indirect genetic method: the elucidation of didactic difficulties, I should say, didactical diagnosis and therapy, on the basis of historical analyses, where these analyses serve only to turn one's attention in the right direction.

I believe, basically, that the principle in this goes far beyond the theme I have set for myself today. Besides, I have treated this theme entirely in a practical framework, inductively, as it were. Therefore, in the end at least, I feel the need not to evade the great philosophical question in the background of all this, the question of the relationship between a genetic and a normative idea of mathematics. Mathematics is under the spell of the normative idea, under the spell of the belief in its objective nature. It suffers under this spell starting from the mode of presentation of mathematical works and books. The reason why, when we openly admit it, all of us are always able to grasp only the smallest fraction of published work is that the motives that give rise to these works are often more hidden than revealed. It is not the style of mathematics to speak about subjective things. The governing ideal is the final, objective grounding of mathematics in a final axiom system, from which everything, one after the other, inevitably falls out as a matter of course, so to speak.

This mode of looking for the true foundation of mathematics, I hold, is a chimera, an attempt to find the intersection of two parallel lines merely by going out further and further along them. I would like to contrast this mode [of approaching the problem], which Plato already sketched out in the sixth book of the Republic, with the beautiful manner of thinking adopted by his student Aristotle who laid out with the greatest care how thinkers before him thought about all the matters he was dealing with, and with this kind of comparative study saw how he might make a better start towards obtaining general norms. In principle, it was in this spirit that we brought the indirect genetic method to didactical matters above. Transferred from didactics to this other sphere, it [the indirect genetic method] allows one to approach the fundamental norms of mathematics in a very similar manner from a completely different direction. Of course invariant norms cannot be deduced logically from subjective variants of human events, not even from intellectual ones. But the analysis of these [subjective variants of human and intellectual events] can steer one's attention to differences and conceptual distinctions which would never come to mind by way of direct, abstract considerations – one need only think of the analysis of the number concept and the function concept discussed above.

This would be the germ of a change in the very way one conceives of what it means to be mathematical. Hilbert's foundational studies are far from this kind of thinking; Weyl's works already contain the beginnings of an approach marked by attention to historical matters to which he wishes to build a bridge from the other side, from the normative side of mathematics. In this paper, it was the genetic side which was our starting point.