Archimedes’ legacy for early modern science: Historical-philosophical reflections

The core of Greek mathematics

Reviel Netz's paper on ‘The place of Archimedes in world history’ is an ambitious project not only in terms of space and time, but also in its attempt to elaborate historical as well as philosophical/epistemological insights. It tells the story of Greek mathematics as a contingent ‘individual event’ whose main protagonist was Archimedes (d. 212 BCE), considered here to be the most prominent mathematician of the Greek world. In this story, Archimedes’ work opened the door to Galileo, Kepler, Descartes and Newton, hence to the scientific and industrial revolutions – the main building blocks of the modern world.

Netz divides his project into three parts:

(a) The first and longest part presents the historical kernel of Greek mathematics as the product of two generations of prolific mathematicians: the first emerged in fourth century BCE Athens, among whose main figures were Archytas (d. 347 BCE); Theatetus (d. 369 BCE) and Eudoxos (d. 337 BCE). One remarkable feature of the first generation's discourse that separates it from the second generation was its involvement in dialogue with philosophers (Zeno of Elea; Leucipus; Democritus).

The second generation of Greek mathematicians emerged in the Hellenistic period (circa 323 BCE-30 CE) and included, among others, Archimedes (d.212 BCE) followed by Apollonius of Perga (d. 190 BCE) and Hipparchus (d. 120 BCE). The mathematicians of this generation seem to have developed a tradition of critical exchanges among themselves, thus responding to the work of their fellow mathematicians and establishing an internal mathematical discourse. This first part of the project, however, also aims at tracing the wider context of Greek mathematics, especially in comparison with old, non-Western mathematical traditions such as Mayan, Chinese and Mesopotamian. Thus, Netz delineates the universal dimension of mathematical activities but also the differences between their particular cultural contexts and the style and contents of the mathematics they developed, partly resulting from such differences. Against this background, he emphasizes the uniqueness of Greek mathematics.

(b) The second part of this project deals with the historical narrative concerning the legacy of Greek mathematics and Archimedes’ role in it in the transition from late antiquity to early modernity. Here, the effort is dedicated to pointing out the dynamic nature of Greek mathematical discourse, its potential mobility and the developments and modifications it underwent while travelling, especially to the Islamic world, to Renaissance humanists interested in its reconstruction, and to seventeenth century scientists like Galileo and Descartes. Against this background, Netz offers his readers a modified view of 16th–17th new astronomy and the physico-mathematical science of mechanics that he conceptualizes not in terms of revolution, but in terms of transformations within the old Archimedean framework.

My comment on Netz's rich and provocative project will follow his three parts division. Thus, I shall first relate to his characterization of the mathematical work of two Greek generations as a clue to its uniqueness. Netz is a world expert on this topic, and in the first part of my review, I shall follow in his footsteps in an attempt to sharpen his main claims about the contents, form and context of the Greek mathematical corpus. My second chapter will be dedicated to some additions to and modifications of Netz’ historical story, while in the third part I shall focus on a new socio/epistemic framework of my own which, I believe, can clarify the mobility of scientific bodies of knowledge throughout historical time and space.

Contents and style of Greek mathematics

As mentioned above, the first generation of Greek mathematicians emerged around the fourth century BCE and hence should be seen as the offspring of the Athenian Classical Age. The protagonists of this generation were mainly located around Athens, where they established their networks of communication. Netz emphasizes the uniqueness of their contribution by drawing attention to Archytas's discovery of the numerical pattern behind musical harmony and to Eudoxus, who ‘found the hidden geometrical pattern underlying the motions of the planets’ (Netz 2022, 304). Both inspired Plato and Aristotle, thus giving birth to the dialogue of mathematicians with the philosophical culture around them. Their work survived mostly in fragments, and much of our knowledge about them depends on Euclid's Elements. Netz emphasizes their outstanding originality and creativity, which he sees not only as the outcome of their special areas of interests, but also of their social context. Rather than focusing their attention on obvious aspects of mathematical practices such as measuring – in numerical terms – a field, or a pile of grain, Greek mathematicians tended to spend their time in comparing – in geometrical terms – the areas of curved figures with simple rectilinear figures. Such interests gave birth to surprising, counter-intuitive results that Netz sees as the core of Greek mathematical thought. Moreover, Netz tends to relate the status of Greek mathematical practices to the cultural position gained by Greek mathematicians in the Classical Age (between the beginning of the fifth century BCE and circa 323 BCE). Thus, he argues that classical Greek culture around Athens was mainly a literary culture that tended to attribute the poetic and epic products of artists to the specific private names of their composers: poets, narrators, and authors. Such practice completely changed the position of literary producers, endowing them with a special authority that was not common in earlier time, tending to see the artists as expressing a divine wisdom or voice. According to Netz, the first generation of Greek mathematicians acquired a position of authority following poets and artists: like them, their achievements were recorded under their private names. Thus, both the objects of mathematical discourse – numerical or geometrical patterns underlying musical harmonies or the motion of the planets – as well as the authority of mathematicians assumed particular nature that further crystallized with the second generation of mathematicians of the Hellenistic world.

The figure of Archimedes shines through as the most outstanding mathematician of the second generation. Two of Archimedes’ contributions deserve special attention. The first concerns the particular kind of the mathematical objects he invented such as the ellipse, parabola or hyperbola, produced by cutting a cone in particular ways. Even more outstanding were the paraboloids that emerge while rotating such curves around their axis and whose volume is found following the rules discovered and proved by mathematicians. It is hard to imagine Greek mathematics outside the realm of such ‘constructed’ objects (not merely given objects like a field whose area is to be measured). Rather, they constitute the basis of a special mathematical discourse of propositions, diagrams and deductive proofs. Netz relates the rigorous nature of Greek mathematics to the need for firm, deductive proofs that necessarily accompanied the challenging mathematics of counter-intuitive and witty surprise. Naturally, surprising results needed to be rigorously proved if they were to support unexpected truth claims. Not least, the mathematics of surprise, developed as the legacy of Archimedes, cannot be understood without realizing the extraordinary structure of authority of Greek mathematicians, mainly of the second generation, who thrived on their specially named contributions that pushed them to pursue fame as well as prizes.

The second contribution made by Archimedes and other Greek mathematicians of the Hellenistic era concerned their tendency to mix genres of discourse, the discrete (numbers) and the continuous (lines and areas), or the physical and the mathematical, as found in Archimedes’ On Floating Bodies. Such work was based on the study of the center of weight of bodies set in liquid, culminating in research of segments of paraboloids that are lighter than the liquid in which they are set. Thus, Archimedes showed the way to treating physical problems with mathematical tools. This can be seen as a breakthrough towards physico-mathematics that only developed much later, during the early modern period.

The originality of Greek mathematics, Netz claims, lies in both its contents and its form. In terms of its contents, Greek mathematicians enriched geometry with the new kind of constructed objects that enabled the Greeks to create sophisticated astronomical models beyond those developed by Eudoxus (Netz 2022, 314). They also managed to approach problems equivalent to equations of higher degree than the first and the second degree. And, Greek mathematicians started to move from pure geometrical questions to physico-mathematical science such as Archimedes’ science of statics and hydrostatics. While developing these new areas of interest, they crystallized the idea of the special epistemic status of mathematics.

Certainly, there was a price to be paid for such achievements. For, in order to treat physical problems with mathematical tools there was a need to idealize physical reality by treating it in terms of points, lines and arcs. Thus, solids were treated by Archimedes in terms of single points based on the concept of center of gravity, while motions of objects or rays of vision were thought of in terms of combinations of straight lines or circular arcs.

Summing up the outstanding position of Greek mathematics in world history – according to Netz – one may emphasize the following points:

First, it is important to realize the breadth of the corpus left by Greek mathematicians that consists of thousands of papyrus rolls (by 150 BCE), written by approximately 150 creative and original mathematicians, a number that exceeded that of previous time in a variety of cultures of the ancient world, but is still quite restricted. Second is the systematic nature of this corpus that Netz analyses in terms of both contents and form. Furthermore, unlike the case of ancient Mesopotamia, where mathematics was relevant to state affairs, and mathematicians were mostly part of state bureaucracy and the educational system, Greek mathematicians were mostly citizens of city states – most prominently Athens and Alexandria – who left behind them written texts. Hence, their later mobility depended more on their texts being read than on oral practices such as face-to-face teaching. The Greeks, then, were able to develop a professional mathematical discourse with an authority of its own that started in dialogue with philosophers, but developed into a conversation among mathematicians who commented, criticized and reacted to each other. It was finally by the rules of such discourse that Greek mathematics developed while traveling to the Islamic world and to Europe in later generations.

The historical narrative of Greek mathematics: between antiquity and early modernity

The second part of Netz's story about the role of Archimedes in world history is dedicated to the history of Greek mathematics between late antiquity and early modernity. Starting from its reception in the very different contexts of the Byzantine Empire, among Islamic mathematicians between the ninth and twelfth centuries and by the Persian Maragha school in the thirteenth-fourteenth centuries, Netz tends to see the Byzantines as interested mainly in curating Greek mathematical works, and sometimes in teaching them. As for the Islamic contributions, Netz's clear judgment is that they were ‘very Greek’ since Islamic mathematicians were mainly interested in emulating the Greeks, including their development of the individual proofs extant in Khayyam's Algebra (Netz 2022, 320). The main focus of Netz's historical narrative, then, is dedicated to mathematics from early modernity up to Newtonian mathematical physics of the seventeenth century. This focus is well chosen from Netz's perspective. Netz suggests that the experience of early modern scholars of mathematics, optics, astronomy and mechanics was far from being expressed in revolutionary terms, as inventive historians like Koyré and Kuhn claimed. According to him, the popular history of the ‘scientific revolution’ – that of Koyré and Kuhn – deviated substantially from the major historical agents’ vocabulary that he strives to restore and interpret while focusing on Copernicus; Kepler; Galileo; Descartes and Newton.

Thus, Copernicus – in Netz's story – appears as the last Ptolemaic astronomer, dedicating most of his life to studying Ptolemy's techniques and trying to re-write him. Copernicus, he argues, did all that by following closely in Ptolemy's footsteps: identifying the relevant circular rotational movements; analysing them geometrically; performing few observations that would allow him to fix the basic parameters of such movements; and finally using trigonometric tables for his final calculations, which he used in order to represent more accurately celestial motions.

And what about Copernicus’ heliocentric turn that has endowed him with the reputation of an arch-revolutionary? Well, in Netz's story, heliocentrism was included in the repertoire of antique cosmologies, and Copernicus, being a trained humanist, saw himself as continuing the classical debate between geocentrists and heliocentrists of the ancient world. He just chose a different option to what became the majority position after Ptolemy. Copernicus, then, was the first among a series of fathers of the new science who actually strove for a Renaissance of Greek mathematics, but certainly not for a ‘scientific revolution’. Furthermore, Galileo, Kepler, Descartes and partly even Newton – the leaders of the ‘new science’ – actually anchored their own innovations in Archimedes’ science. Hence, in Netz's narrative they are represented as Archimedes’ followers, mainly attempting to revise ancient mathematical traditions.

A few more examples would suffice for clarifying this major claim of Netz. As is well known, Galileo opened new horizons towards a modern theory of motion while first using a model of fall based on Archimedes’ hydrostatics. Here, however, one should add that this model actually failed. Later on, Galileo discovered the parabolic trajectory of the projectile, while Kepler developed his new theory of the elliptical trajectories of celestial bodies, both adopting mathematical tools from Apollonius’ work on conic sections that originated in Archimedes’ work on curves.

Among the achievements of early modern science, it was mainly the philosophically inspired mathematics of Descartes that opened the door for the new scientific paradigm of Newton. Descartes read ancient mathematics as disguising a secret method of discovery – analysis – that allowed for solving geometrical problems by means of algebraic equations. In doing so, he probably followed Islamic mathematicians, whose techniques he interpreted as a continuation of Apollonius’ work. On this basis, Descartes pushed forward towards analytic geometry. However, Netz calls the readers’ attention to the fact that Descartes actually believed that he had reconstructed Greek mathematical knowledge and saw himself as the last of a series of great Greek mathematicians: Euclid, Apollonius of Perga and Pappus of Alexandria. In this he was following Cavalieri (1598–1647), who relied on Archimedes’ idea of considering a curvilinear figure as constituted by rectilinear figures made indefinitely smaller. It was by following the Greeks, then, that Cavalieri and also Descartes came up with the notion of an ‘integral’, thus pushing towards Leibniz's and Newton's calculus.

Finally, to sum up Netz's historical narrative, it is also essential to mention the philosophical insights articulated by the fathers of the new science, first and foremost their conviction that mathematical proofs were far more demonstrative than syllogistic ones. Such a claim expressed their changing conviction concerning the epistemic status of mathematics, succinctly present in Galileo's slogan that the book of nature is written in mathematical letters. However, for Netz such a claim does not necessarily manifest a ‘revolutionary spirit’: rather, it underlies the enormous break within Greek mathematical tradition of the second generation, represented by Archimedes’ innovations. In other words, Galileo's claim interpreted by Netz is a further proof that Galileo's was not revolutionary, but rather a revisionist voice.

My reaction to Netz's historical narrative is complex and twofold: on the one hand, I fully agree with him in showing that in general the protagonists he chose for his story were mostly followers of Archimedes. However, I do not see how this fact is sufficient for illuminating, in historical terms, the necessary conditions for the major shift in the development of early modern science, often called the ‘scientific revolution’. Netz rightly refuses to name such shift a ‘revolution’. Instead, he sees the shift as a mode of ‘transformation’ of Greek mathematics, and particularly of Archimedes’ body of knowledge. Continuing his thoughts, I shall add some remarks on the transformations of mechanical knowledge in the early modern period. In addition, I shall present a historical case study concerned with the discovery of sunspots at the beginning of the seventeenth century in order to concretize my conceptualization of bodies of knowledge on the move – hence being transformed – in time and space.

First to mechanics, where I shall be using the definition of Damerow and Renn, according to which mechanical knowledge concerns material bodies in time and space, their motions, and the forces that cause or resist such motions (Damerow et al. 1992; Damerow and Renn 2012). Such a definition perfectly fits Netz's perception of Archimedes’ science as is expressed in the following citation that continues a thought about a breakthrough in pure geometry brought about by the second generation of Greek mathematicians:

‘Greek geometry’, especially that of Archimedes, was not ‘pure’: there was a premium on the combination of scientific projects and so Archimedes pioneered several such studies, of which we have extant the study of statics and hydrostatics. (Netz 2022, 310)

Whereas it is clear that Archimedes’ science was not pure, being concerned with real material bodies, it was definitely conceived as a ‘mathematical’ science, dealing with the equilibrium of physical bodies, for example, but not with motion. In medieval schemes of knowledge organization, however, mechanics and astronomy – among other mathematical sciences – were known as ‘scientiae mediae’, namely existing ‘in between’ mathematics and natural philosophy, to which, according to Netz, Archimedes and his followers extensively contributed. In contradistinction, for the protagonists of the ‘new science’, mechanics and astronomy were explicitly seen as physico-mathematical sciences. Taking the long-term perspective of Netz's essay, but also relying on other insights (Damerow and Renn 2012; Henninger-Voss 2000; Valleriani 2010), implies that theoretical knowledge about mechanical phenomena emerged from everyday experiences and the accumulation of practical know-how concerning tools and machines. These had a major role in leading to reflection, conceptualization and codification of mechanical intuitive explanations. Moreover, it seems that already in antiquity, and later on in the Islamic context as well as in medieval universities, mechanical issues brought about encounters and debates between philosophers of nature and mathematicians. Such debates preceded the intense conversation between practitioners dealing with machines, fortifications and architecture and other mechanical arts – construction, metallurgy, agriculture – and theoreticians focusing on the interpretation of texts that took place during the Renaissance. In my historical perspective, such encounters, debates and conversations were at the heart of the shift that led to modern science, and thus in need of elaboration.

I shall start my schematic narrative line about the transition and transformation of mechanics from a mixed mathematical science into physico-mathematics with a few relevant facts in need of elaboration here. I shall first refer to the project of revival of Greek mathematical texts that stands behind the theoretical shift from mechanics as a ‘subalternated’ mathematical science into physico-mathematical mechanics (Rose 1975; Laird 1986). Such shift also demanded an account of motion differing from the one so popular in ancient and medieval discourse on natural philosophy. I shall then refer to the changing status of practical knowledge in the Renaissance that brought together practitioners and theoreticians and contributed immensely to the conversation and feedback-loop between them. All these, I am arguing, will throw light on the transition from Archimedes to pre-classical and classical mechanics as I understand it.

The sixteenth century saw the completion of a huge project of editions, translations and commentaries of mathematical and mechanical texts. (Pseudo)-Aristotle's Mechanica (Mechanical Questions) was the first of these texts to be translated into Latin and then into Italian and became a focus of discussions about the balance and the five simple machines, about its relation to Aristotle's theory of motion, and the relationship between philosophy and mathematics (Aristote and Bade 1517; Tomeo et al. 1525; Piccolomini and Blado 1547). After the translation into Italian by an engineer (Guarino 1573), it was read both by practitioners and by university mathematicians. In addition came the translations of Euclid and Jordanus into the vernacular, done by Tartaglia (Euclid et al. 1543; Jordanus and Tartaglia 1565). These were followed by the publications of new and authoritative Latin editions by Frederico Commandino of the works of Archimedes and his successor Pappus with the fragments from Heron's Pneumatica that was then edited, published and translated by Aleotti and others. These were followed by the translation of the first book of Apollonius of Perga, done by Commandino, and edited by Guidobaldo del Monte (Commandino et al. 1565; Pappus et al. 1588; Apollonius, Pappus, and Serenus 1566). Among the most significant publications within the project of revival was the elaborate modern commentary to Euclid's Elements by Christopher Clavius (Euclid and Clavius 1574). as well as Clavius’ commentary on the Sphere (Clavius 1585) that contained passages relevant to mechanics: for example, Aristotle's account of what would happen if the earth was displaced from the center of the world.

Establishing a rather complete, reliable body of texts was one element needed to ground the idea of a Euclidean-Archimedean tradition to which meaningful groups of sixteenth century mathematicians belonged and contributed. People like Alessandro Piccolomini (1508–1579) and Frederico Commandino (1509–1575) – translators and editors of the pseudo-Aristotelian Mechanical Questions, as well as of Archimedes’ writings on equilibrium of planes, centers of gravity and floating bodies – were followed by practitioners and mathematicians of the mechanical arts. Among the latter were Niccolò Tartaglia (1499–1557), Giovan Battista Benedetti (1530–1590) and Guidobaldo del Monte (1545–1607), all able to create a dialogue between mechanical practitioners becoming aware of, and interested in theoretical mechanics.

One of the drivers for the growing diffusion of mathematical knowledge by the end of the fifteenth century was the political-economic and technological conditions in the Italian territories. The fifteenth century saw the re-organization of communes into greater and more centralized city-states, some maintaining a form of a republican regime, others assuming the form of duchies and principalities. The re-organization also meant intensification of war, hastened by the introduction of firearms to Europe just before that period. The building of new roads that allowed for better transportation of canons; the reconstruction of city centers, and the development of new ways of defending cities through fortification, all demanded the services of architects, cannoneers, military engineers all applying practical mathematical knowledge in fulfilling their tasks, and raising the curiosity of professional academics.

Enhanced interest in mechanical arts was first advanced by a new class of engineer-scientists, many of them Italian (Bennett 2011; Valleriani 2010; Valleriani 2017). It was diffused throughout Europe and beyond, through centers of knowledge such as building sites, workshops and arsenals, universities, academies and courts. Some of them – especially universities – were traditional centers for the production and diffusion of knowledge, which nevertheless underwent transformations in the early modern era (Feingold and Navarro-Brotons 2006). Others, such as the academies, were new types of institutions (Boschiero 2007). Still others, such as the court or the arsenal were, for the most part, not conceived as sites of knowledge by traditional historiography of science, yet came to assume importance in more recent forms of that historiography (Biagioli 1989; Biagioli 1993).

At this stage, I shall invoke just one example that may further illuminate the ‘practical turn’ that has characterized the historiography of science in the last decades, arguing that such turn has had the potential to modify the image of the new scientists as ‘revolutionists’. In this context, the fathers of the new science were now perceived as ‘experts’ capable of ‘transforming’ the field of the mechanical arts into a new physico-mathematical science by combining Greek mathematics – especially that of Archimedes and his followers – with the practical knowledge accumulated in the fifteenth–sixteenth century. I shall exemplify this using Matteo Valleriani's interpretation of Galilean science in his book Galileo Engineer (Valleriani 2010).

When Valleriani's book came out, the field of the history of science was still enamored of an image of the new science as wholly innovative and original, revolting against the traditional discourse embedded mainly in Aristotle's writings on natural philosophy and adapted to Christian theology. One indication of the changing but still prevailing state of the art at the time relates to recurrent attempts to distinguish between traditional accounts of ‘experience’ as a starting point for analysing and representing natural processes, and ‘experiments’ – characterizing the new science. Within such a narrative, ‘experiment’ – unlike experience – was interpreted as an artificial construct, a site of testing mathematically formulated hypotheses against experience. This is indeed part of the story of the shift to the ‘new science’ but it is not the whole story. Under the old assumptions, Galileo's claim to fame as one of the five major philosophers/scientists of the period, together with Kepler, Descartes, Leibnitz and Newton, was his critique of practical experience and his expertise in the new experimentalism. Within such framework, being a ‘scientist’ meant ‘overcoming’ ancient practices of generalization from experience that characterized traditional natural philosophy.

Now, by calling Galileo ‘Engineer’ and historicizing his writings, Valleriani has given the ‘Galileo industry’ a new focus. Galileo's work is now seen not just as mainly experimental and thus ‘wholly revolutionary’. Rather, his ‘oeuvre’ is now distinguished by his ability to ‘transform’ the kind of practical knowledge he was interested in into a modern, physico-mathematical scientific body of knowledge. Thus, Valleriani focuses on Galileo's early work based on his apprenticeship together with artists; on his functions within the Venetian arsenal; on his interest in machines, in constructing and inventing instruments that eventually led to the telescope; in gunpowder, cannonry and fortifications.

While writing his book on Galileo, Valleriani was already swimming against the tide of ‘classical’ history of science, so he chose to join an innovative current of history of practical knowledge, insisting on studying the different ways of its codification. This condition enabled the integration of practical knowledge into ancient theoretical works – (Pseudo)-Aristotle's Mechanical Questions, Archimedes’ On the Equilibrium of Planes and others. Most significantly, Valleriani managed to show how Galileo was among the first scholars capable of providing a feedback loop between practice and theory that led to his later well-known experiments, to the construction of the telescope and its significance for the new cosmology, and to his new science of motion.

Philosophical aspects of the mobility of Greek mathematics: a theory of migrating knowledge

The third and last chapter of my paper is an attempt to draw a theoretical outline that would allow for a richer historical-philosophical account of the transition from Greek mathematics via medieval natural philosophy, Renaissance astronomy and mechanical arts and up to the ‘new science’. Two quotations from Netz's paper express, in a nutshell, the kind of analysis with whose conclusion I mostly agree. Concomitantly, I still think that the narrative presented by Netz needs to be deepened both historically and theoretically. Throughout his paper, Netz tends to emphasize the centrality of mathematical demonstrations and the experimental spirit to Greek traditional thought about nature. It thus seems natural to him to argue that all modern scientists had to do was to adopt ancient resources in order to continue treading the path laid down in antiquity. Hence, he says that ‘Faced with a scientific question’ scientists ‘ … apply to it whatever resources, material and intellectual … [that] are available’.

Indeed, there is no doubt that the availability of theoretical tools is a meaningful condition of possibility for the shift that occurred in the sixteenth–seventeenth century. However, I believe these are not sufficient for an analysis of the historical process that led to modern science. In fact, there is much more to say about bodies of knowledge and the way they travel across space and time until they prove relevant – two millennia later – in a completely different historical context. Moreover, innovation, in my view, is never innovation ‘ex nihilo’. Thus, there is no contradiction between speaking about innovation through transformation, which is how I (together with Netz) see the development of early modern science.

Let me, then, further develop this argument by first presenting a historical case study that refers to a scientific debate between Galileo Galilei and Jesuit mathematician Christopher Scheiner that took place in the second decade of the seventeenth century. While referring to this debate, I shall then present and elaborate my conceptual–theoretical framework concerning transformations of bodies of knowledge in time and space.

Conclusion

This paper is an attempt to seriously cope with Reviel Netz's radical view concerned with the origins and historical development of (physico-)mathematical science from the Greek classical age and up to the birth of modern science. From its inception, Greek mathematical science is perceived by Netz as unique in its objects, its potential to spread, its originality and creativity, as well as its authority. These are presented as the sources of its longevity and ability to give birth to the new science of Western modernity. In my paper, I mostly agree with Netz's conclusions. My suggestions, however, concern places in his paper that need elaboration, the outlines of which I tried to articulate here. These are: (a) The need to enrich the historical narrative by paying more attention to the feedback loop between theory and practices that assumes shape and impact in the process of knowledge mobility, allowing the potential of change to be realized in different ways. (b) My second suggestion touches on the need to develop a set of parameters for analysing the mobility of knowledge in time and space, such as the use and transformation of the objects of knowledge, the boundaries between fields of knowledge, the authority of knowledge and the place of knowledge.