Fokko Jan Dijksterhuis, Universiteit Twente
When I started my PhD research some ten years ago, the topic was Huygens’ wave theory of light. One of the central questions was how he brought about the mathematization of the mechanistic nature of light, that is: how he fitted the motions of imperceptible corpuscules into a mathematical conception that enabled the derivation of the laws of optics, thus explaining the phenomena of reflection, refraction, and so on.
Huygens’ wave theory is a text book case of ‘the mathematization of the world-picture / nature’. That is: of the crux of the scientific revolution. The unfolding of a universe of precision in the hands of Galileo, Descartes, Newton. These guys thought of nature, observed nature and applied their versatile mathematic thinking to fit in the phenomena; no: the essence of nature. This view raises a range of questions like: what mathematics did they apply, where did they get it, what exactly is applying, and so on, and so on. Questions like these I learned to phrase only after I found out that Huygens’ mathematization of light took a very different sort of course.
What I found out when studying the development of Huygens’ optics, most of it in writings he did not publish, was the following. The seed of the wave theory was the problem of strange refraction, he stumbled upon in the early 1670s. Strange refraction is a phenomenon displayed by Iceland crystal. Basically, it comes down to rays of light not being refracted according to the sine law of refraction. In particular, a perpendicularly incident ray is refracted from the normal. To Huygens this posed a problem for his understanding of waves of light, which drew on father Pardies’ theory. It presupposed that rays are always normal to waves. Well then, if a perpendicularly incident ray is refracted this can no longer be the case as Huygens noticed in his manuscript notes (and kept on puzzling over).
By the way: Huygens’ problem with strange refraction is a fine example of circulation of knowledge. He was in Paris at the time; the crystal (plus Bartholinus’ book describing the peculiarity of its refraction) had been brought to Paris from Copenhagen by Picard on his geodetical expedition; the crystal had been brought to Copenhagen from Iceland for the cabinet of curiosities of the Danish king, where Bartholinus was asked to study it.
Aside from the question why he thought it was a problem and why he wanted to solve it, what is fascinating is the way he tried to solve it. Huygens analyzed the behavior of rays in Iceland Crystal and took Descartes’ analysis of ordinary refraction to extend it to strange refraction. (While ordinary refraction affects the perpendicular component of an incident ray but doesn’t alter the parallel component, strange refraction adds a strange, constant measure to the parallel component.)
What is interesting about this is not so much that Huygens borrowed a theory he had a little earlier explicitly rejected. And neither that he considered the behavior of rays rather than waves, although strange refraction was a problem of waves. What is interesting is that it gives clues to the questions just phrased. It gives an idea what mathematical thinking or the application of mathematics in early modern times might means. We see that Huygens appropiated a piece of mathematics – Descartes’ analysis of refraction – and employed it in a new context – Iceland crystal.
This is important, because it sheds light upon the next step of Huygens’ struggle with strange refraction. Or struggle ... once he conceived of a new understanding of ether waves and their propagation the solution to the problem of strange refraction presented itself almost effortless. He supposed waves of light to propagate in the crystal at different speeds in different directions – which is not normal – resulting in elliptic waves. Getting the parameters right, the refraction of the perpendicular ray follows immediately. The refracted ray is indeed not normal to the wave, but with Huygens’ new principle of wave this did not pose a problem anymore.
What is interesting about this solution is that it was not so much about ether, corpuscules, the crystal, but a clever tinkering with circles and - eventually – ellipses. Leafing back through Huygens’ manuscripts, we see that his new conception of wave propagation likewise was a matter of mathematics. It came up when he was tackling the rather technical problem of caustics. What he had done was transferring his ideas on linear motion and impact to ether waves and reducing the propagation of those waves to one defining characteristic: velocity. What he did, in other words, again was borrowing some mathematics from a different context to attack a particular problem.
So, we now have a better idea what mathematization might be. It is interesting to notice that Huygens did not bother about nature so much – let alone the universe – nor about the mechanistics of ether motions. What he bothered about was the mathematics of the problem and he wanted to get the mathematics right. Find out where the relevant mathematics was – rays, waves, ... – then how to phrase the matter exactly – components, velocity.
The problem of strange refraction did not arise out of the blue. In 1672, Huygens had been working on the mathematics of refraction for some twenty years. The mathematics of refraction, that is: dioptrics, the analysis of the imaging properties of lenses and their configuration. He knew a lot optics and he had developed a lot of optics [=math]. And what he did when dealing with strange refraction and eventually renewing the ether waves of light, was extending his work in dioptrics to the properties of waves.
The story about Huygens’ wave theory can be generalized to create a different view of the history of seventeenth-century optics. If we begin with Kepler – who else – you see a well-developed body of optical knowledge, albeit of a very different nature than modern optics. Perspectiva, the Medieval designation, was the science of mathematically reconstructing visual appearances. ‘Optica’ to some extent can be called geometrical optics, although questions of perception and cognition were central. What happened was that within this ancient field new questions arose in the seventeenth century, questions regarding the physical nature of light. And these questions were approached from within this existing field. These were questions about the mechanics of refraction and the like. Descartes notorious explanation of refraction is a perfect example of the dynamics I have in mind. Tinkering along the mathematical lines of ‘optica’ people like Kepler, Descartes, Hobbes, Barrow, Huygens, Newton tried to tackle the new questions of the physics of light, analyzing old phenomena like reflection and refraction as well as new phenomena like strange refraction, dispersion and diffraction.
And the funny thing is: when struggling with dispersion Newton did exactly the same as Huygens when struggling with strange refraction. He analyzed the behavior of rays, borrowed Descartes’ derivation of refraction and extended upon it to formulate a regularity of unqual refraction. I have elaborated this in my article ‘When Snel breaks down’, in which I have tried to show the importance of mathematical practice in the development of optics.
My point is: mathematization of the physics of light was not so much some kind of mathematical contemplation of etheral corpuscules and the like. Some kind of mathematically philosophizing. There was much more doing involved: tinkering with mathematical objects, using expertise, skills and strategies developed in the existing field of geometrical optics. More important, it provided the conceptions of valuable knowledge that directed these efforts. This resulted in an optics in which the nature of light was central and the goal was to explain phenomena by deriving the laws of optics.
My experiences with Huygens’ optics led me to question the concept of mathematization and its historical elaboration.
Mathematization is an old theme in the history of science and when you look at recent literature it seems to be out-dated. It has disappeared from the big pictures of the scientific revolution. Even Peter Dear, who has innovated the theme like no-other, pushes it to the background of his well-known textbook. Although the rise of mathematics seems to form the pivot of his renaissance and revolution account, in the conclusion the theme of ‘operationalization’ figures as the defining characteristic of the new science. (Mathematization has not disappeared from narratives on 19th and 20th century science, see for example Porter of which I draw a lot of inspiration.) I think the theme of mathematization has erreneously disappeared from the narratives of the scientific revolution. (I leave aside the question of the historical importance or even existence of the scientific revolution.) I think mathematization is historically important. It is one of the most far-reaching feature of modern technoscience. We must, however, breathe new life into it. That is: we must ask ourselves what was mathematized, how did went about, what did this mean, and so on, and so forth.
Why did mathematization fall out of favour? It has to do with the way it has been interpreted and applied historically. It was the linchpin of the classical history of science with its focus intellectual history and it philosophical understanding of science. Mathematization was understood in a basicly platonic way, in which a concept of ‘application’ of mathematics was used more or less implicitly. Application is of course a very troublesome concept. Historically it is a byproduct of nineteenth-century ideological discussions about ‘pure’ mathematics – starting in Enlightenment – and, more important, political and institutional discussions about education, organization and the like. As Gerard Alberts has shown, the idea of applied mathematics was invented in the nineteenth-century university. It remains to be seen whether it has ever existed at all. At least we may conclude that it is ahistorical to speak of application of mathematics when dealing with the seventeenth-century.
I think it is more fruitful and historically more sound to speak of mathematization as the production of mathematics in new domains, involving transfer of practices from other domains in which mathematics is already practiced. These can be knowledge domains, domains of crafts, but also social domains, institutional domains. The creativity of introducing mathematical practices in new domains then consists of ingeniously appropriating tools and methods developed elswhere. And appropriating, as we all know, is creating in itself.
The example of Huygens I just gave shows a transfer from mixed mathematics to natural philosophy in which a new kind of dealing with light was created. One might also think of the transfer of generalizing formal approaches to practical domains – for example when tacit knowledge of telescope manufacture is translated into mathematical language – or vice versa – when problems of telescope manufacture set the agenda of abstract analysis.
I can think of a wealth of examples.
Coulomb who transferred his engineering expertise to the measure of electrostatic and magnetic forces and built an instrument – the torsion balance – that materialized his analysis of those forces. An instrument, by the way, that was nothing else than a reversal of his earlier torsion compass. And Coulomb did not just transfer his expertise, but also transferred himself and his engineering persona to the world of academic savants.
Kepler mathematized penumbra by analysing the passage of rays through pinholes in a way in which his optical axioms were translated into tangible set-up of threads and frames, much like Dürer’s instruments of perspective drawing.
Helmholtz mathematizing tones by analyzing sound in a siren.
Etx.
Examples like these draw our attention to a particular characteristic of mathematics: its operative nature. Knowing in mathematics is often knowing how: to know calculus is to know how to make a derivative, to know a equilateral triangle is to know how to draw it. Of one looks at seventeenth-century publication on mathematics a lot of it deals with how to solve, construct, or make something. A fascinating thing about mathematics is that the distinction between mind and body is not very clear. A lot of mathematical doing takes place in the mind and often thinking can be realized by the hands, with compass and ruler. Or threads and frames as with Kepler.
It is very important in this regard to realize that mathematics before the Enlightenment was essentially mixed mathematics: to various degrees of abstraction mathematics was the analysis of concrete things. We should not make the mistake of seeing for example optics as mathematics applied to rays. Rays were the object of mathematical inquiry, namely being light considered in its mathematical aspect. Consequently, a ray is mathematical. And optics is mathematics. Thus the subject matter of mathematics was much broader than we see it today. And much more wordly, even in the cases of geometry and algebra. If we look for a hiërarchy in mathematics, the latter will not appear on the vertex but rather astronomy to which the other mathematical disciplines were subservient.
So, we have to look at different mathematical practices, at different mathematical domains, at different – I would say mathematical practioners but that term is reserved for a very specific group of seventeenth-century mathematicians – at different mathematicians. And this is a second thing classic history of science did not recognize: that mathematics is heterogeneous. If you look at the domains in which things mathematical were practiced, you will see that a single, bounded field did not exist. Already on the level of the general word a distinction was made between ‘mathematique’ and ‘géométrie’. The latter being the scholarly, academic pursuit, the former the practical, crafty. First of all, it was a social distinction between the learned elite and the working engineers.
If you look further, it turns out that a multivariety of activities existed that can be called mathematical, and were called so, of which the mutual connections are not self-evident. Counting and reckoning, constructing and measuring, stargazing and mirrormaking, encryption and analysis. But also philosophizing, see how Molyneux praised Descartes’ Meditations first of all for its mathematical character. This leads to the conclusion that mathematics as such is heterogeneous and that historical caution should be taken when humping together all kinds of mathematical activities – be it as a ‘universe of precision’, or a mechanization of the world-picture, and so on. The question that needs to be asked first is which connections existed between the diverse activities we connect by our understanding of mathematics. Did a mathematics exist in the early modern time that had all kinds of manifestations, or should we ask ourselves how those heterogeneous activities were connected?
Assuming that our idea of mathematics is a nineteenth-century invention, we should for the time being approach those practices as unrelated in order to look anew for the connections that were made at that time. We may in the meantime keep in the back of our minds the surprizing circumstance that connections existed between all those mathematics at all. It is not a coincidence that misunderstandings could arise about the status of Christiaan Huygens as a ‘mathématicien’ or ‘géomètre’. And it is no coincidence that Descartes engineer could embark upon the creation of a mathesis universalis. Apparently there is something mathematical that enables contact and transfer. That enables Huygens to deploy his expertise in dioptrics to work on ether waves.
What will be a good approach? The problem of programmatic texts from the period, like Barrow, create an intellectual arrangement, often aimed at philosophical questions of mathematizability. If you want to start with practices it is better to look at the practitioners – including a Barrow. What did they do, how did they come in contact, what did they exchange and how, how did it transform?
The Dutch Republic is interesting because one finds here a broad spectrum of practitioners with close links, an assembly in which all and sundry moved about. Mathematics was a thriving enterprize in the early life of the Dutch Republic. Practitioners found employment with towns and provinces in the development of the new society, savants cultivated the metamorphozing mathematical ‘scienze’, the cultural and political elite appropriated the new ‘esprit géométrique’. People engaged in mathematics were a motley company, ranging from arithmetic teachers like Bartjens, surveyors like Wassenaer, professors like Metius, ‘amateurs’ like Christiaan Huygens, statesmen like Johan de Witt, and so on. Mathematics in the early Dutch Republic was a multifaced enterprize that yielded a large variety of intellectual and material production.
Take again our Chistiaan. He had been educated by Stampioen and Van Schooten, who both moved about the elite circles in which Constantijn Huygens figured prominently. They engaged in exchanges with mathematicians like Descartes, aspired a place in the Duytsche Mathematicque of training surveyors. A lot is known about this network and the mathematics it produced. But mostly the mathematics has been discussed ‘qua’ mathematics: distilling the mathematical portions and leaving aside the heterogeniety of content and context. What did its creators have in mind?
Take Van Schooten. He is well known for his translation of Descartes’ Géométrie which is generally acknowledged as a highpoint of mathematics in the Dutch Republic. His previous work, however has hardly been considered, let alone its chronological development. When you consider the Geometria as the coping stone of a lifelong career, a nice picture arises of someone he employed his mathematical activities to move up step by step in both the patronage and the intellectual hiërarchy of the Dutch Republic.
Frans became professor at the Leiden engineering school instituted by Count Maurits to provide formal teaching for surveyors: mathematics in the vernacular, or ‘Duytsche Mathematicque’. Frans’ father – Frans the elder – had established a firm tradition in Duytsche Mathematicque, closely linked with actual practice. In the winter he taught the theory of fortification, while attending field practice in the summer. He published on trigonometry, including a book on sine tables that went through numerous editions. The Van Schooten sons continued their father’s curriculum and thus their classes at the Engineering School became rather conservative. The lessons on fortification remained to be based on the old-dutch system as it had been codified by Stevin and Marolois. Around the middle of the century complaints were voiced that the education had not kept up with recent developments. Next to his lessons in Duytsche Mathematicque, Frans the younger, took his mathematics higher up from the theory of measuring. He redirected his attention to recent developments in ‘higher’ mathematics that presumably were more prestigious.
There are several reasons to think of, why Frans the younger took his mathematics interests higher up. In the first place, he had studied at Leiden University, with Jacobus Golius (or Jacob Gool) (1596-1667), professor of Arabic and successor of Snellius at the chair of mathematics. Secondly, through Golius he came in contact with recent developments in geometry and got access to Holland elite circles. Thirdly, he remained free of obligations until a relatively late date. Although he started replacing his father at the Engineering School in 1635, from 1641 to 1643 he traveled to France (and allegedly England). In this period, he gradually developed his acquaintance with the new theories in mathematics. Or rather geometry, as the word mathematics was used in the seventeenth century for the less lofty practices of measuring and calculating.
With professor Golius Van Schooten first met Descartes, who had come to Leiden in 1630. He quickly became one of Descartes’ favorites and assisted him on several projects. He made the illustrations for the essays of Discours de la Methode and drew a template of a hyperbola for the grinding of a non-spherical lens. This latter project was organized by Constantijn Huygens, who was introduced at Golius to Descartes in 1635. To Van Schooten the participation of Huygens meant a direct access to the Holland elite, for Huygens was a prominent figure in the highest political and cultural ranks. He was secretary to the Stadholder and a renowned poet and composer.
The association with Golius created opportunities for Van Schooten to go beyond his milieu of Duytsche Mathematicque. Around 1639 he wrote an introduction to Descartes’ geometry, a basic exposition of the new method of letter calculation. Sending it to Mersenne, Van Schooten used it as his introduction to the Republic of Letters. He later published it as Principia Matheseos Universalis (1651) and included it in volume two of his second edition of the Geometria. Around the same time he stroke a deal with the Leiden publisher Elzevier to collect writings of the new French mathematicians. He traveled to France in 1641, where he copied several manuscripts of Fermat and Viète. It resulted in the publication of Francisci Vietae Opera mathematica with Elzevier in 1646. The same year Van Schooten had published his first original work, De organica conicarum sectioneum in plano descriptione, an exposition of the kinematic generation of conic sections in the manner of Descartes and Mydorge. It mirrored the gardener’s method of drawing ellipses and hyperbolas Van Schooten had illustrated for La Dioptrique, while elaborating such procedures in a general, geometrical exposition.
So, in 1646 Van Schooten had established himself as a teacher of the new geometry. He gathered the writings (and acquaintance) of prominent mathematicians and rendered them into a didactically appropriate form. In this sense, he continued the tradition of Duytsche Mathematicque of providing textbooks of mathematics. However, Van Schooten’s books went far beyond the original Duytsche Mathematicque. They led abroad to the new geometry in France in the academic tongue of Latin. The project for a translation and adaptation of La Géométrie, that would become the acme of his work, perfectly fitted this purpose.
With Geometria, Frans van Schooten transported and manufactured Descartes’ Géométrie. He did not just translate it – thus making it accessible to the learned world, Cambridge in particular – he also transformed it by commentaries, explanations, elaborations – thus refining it to the educational context he was in. He did so by mobilizing his patrician students: Huygens, De Witt, Hudde. So his teaching was a means of manufacturing the text. How Dutch! From France via the Duytsche Mathematicque of Holland to the world. Rags to paper. This produces not only a nice historical story of the development of mathematics from the Duytsche Mathematique to calculus, it also shows how the specific instances of mathematics he worked on were linked to persons increasingly higher in the hierarchy.
Van Schooten’s strategy was succesfull. In the 1650s the surprizing thing happened that patrician sons went to study with the professor of Duytsche mathematicque. Why didn’t they go to the real professor, instead of this teacher of the masses?
“And in this church, where the English preach nowadays, in this beguinage, all days (except Wednesday and Saturday) from 11 to 12 o’clock, public lessons are given in the Dutch language, on the mathematical arts, for the convenience of the unlettered, like bricklayers, carpenters, and the like; who at that time find themselves here in crowds without coats but equipped with their sticks, aprons, etcetera; which then is very farcical to see. The professor, who gives Dutch lessons, nonetheless in his usual distinguished professor gown, or coat, (like al the other Latin professors do theirs,) is the very learned, and widely renowned sir Franciscus van Schooten.” 1
One way or another Van Schooten had managed to acquire enough status to have the young Huygenses, the young De Witt, the young Hudde, the young Heuraet come and study with him.
In this way Van Schooten’s mathematical activities uncover a whole network of mathematically interested and active people. And this is not all. Via the engineering school he dissiminated his mathematical teachings to a whole world of practitioners that set out – after they had passed the provincial exams – to lay out and plan Dutch society and thus create the landscape of the Republic. this is a process that deserves further study.
Now back to upwardly moving Van Schooten. Why did those patrician boys wanted to study mathematics? Or – a question that to my surprize has not yet been asked – why did Jan de Witt write an Elementa?
This brings me to my main interest in mathematization: how come mathematics turned valuable in the early modern time? If you look at the reasons why people pursued mathematics, or said it ought to be pursued, the same heterogeneity appears. All kinds of values are linked to mathematics: it is useful, it is a source of lucid thinking, it is a model of rationality, and so on. A whole range of uplifting features.
The engineering school renowned as a pioneering effort to institutionalize technical schooling and Maurits cum Stevin are praised for their vision. However, why would someone who was going to lay out urban developments, build bulwarks, design polders, or manage rivers, need to read up on Euclid? Maurits and Stevin merited mathematics in particular ways and interwoven with their vision of a new society. They were not the only ones, at the same time similar things went on in the province of Friesland and later on in the seventeenth century the social, political and cultural values of mathematics is a recurring theme in Dutch life.
When speaking of mathematical production in the Republic you find a broad and heteregeneous site at which a lot was going on. For example the odd circumstance that patricians started to value mathematics particularly. This is a site where mathematization can be studied as a transfer of practices between domains, between people, between groups.