Purpose and method of the history of the exact sciences.
Madam and my gentlemen curators,
Ladies and gentlemen professors, lecturers and private lecturers,
Ladies and gentlemen members of the academic staff,
Ladies and gentlemen students and furthermore all of you who honour this ceremony with your presence.
Much appreciated audience,
As long as the history of mathematics and the natural sciences was a hobby of some practitioners of those subjects, it found its full justification in the interest it aroused among these individuals. However, now that it is going to be recognized as an official component of university education in various places in numerous countries, not only demanding attention in the Faculty of Mathematics and Physics, but also addressing other categories of students, it is obliged to demonstrate its right to exist for its purpose, to explain the reasons why it pursues this objective, and to identify the ways in which it intends to pursue that objective. It is about this general problem of purpose and method of the history of the exact sciences that I want to consider briefly in this hour.
I assume in this regard the fundamental fact that this history is about a phenomenon that, as far as we know, has not occurred in cultures other than what is commonly called Western. Formation of human communities has taken place everywhere and at all times; conflicts between those communities have arisen again and again; man has always expressed himself in religion, in philosophical thinking, in visual arts, music, dance and poetry; comparative study is therefore possible in all these areas. Mathematically-empirical natural science, which leads to technical control of nature, is, however, a unique phenomenon that has become increasingly characteristic of Western Europe and, historically speaking, its culturally dependent territories in the course of the last four centuries. The preacher's word that there is nothing new under the sun is seen as a lie here. There is something new: man has embarked on an awesome adventure that is unparalleled in history and the effects of which are already being felt in all areas of life and will become more and more noticeable. Is it surprising that the historical sense is affected by this unique process, that there is a need to establish the lines along which it has so far gone, that attention is paid to the influences it has so far exerted and undergone? The enlivening of interest in the history of mathematics and natural science in our time is therefore not a coincidental phenomenon: it is a natural consequence of the growing awareness of the essential significance that these subjects, in both material and spiritual terms, have for owning society.
If mathematics and natural science on the one hand, and their history on the other, are closely linked to each other as long as there is meaning, then that link no longer exists, or at least not substantially, once a value judgment has to be pronounced on that meaning. The mathematization of natural science and the mechanization of society, these two extremely important characteristics of our contemporary culture, can only be summarized under the one name of mechanization at the expense of clarity of thinking, because of the consequences they have for bringing woe to human society, no neutral phenomena that can be objectively and emotionally studied without determining one's personal position. Experience shows that that position varies between the extremes of unconditionally consenting acceptance and uncontaminated rejection. Should these extremes practically not occur, the spread in the value judgment remains strong enough: where one feels and expresses profound satisfaction with the triumph of science and technology, the other expresses deep concern about the future where that triumph can still lead us. However, this difference in appreciation of the flourishing of the exact sciences itself has no consequences for the assessment of its history: understanding how it came to be so impressive is just as important for those who consider it a blessing to society and on the basis of this, advocates its more powerful practice, as for those who see nothing less than a disaster for mankind and longing to return to times when people knew less about nature and had less of its powers in their service.
As a result, it will already be clear why the history of the exact sciences can never be regarded as a matter that concerns only practitioners of these subjects; they intervene too deeply in everyone's life for that, they penetrate everyone's mind too deeply. Its independent practice, which requires mathematical and natural science training, will as a rule have to be reserved for the Faculty of Mathematics and Physics; the knowledge and consideration of its results, however, is important for a much wider circle: for the professional historian, who cannot remain indifferent to the way in which a power that increasingly controls world affairs has come into existence; for the classicist, who, with a sufficiently broad view of the study of ancient culture, cannot ignore the streams of mathematical thought and scientific view of nature that form both a characteristic feature of the Greek spirit and one of the main sources of contemporary culture; for the philosopher, who finds the older phases of his own subject inextricably entangled with those of mathematics and physics and can notice many important points of contact in the younger ones, while the data on mathematics and mathematical methods revealed by historical research about natural scientific thinking provide him with the most important material for his cognitive considerations; for the sociologist who, in one of the branches of his richly varied activity, recognizes the essential relationship between science and society for both parties; for the theologian, who must feel enthralled by the frequent contacts, tensions and conflicts between religion and science, the history of which can be traced to almost every branch of scientific thinking and which in our time is increasingly forced to determine its position regarding the growing dominance of human life, feeling and thinking by determining natural scientific and technical influences.
However, the greater the diversity of the categories of students becomes to whom the history of the exact sciences thinks it has something to say, the more urgent it becomes to give clarity to the many questions concerning the purpose and method that can be asked and to refute the objections that have been raised against its claims for so long and which in our time, although they are spoken less loudly, have by no means been overcome.
Among those objections there is one that deserves very serious consideration and with which every practitioner of the history of any science must necessarily try to come to terms with himself. It is perhaps best to make it clear by referring to one of the many brilliant aphorisms about the practice of science that can be found in Blaise Pascal's De l'esprit géométrique and Fragment d'un traité du vide. Here as elsewhere giving an unforgettable form to thoughts that one can also find with other thinkers of his time and thereby unintentionally usurping them, in the second of these writings he points to the fundamental difference between man and animal that consists in that the results of the human thinking accumulates, while animal instinct always remains in the same state. The hexagonal cells of the bees were as well built a thousand years ago as they are today; nature teaches the animal when the need compels; but the fragile knowledge that it bestows goes hand in hand with the need for it; it gives it to prevent the animals from perishing, but it does not allow anything to be added to it, so that it will not exceed its limits. "Il n'en est pas de même de l'homme, qui n'est produit que pour l'infinité', that is to say, man's spiritual development has no limits. Beginning in ignorance, his mind develops in steady growth because, apart from his own experience, he can take advantage of what he learns from his predecessors and can retain the knowledge that he has once acquired in his memory. What happens in everyone's life, progressing from day to day in knowledge and insight, also takes place in the succession of people through the ages: 'De sorte que toute la suite des hommes, pendant le cours de tous les siècles, doit être considérée comme un même homme qui subsiste toujours et qui apprend continuellement' [So that the whole succession of men, during the course of all ages, must be considered as the same man who always subsists and who continually learns].
The insurmountable insight into the cumulative-progressive nature of the practice of human science lies like a rock in the way of every science historian who tries to account for the right to exist of his profession. If Pascal's view is correct - and who would doubt it if he sees how all the scientific results that are achieved in any period of development, the extent to which they prove useful, live on in the minds of future generations and thereby take on an increasingly simpler and clearer form? - there appears to be no justification for the practice of the history of a science to arise, even if the individual has a motive for himself, once he has acquired an insight or a skill, once again to empathise with the time when that insight was still hidden from him whether he had not yet mastered that skill. What then is the history of science, where does the superfluity of its history seems to arise from the specific character of science?
One can respond to this indeed serious objection in two different ways. One can first of all ask the question: what is the history of science for? give one of those general answers that are no less valid because they actually do no more than re-establish the factual nature of the phenomenon to be explained. It is: because the intellectual man in every field he moves possesses the historical need, because he simply wants to know how everything has turned out the way he finds it, which desire cannot be dismissed by the intellectual consideration that this does not actually have to interest him in his field of science and that he will not expect any demonstrable benefit from it in the practice of his profession.
The question: what is the history of science for? is therefore really just a special case of the more general one: what is history for? And when one gives an answer to this: man is like that, at the same time one has discharged himself adequately from his task and indicated one of the deepest human traits; it is his ineradicable sense of being a link in a chain, a binding member between those who preceded him and those who will follow him, at the same time seeking the end and beginning, with the resulting assignment to, where the future is unknowable, at least the past memory is left to help maintain that memory. Schopenhauer has described man as a metaphysical animal for his unquenchable metaphysical need; with equal rights, however, one can call him a historian animal and briefly state the quintessence of the difference between man and animal indicated by Pascal: man knows about his past and is aware of this knowledge.
The second means of escaping the confusion into which Pascal's argument has taken us consists of denying the consequence to which his consideration appears to lead. First of all: admitted that everything that had scientific value in the past lives on naturally in contemporary thinking, does it follow that one can therefore forget that past? After all, it is the past that lives on and co-determines current science in terms of content and manner of expression, so that anyone who wants to fully understand the latter in its objective, its methods, its terminology and its notation, can only benefit from a careful study of the former. And what about those components of the past of a science that no longer have any value for its present phase in the sense that they are no longer recognized as correct or important and therefore do not live on? Would the history of science not already be justified if it only made clear to us the ways in which people could not make progress and how often it has happened that earlier results have lost their value and have been forgotten due to a change in direction?
The history of science should not be limited to merely recounting successes; it must also be aware of errors. First of all, the judgment of what was success, what was error, can change over time; but, moreover, even something that has proved to be irrevocably worthless may well be worth studying. This is in particular the case when we learn a way through which human thinking had to get lost spontaneously in its first confrontation with nature, of which there is no innate knowledge, and thus a deeper understanding is gained of the psychology of thought. That the axiom cessante causa cessat effectus (if the cause ceases also the effect ceases) with the consequence that at the same time when the cause that has set a body in motion ceases, also the obtained speed, interpreted as the effectus of that causa, must disappear, has been able to control the mind for twenty centuries; whereas for the same length of time the unanimity and circularity of the actual movements of the celestial bodies could have been regarded as an unassailable axiom and could have remained convinced of the composition of all substances from four elements with characteristic movement tendencies; these and countless other views that have proved to be untenable are important historical facts for those who want to get to know the relationship between human thinking and nature, the value of which is further increased by the consideration that often the historical process of development of natural science is repeated at an accelerated pace in the individual thinking of current beginners.
After thus triumphantly overcoming the Pascal aporia, the defender of the history of science opens himself full of confidence to all questions that can still be asked: what do you mean? What do you promise? How do you want to reach your goal?
Not to be touched by contemporary historical criticism, he cannot give a single answer limited without restrictions to the first question. But the first thing that makes a big impression on him in full awareness of the criticism it will provoke is the so often repeated but in his sublime simplicity still so striking word of Leopold Ranke: 'sagen, wie es eigentlich gewesen' [tell how things actually were].
The criticism starts immediately: do you really imagine that the past can be reconstructed exactly? Has it not already been said when it is called approachable? Do you truly believe that, even though an abundance of sources were at your disposal, in which you have good faith, it would be possible to influence those of your own personality, your upbringing, your individual interest, your time, your social class based on the choice and interpretation of the material?
Again, not a single answer fits this complex of questions. It can, however, be noted immediately that in the history of exact science there are cases in which a piece of the past is really precise, that is to say without recourse to the imagination and without having to take into account the possibility of a difference in interpretation, it can be experienced again. This is the case where we have the complete text of a mathematical work, the reasoning of which we can follow perfectly and it is correct. It is not believed that this is an exceptional case: the works of Euclid, Archimedes, Apollonius and numerous other Greek mathematicians are examples of this. Philologically speaking uncertain places may occur in the text, but as a rule they do not stand in the way of understanding the mathematical content, because it is determined by the starting point and the unchanging structure of logical thinking. The famous word that Pascal wrote to Fermat when it turned out that he had come up with the same answers to De Méré's questions about the calculus of probability as he himself had: 'Je voy bien que la verité est la mesme à Tolose et à Paris' [I see that the truth is the same in Toulouse and Paris], also applies to the mathematical truth of the Babylonians, Egyptians and Hellenes and ours. It is precisely one of the charms of the study of ancient mathematics that, unlike the natural science of the same time, no dislocation of the mind, at most an adjustment to certain formal peculiarities, is required for a historian to be able to read work and to think along with the author.
The conviction of the essential correspondence between the thinking of mathematicians in all times and in all places even goes so far that mathematicians engaged in ancient mathematics, in the event of an incorrect conclusion in the work fully guaranteed by the texts or by the communications of others of a mathematician who has abundantly demonstrated competence, is more inclined to reject the authority of the sources than to admit that the author has made a mistake. A typical example of this is the evaluation of the work of the Greek mathematician Hippocrates of Chios, who, probably towards the end of the fifth century BC, wrote a treatise on moon shapes, who described his sickle-shaped figures bounded by two circular arcs, in which he demonstrated the possibility of quadrature, that is to say, the construction of a square with the same surface area, for three different ones. Of those three special moon shaped, the outer circumference was successively a semicircle, an arc larger and less than a semicircle. From a statement in the Physica of Aristotle, which prompted the commentator Simplicius to report extensively on the investigations of Hippocrates and thus preserve them for posterity, it now seems to be deduced that if Hippocrates also succeeded in squaring a figure consisting of a circle and a certain moon shape, he would have thought that he had now also succeeded in squaring the circle. That conclusion would of course have been incorrect; after all, his introductory propositions related to three special moon shapes, none of which were identical to the equally special moon that had been turned into a square together with a circle. In general, however, historians of mathematics have simply refused to attribute such a grave fallacy to such an outstanding professional as Hippocrates shows himself to be in his other propositions.
Here, collegiality goes too far. However, when Archimedes, in the proof of the twenty-first proposition of his work On conoids and spheroids in the middle of one of those always flawlessly complicated double indirect proofs, which fulfil the function of a transition to a limit, it is claimed that two quantities are equal, while if a superficial consideration already makes it clear that this is not true, no one will want to believe that he actually wrote it that way, all the more so because, as could be shown recently, the matter can be corrected by a slight change of text. By the way, it will be clear on which dangerous terrain we find ourselves here, and the historians of mathematics have not always succeeded in avoiding the methodological errors that are looming here. In addition to the tendency to clear away obvious inaccuracies on the basis of proven competence, we can also mention the light-heartedness with which an author who proves to know a certain proposition is often attributed the knowledge of other results that appear to us simple corollaries of it or to be an essential link in its derivation.
So if the identity of mathematical thinking generally guarantees the exactness of a reconstruction of the past where the actual content of the works studied is concerned, then the matter will of course be very different if we want to understand the motives of the Greek mathematicians that may have driven them to build up their mathematics in such a way and not otherwise, to deal with these issues in particular, and to neglect others that seem more important or more obvious to us, or when they are asked what caused their profession, after having reached an awe-inspiring height in a relatively short period of time, not to develop further and not to be able to retain what had been achieved. We are then dependent on hypotheses and it does not require so much self-criticism to see that these hypotheses are necessarily determined by our own point of view and by the state of our own mathematical knowledge.
The value judgment that one makes about an ancient mathematical achievement also depends to a large extent on this: after twenty centuries of Euclid's elementary work being regarded as the inviolable foundation for the whole of geometry, marred at most by the parallel postulate in a single place, in the light of non-Euclidean geometries and modern axiomatics, we now see things quite differently: as a work of imperishable historical significance, but that the fundamental function for which it was intended it can no longer fulfil in contemporary mathematics.
Much less than for mathematics, of course, is the possibility of exact reconstruction of the past when it comes to following and displaying scientific reasoning that modern science no longer recognizes as correct, as is so often the case in Greek and medieval physics and chemistry. Here the tool of internal verification, which is available in the history of mathematics, falls away, and in spite of all the effort one has to put oneself in the way of thinking of the period considered, the interpretation becomes uncertain.
After thus cleansing himself of the suspicion of critical naivety, the historian of mathematics and physics still stubbornly sticks to the description of his first task: 'sagen, wie es eigentlich gewesen' [tell how things actually were], which then amounts to research and explanation for his special field, more extensive and comprehensive than has been done so far, what was previously suspected, asserted, thought, found, proven, and thus if not to lay the factual foundation, then to expand and strengthen those philosophical reflections on the development of the exact sciences and think about their significance for society will first be truly fruitful. This task will keep him busy for a long time to come: there is still a lot of purely descriptive and interpretative historical work to be done in which there is very little cause for subjective differences of opinion.
Determining the course of historical development inevitably leads to a critical assessment of that development. The historian of the exact sciences does not want, like Ranke, "bloss sagen, wie es eigenlich wesenen" [just tell how things actually were]; he will gladly take the first of the other two statements mentioned by him but rejected, 'die Vergangenheit zu richten, die Mitwelt zum Nutzen zukünftiger Jahre zu belehren' [To judge the past, to instruct the world for the benefit of future years] he will gladly take on the first, provided that he may replace the metaphor of the judiciary with that of critical judging. The characteristic of historical research in the field of exact sciences is, after all, that one only comes into contact with important personalities, in many cases even with the greatest geniuses mankind has possessed, with the result that judging repeatedly gives rise to profound admiration and that criticism should only be exercised in a limited sense and with great caution. However, he will reject the pretension of being able to teach the world something that could benefit it for its future actions as decisively as Ranke did. The exact sciences are not advanced by those who look at the past; if a pioneer is also historically interested, this is only by accident.
Unanimity can be expected from all historians about what has been said so far, but with regard to the consequences that the critical assessment may and must have on the choice of material to be treated, the paths are separate. Although acknowledging that the identification of stray paths walked in the past is still a task of the historian, many are of the opinion that one should suffice with that designation, but above all not having to go astray again. On the other hand, others, who for the time being will probably still be the minority, argue that those who take a sufficiently broad view of their task should also make this a duty, so that they should try to imagine themselves as much as possible in lines of thought that did not lead to an acceptable result if he can thereby contribute to a more precise knowledge or understanding of a line of thought, to sharpen the characteristic of a historical personality or a period of the past.
The very considerable difference between what is the legitimate object of historical research according to one view and the other is perhaps the most obviously demonstrated of the example of the astronomer Kepler. According to the first view, Kepler is important for the history of astronomy because he has pronounced three laws about planetary motion, the first two of which describe the movement of each planet, while the third expresses a certain order in the planetary system. The first two laws are Astronomia nova and the third are Harmonice mundi in high honour. However, as a rule the appreciation does not extend to the point that one finds it necessary to accompany him on the wonderful wandering that in the first work after endless searching and trying leads, as it were, to the insight that a planet describes an ellipse of which one of the focal points is occupied by the sun, and that the sun-planet radius passes through equal limits in equal times; and the second work is consulted no more than at the place where it is reported relatively casually and without much coherence with the other contents that the squares of the orbits of the planets relate as the third powers of the semi-major axes of their orbits. What is further contained in it, extensive Pythagorean speculations about the relationship of the planetary movement with the arithmetic of music, is left unassailable for a modern scientific person.
From the second point of view one sees it quite differently: here one is not only interested in the correct results of Kepler's work, but also in the way in which he achieved it and, above all, for his scientific and human personality. Attention is paid to the exuberant fantasy that constantly conjured up new ideas about world harmony for him, for his tireless efforts to give those ideas a mathematical form, for his unconditional willingness to see the results of that design, as far as possible, for the test of precise experience gained from observation. One sees how he achieves great results of lasting value where these three qualities work together, but how he loses himself in barren speculation when the first works alone or unites with the second without the third. But they continue to follow him in the latter case, perhaps partly from the methodological consideration that the insight into the way in which natural science has grown, by what Bacon calls a negative body, also benefits from the knowledge of cases in which a great spirit failed to contribute to that flourishing, but in the first place because of the remarkable personality with which one comes into intellectual contact.
A second example in which the contemporary practitioner of natural science often appears to be inclined too quickly to turn away from periods in the history of his profession that, according to current insight, have yielded nothing of any value, is astrology. It is always a bitter pill for a contemporary astronomer that his great ancient teacher Ptolemy, after creating the centuries-old handbook of mathematical astronomy in the Almagest, placed an equally fundamental work for astrology in the Tetrabiblos. But he can imagine even less that contemporary philologists and science historians make the enormous effort to organize and interpret the very extensive astrological literature from Hellenism and the Imperial period. However, whoever becomes acquainted with the results of these investigations, as laid down, for example, in the works of the great Belgian philologist and religious historian Franz Cumont, will soon judge otherwise. He will notice what a rich source of information about daily life and about the religious and cosmological representations of various ancient peoples that form astrological writings and what invaluable services they provide in determining how the transfer of ideas from one people to the other has taken place. Comparing the two points of view outlined above, I would not like to place them as equal views, but to distinguish them as a lower and a higher phase of historical consideration. It is logical that historical interest initially only focuses on the observation of progress, that it asks for achievements that have been important for the development of our current science, that it only takes cognizance of works that contribute to the growth of that science as far as those contributions are concerned. But the history of the mathematical and physical sciences will never work itself into a fully-fledged branch of general historiography if it perseveres in this essentially unhistorical attitude, when it continues to consider the past important only if and insofar as it has helped prepare the past instead of considering it in its own essence and worth, if it does not realize, to put it briefly, that it is equivalent to Ranke's other famous word for every branch of historical science: "Jede Epoche ist unmittelbar zu Gott" [All ages are equally God's children].
The light from this insight will purify the historical endeavour in the field of mathematics and physics from numerous imperfections that are so often attached to it. It will put an end to the tendency of contemporary practitioners of those sciences to regard the past of their profession as a partly entertaining, partly annoying spectacle, that in the condescending superiority attitude of the man who understands everything so much better and knows that one sometimes wants to take a moment to look into it, but that one will soon turn his back again to devote himself to more serious matters.
Moreover - and this is certainly not the least value - the practitioner of history of science - taken this word in the broadest sense that can be attributed to it - will make it possible to share in the intellectual gains already made in the pursuit of another's standpoint and to those who succeed in this endeavour strive abundantly. The historian who feels it is an undeniable obligation to put aside contemporary views, views and beliefs in the spirit of the time he is studying, to approach and understand other types of different-thinking encountered in it and to include that contact and understanding and to record it in the judgment; the phenomenological practitioners of religious studies, who manage to maintain their own point of view but temporarily deliberately forgets this, and deviates into religious representations of other peoples and other times, are not only admirable, they are also enviable. They have already received their wages in the enrichment of their own personality that comes with their mental dislocation. It is a hopeful thought for the historian of mathematics and natural sciences that his profession offers him ample opportunity to attempt to follow their method.
In order to revive the abstract sphere of general contemplation by a concrete example, I would like to point out the considerable difference in the assessment of medieval physics, depending on whether one takes the first, evolutionary, or the second, phenomenological, standpoint. Anyone who regards it as merely a link in a development chain of which the provisional last member lies in contemporary science, will undoubtedly want to admit that scholasticism fulfilled an important mediating function between antiquity and the reviving science of nature in the sixteenth century; he may also want to recognize a number of fourteenth-century philosophers, such as Bradwardine, Buridan and Oresme, as precursors of that revival; but he will regard it as a waste of time and effort to really delve into the widespread medieval speculation about falling objects and throwing objects, the structure of matter and intensity of change of qualities that have occupied such a considerable part of the philosophers' thinking power for several centuries. Completely different from the phenomenologically oriented historian: using the opportunity offered to him by these considerations to imagine himself having a completely different way of experiencing natural phenomena than that in which he himself was brought up, he will do his best to follow the reasoning made by a participant instead of rejecting them with a shaking astonishment, hilarity or annoyance as a barren play on words. It is not inconceivable that he might also come to the conclusion that compared to the much more successful newer development of natural science, the dead-end scholastic physics sometimes has aspects that he needs in our time. I give as an example the question of the essence of chemical bonding that was discussed so extensively in the Middle Ages: are the elements of which something is composed still present in a chemical compound and, if so, in which way? The medieval thinkers have not succeeded in giving a satisfactory answer to this undoubtedly legitimate and spontaneous intrusive question, and we even see Oresme ending his investigation with the appeal to God in which a medieval philosopher tends to confess his scientific impotence. However, it would now be a mistake to believe that the problem posed in the later development of chemistry would have found its solution; it is rather ignored, but this has not prevented chemistry from becoming a science of great allure. We are faced here with the remarkable fact that what is commonly referred to as a philosophical attitude in the path, an attitude that drives man to penetrate what he calls the essence of things, is not the most conducive to the growth of natural science. A certain philosophical superficiality or the possession of a philosophical conception that enables us to expose what has been felt for centuries as a serious philosophical problem, as an apparent problem and thereby to eliminate it, offers certainly better chances of success.
Of course, the situation just described can also be viewed very differently: the lack of success of medieval natural science, one can say, proves that it was on the wrong track at the time; the triumph of the modern is an indication that we have now found the right path. We have learned through the course of events that we do not do well to ask about the so-called essence of things, that the purpose of science is not to know what things actually are but how they behave, so that the older substantive way of thinking has been replaced by the newer functional one. The more fully the latter works, the weaker the need to apply the former. The more we have become familiar with electrical phenomena, to give an example, the better we know how electricity works, the less it comes to us to ask the formerly not unusual question what electricity actually is.
Representatives of contemporary natural science do not wrongly commit to take this position to praise its modesty and sobriety. It is, however, thoughtful that the more it exerts its influence in education, the more and more often there is a phenomenon among students that can be described as homesickness for philosophy. They often have the feeling that they remain focused too much on the practically useful and do not penetrate enough into the depth. There is not much that can be done against this by the way of intellectual reasoning; however, it does not seem impossible that a careful study of the course of development of natural science will be able to contribute to the cure of this nostalgia.
With the distinction between an evolutionary and a phenomenological conception of the history of the exact sciences, we have already moved into the field of methodology, where the historian of science has numerous problems and where he sees himself threatened by serious dangers. For a few of those problems and dangers, may I ask your attention for a moment.
When the historian of a science is all too one-sidedly evolutionary, he is constantly exposed to the temptation to see the entire history of his profession as a process that from the outset has been aimed at achieving the present condition. On the one hand this leads him, as we have already seen, to justify a small part of the past to be worthy of his attention, but on the other hand it drives him to construct reasonable relationships that in reality may not exist, at least not as clearly be present as would be desired. The history of Ptolemy, Euclidean geometry and astronomy and the question of a possible connection between Parisian scholastics of the fourteenth century and the revival of mechanics by Galileo through the Averroes oriented North-Italian universities provide striking examples of this. It is sometimes difficult to remain aware in such cases that the plausible-looking construction means no more than a work programme, the implementation of which has yet to begin.
Anyone who is going to write about such things and, in particular, who is aiming at a broader circle of readers, may well be wary of the danger that, for the sake of a clear and simple representation of the course of events, he will explain his ideas with a higher degree of certainty than they in fact actually possess. It is simply the case that a historian, apart from being a man of science, is always to a certain extent a literary author; this entails for him the temptation to more or less sacrifice the requirements that he sets for himself as a historian to the effect he hopes to achieve in literature. It is of course pleasant for the reader if he is offered a well-composed, fascinating history. But if it becomes very fascinating and even compelling, people are wary. Is the love of truth still primary here, as it should be, or is it already the beautiful form? Perhaps the author is taking a first step on the sloping road that leads from bona-fide historiography to romanticized life and from there to the historical novel?
The indicated danger of incorrect dosing of the nuance of certainty in a historical argument, however, does not only stem from the ambivalent nature of the historian; it also occurs in the following form, that the degree of certainty tends to become stronger with increasing distance from the source of the given representation. Anyone who first conceives a certain view of a historical context will be surrounded by all of the cautious restrictions that the hypothetical character of his opinion is all too well known to him. The science historian, however, who refers to the subject without his own source study, and first of all the cultural historian who uses it as one of the elements of his own general view of a certain era, often only takes over the core and forgets the limiting conditions. And that is how those countless scattered and widely accepted criticisms arise, at best half-fiction about the development of the exact sciences, against which the historian of the sciences so often has to resist the dismay of his hearers.
Of course, it is often difficult for him to escape the suggestive influence of the historical vision in which he was brought up. Regarding surviving historical representations, one needs a duty of Cartesian doubt, but then the practical impossibility of fully fulfilling that duty is soon experienced. A special complication is formed by the after-effects of the polemic atmosphere, which was very strong in the nineteenth century and did not disappear in the twentieth century, in which the emergence of new natural science in the sixteenth and seventeenth centuries took place and in which the transition from classical to modern natural science in the decades that lie just behind us. This after-effect has long obscured the vision of the historical significance of figures such as Aristotle and Ptolemy, and even obscures them, while they also hamper the relative appreciation of the function that scholasticism and humanism have played for the emergence of new natural science.
With the same concern as for the uncritical acceptance of traditional but inaccurate historical representations, the historian of science ought of course to guard against cultivating or helping to spread new errors. The danger of this, as experience has repeatedly shown, is not imaginary. It is especially the case when a contemporary practitioner of the exact sciences starts to deal incidentally with the history of his profession. One can then establish a strong tendency to detect so-called precursors and a need to formulate statements in which the ominous word combination 'has already known' it is argued that some insight is of a much older date than had always been believed to be true and that the thinker to whom it is now ascribed has therefore always been misunderstood. When this is done correctly, one must of course embrace it as a welcome addition to our historical knowledge and change its representations accordingly. It also happens, however, that the new find in a wider historical context immediately turns out to be untenable; by way of a recent example, I mention an attempt to show that Sadi Carnot already possessed the concept of entropy; the reasoning contained an elementary fallacy, but the result could be rejected in advance as it would turn the hitherto quite understandable developmental history of thermodynamics into an inextricable chaos.
It thus appears that there are still many methodological difficulties to be overcome in the practice of the history of the exact sciences. As a result, and because of our still inadequate factual knowledge of the past, the profession is still far from the highest rank that science can achieve. Is it surprising? Born out of the interference of amateur historians, long susceptible to denial and combat, or, worse, benevolent indulgent disagreement, it has to catch up with the other branches of historiography. The satisfaction with the opportunity now opened at this university to be able to cooperate in this can only be all the greater.
At the end of this speech I may first of all express my respectful thanks to Her Majesty the Queen for having pleased me to appoint me as an extraordinary professor at this university.