Otto Neugebauer: The History of Mathematics

This article, originally written in German, was translated into English by G Waldo Dunnington.

The History of Mathematics, by Otto Neugebauer

The historiography of a science usually does not enjoy any too high an esteem among its productive representatives. The reasons for this are several and they are not difficult to recognise, especially in a science like mathematics, which can distinguish with such precision between secured possession and unsolved problem. Mathematical problems and methods are indeed, like every other element of existence, historically conditioned, but that portion of the way already traversed - which for their continuation one must know and as such survey - is a relatively short one. Probably long centuries worked only in closest connection with antiquity, but the great rise of modern science begins with the appearance on the scene of entirely new ideas whose import lies in the opening up of hitherto unknown questions rather than in the settlement of old ones. In all natural sciences definitive results may have an absolute value - in mathematics a settled theory (the technical term for this is a "classical theory") is something dead, which cannot captivate fresh forces.

With this set-up the non-historical character of mathematics is not to be wondered at. In addition however, or rather connected with the above is the fact that the existing historical presentations of this science cannot interest the professional because their authors, as outsiders, with all their formal knowledge of subject matter, do not touch the real essence and interest of the problems. Instead one finds things treated such as the subject "Geometric forms that were in existence before the advent of life on the planet" (in a "History of Mathematics" appearing in 1923), which have nothing at all to do with mathematics. And only too often an anecdote collection, or even worse, an endless chain of priority questions must replace history.

It would not be worth while to write about such things, if one had to regard them as irrevocably united with the substance of the history of mathematics. Indeed I do not believe that the above mentioned purely objective relationship between progressing research and its history can be changed; but I regard it as an absolutely attainable goal, so to recast the history of mathematics in itself, that, like the history of philosophy, it will become an integral member in the series of modern sciences and not lead to an entirely meaningless existence untouched by mathematical as well as historical spirit. Then one will again dare to hope that even the purely professional mathematician, from this broader viewpoint, can profit by occupation with it.

I have already touched on the point where according to my opinion the chief deficiency in the present condition lies: in the lack of an historical-problem attitude. I should like to explain somewhat more in detail by a very special question the way I desire this to be understood: What is the treasure of mathematical knowledge which the Greeks took over from the Orient?

Immediately an objection: What interest at all does such a question have? Indeed it is quite immaterial to know whether some Egyptian or this or that Greek possessed a formula for the volume of a truncated cone or not. And such an objection really exists quite properly, as long as one contemplates such knowledge only on account of its absolute content, but not as points of demarcation on a scale which one needs in order to be able to draw any sort of historical comparisons at all. However immaterial it may be in itself, whether these points of the scale are constituted by propositions of "elementary mathematics" or by propositions of an optional brand of modern analysis - that we have to deal with the one or the other is historically accidental - nevertheless the gaining of reliable factual material becomes important as a preliminary labour in order to have safe ground under our feet, and not to go to seed by merely attitudinising aesthetically.

Here I must interpolate a purely methodical remark which refers to the securing of the foundation for answering the question asked above. Our entire tradition from the ancient Orient exhibits one great advantage: we scarcely know one name of an artist or scholar. The entire cultural evolution of those periods is from the very beginning most closely united with the national unit, its picture is presented in much more tranquil lines before a larger background, than in an epoch when the struggle for "master or school" or for "genuine or false" distorts the perspectives. The chronological arrangement necessary for every historical comprehension must of itself ensue in a much broader framework: in the framework of general history. Thus with all the scantiness of our knowledge the history of Egyptian art, religion, and indeed of linguistic history forms with the "pragmatic" history a much more closely knit unit than is the case for analogous Grecian conditions. Directly therefrom arises however the demand to connect the utterances of mathematical thought with these general viewpoints. Not until the investigation of purely objective questions takes place strictly on the basis of the history of civilisation can one expect to attain a relatively correct evaluation of the separate problems. Then too such an investigation obtains, on the other hand, a much more general meaning because it is able to bring to light in a quite precisely comprehensible manner very characteristic features of a people.

The disappearance of the individual in the history of Egyptian civilisation has not always been regarded as an advantage. Thus the poor scribe "Ahmes" who immortalised himself as the copyist of the mathematical Rhind papyrus (the most important monument of Egyptian mathematics), has had to assume all possible titles from "king" or "teacher in an agricultural school" down to unskilled "pupil". Or however one has taken refuge in an intentional concealment of the individual behind the very popular "priest castes", although they do not exist, at least in the genesis period of all Egyptian sciences, thus it is maintained: "Mathematics as a science was in Egypt the exclusive possession of the priest caste and was carried on in the priesthood as an occult science and concealed from the people," - with such success indeed, that not the slightest vestige of this "occult science" has been handed down to us!

To the demand for arrangement of historico-mathematical investigations in the scheme of general history of civilisation is joined a sphere of questions much more difficult of access, as soon as it has to do with ancient history: the connection with philology. Such a fundamental investigation as Sethe's book "On Numbers and Numeral Words among the ancient Egyptians" shows how much can still be summoned from these things for the beginnings of mathematical thought. One must not carelessly pass by these things, as soon as one asks questions about the historical development of the most important categories of thought, especially in a period of mathematical research like the present, in which the question about the logical foundations of mathematics assumes a central position. Here is really one of the points where the most un-cognate sciences encroach quite directly on each other, so that it is not pertinent to grant philosophical speculation the only decision.

If one turns to the actual content of pre-Grecian mathematics, the first impression is that it has an exclusively "elementary" character, and is in itself rather homogeneous: simple problems of calculation, executed in part with the help of numerical tables, or problems involving the calculation of areas and volumes of geometrical figures such as those demanded by agriculture or at the most stonemasonry. But if one observes the role which these things played in their own cultures, this picture is quite essentially changed and offers problems which are by all means worthy of historical investigation. The deep-reaching distinction between the two great cultural units Egypt and Babylonia, even in simple numerical notation and application of the first calculation exercises, asserts itself here quite essentially. The beginnings are indeed in both districts the same: Hieroglyphics with special signs for the most important numerical values 1, 10, 1/2, 1/3 etc. and a purely additive counting foundation of all calculation. Now however the individual development sets in. The fundamental problem of Egyptian mathematics (which bears a purely "arithmetical" character) can be bluntly formulated: to provide those oldest additive methods with as sizeable a domain of operations as possible. That which we today call "multiplication" is in Egypt repeated addition (by continued doubling and suitable collecting); indeed the entire fraction calculation, which in this form extended its influence over all of later antiquity far into our Middle Ages, owes its externally very intricate methodology only to the consistent execution of the same fundamental principle. We have here in abstract pure culture so to speak the same tenacious adherence to old traditional forms which characterises all other portions of Egyptian life, which has made its theological systems a chaos so difficult to unravel. The inevitable transformation of religious concepts does not ensue by the formation of clear, new systems but by artificial interpretation of the oldest texts, whereby their simple meaning is distorted and (from our viewpoint) the most contradictory statements are entangled, rather than give up the fiction that everything has been thus from olden times.

Quite different in Mesopotamia with its much more stirring history. Even the further development of the hieroglyphic system created by the Sumerians ensues in an essentially different direction. While the hieroglyphic system in Egypt, at least as a system for inscriptions, was preserved to the latest period and while the hieratic writing represents only a levelling-off of it, nevertheless in Babylonia the hieroglyphic symbols, already of a marked linear style, were finally replaced by a number of pure "cuneiform symbols" which give up every conscious connection with the old hieroglyphics. Of course this is also tied up with external influences such as the inconvenience of clay as a writing material and the rise of new elements in the population. Consequently for numerical notation a system of figures very deficient in symbols is formed, which approximate closely our present notation by "local value". However the latter is again conditioned by a very early creation of independent multiplication (as is shown by multiplication tables and tables of squares preserved from a very ancient period) and a strong influence on the entire system of notation due to the standardising of weights and measures. The details of this process are of course much too complicated for me to discuss here. As to general history it is stimulating to remark that the Babylonian system of writing ran into a cul de sac by preventing the gradual transition from hieroglyphics to "uni-consonantal symbols" and finally to "letters", that mathematics however from the beginning on pointed in a direction which in its consistent expansion by means of the Hindu local value system (with which our present notation by digits is identical) has furnished one of the most important supports for modern further development.

I hope one will be able to see even from these hasty references that the elementary, indeed frequently primitive character of the mathematical problems and methods of this early period, which has never been calmly admitted as such and efforts made to conceal it by phrases like "primeval Egyptian wisdom", results in quite the opposite by granting us an especially clear insight into the historical beginnings of mathematical thought. To be sure, one must waive the desire to build up a linear chain of mathematical knowledge from earliest antiquity to our time, but one must learn to place independent cultures like separate personalities beside each other. Then it will be recognised that every people and every epoch seeks in its own manner to meet the problems presented to it and the comparison of these various phases receives a new meaning. Not until Oriental mathematics in its singularities is really known, will one know how to evaluate properly those of Grecian mathematics and look for those points where specifically Grecian problem-attitudes set in.

But historical processes do not originate only by the encampment of various cultural types beside each other, for beside this "horizontal" articulating of cultures a "vertical" stratification plays an ever recurring role. The desire to regard such a complex structure as Grecian civilisation as a unity would be a very fundamental mistake. Between the influences from outside enter the great differences between groups of the most varied intellectual tendencies. Pythagoreans, the Academy and the Sophists cannot be brought into one line of development, but stand in their mutual influence as independent simultaneous factors beside each other. Thus one cannot coordinate the attitude of all these tendencies on mathematical problems, even though the external result of occupation with mathematical questions is a permanent increase of objective knowledge. The really essential thing however is not the question whether one multiplies the square of the radius by 3.16 like the Egyptians or by 3.14 in order to determine the area of a circle, but to fathom the collective attitude on the problem of measuring the area of figures bounded by curves and the meaning of infinite processes to which this leads. And in the answering of these questions one will again have groups quite separated by principle to differentiate, groups whose transformations are to be pursued in detail, in order to gain a true-to-life picture of the entire process. And when the Arabs or West European civilisation later tie on to the Grecian acquisitions, this occurs every time in a special manner and with preference for quite a definite method among these various currents, in spite of all continuity with reference to mere content. Thus the history of mathematics suddenly reaches out far beyond its narrower framework and offers no end of the most interesting questions, which reward the effort to review steps long archaic in mathematical thought.

And beside this general historical significance here comes to the front yet another which refers to mathematics in the narrower sense, but on that very account must not be forgotten. I mean, viz., that only historical thinking can possibly form a balance to the much deplored specialisation. The last phase of our science inaugurated in grand style at the passage of the eighteenth into the nineteenth century, of whose universal character the great French scholars and the Humboldt brothers may serve as an example, has not been able to maintain this initial level. It is clear that a rigorous establishment of the newly unlocked sciences is to be accomplished only by the greatest division of labour in careful separate investigation. A consequence thereof was however not only a separation of the single sciences from each other, but also a crumbling of these disciplines themselves into divisions scarcely understandable or interesting to each other. There is no doubt that a serious reaction must be set in against this condition and in part already has set in in a very perceptible manner. The question about the whence and whither of a science, about its place in the broader sphere of our entire civilisation, is being asked more and more decidedly. In all fields it is being shown that only in the synthesis of modern research methods with the less hampered perspectives of a deeper intellectual content can a guarantee for restoration of the unity of all sciences be found. The work by Felix Klein "Lectures on the Development of Mathematics in the Nineteenth Century" shows as does none other what the historical view in this sense can mean for mathematics. Truly historical thinking united with the most intimate research activity speaks to us here, reminding each one to understand and evaluate his own research tendency as an element of a great historical process.

It will not be vouchsafed to many to write the history of a science in this sense. However every single historical investigation can count as a usable preliminary performance toward further synthesis only if it is guided by two viewpoints: to see the history of mathematics in the framework of general history and to understand mathematics itself not as a collection of formulas to be continually increased, but as a living unity.

Note by G Waldo Dunnington.

Professors Neugebauer, Richard Courant, and Erich Bessel-Hagen (Bonn) prepared this volume for publication. It is entitled Volesungen über die Entwicklung der Mathematik im 19. Jahrhundert and appeared with the imprimatur of the publishing firm Julius Springer in Berlin, 1926, a year after Klein's death. These lectures cover a period of approximately 1914-1919 and were delivered by Klein to a small circle of students in his home. His death in 1925 brought to nought his plan to issue a more voluminous work on the subject, although this presentation fills nearly 400 pages. These lectures are especially charming because they are published just as he gave them, and never received literary "finishing touches". The book is a history of mathematical ideas rather than mathematicians, and an amazing example of his power of presentation; he portrays the train of thought leading to some discovery and as Professor Neugebauer has so well said above, he views isolated pieces of research within a larger framework. Klein's main thesis, running through the entire book, is that regardless of this or that genius a fixed and definite evolution of mathematical ideas exists. The science moves and must move ahead continually on this stream of development. The appearance of a genius merely hastens the current.

Last Updated July 2026