Space and Place: A Scientific History – Part One

With the present work, I begin a series of two articles which deal with the scientific perspective on questions of space and place, through the presentation of the following historically-based texts written by three physicists: the first is ‘The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories’ (2001), written by the British theoretical physicist Julian B. Barbour; by means of this book, we will mainly cover the period from Aristotle to Newton, with some successive clarifications of dynamical concepts. In the following article I will present two texts: the first, is the well-known ‘Concepts of Space: The History of Theories of Space in Physics’ (third enlarged edition, 1993), written by the Israeli physicist and philosopher of physics Max Jammer; then, I will present the article ‘The search for Unity: Notes for a History of Quantum Field Theory’ (1977), written by the American theoretical physicist and Nobel laureate in Physics Steven Weinberg. With the latter two texts, we will take into consideration more recent scientific perspectives on questions of space, place and related concepts (Weinberg’s text is specifically focused on the field concept, which is a necessary scientific concept to understand the evolution of the relation between the old concepts of matter and space, or place).

Both articles are complementary to the previous one – Place and Space, A Philosophical History – which was dedicated to the introduction of the thesis contained in the book ‘The Fate Of Place’ (1997), by the American philosopher Edward S. Casey. For Casey, the direct topic of inquiry was the concept of place, while space appeared as a secondary, but no less important character which accompanied the vicissitudes of place. Conversely, with Barbour and Jammer, the situation is quite different. Even if there are evident differences between the approaches of the two physicists, space occupies the greatest part of their works, either directly (Jammer) or indirectly (Barbour); the subtle but important difference between place and space is sometimes underestimated (it seems to me this is especially evident in Jammer’s book, Concepts of Space).

Barbour deals with questions of space and/or place in a seemingly indirect manner: as the title suggests – The Discovery of Dynamics – his primary concern is the dynamical question concerning the motion of bodies and the way it was interpreted over the centuries. However, questions of place and space are intrinsically related to questions of motion (together with questions of matter and time). After all, where does motion takes place? How can we discern motion? Then, the debate on the absolute or relative nature of motion – which is Barbour’s primary interest – is another side of the same debate regarding the absolute and relative (or relational) nature of space and/or place. Is the ultimate framework of all motion absolute or relative? Or, to state it differently: does the concept of (absolute) space really exist, or do we have to analyse motion with respect to concrete matter, as the extended title of Barbour’s book suggests, by embracing a Machian perspective? It may be the case that our future understanding of the concepts of space and place greatly depends on the correlation between philosophical inquiries and scientific evidence. Or, at least, this is the direction that I’m pursuing with my present research on the concepts of place and space.

The will to understanding the basic structure and the basic phenomena of reality is where all discussions on place/space and motion first aired.  As I’ve already said in the Preliminary Notes, there’s a lot at stake behind concepts of space and place, since behind these notions there are ‘our deepest conceptions of things’ (to borrow an expression used by Barbour) that is, our deepest understanding of reality.[1]  If we want to find the source where such debates about the nature of reality stemmed from, we necessarily have to inquiry into the origin of Western philosophy, physics and metaphysics. That’s why Casey’s and Barbour’s texts are especially relevant: Julian Barbour’s historico-scientific inquiry is tantamount to Casey’s historico-philosophical inquiry. Indeed, I see a fundamental convergence – and it couldn’t be otherwise – between Casey’s historico-philosophical analysis of place and space, and Barbour’s implicit pronouncement about the three watersheds around which the vicissitudes of motion, space, place and matter turn: ‘Newton is truly a watershed – Barbour says in his introduction; forward, there is no stopping until you get to Einstein (and no doubt we shall go further; indeed there are unmistakable signs that the caravan is already on the move); backward there is no logical stopping place before Aristotle or even a little earlier.’[2] The first watershed, which corresponds to Aristotle, refers to a relational understanding of reality through the intimate association between place and matter (this period refers to the triumph of the Aristotelian concept of place – topos – over the concept of the void – to kenon – and corresponds to the first and the second part of Casey’s book); the second watershed  is represented by Newton, to which the triumph of (absolute) space, and the demise of place, corresponds (this period refers to ‘The Ascent and Supremacy of Space’ as Casey said in Part Two and part Three of his book); while the third watershed, which is yet to be determined after the important contribution of Mach and Einstein, seems to go in the direction of a relational understanding of reality that overcomes the traditional concept of space as an absolute entity; this fact implies the reconsideration of the concept of space (and/or place) in its intimate connection with matter and time.

Isn’t Barbour’s proposed image of ‘the caravan on the move’ (with which the author metaphorically describes the ongoing search for a new understanding of reality which is able to make a synthesis between general relativity and quantum mechanics) very appropriate to meet the requirements of what Edward Casey called ‘a third peripeteia’ concerning the reformed understanding of the concepts of place and space?  I believe this ongoing journey will lead us to what I have called the third grand systematization of knowledge, that is, it will lead to a new understanding of the nature of place and space (and of matter and time, of course), or, to put it another way, it will lead us to a new understanding of the framework that keeps all of the aspects of reality tied together and ordered (physicochemical, biological, social and symbolic aspects).

Given that, until today, Barbour left unfinished his initial plan (which should have been divided into two Volumes: Volume I – from antiquity to the clarification of dynamical concepts after Newton – was published in 2001 and is the topic of the present article; Volume II – focused on the clarification of the Machian and Einsteinian perspective on motion was never published), I also had to rely on Jammer’s and Weinberg’s writings to find alternative scientific views on more recent perspectives on space, place and related concepts. I believe that from the confrontation of the philosophical and the scientific perspectives on such fundamental concepts, we could be in a position of vantage when we will make incursions into other fields of the human knowledge for which concepts of space and place are particularly important (architecture, to begin with, which is my direct field of competence).

So, I now start with the presentation of Julian B. Barbour’s The Discovery of Dynamics; just like the book, this article is quite extensive (far beyond my initial intention). My aim is twofold: on the one side, I want to point out the many interesting historico-scientific passages that the author deals with and which influenced the way concepts of space and place developed over the centuries. On the other side, in those passages where my thought on questions of space and place diverged from the opinion of the author (for instance, see the paragraphs on Aristotle, Galileo or Descartes), or where I believed some passages were critical to understanding the history, the meanings and the differences between those two concepts, I have made some comments that I hope can contribute to enlarging the debate and, at the same time, clarifying my position on such questions. The images I have used along with this article are not included in the book, even if, sometimes, I drew inspiration from the original images or drawings that accompany Barbour’s text.

Barbour begins his intellectual journey through centuries by introducing a fundamental question which contributes to illustrate his overall plan to the readers; Barbour writes: ‘If a stone is thrown at the stars with sufficient force it will travel through the universe forever […], but we do not know what determines the path. Is it space, or is it matter or some combination of the two, or what? […] Newton identified absolute space and time as the ultimate framework of all motion. […] Newton’s concepts of absolute space and time were severely criticized, above all by Huygens, Leibniz, and Berkeley. For, space being invisible, how can one say how a body moves relative to space? And, space being nothing physical, how can it influence actual motions of physical bodies? But Huygens died soon and neither Leibniz nor Berkeley could produce any sort of theory to rival Newton and their objections were gradually forgotten, until they were rediscovered by Ernst Mach, [who] proposed one of the most radical ideas in the history of science. He suggested that inertial motion,[3] here on the earth and in the solar system, is causally determined in accordance with some quite definite but as yet unknown law by the totality of the matter in the universe. Mach asserted that motion does not exist except as a change of position relative to other bodies and that the law which governs the changes in relative position must be expressed directly in these same relative terms. In the period from 1907 to 1918, Einstein worked with feverish enthusiasm to discover the mysterious law of nature that Mach had postulated. He coined the expression Mach’s Principle for the conjecture that the inertial properties of local matter are determined by the overall matter distribution in the universe.’[4] However, Einstein’s attempt to fully incorporate Mach’s principle within his General Theory of Relativity failed: ‘Einstein was forced to conclude that although matter in the universe did clearly influence inertia within the framework of his theory, his theory was nevertheless unable to demonstrate that inertia is completely determined by matter […]. Einstein reluctantly concluded that his attempt had failed [while] the present status of Mach’s Principle can only be described as confused.’ [5]

Through these words, Barbour illustrated his overall plan, which, as I have anticipated, was left unfinished with respect to the elucidation of Mach’s and Einstein’s attempt to overturn Newton’s deepest conviction – the existence of absolute space. After having expressed the basic reasons for which he embarked on such vast program of rethinking ab initio the theory of motion (and, therefore, of space), and after making a digression introducing his own attempt to reformulate general relativity according to Mach’s Principle, Barbour directly presents Newton’s laws and the conceptual framework behind those laws.

1. Julian Barbour’s The Discovery of Mechanics: Preliminaries

The concept of space – absolute space – is the conditio sine qua non for Newton’s dynamics: as Barbour explicitly says ‘of all the Newtonian concepts, those of absolute space and time are the most important, for they provide the framework of everything else.’ [6]

Now, we understand why questions of motions imply space and time: according to Newton’s perspective, they underpin the possibility to unambiguously understand and explain motion. It was through an accurate analysis of space, time and the motion of bodies that the three laws of motion and the law of universal gravitation could be stated. If a body is subjected to the influence of space only, then the First Law results: ‘Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.’[7] If a body is subjected to the influence of another body by way of a direct collision or by way of an indirect influence exerted through space, then Newton’s Second Law, Newton’s Third Law and Newton’s Law of Gravitation can be extrapolated. The Second Law ‘tells us how the momentum of the body we consider is changed by other bodies’ – the familiar expression of this law in terms of accelerations  can be stated as follows: f=ma; the Third Law tells us that ‘to every action there is always opposed an equal reaction’, while the Law of Gravitation says that ‘if there are two bodies of mass m1 and m2 separated by a distance r then each exerts a force on the other which is proportional to the product of the two masses, is inversely proportional to the square of the distance between them, and acts along the line joining them.’ [8]

Among the important features of these laws (they are the same throughout the Universe, they are invariant with respect to place and time, they put the focus on the importance of physical quantities like mass, force, acceleration…) Barbour points out the ‘distinguished role’ of the First Law, which unveils the fundamental framework upon which the Laws are constructed: absolute space and time. This fact gives Barbour the possibility to make some general or preliminary ‘philosophical’ comments on the absolute/relative debate; he introduces what he calls (i) the ‘epistemological imperative’ regarding motion: ‘any statement about the motion of a body will be meaningless unless it is simultaneously stated with respect to what that motion takes place.’[9]  Then, concerning questions of motion, this seems to imply a dichotomy between space and matter: either movement is with respect to space – this is the Newtonian hypothesis -, or with respect to other bodies, that is with respect to matter – this is the Machian hypothesis. Barbour takes sides on the absolute/relative debate by introducing the so-called ‘detachment hypothesis’ which, according to Mach, regards Newton underlying framework, that is, ‘that a phenomenon observed within a given environment is in essence independent of the environment.’[10] Barbour is contrary to that hypothesis and says: [11]

[detachment hypothesis] we never actually see space; all we ever see is matter against the background of other matter. If we were to stick to the bare facts, all our talk would be relative – how certain matter stands with respect to other matter. With the assistance of the concept of space, the atomic hypothesis breaks the bond between matter and matter, detaches individual parts of matter from the overall material concatenation, and sets them loose in conceptual space.

Concerning that hypothesis, it is in agreement with the overall argumentation sustained in this website; thus, I side with Mach and Barbour’s perspectives; I would say that the ‘detachment hypothesis’ corresponds to understanding space the way it should be: an abstract concept, a figment of the imagination. Whenever we speak about something material, whether a celestial body or our own body, such body moves with respect to other bodies: a planet, a star, a wall, the ground, the ceiling, etc. A physical body always moves from a certain place to another place, where ‘place’ should be understood as an entity which is not detachable from matter (and from time understood as the duration of a process or event). An object-place is the referential background for the movement of other objects-place understood as physical bodies.

In this preliminary chapter, Barbour also introduces two important and related questions that will give a definite direction to the development of the book until we get back to the discovery of Newton’s Laws: (i) the reason why it took so long to find the laws of motion and (ii) the driving role of astronomy for the discovery of such laws (or for ‘the discovery of dynamics’ to quote the title of the book itself). As to the first point, Barbour enumerates some of the obstacles: the presence of ‘air resistance’ – an element which was not so intuitive to consider and which was not easily amenable to mathematization; the difficulty to realize and sense ‘inertial motion’ as characteristic and primary feature of any physical body; the difficulty to acquire the notion of ‘decomposition of motion into parts’ (a horizontal inertial component and a vertical component affected by gravity – this was due to Galileo); the difficulty to appreciate ‘the role of acceleration’ and of isolating ‘the concept and the role of forces’ (nuclear forces were obviously out of reach – Barbour reminds us -, while electrical, magnetic and gravitational forces were not so easily detectable and traceable to those Laws, especially to the second and the third law); the difficulty to see and analyse ‘quantitative change’ because of the lack of ‘suitable clocks’: this specific fact was a practical obstacle for the necessary passage from an analysis of motion based on quality (as it could be considered the analysis made by Aristotle and his contemporaries) to an analysis based on quantity (again, here, Galileo deserves a special mention). And this takes us back to the decisive role of astronomy: on the one hand the ‘extremely powerful impression of the complete immobility of the Earth […] was a powerful factor that held back the discovery of dynamics;[12] on the other hand ‘the cyclic regularity of the celestial motions, [and especially] the rotation of the earth and the apparent motion of the sun and the moonwas probably the single most important factor in the discovery of dynamics if not the entire scientific revolution’,[13] since it allowed the mathematization of actual phenomena by offering natural clocks, and, with them, the possibility of prevision and verification of empirical facts and data.

The following chapter – chapter two, which is focused on Aristotle – is particularly important: first because with Aristotle we have the first airing of the absolute/relative problem – this is the title of chapter two, properly; second, because the limits of Aristotle’s cosmological model are supplemented by the development and progress of Hellenistic astronomy, which played a decisive role for the discovery of dynamics and which is the main subject of the other chapters. Astronomy played a prominent part in the development and systematization of the concept of space.

2. Aristotle: first airing of the absolute/relative problem

Before introducing the work of Aristotle, Barbour briefly deals with those who preceded the life and the work of the Stagirite – the so-called pre-Socratic philosophers. He just offers a very general picture of the Hellenic period – we can divide the Greek History into two periods, as also Barbour says: before and after Alexander the Great (356-323 BC), that is, the Hellenic Period and the Hellenistic Period. Since I believe the generic reader can benefit a recall of what we have learned at the High School concerning the origin of Western Thinking, and since that origin is directly or indirectly connected with the questions of space and place, I will briefly touch upon the argument the same way Barbour did, following the same temporal sequence that is usually presented by any old High-School book.[14] The images I have used are not included in the text.

In these beginnings, we find the Milesians, whose attitude towards things ‘was clearly a development of the genetic or genealogical approach to nature exemplified by the Hesiodic Theogony.’[15] They held that ‘a single elementary cosmic matter underlies all the transformations of nature’:[16] for Thales of Miletus, that single element was water; for Anaximenes, it was air; for Anaximander, it was the so-called ‘apeiron’, an indefinite material principle out of which any known material entity emerged (the ‘apeiron’ is literally that is which is without boundaries – boundary in Greek is peras – therefore, it should be translated as ‘the unbounded’ to keep the original sense of the word).

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Image 2: The school of the Milesians developed in Miletus, Ionia, an ancient region in the central part of the western coast of Anatolia (now Turkey). They held that reality, in its primal state, was made out of a single material entity.

Another group of thinkers that developed in the Italian colonies of Magna Graecia, quite independent of the Milesians, was the Pythagoreans, whose name is due to the founder of the school, Pythagoras of Samos (he left Samos to escape the tyranny of Polycrates when he was forty and he founded his school in the Achean colony of Croton, in the Southern part of Italy).[17] The Pythagoreans ‘held that all matter is literally composed of numbers and that numerical ratios underlie all sensuous phenomena.’ [18]

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Image 3: The school of the Pythagoreans derived its name from his founder Pythagoras who founded a school in Croton after he emigrated from Samos. They believe that matter was composed of numbers and that numerical ratios were behind any sensuous phenomena.

Another mayor thinker belonging to the pre-Socratic period was Heraclitus of Ephesus, who opposed the belief of the Milesians in a single being, or an archetypal principle, out of which everything was created; Heraclitus ‘held that there is nothing in the world that persists. There is merely a constant and ceaseless flux. Change is the only reality.’ [19]

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Image 4: Heraclitus of Ephesus, who flourished in the 5th century BC, held that ‘change is the only reality’.

In reaction to Heraclitus, Parmenides of Elea, the founder of the so-called Eleatic school, ‘by denying the possibility of change, insisted on the supremacy of being… He held that there is only the one being without inner differentiation and that change and the apparent diversity of things is an illusion. He was followed by Melissus of Samos, who believed in a One [being] that is eternal, motionless and without change, and by Zeno of Elea who devised his famous paradoxes in order to prove that motion is an illusion.’  [20]

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Image 5: Against Heraclitus, The Eleatics believed in the One Being, eternal, motionless and without change. 

The reconciliation between Being and Becoming as the ultimate principles of reality, which is at the same time static (the point of view of the Eleatics) and dynamic (the point of view of Heraclitus), was attempted along two different lines of thinking: on the one hand, Empedocles of Agrigentum ‘held that all individual things are produced by the mixing of the four elements: earth, air, fire and water.The elements remain the same, things arise from their mixing in different proportions.’ [21] On the other hand, we find Anaxagoras of Klazomene, who brought philosophy and the spirit of scientific inquiry from Ionia to Athens, and ‘taught that there was an infinity of simple substances, divisible into parts’ [a hypothesis that will also be pursued by the Atomists]. All becoming is due to their combination and separation. To explain the motion of the parts he assumed the existence of a ‘soul-substance’ that is itself in motion and can set the normal substances in motion. […] Anaxagoras accorded purpose an important role in nature.’ [22]

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Image 6: Empedocles of Agrigentum inaugurated the pluralist vision about reality according to which any material thing emerges out of the action of different elements (for Empedocles those basic elements were: earth, water, air and fire)
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Image 7: By introducing the principle of the Mind or Reason – a soul substance – as an active agent for the disposition of things out of infinitesimally small fragments, Anaxagoras inaugurated the detour of the immaterial into the material. [23]

Such a peculiar vision of reality composed by an infinite number of simple substances was the hallmark of the Atomists: Leucippus of Abderaretained Parmenides’ idea that there is just one being, of itself completely homogeneous, but he supposed that it was broken up into infinitely many pieces, or atoms, of different shapes and sizes, and that their combination gave rise to all the variety of individual bodies. This atomic hypothesis was made famous by Democritus (Leucippus’ pupil) and in antiquity, it was further developed by Epicurus.’[24] And, as also Barbour notes, it was especially diffused by the vision of the Roman poet Lucretius in the famous poem De Rerum Natura, a few centuries later.

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Image 8: The Atomists believed in the reality of the void through which indivisible material particles of different shapes and sizes – the atoms – moved, giving rise, through their combinations, to all the variety of individual bodies.

The thinking of the Atomists is particularly important for our debate on the question of place and space, for two reasons: first, since their conception of the voidto kenon – seems to be an anticipation of the concept of space, which, as first approximation and intuition, is understood and imagined as ‘empty’, usually. Second, since it is properly by criticizing the concept of the void that Aristotle (Stagira, 384; Chalcis, 322) arrived at its definition of the concept of place (topos).

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Image 9: Aristotle’s cosmological vision and concepts prevailed over his antagonists for nearly two millennia.

While Barbour is clear and direct in pointing out the conceptual affinity and the historical legacy between the concept of the void and the concept of space, as concerns the second point it seems to me Barbour did not consider it with the importance it deserves: that’s why my account on Barbour’s view concerning the work of Aristotle is not simply a presentation of his argumentation, but I will express my opinion on such decisive question.

With respect to the first point, Barbour says: ‘The atomic hypothesis is important for this book in marking an early step towards the concept of absolute space. The point is that Leucippus not only broke up the homogeneous block of Parmenides’s being into atoms but also assumed that these pieces of being are separated by that which is non-being or void, i.e., empty space. This introduction of empty space, as that in which the atoms can move (mobility is evidently crucial for the atomic hypothesis), was the outcome of a sophisticated philosophical debate. […] The void is not nearly such an obvious concept as it might appear to the modern mind. Leucippus defined the void, or emptiness, by contrast with what is real and tangible. If we stretch out our hand and feel something tangible, then that is matter; but the lack of this positive sensation is also in its way real. Thus, nothing (=no-thing) has a kind of reality too. It should, however, be said that the atomists seem to have developed only the vaguest of spatial concepts, getting little further than the idea of emptiness between the atoms.’ [25] 

We must now introduce what Barbour said about Aristotle, to elucidate the other question. Barbour’s overall judgement on Aristotle is double-edged: on the one side, he was not soft at all when he said that ‘Aristotle’s output was purely qualitative and much was, with hindsight, badly wrong’;[26] on the other side, Barbour is willing to attribute the Stagirite the prominent role of the great systematizer and pioneer of dynamics. [27]

Barbour introduces the work of the Stagirite (Stagira was the birthplace of Aristotle, in the Greek region of Macedonia) presenting his cosmological model, which was characterized for being finite and rather like an onion. The spherical Earth was considered at rest in the centre, while other concentric spheres rotated carrying the moon, the sun, and the other naked-eye planets known to the ancients: respectively, Venus, Mercury, Mars, Jupiter, and Saturn (the total number of spheres carrying the planets and ‘needed to explain the observed facts’ were 55). [28]

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Image 10: Aristotle’s cosmology: a theory of place (topos), matter, and motion.

This model was characterized by the concepts of ‘proper places and natural places’ after which a sort of teleological concept of motion could be envisioned; in fact, as Barbour puts it, ‘everything happens for a specific purpose’,[29] that is, any material entity, from earth to fire had its natural place and strived for reaching that place: so ‘the stone falls for a purpose, to reach its proper place’, or, conversely, the fire move upward to reach its natural place, the place where the sun or, eventually, the distant stars are located. The sublunar region – the region composed of earth, water, air and fire respectively – is characterized by rectilinear motions; conversely the celestial regions, made of an incorruptible and quintessential element – the aether, in Greek aither, -, only perfect circular motions occur. This finite cosmological model had in the divine power – the Unmoved Mover – the ultimate reason for that order, a power which, ‘in Christian theology came to be identified with God’: this was obviously a very good reason for the future success of this model against concurrent models like those envisioned by the Pythagoreans and the Atomists. Then, Barbour specifically deals with Aristotle’s theory of motion and place. Very briefly, as we have already anticipated, by his theory of motion Aristotle observes that either heavy or light matter is subjected to natural motions downward or upward according to the natural place matter belongs to. A rock will naturally move down to the centre of the earth, while the fire will naturally move upwards – this fact establishing a kind of order from ‘heavy’ to ‘light’ places: respectively earth, water, air, fire. Of course, correlated to the analysis of motion there is the recognition of the notion of place: that’s why Aristotle needs to ask and to define what place is, precisely; and he concludes that place is actually ‘the inner surface of the container’ – an envelope in the end. [30]

There are two specific points I wish to comment on with respect to Barbour’s interpretation of Aristotle’s vision: the first tries to cope with Barbour’s question about the reason ‘why Aristotle put such emphasis on motion’;[31]  the second is the sometimes questionable use that Barbour makes of the term ‘space’ (instead of the term ‘place’ as translation of the original Greek term topos) when it is attributed to, or associated with Aristotle.[32]  I need to say that what I consider a questionable attribution of the concept of space to Aristotle is far from unusual; however, I believe that by attributing the use of the term space to Aristotle we overlook the extremely important philological and linguistic question regarding the adherence of meaning to the original Greek terms and concepts (I believe the philological and linguistic question may have impact on the ontological question, which is even more fundamental).

Concerning both argumentations, it is important to discern that Aristotle’s overarching vision of nature and the cosmos was also a reaction against those who believed in the existence of the void, which, for Aristotle, was a logical impossibility, a conceptual blunder. This fact had an obvious consequence: the necessity to introduce another concept that could summon the basic role of the void within an alternative vision of the cosmos (this time, a ‘plenist’ vision). This basic role, for Aristotle, was played by the concept of place, which was, in the most reductive of the hypotheses, the co-protagonist with matter in the dyadic relation between matter and place (actually, for Aristotle himself, place ‘takes precedence of all other things’, and, similarly, according to Damascius place had to be considered ‘superior to things in place’).[33] Motion – through the concept of ‘natural place’-  was the concept that could give matter and place a definite unity, or, to put it differently, motion was that which could link matter to place through a coherent vision and satisfactory explanations of striking empirical phenomena: it is undeniable that a stone falls down to reach its ‘proper place’ if that place is the centre of a massive body like the earth; it is undeniable that fire aims at reaching its ‘proper place’ in the direction above our heads. It was by relying on these simple empirical pieces of evidence – where the notion of place played a basic role – that Aristotle set up his model. That’s why motion was important for Aristotle: it was the squaring of the circle of his ‘plenist’ vision of a cosmos filled with matter, whose existence was (sub-)ordinated with respect to ‘proper places’; the motion of bodies made evident the correctness and superiority (in Aristotle’s view) of the concept of place with respect to the concept of the void; that’s why Aristotle rejected the abstract and homogenous concept of space, namely the atomists’ void: it offered no local discernment to the local movement of bodies; conversely, Aristotle’s concept of place offered such possibility.  As Pierre Duhem said with respect to Aristotle’s conceptualization of place/topos: ‘place must be defined in such a way that it provides fixed points in relation to which one is able to discern local movement. This is the fundamental thought which guides Aristotle in his search for a definition of place.’[34]  That is the reason why concepts of place and motion are so important for Aristotle. Now, that being said, I hope I have convincingly answered Barbour’s question about the reasons why Aristotle put such emphasis on motion, and, consequently why we should not attribute any use of the concept of space to Aristotle (space, just like the void, offers no possibility of discernment to the motion of bodies; therefore, we do not even have to mention the fact that the Greeks had no concept of space). [35]

In spite of these two objections to Barbour’s interpretation of Aristotle, there are distinct merits that Barbour must be credited for: first of all, by analysing Aristotle’s theory of place and motion within an ideally encompassing historico-scientific perspective, Barbour was able to trace back the origins of dynamics down to Aristotle, who, as Barbour himself says, was the first man to have had a ‘potentially complete concept of motion’.[36]  Therefore, in this historical journey, he was able to distinctly isolate the figures of Aristotle, Newton and Einstein as ‘the great systematisers’ with respect to questions of space and/or place, which is a hypothesis that I completely agree with and which marked my way to the acknowledgement of the distinction between concepts of place and space. Second, within such encompassing historico-scientific perspective Barbour unveils some intriguing correspondences between Aristotle and modern physicists, notably Mach and Einstein, as when he said that ‘Aristotle’s discussion of weight and lightness anticipates the methodology of Mach’s eventual clarification of the mass concept’, or, more generally, when he says that ‘in several other respect [Aristotle] was a precursor of Mach’.[37] At this regard, Barbour mentions the fundamental argument regarding the relationist position held by Aristotle (Aristotle’s concept of place is relative or, better, relational, as much as space is relative for Mach or Einstein), resulting from his rejection of the idea of the void; such position anticipates Berkeley, Leibniz, Mach or, to a certain point, Einstein.[38]  Third, even if by indirection, thanks to Barbour’s analysis of motion we understand that Aristotle was the first to have linked – in a questionable manner Barbour would probably say – the Heavens and the Earth with a physical principle stemming from the analysis of place, motion and matter. With this respect, Barbour is a bit more critical than I am, since even if, on the one hand, he gives credits to Aristotle for having ‘attributed to the motion of the heavens pretty well all the correct properties of inertial motion’  – which is a universal property – , on the other hand, the differentiation of the two types of motion – rectilinear motion for happenings on the Earth and in the sublunar regions, circular motion for the Heavens – and, most of all, the endowment of divine principles for that which happens in the uppermost regions of the Heavens, at the same time created ‘an almost unbridgeable gulf between the heavens and the earth’- in the words of Barbour.  All things considered, I believe we can see Aristotle’s tentative description of the different kinds of motions as necessary qualitative anticipation of the mode of thinking that will lead Newton to discover the laws of motion and universal gravitation  (the concept of space in Newton substituted the role that place had in Aristotle’s conceptions).

Finally, Barbour elucidates another important point: the main limit of Aristotle’s approach, that is, the almost exclusive reference to qualitative argumentation and the neglect of a mathematical treatment of motion (yet, in the specific circumstance of his theory of motion and place, I believe this fact should presuppose the notion of a three-dimensional notion of space – or the mathematization of space itself, its geometrization properly – which was something alien to the mind of the ancient Greeks; as a matter of fact, it took almost two millennia of reasoning and discussions before all the pieces of the puzzle concerning the notion of space could fall into the right position). These are Barbour’s words: ‘Where is the flaw in this all but self-contained [Aristotelian] whole? Undoubtedly in Aristotle’s neglect of the metrical, as opposed to topological, properties of space. His concept of place is at once fundamental but primitive. Fundamental because it derives directly from the most basic of spatial concepts – contiguity, coincidence, and inclusion. Primitive because it stops there and takes almost no account of the metrical properties of the world, which although secondary nevertheless have a very real practical existence […]. What was lacking, I suggest, was really hard and ineluctable evidence, empirically based and expressed in mathematical form, suggesting that there was something fundamentally wrong with the Aristotelian scheme. This evidence was supplied by the astronomers.’ [39]

Barbour dedicates the following chapters to the analysis of astronomy, its development from antiquity to modern times: it was (also) through the important contributions of astronomy that a certain disposition of the mind – the scientific mind – could be forged and with it the notion of absolute space concocted (and the laws of motion discovered).

3. Hellenistic Astronomy: the foundations are laid

The death of Alexander the Great, 323 BC, marks the beginning of the Hellenistic period. This period was characterized by some events that have important consequences for our study on space (and place) and the development of scientific methods and concepts. These events are enumerated by Barbour: the leading cultural and scientific role of Alexandria of Egypt where two critical books were published – The Elements of Euclid (c. 300 BC) and The Almagest by Ptolemy (c. 150 AD); the role of the Pythagorean Archimedes (287-212 BC) whose mathematical writings contributed to set up the Science of Statics; and the work of the Hellenistic astronomers: especially, Heraclides of Heraclea (c. 390 BC – c. 310 BC),  Aristarchus of Samos (310 – c. 230 BC), Apollonius of Perga (255 – 170 BC), Hipparchus of Nicaea (c. 190 – c. 120 BC) and Ptolemy of Alexandria (c. AD 100 – c.  170).

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Image 11: Mathematics, Geometry, Astronomy and the road to modern scientific methods and concepts: events, people and places in the Hellenistic Period.

A characteristic aspect of the first astronomical models concocted by the Hellenistic astronomers and mathematicians was ‘the lack of interest in empirical numerical data in contrast to the emphasis on the purely mathematical structure’- to use Barbour’s own words.[40]  The possible influence of Babylonian astronomy – especially the use of tables giving the accurate positions of the celestial bodies – is first clearly visible with Hipparchus (around 150 BC), Barbour says. Then, by the combination of abstract mathematical models and empirical data through which it was possible to verify those abstract models ‘astronomy becomes a real science’  or – as Barbour puts it –‘observable numerical data are made the decisive criterium for the correctness of whatever theory is suggested for the description of astronomical phenomena’.[41] This was a critical aspect for the development of the scientific method and the science of dynamics as well.

We know from Barbour that among the first solid achievements of the Greek astronomy – the spherical shape of the earth, the estimate of the diameter of the earth, the distance of the moon, of the sun and distant planets – those regarding the estimate of distances were probably the most remarkable since they contributed to the diffuse use of trigonometry.  ‘The development of trigonometry – Barbour says – provided a means for man to probe the universe far from the regions to which he had direct access.’[42]  This method to estimate distances (especially the distance of the sun) was used by Aristarchus and Ptolemy as well, and even if it was of little practical importance – Barbour notes – ‘it highlights an aspect of position determination very relevant to the absolute/relative debate.’[43] The observation regarding the use of trigonometry deserves further inspection.  At this regards (the influence that trigonometry may have played with respect to the development of the concept of space – and therefore of place -, or, to put it another way, the influence of trigonometry with respect to the correlation between reality and abstraction), I will now quote what I consider a very relevant passage in the text. Barbour writes: ‘It is clear that the earliest conceptions of space, position, and motion were strongly influenced by the (fortunate) accident of our happening to live on the earth. Geometrical relations were discovered on its remarkably stable surface. Thus, we walk around on a part of the lawn of a garden and rapidly form a three-dimensional picture of the disposition of the bushes, trees, and borders distributed over the lawn. We can then choose a point that we have never previously visited, let ourselves be guided there blindfolded, and predict the two-dimensional picture we shall see when the scarf is taken from our eyes. We think of the trees and bushes as having a definite place on the lawn. We ourselves move across the lawn. Not surprisingly, man hypostatized space, took it to be a real thing with all the geometrical properties of the lawn, removing only the tactility and visibility. The environment was divided into two – the lawn and the objects on it. The concept of intuitive space, the container of the material objects in the world, developed very naturally as a consequence of the very special nature of our environment. All the major contributors to the development of dynamics, above all Galileo and Newton, worked with an underlying conception of motion as taking place in space. Even Einstein […] could not entirely shake off this way of thinking. But the real lesson of trigonometry is that the establishment of spatial relationships is in no way dependent on the lawn; the trees and bushes, and, very important, the light by which we see them, are sufficient in themselves. Figure [see image 12, below,  which is a modification of Barbour’s original drawing] contains just three bodies – the earth, sun, and moon; the relationships of relative size and mutual distance which Aristarchus deduced for them owe nothing to ‘space’. Even the backcloth of the stars is not required (the observation of the angle MES must of necessity be made in daylight, when the stars are not visible). Why then must we conceive of them as moving in a space that plays no part at all in the operational determination of these relationships? Should we not do better to concentrate directly on what is objectively given? It is the ascent into the heavens that forces upon man a revision of his concepts of space. The further we progress, the more the old familiar framework recedes. The surface of the earth is far below us, but somehow we seem to feel the need to take it, or rather its surrogate, space, with us – a safety net for the first hesitant attempts in celestial acrobatics. But the acrobat keeps his eye on the trapeze and on his partner; he cannot even see the net. Aristarchus pointed the way. Kepler learnt the trick. But not until Poincare perfected the art of celestial dynamics at the end of the nineteenth century was the ‘lawn’ revealed for what it was: a convenient but dispensable aid to conceptualization.’ [44]

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Image 12: Trigonometry was used by the ancient Greek astronomers to estimate astronomical distances, especially the distance to the sun.

There are aspects of this interpretation I do not agree with (one for all the description of ‘intuitive space’ which – according to me – is a flawed notion if compared to the accurate description that the notorious Swiss psychologists and epistemologist Jean Piaget gave with respect to how the notion of space is grasped by the human mind),[45] but, in the overall – and in agreement with Barbour – I believe the substitution of concrete entities by means of progressively more abstract notions that try to cope with those initial concrete entities is the key to understand the notion of space (and the correspondent/correlate oblivion of the notion of place); that’s why I believe Barbour’s statement – ‘it is the ascent into the heavens that forces upon man a revision of his concepts of space’ – tells a profound truth if we admit that what was actually going to be revised – or under process of revision – was a concept of spatiality deriving from old Greek conceptualizations influenced by, or expressed through the terms topos, chōra, and to kenon, which cannot be simply reduced to the formula ‘concepts of space’ (unless we consider concepts of place or the void as mere typologies of space); a process of revision that was completed and formalized when Newton proposed his concept of absolute space (and of place) in the famous Scholium. What Casey explained from a philosophical perspective (see the previous article), Barbour is explaining from the perspective of the scientist. Astronomy offers privileged access to this phenomenon of transition regarding the understanding of spatial and placial concepts that we are particularly interested in.

This process of progressive abstraction I’ve spoken of by pointing out a precise statement is exemplified few pages later when Barbour speaks of the parallel transition (this time a cosmological transition properly) that occurred when men passed from the earth frame (the centre of the earth considered as the origin with respect to which all of the planetary motions happen) to the heliostral frame (the sun is the origin): this passage was mediated by the reference to the geo-astral frame (the origin is still the earth, but the markers of angular positions are the fixed stars of the distant celestial sphere, with particular reference to the work of Ptolemy). The distant stars, by providing points of reference far removed from our immediate experience (the static earth) ‘made possible that first loosening of man’s imagination from the chains of preconception imposed by the immediate vicinity; the first step away from the primitive earth-based viewpoint occurred unconsciously and unwittingly. As the ancient astronomers immersed themselves more and more in the planetary phenomena, the diurnal motion was gradually lost from sight: they paved the way for the ultimate transition to the helioastral system’.[46] I believe there is a fundamental correlation between ‘that first loosening of man’s imagination from the chains of preconception imposed by the immediate vicinity’ and the loosening of the concept of place-as-topos (which in the end, if we consider Aristotle’s definition of place, is the literal representation of that which is in the ‘immediate vicinity’) to the benefit of the concept of space (which, at the time of the ancient Greek astronomers we are dealing with, was understood more as simple extension, or distance – ‘diastemic space’ – rather than as fully-fledged three-dimensional concept in the modern vein). [47]

There were two important phenomena that the first Greek astronomers dealt with, Barbour says: first, the rotation of the earth, which ‘coupled with the complete absence of any apparent dynamical effects of the rotation for an observer on the earth rotating with it, created the powerful impression of a stationary earth about which the stars and other celestial bodies appeared to move in perfect circles’ (and this fact, as Barbour explained, contributed to realizing the powerful idea of how special the notion of perfectly uniform circular motion was);[48] second, the apparent motion of the sun against the background of the stars, that is, to use a technical term, ‘the ecliptic’. [49]

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Image 13: Given the visible effects of the rotation of the Earth, ‘it is not difficult to realize how the notion of perfectly uniform and circular motion made such an impact on the ancient imagination’ Barbour says (photograph: A.Duro/ESO)
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Image 14: The ecliptic – the apparent circular motion of the sun projected onto the celestial sphere – offered an alternative frame of reference with respect to the celestial poles and equator. The attempt to solve a flaw in the ecliptic – the apparent non-uniformity of solar motion – pushed the development of dynamics ‘as a quantitative discipline controlled by empirical observation’, Barbour says. [50]

These two phenomena offered key instruments to decipher the secrets of nature (and especially the secrets of motion): while ‘the rotation of the earth suggested to the ancient astronomers the notion of perfectly circular and uniform motion as the paradigm of celestial motion, provided an effective clock, and strongly suggested a method of position location on the sky’ (the celestial poles and the celestial equator suggested a convenient reference system),[51]  the other phenomenonas steady and regular as the formersuggested the introduction of an alternative reference system – ‘in which the analogue of the celestial equator was played by the ecliptic’- [52] so that two sets of techniques could be used and developed to analyse celestial patterns.

The development of dynamics ‘as a quantitative discipline controlled by empirical observation’ was pushed by the attempt to solve a flaw in the ecliptic of the sun: that flaw regarded the apparent non-uniformity of solar motion. This observed fact was quite contrary to a paradigm taken as an axiom since antiquity: the divine and perfect uniform circular motion of the celestial bodies in the Heavens. How to reconcile such difference?  ‘The first attested attempt at a theoretical explanation of reasonably accurate observations is due to Hipparchus’, Barbour says by introducing the paragraph on Hipparchus of Nicaea (190- c. 120 BC). In Barbour’s own words, ‘Hipparchus’s work is to be seen as a most significant step forward in the Greek programme of finding geometrokinetic explanations for why the observed motions of the sun, moon, and planets did not fit the divine paradigm of perfect uniform circular motion. The divinity of the celestial bodies had to be saved by a geometrical explanation. This was the programme, begun by Eudoxus in the time of Plato, that the Greeks came to call saving the phenomena (or appearances)’.[53]  We know from the Almagest by Ptolemy that Hipparchus was aware of the non-uniformity of the solar motion and that he tried to ‘save the appearances’ by hypothesizing the motion of the sun was perfectly uniform running on a circular path (orbit), but the centre of such orbit was not the earth but an eccentric point with respect to the earth. Apart from Hipparchus’ theory of motion, the other known attempt to solve the ‘problem’ regarding the non-uniformity of celestial motions was the so-called ‘epicycle-deferent theory’, which was probably due to Apollonius of Perga (255 – 170 BC). Through this method, the irregular motions of each known planet as observed from the earth were decomposed into two perfectly uniform circular motions, so that one motion along the main circular orbit – called ‘deferent’- could explain the apparent change in velocity of the celestial body with respect to the earth at rest in the centre of the cosmos, while the other motion, along a second circular orbit – the ‘epicycle’ -, could explain the apparent retrograde motions of such celestial bodies. According to Barbour, the epicycle-deferent theory was an important finding that greatly contributed to the discovery of dynamics. In the course of the text Barbour will highlight a precise number of discoveries (Barbour calls the total number of those discoveries or insights, ‘the baker’s dozen’:  six are associated with the study of celestial motions, six with the study of terrestrial motions and one mathematical insight favoured the possibility to link the two different types motions); the epicycle deferent theory was the first in the list, since ‘it gave a tremendous boost to the faith in a rational explanation’ of celestial phenomena.[54] One of the most important implication that the epicycle-deferent theory had was that it prompted to put focus on the role of the sun with respect to the motions of the other planets; in the words of Barbour, ‘no matter how the decomposition of the motion is made, one leg of the decomposition – either the deferent motion or the epicycle motion – always marches in phase with the sun. It seems that the sun exerts a partial control on the planets in a most mysterious fashion.’[55] Also important from a symbolic point of view, the fact that the discovery of different centres of revolution began to mine the solid ground which the Aristotelian cosmology was built on; as Barbour says ‘the world no longer had a single centre. The Aristotelian unity was falling apart.’[56]

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Image: 15  Both Hipparchus’s theory of eccentric motion and Apollonius’s use of the epicycle-deferent system aimed at explaining the non-uniformity of motions of the sun and the other planets with respect to the earth.

The next astronomer taken into consideration by Barbour was Ptolemy of Alexandria, (AD 100 – c. 170) author of the famous treatise on astronomy ‘the Almagest’ (c. AD 150). Hipparchus’s theory of eccentric motion and the epicycle-deferent scheme were not sufficient to explain the planetary orbits because of the eccentricity of the orbits, the non-uniformity of motions and the ellipticity of such orbits. Therefore, again, the task of Ptolemy was to eradicate anomalies and inequalities from the theory of celestial motions. Ptolemy introduced two significant innovations: the first was in the theory of the moon for which he discovered the so-called evection – a perturbation on the motion of the moon exerted by the sun. The second – Ptolemy’s biggest innovation – was the discovery of the equant (this is the second insight of ‘the baker’s dozen’ according to Barbour), ‘that is, he [Ptolemy] found that all the observations […] could be explained by assuming that an observer stationed on the opposite side of the centre of the deferent from the position of the terrestrial observer and at an equally great distance from the centre would, if the guide point were visible, observe that it moves around against the backcloth of the stars with a constant angular velocity […]. Without knowing it, Ptolemy had found the two foci of the planet’s elliptical orbit together with the circle, correctly positioned with its centre at the centre of the ellipse, that gives the best approximation of the elliptical orbit.’[57] All of the planets but Mercury fit into Ptolemy’s new patterns of motion.

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Image 16: Ptolemy’s model. The equant was Ptolemy’s greatest insight: with it he was able to explain the apparent motions of the planets, keeping the ancient paradigm of perfect uniform circular motion alive.

According to Barbour, three main considerations can be done regarding Ptolemy’s model. First, his system was ‘good but not quite perfect’, Barbour says; the defects were not so manifest to suggest that it was ‘drastically wrong’. The second consideration regards the lack of physical concepts: as with the past astronomers, the scheme was fundamentally ‘geometrokinetic’, that is, it described the motion of planets without a hint of the possible causes of such motion. This was the reason why – third characteristic aspect of the Ptolemaic scheme – a prominent place in that scheme was reserved to ‘void points’ (the equant, the centre of the epicycles, the eccentric – or the centre of the deferent).

With a small digression regarding the main question we are interested in – the concepts of place and space –  I would say that the attention astronomers paid to the descriptive aspects of motion (descriptions expressed through abstract geometrical schemata) may have favoured the shift of interest from place to space, which we have already analysed from a philosophical perspective in the previous article  (the conceptualization of space – which in the Hellenistic period was still under construction – allowed imaginary scenarios or a sort of freedom of thought that the traditional Greek concept of place/topos – already defined by Aristotle – did not allow). The attention to ‘void points’, that is the attention to the location of astronomical void points without reference to the ultimate reason for those locations (or, to put it another way, the exclusive attention to geometrical rather than to dynamical/physical reasons) was an important passage towards abstraction; a process of knowledge that may easily lead to committing two accidental and related epistemological errors: the fallacy of misplaced abstractness (that is, the unwarranted abstraction of intrinsically material characters of entities, either place or objects, which is the error of mistaking the concrete for the abstract) and its correlate, the fallacy of misplaced concreteness (e.g. the hypostatization of space, that is the attribution of a concrete nature to what is abstract, which is the error of mistaking the abstract for the concrete). [58]

Coming back to astronomy and the mystery of those ‘void points’, as Barbour notes, while Ptolemy didn’t even see the mystery behind those points, Kepler, who tried answer to those mysteries, made conspicuous advancements in astronomy and left his mark in the discovery of dynamics. Another important insight concerning Ptolemy’s astronomy and, more generally, ancient astronomy, was ‘the clarification of the relationship between time, or rather the measurement of time, and the mathematical description of observed motions.’[59] There is a significant section in the Almagest in which Ptolemy discusses the practical measurement of time and identifies what should be considered the one true source for measurement of the passage of time: that source, according to Ptolemy, was the sphere of the distant (fixed) stars. It is the fixed stars that provide the so-called sidereal time, that is, time measured relative to the backdrop of the fixed stars.[60] There were different possible choices to measure the passage of time at disposal of the ancient astronomers (the motion of the sun around the ecliptic, the motion of the moon, the motion of the planets, of course, the apparent diurnal rotation of the stars), but the choice of sidereal time as the ‘true measure of time’ directly passed almost unchanged into Newton’s Principia and allowed the transition to the current ephemeris time (for astronomical purposes) relatively painlessly – Barbour says. This fact was remarkable, and – Barbour says – deserved a third ‘award’ in the list of ‘the baker’s dozen’. In the overall, according to Barbour, the work of Ptolemy and the other Hellenistic astronomers was particularly important for the following achievement: they offered an ‘anticipation of the genuine scientific method, in particular, the use of theory controlled by observation.’ [61]

In conclusion of this section, Barbour noted that for the transition from the old to the new astronomy four major steps were necessary: ‘liberation of astronomy from the diurnal motion of the earth; liberation from the annual motion; liberation from uniformity of motion; and liberation from circularity of the motion.’[62]  While the first two aspects were the result of the joint contributions of ancient astronomers, and the third was Ptolemy’s contribution; unveiling the latter aspect was a task that needed the joint efforts of medieval and modern astronomers to which the next chapters are dedicated.

4. The Middle Ages: first stirrings of the scientific revolution

In the introduction to Chapter 4 Barbour makes what I consider an important reference to the monumental works of Pierre Duhem and a quotation from Alfred N. Whitehead: the French theoretical physicist believed there was not a true single revolution in the passage from the ancient cosmology to the new model offered by Copernicus, but ‘a long series of partial transformations’ that contributed to change the big picture;[63] this vision was consistent with another meaningful passage by Alfred N. Whitehead contained in Science and the Modern World. [64]

As regards the important contribution of the scholastic tradition – which favoured the synthesis of Aristotelian philosophy and Christian theology – is only briefly mentioned here. Nonetheless, it was the rediscovery of Aristotelian theories that contributed to a new critical attitude towards the traditional cosmology and the traditional Aristotelian concepts of place and motion. Barbour introduces the effects of this critical attitude towards those important Aristotelian notions, in two paragraphs on the development of kinematics and dynamics. In fact, the advancements in astronomy and the development of a real science of motion through the findings of kinematics and dynamics, (advancements pertaining the sublunary sphere, that is the terrestrial realm – the reference is to the works of Bradwardine, Oresme, Buridane, the subject of the present chapter), were the necessary steps for Newton’s dynamics and the development of classical physics.

Concerning the paragraph on kinematics, Barbour takes into consideration the works made at the Merton College, Oxford, – in the period 1328/1350 – by Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton; they were interested in the intensity and quantity of movement, and the subject of their studies was the problem of how qualities change in intensity. To their works is due: (i) the distinction between kinematics and dynamics (the former is the study of the effects of movements through the concepts of position, velocity, acceleration; the latter regards the study of the causes of movements, involving the notion of forces, a conceptualization not yet fully-fledged, at that time; (ii) the development of the concept of instantaneous velocity; (iii) definition of uniformly accelerated movement and (iv) the kinematic theorem known as ‘the Merton rule’, which confronts the distance travelled by a uniformly accelerated body with that of a body with uniform speed. In the overall ‘the Calculators’ – this is the way they were called – highly contributed to the application of mathematics for the problems of motion (especially for non-uniform motion – the distinctive character of the motion of celestial planets); as Barbour says,  ‘they prepared the ground for several of Galileo’s most important results’ and even more important for questions of methods ‘we have here a classic example of the mathematics necessary for a physical discovery having been created long before it found physical application.’ [65]

I make another short digression to say that the works and attitude of these authors very likely contributed to the development of the modern concept of space; that’s why the origin of the concept of space has to be found in the domain of mathematics – an abstract domain, indeed –  rather than in the physical realm of concrete phenomena (that is to say: it is dubious to affirm that ‘intuitive space’ is an emergence from the realm of concrete phenomena without the necessary abstract conditions mediated by mathematics and/or geometry). In a certain sense, they anticipated the Cartesian method. In fact, by following Barbour’s own words, we discover that ‘the spread of Merton kinematics to the continent led to an important event, namely, the application of graphing or coordinate techniques to the English concepts dealing with qualities and velocities. This was, in fact, a partial step towards the analytic geometry of the seventeenth century initiated by Descartes, though it did not go so far as to translate algebraic expressions into geometric curves and vice-versa, which is the essence of that method…’ [66]

The argumentation concerning kinematics is closed by a brief analysis of the work of Nicole Oresme (Fleury-sur-Orne, France, c.1320; Lisieux, France 1382): to his work is ‘normally attributed… dated around 1350’ the aforementioned analytical method that was later developed by Descartes, even if some scholars argued that that method ‘arose slightly earlier in Italy’ and that, more generally, ‘the graphing concept had been applied from antiquity to cartography and astronomy.’[67] Like the Mertonians, Nicole Oresme was interested in the variations of qualities, which he treated by geometrical methods. This is the conclusive statement of the paragraph on kinematics‘it was, then, in the area of kinematics, and particularly in the geometrical analysis of uniform acceleration, that the medieval tradition was to be most significant for the development of modern mechanics.’ [68] Modern dynamics and the modern concept of space, I would add.

As regards dynamics – the study of the causes of movements – it was this new strand of study, especially concerned with sublunar motions, ‘that finally broke the tyrannical grip of the Aristotelian notion that terrestrial bodies can only move if they are constantly being pushed by something else and that if the cause of motion ceases then so too will the motion’, Barbour comments.[69]  While the reason for motion in the celestial spheres was God – the unmoved Mover – and this description was enough for the ancient astronomers (who were especially interested in the description of motion), for terrestrial motions the main problem was to locate ‘the pusher’.

At this point we confront with a question that – I believe – had remarkable consequences on the development of the concept of space (emergence and supremacy of space to stick to Casey’s terminology) and the correspondent fall into oblivion of the concept of place: we are directly confronting here with the question of the medium into which motion occurs. The question is introduced by Barbour as follows: ‘the concept of a plenum was a severe hindrance to the recognition of inertial motion.’[70] For Aristotle movement was a change of place: he did not admit the possibility of the void (as a matter of fact I also believe the void is a logical impossibility, before than physical), therefore – I say – the where of motion could be ultimately be reduced to an ultimate place. The impossibility of the void implicated the triumph of Aristotle’s plenist vision of the cosmos, for many centuries. In spite of that, Aristotle’s vision and especially some of the concepts that offered the stable ground to his vision (concepts of place and motion above all), were harshly criticized from the very beginning. So, for instance, as Barbour says, ‘in Philoponus is the clear tendency to thinking about motion in the first place in a void rather than in a plenum.’[71]  At this regard, it is interesting to note what Galileo’s position was with respect to the question of the medium; according to Barbour, ‘whereas for Aristotle the medium is the sine qua non of the body’s motion, determining directly the most fundamental features of the motion, for Galileo the medium is merely a source of disturbance and resistance that causes a body to move otherwise than it would. In Galileo’s view it is the idealized motion that the body would follow in the absence of a medium that is truly significant.’ [72] It seems Galileo’s position is reversed if compared to Aristotle, almost opposite (the same way the concept of space is ontologically opposite with respect to place: one refers to the abstract, the other to the concrete): one looks at an idealized motion; the other at the concrete and visible phenomenon. Coming back to Barbour’s text, Aristotle’s explanation of the motion of projectiles was contrasted by Philoponus and, with modifications, by some Islamic commentators; this strand of though against Aristotle’s theory of motion regarding projectiles was also pursued by Buridan.

Jean Buridan (Béthune, France, c.1295; Paris, France, 1361) was a key figure for the development of dynamics: he was the founder of the Paris school of dynamics. He is especially famous for the impetus theory.[73]  With the increasing interest in questions of motion – not just celestial motions but also terrestrial motions – early ideas about the relativity (of motion) emerged; and with them, the possibility to extend such ideas to astronomy. It is highly plausible that, among others, these ideas facilitated the decisive step that took Copernicus to propose the helioastral frame as the correct explanation for the apparent motion of celestial bodies. The two types of motion that made Copernicus famous for the eternity – daily rotation of the earth about the axis and annual revolution around the sun – were considered since antiquity, but no explanations were given (for instance, see Aristarchus). Buridan and Oresme were certainly conscious about those celestial motions and, more generally, they had ideas about the relativity of motion. As Barbour shows in his text, there were several comments and assertions made by Buridan and Oresme that will have significant development in Copernicus, but Barbour’s concluding remark is that ‘genuine discovery in dynamics requires not only the recognition of an idea but nontrivial demonstration of how it can be used. […] The ideas were there and the mathematics had been developed. But for some reason or other, they were not put together.’ [74]

With specific reference to the forthcoming work of Copernicus – the subject of Barbour’s next chapter – he says: ‘it was Copernicus who revolutionized the world, not Oresme. He did it by an insight that no one (with the possible exception of Aristarchus) before him ever had. And because it involved nontrivial mathematics and empirically observed motions Copernicus spoke with urgency and the authority of one who knows. Thus, Copernicus stated that the earth does truly move whereas Oresme merely advanced the idea as a diversion.’ [75]

5. Copernicus: the flimsy arch (between the heavens and earth)

What did Nicolaus Copernicus (Torun, Poland, 1473; Frombork, Poland, 1543) propose? With this introductive question we directly go to the core of Copernicus’ innovations: as everyone knows, he proposed ‘ [i] that the earth rotates around an axis and [ii] simultaneously travels around the sun; he also proposed [iii] a third motion of the earth to account for the precession of the equinoxes’, Barbour says.[76] Yet, probably, not everyone knows that the main intent of Copernicus’ opus magnum – ‘De Rivolutionibus Orbium Coelestium‘, published in 1543 – was to undo Ptolemy’s theory of the equant and non-uniform motions, while only a small part of the text – ‘a fifth…’ Barbour says – was about the true Copernican revolution. With those two insights – rotation of the earth around its axis and revolution around the sun – Copernicus revolutionized the motions of the planets. In fact, ‘he found an alternative explanation, as precisely mathematical as that of the ancients, of the non-uniform motion of the planets, above all the retrograde motions […], the apparent diurnal rotation of stars and the apparent motion of the sun around the ecliptic.’[77]   Barbour adds: ‘to explain the apparent motion of the sun and stars and simultaneously, without having to introduce any extra motion, to explain all the retrograde motions of the planets – that was a real coup and a nontrivial advance […]. There is nothing like this in the writings of Buridan and Oresme; it was the first sophisticated exploitation of kinematic relativity.’ [78]

After all, Copernicus was able to explain all these apparently unrelated celestial phenomena by a simple change of place (the sun in the centre instead of the earth), that is, by a shift of perspective regarding the referential frame of motions (this is after all the meaning behind the principle of kinematic relativity – or optical relativity as we could also call it)[79] where the same apparent motion of distant celestial bodies could be explained with a single motion – the rotation of the earth – rather than with the revolution of a multitude of celestial bodies around the earth itself; as Copernicus says,  ‘every observed change of place is caused by a motion of either the observed object or the observer or, of course, by an unequal displacement of each.[80] Analogously to the example of the ship already made by Buridan, Copernicus notes: ‘when a ship is floating calmly along, the sailors see its motion mirrored in everything outside, while on the other hand they suppose that they are stationary, together with everything on board […]. Why should we not admit, with regard to the daily rotation, that the appearance is in the heavens and the reality in the earth? This situation closely resembles what Vergil’s Aeneas says: “Forth from the harbor we sail, and the land and the cities slip backward[Aeneid, III, 72].’ [81]

It is properly in virtue of the first exploitation of kinematic relativity that Barbour awards Copernicus with ‘the extra odd ‘bun’ that makes up the ‘baker’s dozen’ and […] provides the crucial link between the heavens and the earth.’[82]

Another remarkable consequence of Copernicus’ great insights regarded the position and the distances of the planets, which definitely changed the image that man had of the cosmos: ‘the earth’s motion around the sun led immediately to an unambiguous ordering of the planets and to a fixing of all distances apart from one common scale factor, the radius of the earth’s orbit.’ [83]

In consequence of the new scheme, we have the first clear statement regarding the organic nature of the Universe – which, we currently say, could be seen as a system, properly: ‘in his preface to De Revolutionibus, he said that the principal failing of the astronomers was that they could not deduce “the structure of the universe and the true symmetry of its parts. On the contrary, their experience was just like some one taking from various places hands, feet, a head, and other pieces, very well depicted, it may be, but not for the representation of a single person; since these fragments would not belong to one another at all, a monster rather than a man would be put together from them”.’ [84]

Coming back to the important argument of the distances between celestial bodies, how could such distances be assigned? Employing trigonometric techniques, as Barbour explains: ‘if the earth is assumed to move around the sun, the diameter of the earth’s orbit can be used as a trigonometric baseline’. [85]

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Image 17: Example of trigonometric parallax diagram to measure the distance to stars (original drawing from the Australia Telescope National Facility website).

The use of a trigonometric baseline to determine distances had some immediate consequences: first, and for the first time in astronomy, the idea of an astronomical unit (AU) could be introduced and with it the distances between the various planets determined (as a matter of fact, Barbour says, ‘it was Copernicus who measured the heavens […] even with the too small value for the astronomical unit that Copernicus accepted, the distance to the stars was increased at least two or three thousandfold’).[86] There is another major consequence, which directly deals with the questions of place and space we are interested in. Let’s see what Barbour says in this respect: ‘Although, as we shall see, Copernicus retained an essentially Aristotelian cosmology, his dramatic magnification of its size was almost certainly one of the factors that led to its ultimate replacement, about a century after Copernicus’s death, by the notion of an infinite space. Note how the interstitium between Saturn and the fixed stars is described as ‘immense and even like the infinite.’[87] (see the image below). Barbour continues:  ‘This was only one of the ways in which Copernicus forced upon thinking man a radical revision of basic concepts. […] it is interesting to see here explicitly how the practical astronomers imposed the trigonometric viewpoint at the expense of the philosophers.’ [88]

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Image 18:Interstitium inter Saturnum et stellas fixas, Immensum, a deo infinito simile’ (image adapted from Kepler’s Mysterium Cosmographicum, 1596).

If we consider the emergence of the concept of infinite space as a phenomenon that could have been ‘imposed’ (I’d prefer to say ‘favoured’) by trigonometric techniques (used by mathematicians, astronomers and geometers) and the parallel ‘downsizing’ of the concept of place (which is representative of the traditional ‘viewpoint of philosophers’  to stick to Barbour’s words), then we could appreciate a convergence of perspectives between Barbour and the philosophical history outlined by Casey in The Fate of Place (see the article Place and Space: A Philosophical History). More generally, I believe the slow but progressive success of mathematical techniques applied to decipher or represent phenomena of reality contributed to the shift of interest from place to space (and the revision of their meanings). There is the case of trigonometric techniques used by astronomers, as Barbour explained, but, in parallel, architects, artists and mathematicians were developing sophisticated perspective techniques to represent real or imaginary places; it means that a scientific mode of thinking was progressively changing people’s minds since the medieval period, at least: that’s the ultimate reason for that change of meanings – or revision – of place and space.

Coming back to the Copernican system, even if, on the one hand, that system proved to be a beautiful, elegant and simple model that accounted for different celestial phenomena – ‘it gave the planetary system a far greater coherence and intrinsic symmetry’-[89], on the other hand, it was not free from incongruities and limits: the constant appeal to perfect circular motions, and the fact that the Copernican system ‘remained to a remarkable degree geocentric’.[90] These are the remnants of Aristotelian cosmology. I believe this legacy is critical to understand the aforementioned shift of interest from place to (infinite) space, as I’m going to show soon. Barbour devoted one paragraph – Copernicus’s Concept of Place and the Ultimate Frame of Reference – to the influence that Aristotle’s vision and concepts still exerted on Copernicus. He writes: ‘the Aristotelian element in Copernicus is particularly interesting. Copernicus is very concerned that his concept of motion should be epistemologically sound. It is clear that he rejected the idea that motion takes place relative to space – it must be relative to observable matter.’ [91]

In my opinion, Barbour presented quite well the co-presence of two different spatial conceptualizations within the mind of Copernicus. Here, in the context of astronomical discussions, the concept of place merges with space. This is Copernicus speaking (I have added some quick notes, between square brackets, regarding the questionable choice of terms used by the English translator of the Latin text): ‘since the heavens, which enclose and provide the setting for everything, constitute the space[*] common to all things, it is not at first blush clear why motion should not be attributed rather to the enclosed than to the enclosing, to the thing located in space[**] than to the framework of space[***]  [* I have verified that the terms used by Copernicus in his original Latin text are, respectively:  *communis locus – then ‘space common to…’ is the English translation, but in this case I believe a literal translation ‘place common to…’ would have been more appropriate and less misleading;  ** locato is the term used by Copernicus, which can be literally translated as ‘that which is located‘; *** locanti is the Latin term used, which is literally, ‘that which locates’. In all of these cases the term space is an addition – a misleading addiction I would say – made by the translator; the ‘ultimate where’ of Copernicus – that in which things are located, or even that which locates things – is place, not space].  Barbour continues: ‘By the “heavens” Copernicus meant, of course, the immensely distant sphere of the fixed stars. The above concept of position, and hence motion, clearly comes from Aristotle. Nevertheless, despite this strongly Aristotelian element, the centuries of practical astronomy left their mark. Copernicus needs a container, but one is sufficient; his world is not an onion in which a succession of layers are needed to give meaning to position all the way from the outermost layer right through to the centre. Copernicus combines Aristotle’s ultimate ouranos [which, according to Aristotle, is ‘the place’ of the fixed stars, or even the place of the entire cosmos, actually] with the Platonic concept that Aristotle rejected, namely, that space[*] is some kind of dimensional extension lying between the points of the containing surface’  [* it is everything but immediate, or mechanical, to say that that the ‘kind of dimensional extension…’ is space since the Platonic notion Barbour refers to is ‘chōra’ – see my previous item Back to the Origins of Space and Place]. ‘The space within the container is no longer a nest of topological envelopes but a tautly spanned region of trigonometric relations, an invisible membrane spanned by the rim of the drum provided by the fixed stars. This concept of space is rather well illustrated by Fig… [see below,  image 19, which, apart from the text that I have added and the ‘rim of the drum’ that I have highlighted, is analogous to the image used by Barbour],  which actually is due to Kepler but this is not relevant since on this particular point his views were almost identical to Copernicus’s. The stars (‘studs’) are represented by the signs of the zodiac around the rim of the ‘drum’. The successive points on the rim represent successive conjunctions of Jupiter and Saturn. The lines within the rim highlight the way in which metrical geometry is used within an ultimate frame of reference provided by the stars […]. With Aristotle, he is prepared to accept that “beyond the heavens there is… no body, no space, no void, absolutely nothing”. He is only interested in what is inside the outermost rim and he opens the account of his cosmology with these words: “The first and highest of all is the sphere of the fixed stars, which contains itself and everything, and is therefore immovable. It is unquestionably the place of the universe, to which the motion and position of all the other heavenly bodies are compared”.’ [92]

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Image 19: I believe the trigonometric relations astronomers derived from the observation of celestial positions may define an abstract space of mathematical origin, within the concrete place of the cosmos. This is an example of conflation between extensive concepts – place, space – which, at the time of Copernicus and Kepler, were still in the process of being rethought or elucidated (background-image: illustration of successive conjunctions between Jupiter and Saturn, from Kepler’s Stella Nova, 1606).

Apart from the philological problem pertaining the questionable choice of spatial terminology attributed to Copernicus (of course, here, Barbour only reports some passages from an existing translation of De Rivolutionibus),[93] in the overall, I believe Barbour has seized the essence, going directly to the core point: in Copernicus there is a co-presence of ‘spatial’ concepts: he is still influenced by the conception of spatiality devised by Aristotle (maybe ‘placiality’ would have been a more pertinent term than ‘spatiality’, but the use of this term needs another article to be elucidated), which is literally a conceptualization very close to matter (place as the immediate container of matter) but, at the same time, as Barbour acutely observed ‘centuries of practical astronomy left their mark’, this fact meaning that another conception of spatiality was gaining ground in the mind of Copernicus (and of his contemporaries): something like an ‘empty space’ similar to the void  – to kenon – of the ancient Atomists, a concept of ‘spatiality’ more psychologically distant from matter, or even totally detached from it; more appropriately, as it is probably the case for the astronomers who made extended use of trigonometry, a concept of space as container of (‘abstract’ – we should now add) geometrical relations, which was coexistent with a more traditional concept of place as locator/container of matter (i.e. the concrete place of the cosmos, the concrete place the fixed stars, which were supposed to be located on a distant sphere, etc.). That’s the other concept of spatialization present in the mind of Copernicus. A posteriori, we should say that there is here an evident conflation of different realms: while the traditional concept of place belongs to the realm of the concrete (the actual place of the cosmos, the place of concrete bodies, etc.), the space of trigonometric relations belongs to the realm of abstract geometrical relations (and it was not by chance that trigonometry, which was especially used to find distances and linear extensions for practical purposes, influenced the formalization of the concept of space: as a matter of fact, in the origin,  space – the Greek stadion/spadion – was truly a simple distance or linear extension, as I have shown in a previous article). We have to wait for Newton before this conflation of ambiguous spatial terminology could be ‘solved’ (in a very questionable manner indeed) transforming or, better, reifying the space of abstract geometrical relations into a concrete entity – absolute space – so that place could be finally incorporated within space (which is a conception that turns upside down the Aristotelian conception of ‘spatiality’) .

Then Barbour engages with the other important argument: the relation between motion and the frame of reference where motion occurs. With a swift and synthetic overview through the centuries, Barbour says that motion takes place relative to matter according to Ptolemy, Copernicus, Kepler and Poincaré; relative to space according to Galileo, Newton, Maxwell and Einstein.[94] And he goes on introducing the very meaning of the frame of reference with respect to which referring motion:  the surface of the earth for the primitive men, the limitless void for the Atomists, place for Aristotle, the earth and the fixed stars for the early astronomers, until we come to Copernicus for whom the fixed stars represented ‘he only appropriate frame of reference’; then he asks the central question: What is motion? And precisely, what is motion for Copernicus? Motion had an explicatory function for Copernicus since through motion he could explain the observed celestial phenomena; Barbour observed that Copernicus had a mystical attitude toward geometry. According to Copernicus, the observed celestial motions had to be explained by a superposition of exactly circular  uniform motions and this fact still demonstrated his ‘adherence to the archaic geometrokineticism.[95] ‘For Copernicus – Barbour continues – the illusion of the seeming fixity of the stars spread, as it were, a sheet of ice over the abyss and tended to make him content with a geometrokinetic concept of motion. For on ice you can skate and trace the most exquisite patterns. Copernicus looked down from his newly-won high vantage point and had no reason to suspect that God had anything more in mind than the tracing of beautiful circles in the ice.’[96]  And although Copernicus gave the first example of exploitation of kinematic relativity (passing from the earth-centred frame of reference to that of the fixed stars) he only made ‘the first step in the direction of the thoroughgoing relativity of motion.’ [97]  It is Mach who will take to the logical conclusion this sort of reasoning, by saying that position and motion are actually determined by the existing matter in the Universe. Then, we learn from Barbour, in the overall, Copernicus’s concept of motion didn’t change with respect to the tradition (at least it didn’t change with respect to the way celestial motions were understood: perfectly circular and uniform).

Before following Barbour’s exposition, in the lines that follow I will give my interpretation of these facts since this is another critical passage to understand the vicissitudes of space and place in the passage from the Middle Ages to the Early Modern epoch. Copernicus’s conception of motion didn’t change precisely because the Copernican system ‘remained to a remarkable degree geocentric’, as Barbour said. I believe this fact – this permanence of an ancient mode of thinking about celestial motions and their relation to matter – was the inevitable result of the remnants of Aristotelian cosmology expressed through precise conceptualizations: I’m speaking of the almost inextricable relation between place and matter, and of celestial motions, which still survived in the mind of the medieval and the early modern people. In a certain sense, by appealing to conceptualizations taken from the Aristotelian and the Ptolemaic tradition and, at the same time, by appealing to new modes of thinking about astronomy (trigonometric techniques played a major role, and contributed to the emergence of a concept of space out of geometrical considerations – at least, this is my opinion)  we could say that Copernicus was able to spatialize place (within the pre-existing place of the cosmos he abstracted spatial trigonometric relations) or, if you prefer the other side of the coin, he was able to placialize space (the abstract spatial relations that he found were grounded on the reassuring presence of the cosmos as finite place). In the end, what Copernicus did was to change the frame of reference with respect to which motion happened (the sun in the centre and the distant stars as reference instead of the static earth in the centre), but the concept of motion in the ‘celestial spheres’ remained unchanged; even more significant, the understanding of motion with respect to matter remained unchanged: place (the place of matter, actually) is still the referent for motion, not space. While the slow affirmation of the concept of space out of mathematical considerations (in the mind of the astronomers, before all) could afford mental journeys through which a new system of reference could be figured out (a shift of the point of view from the earth in the centre to the sun in the centre could be, at first imagined, then verified by real observations), the contemporaneous permanence of certain aspects of the traditional Aristotelian/Ptolemaic vision didn’t allow to completely change the concept of motion. With respect to the new possibilities offered to the mind by trigonometric consideration let’s see what Barbour says (this is another critical passage to understand how the concept of space was penetrating the mind of people in that precise historical epoch): ‘If you had asked Ptolemy what the universe looks like from Saturn, he would have been quite unable to give you any answer based on a sound theory. He did not know how far away Saturn is; he had no way to overcome the scale invariance of his solution to the problem of planetary motions; his trigonometry failed him. By his extension of trigonometry, Copernicus put in man’s hand a tool that, in his mind’s eye, enabled him to travel in space. He could deduce what the world looks like from unvisited points at vast distances from the earth. Historically, Copernicus performed this task. He recounted his insight to his contemporaries in lucid language and, for the experts, provided the solid mathematical arguments which substantiated his vision. For the first time in history, man had a tolerably accurate picture of the solar system and at least a comprehension of the immensity of the universe. Moreover, unlike the early Greek hints of heliocentricity, Copernicus’s picture was based on solid observation and theory.’ [98]

It is quite clear that, before all, that space was an abstract, a mental space. Here, there is a vision (a mental space, properly) to substantiate, to use Barbour’s own words: a process initiated by Copernicus and formalized by Newton. Only with the definite abandonment of the old concept of place and the progressive affirmation of the concept of space – severed from any causal relation with matter and from the supposed limits of the material cosmos – could the concept of motion be definitely changed, and with it the definite affirmation of a truly new cosmological vision be affirmed: this final step was made by Newton. Then, the function of the Aristotelian concept of place was completely replaced by the concept of absolute infinite space, and place could be reduced and subsumed within space. What I have just said is somehow related and in agreement with Barbour’s hypothesis regarding the division of the Copernican revolution into three phases, or stages: the first initiated by Copernicus; the second prosecuted by Kepler and Galileo; the third, completed by Descartes and Newton, for philosophical and physical aspects.[99] I believe these three stages hypothesized by Barbour can be associated with the progressive modification of the concepts of place, space and motion, which finally led Newton to finish what Copernicus began: a bright new Cosmology. In Copernicus we have the permanence of the concept of place as the material referent for celestial motions; yet, the progressive emergence of a concept of space out of mathematical, geometrical and astronomical considerations led to the possibility of imagining another material frame of reference, different from the earth. In turn, this led to the abandonment of the Aristotelian theories regarding sublunary motions (if we change the place of the earth from the centre of the Universe, as immediate consequence Aristotle’s theory of motion and proper places, becomes meaningless) and, most important, this probably was the coup de grace for the ancient faith in the Aristotelian system and conceptualizations. In the second stage – this is in anticipation of the following two chapters dedicated to Kepler and Galileo – while Kepler still kept place/matter as the referent for motion, he contributed to dismissing the validity the second pillar of the ancient cosmology – the perfect circular and uniform motion of celestial bodies. In parallel, Galileo contributed to the development and use of the concept of space, while he was working on the laws for terrestrial motions. Finally, Newton reified those different shades of spatial conceptions, from antiquity down to Descartes, into absolute space – a true physical entity. He reduced the Aristotelian concept of place into simple location and, finally, subsumed locations within (absolute) space. Motion could be finally freed from any influence from matter and the Laws of motion could be discovered unifying the heavens and the earth. Any remnants of Aristotelian theories (that is, his cosmology founded on the concept of place and the division between terrestrial and celestial motions) could be definitively forgotten.

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Image 20: Synoptic Table concerning my interpretation of the concepts of place, space and motion in the passage from Copernicus to Newton (red and green marks mean the refutation of long-established conceptualizations or the introduction of new ones).

Coming back to Barbour and Copernicus, the attitude of Copernicus towards motion was descriptive (it was the same for Ptolemy): he aimed to describe the motion of planets, and not to explain the reasons for them. As Barbour says, ‘if he has any concept of motion that goes beyond geometrokineticism or ballet, it is mechanical.’[100] Both Copernicus and Ptolemy were interested in the mathematical expression of motion rather than in its physical explanation: ‘My own impression  – Barbour says – is that in both books the mathematical description is paramount; after the mathematics has been sorted out, one may then put one’s mind (if so inclined) to the elaboration of a mechanical model that reproduces the mathematics.[101] The use of the attribute mechanical for the description of motion allows us to lay our eyes on an argument that will haunt the debate about the vision of the world for the coming centuries: the emergence and affirmation of a mechanical and deterministic world-view. In his introduction to De Rivolutionibus Copernicus explicitly speaks about the cosmos as a ‘world-machine‘.  Copernicus says:  ‘I began to be annoyed that the movements of the world machine, created for our sake by the best and most systematic Artisan of all were not understood with greater certainty‘, announcing, at the same time, the reasons for his work.[102] I believe the diffused insistence on the systematic and mechanical movements of the celestial spheres – explained through the use of astrolabes and armillary spheres – coupled to the metaphor of the world like a perfect clock (the repetitive and predictable motions reminded those of a clock) were strongly influenced by Copernicus’s work.

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Image 21: In antiquity, astrolabes and armillary spheres were used for describing the movements and positions of celestial bodies.

Who or what does control the cosmic machine according to Copernicus? The machine is controlled ‘by the best and most systematic Artisan of all: God. That which is systematic (the Copernican scheme is probably one of the first acknowledged systems in the history of human knowledge) coupled to that which is mechanical is the distinctive mark of the new epoch. Copernicus played a decisive role in the diffusion of that image of the world.[103] This question about the ‘power’ or ‘the engine’ of the world – that is the ultimate cause of all motions – allows Barbour to point out an important character – a limit, indeed – of the Copernican scheme: the attribution of that ‘motive power’, or ‘driving force’, to God rather than to the forces exerted by the sun, which, in the new scheme has almost the same negligible role – indeed with a different location – that it had in the Ptolemaic scheme: a lantern that illuminates the ballet of planets; this is the image used by Barbour. [104]

One of the concluding remarks in the paragraph about Copernicus regards the consequences of the Copernican discoveries; and especially the consequences for motion and the discovery of dynamics. This allows introducing the remaining chapters: what does remain to fix in the Copernican scheme? The amendments will be done especially, but not exclusively, by Kepler and Newton. That’s why, as we have already seen, Barbour suggests to interpret the Copernican Revolution in three distinct stages. The first – the one we have just analyzed – inaugurated by Copernicus; the second – which is the subject of the next two chapters of the book – prosecuted by Kepler and Galileo; the final stage completed by Descartes and Newton. Specifically, it was Kepler who will attribute the sun an appropriate motive power: not just a lantern, but an active motor influencing the motion of planets.

6. Kepler: the dominion of the sun

Since the life and work of Johannes Kepler is intimately connected to the Danish nobleman and astronomer Tycho Brahe (Knutstorp Castle, now in Svalöv, Sweden, 1546; Prague, Czech Republic, 1601), Barbour begins the sixth chapter by introducing Brahe himself, a ‘Danish nobleman…an astronomer with talent stimulated to carry out a really extensive programme of observations and comprehensive testing of the models of the celestial motions’,[105] which contributed to change the history of astronomy and also contributed to the emergence of a specifically-modern scientific attitude. As Barbour explains, Brahe made significant advancements in the theory regarding the motion of the moon, but his prominent position in the history of astronomy is due to his extensive programme of observations of celestial motions and positions that were decisive for the amendment to the Copernican system made by Kepler. As Barbour says: Brahe ‘was aware of the need to make observations over very long periods of time and at all positions of the orbits of the planets and moon.’[106] After Brahe was appointed Imperial Astronomer by the Emperor Rudolph II, and after joining him at the Benatky Castle, in Prague, Brahe invited the young Kepler to work as his assistant: the relationship between the two men was a tug of war and reconciliation, as Barbour showed; both needed the help of the other to complete their works and ambitions.

The German astronomer, mathematician and astrologer Johannes Kepler (Weil der Stadt, Germany, 1571; Regensburg, Germany, 1630) was truly a watershed between ancient and modern astronomy: this is the image Barbour used referring to a concept developed by the Hungarian-British author and journalist Arthur Koestler in the book The Sleepwalkers: A History of Man’s Changing Vision of the Universe, to which I also refer in this specific context.[107] Why Kepler was considered a watershed? Because in his works we find the co-presence of elements which are the remnants of an old way of approaching astronomy – like the interest in numbers, geometry and abstract mathematical relations (as in the Pythagorean and Platonic traditions) – coupled with an unprecedented and specifically modern attitude and interest towards physical concepts used to explain the celestial happenings, other than a thorough interest for historical records and data regarding the positions and the movements of planets (that is an interest in the observation of facts as a decisive moment to prove abstract theoretical models). Kepler’s opus magnumAstronomia Nova – written in the period between 1600 and 1606, published in 1609 – ‘contains the account of how, using Brahe’s wonderful treasury of observations, he found his first two laws of planetary motion by studying the motion of Mars’.[108] Indeed, Kepler’s prominent role in the history of astronomy is due to the discovery of three Laws of planetary motions: what we call The First Law, which, actually, Kepler discovered after the second, says that ‘the planets travel round the sun not in circles but in elliptical orbits, one focus of the ellipse being occupied by the sun’. The Second Law says that ‘a planet moves in its orbit not at uniform speed but in such a manner that a line drawn from the planet to the sun always sweeps over equal areas in equal times.’[109] The Third Law, which is about the influence that the sun exerts on planets according to their distance – the closer the planet is to the sun, the faster it moves along its orbit – was discovered later, and was published in the book Harmonice Mundi (1619). Indeed, the three Laws of planetary motion discovered by Kepler exerted a strong influence on the solution of the problem of motion given by Newton, and, Barbour notes, they certainly belong the so-called ‘baker’s dozen‘ of insights necessary for Newton’s synthesis of dynamics.[110]  According to A. Koestler, ‘the promulgation of Kepler’s laws is a landmark in history. They were the first “natural laws” in the modern sense: precise, verifiable statements about universal relations governing particular phenomena, expressed in mathematical terms. They divorced astronomy from theology, and married astronomy to physics. Lastly, they put an end to the nightmare that had haunted cosmology for the last two millennia: the obsession with spheres turning on spheres, and substituted a vision of material bodies not unlike the earth, freely floating in space, moved by physical forces acting on them.’[111]

To come back to the crucial element  regarding Kepler’s modernity, the distinctive modern attitude towards Science is especially evident in the pages of the Astronomia Nova: here, a specific attitude marks a line of demarcation with respect to the previous epoch, that is, ‘his refusal to accept a model of planetary motion unless it matched the observations perfectly and comprehensively’;[112] most of all, ‘what distinguishes the Astronomia Nova most clearly from the Almagest and De Revolutionibus is its intimate blending of physical reasoning with purely astronomical argument based on massive and brilliant application of trigonometry to Brahe’s observations.’[113]  This is after all, what the very extended title of the book Astronomia Nova suggests, exactly: ‘A New Astronomy – Based on Causal Explanation of Celestial Physics Treating the Motions of the Planet Mars Deduced from the Observations of Tycho Brahe’‘. In this book, for the very first time, we assist at the introduction of a truly modern conception of force as the physical reason that may explain celestial happenings.

Which were the starting points for Kepler’s astronomical journey? (i) The incomplete heliocentricity in the Copernican model – as Barbour says: ‘Kepler just could not accept that the sun was not exactly at the centre of the world’;[114] (ii) the now-dismissed theory of the celestial spheres carrying the planets (due to the thorough work of observation made by Tycho Brahe); (iii) the increasing importance and focus on physical concepts derived from the English physicist and astronomer William Gilbert (1544 – 1603). [115]  In broad outlines – Barbour says –  ‘it was necessary to rethink completely the reasons why the planets moved at all. This was where Gilbert and his physical forces came in. If the planets were not moved by crystal spheres, they must be moved and directed by something else – why not by forces? Finally, everything was knitted together in Kepler’s mind by the rather natural idea (or, at least, so it now appears) that the forces which moved the planets originated in the body of the sun. In such a case the sun would naturally be the centre of the planetary motions.’ [116]

In describing the discovery of the three Keplerian Laws of planetary motion, Barbour follows Kepler’s own convoluted path narrated through the pages of the Astronomia Nova. At this point, my aim is not simply to repeat Barbour’s exposition (or, in parallel, Koestler’s exposition) with respect to the discovery of those laws (I redirect the reader who is well-versed in geometrical argumentation to the pages of Barbour’s own book, or, alternatively, I redirect the reader less inclined to technical explanations to Arthur Koestler’s widely-accessible book ‘The Sleepwalkers‘); rather, my point is to put the focus on those elements which are significant for our overall discourse on space and place. Very briefly, Kepler made some assumptions – or preliminary steps – which were necessary before he could devise those Laws. First of all, Kepler advanced arguments for taking the sun as the true centre for the planets’ orbits; then, he considered the orbits of the planets laying in very nearly planes passing through the centre of the sun, and demonstrated that Mars’s plane was inclined with a certain fixed angle to the plane of the earth’s orbit; and, most of all, he rejected the ‘basic axiom of cosmology since the times of Plato’, namely the idea that planets move in uniform motion in perfect circles. Actually, in the first stages of his research on planetary motions, he retained the idea of perfectly circular motions but he threw out uniform speed. What is important to point out is that Kepler was guided mainly by physical considerations: ‘if the sun ruled the motions, then his force must act more powerfully on the planet when it is close to the source, less powerfully when away from it; hence the planet will move faster or slower, in a manner somehow related to its distance from the sun’– it is this idea that will lead him to the discovery of the third Law, properly.[117] By relying on the previous convictions, he tried to determine the precise orbit of Mars, whose trajectory had a more evident eccentricity then the other planets (this fact meant that the orbit of Mars couldn’t be a circle; had he proved that Mars orbit was not a circle, then the orbit of the other planets couldn’t be a circle either: this way he could have dismissed once far all the old convictions regarding the perfect circular motion of the planets). But his first assault to Mars failed, even if for very small minutes of arc. So, we come to a critical moment for the history of astronomy and, I believe, an important moment for the modes of thinking about the phenomena of reality and for the concepts of space and place as well (in the middle ground between the real and the imaginary): this moment regards the new attempt, made by Kepler, to determine the orbit of Mars, starting from scratch. Since the Earth is our observatory, Kepler thought that he had to be sure that what Copernicus did was right; but he did not take that for granted. As Koestler explains: ‘Copernicus had assumed that the earth moves at uniform speed-not, as the other planets, only “quasi-uniformly” relative to some equant or epicycle, but really so. And since observation contradicted the dogma, the inequality of the earth’s motion was explained away by the suggestion that the orbit periodically expanded and contracted, like a kind of pulsating jellyfish. Kepler rejected it as “fantastic”, again on the grounds that no physical cause existed for such a pulsation.’[118]  Now, we come to the critical point; to be sure about the motion of the earth he had an insight nobody had before him: ‘it consisted, essentially, in the trick of transferring the observer’s position from earth to Mars, and to compute the motions of the earth exactly as an astronomer of Mars would do it. The result was just as he had expected: the earth, like the other planets, did not revolve with uniform speed, but faster or slower according to its distance from the sun.’[119]  Let’s see how Barbour describes this critical passage regarding Kepler’s insight: ‘His idea was as simple as it was beautiful. It shows dramatically the liberation of Kepler’s mind brought about by the Copernican revolution. He had learnt to wander freely in helioastral space and solved the problem by transporting himself conceptually to Mars. The space journey was done as follows…’[120]  Very appropriately, Barbour points out the significance of this moment which goes beyond Science: Kepler’s is probably the one who inaugurated the literary genre of science fiction, and specifically of  space travels (this fact implying the ability to replace the concrete place of the cosmos with its mental image, which also affords the possibility to freely roam within an unlimited space – this new abstract extent cannot be certainly described by using the traditional term ‘place’): ‘there is something very appropriate about the fact that Kepler, quintessentially a child of the heady Renaissance times of new endeavour and breaking free from old bonds, was the author of an early work in the genre of scientific fiction – a dream of a journey to the moon. He was indeed the first man whose spirit roamed freely in space […]. In his ‘Epitome of Copernican astronomy’ […] he even advanced ”space travel” as one of the reasons for believing in the Copernican world system. For it was not fitting that man, who was going to be the dweller in this world and its contemplator, should reside in one place of it as in a closed cubicle: in that way he would never have arrived at the measurement and contemplation of the so distant stars, unless he had been furnished with more than human gifts […] it was his office to move around in this very spacious edifice by means of the transportation of the earth his home and to get to know the different stations, according as they are measurers – i.e., to take a promenade – so that he could all the more correctly view and measure the single parts of his house.’ [121]

This is a brand new world, and a brand new horizon is disclosed, or, even better, a world beyond the known horizon is disclosed: this is space, a pure abstraction, a product of the imagination, properly. With it the new ability of the mind to roam freely in space. It is this new possibility which allowed Kepler to find the right way to determine the correct motions of the planets. If we come back to the synoptic table I illustrated above (image 20), we see why, with respect to Kepler, I have spoken of a co-presence between a conception of place and matter inherited by the tradition (which takes account of the physical aspects related to motion, and probably pushed Kepler toward the adoption of concrete physical variables) and a conception of space where the mental aspect has a prominent role (space as the product of our imagination), so that no confusion between place – the concrete –  and space – the abstract – can be done (it is this abstract concept of space as the possible/potential reconstruction of an actual place – the place of the cosmos -, that which allows imagination to make new hypothesis, forgetting the actual restraints of matter and place; in this way we can understand space as an abstract place, and find a correct answer to a real problem from an unexpected point of view).

Finally, Kepler made the most important step to solve the problem of celestial motions: he introduced physical variables to sustain his geometrical argumentation (this is a specifically modern attitude); as Barbour said with respect to this important question of method: ‘he correctly sensed that there were deeper and, so to speak, invisible or transcendent principles at work behind the beautiful geometrical structures that the traditional astronomical techniques […] had revealed. Most striking of all was his conviction that these deeper principles must be manifested in precise mathematical relationships.’ [122]

This topic is just one short step from the even more explicit contiguity between Kepler and Mach regarding the relation between motion, matter and place, rather than space (with respect to the interests of this website, this point is also important because it may give some hints about the relationship between place and matter understood as concrete notions, and space understood as an ideal notion). In terms of novelty with respect to the Copernican system and motions, and, more specifically, ‘in terms of the size of corrections that Kepler made to the Copernican motions, his innovations do not seem all that remarkable. However, what made the true Keplerian revolution was the fact that his corrections brought all of the planetary motions into quite perfect alignment with the sun.’[123] This was possible because of two innovative attitudes of the mind, which look into a distant future (Mach and Einstein): (i) ‘The first was the remarkable degree to which Kepler, acting on his gut feeling that position is ultimately defined and determined by matter and not void space, raised his level of reliance on matter to define the position and motion of other matter.’[124] At this regard, we have to remember that, before Kepler, anomalies and inequalities regarding the motion of planets were solved on the base of abstract geometrical argumentations that could save the appearances of phenomena without no hint at actual physical explanations: so planets or the sun itself were hypothesized as moving around or revolving around void points in circular orbits; Kepler himself is quite explicit at this regards as we can read in the following passage from the Astronomia Nova as reported by Barbour: ‘I do not deny that one can conceive a point and around it a circle. But I maintain that if the centre point exists only in thought, timeless, without outer sign, then, in reality, no mobile body can form a perfectly circular path around it.’[125] Then, for Kepler, the ‘control of the planet’s motion was linked explicitly and directly to the real sun and the real stars.’[126]  A position which has an evident echo in the work of Ernst Mach.

I make a brief digression, since, for me, this is a critical passage for understanding the difference and the continuity between the concept of space and the concept of place; a passage hinging on a distinction that I believe Kepler has instinctively grasped, but that he did not elaborate explicitly: the distinction between space as a mental notion (this is when Kepler refers to void points: a void point can only exist in a void extent or space, that is in thought, and not in an actual place) and place/matter as concrete notions (that which exists in reality and have causal effects on the motion of bodies). I believe we can find in those words pronounced by Kepler’s the perfect realization of the required balance or correspondence that should exist between abstract thinking and the actual reality; such awareness is at the base of the correspondent correlation that I think should exist between concepts of space – that which is abstract – and concepts of place/matter – that which is concrete. [127]

This brings us to the second significant shift of attitude with respect to the past thinkers: (ii) while in the Copernican scheme ‘the sun had an entirely passive role – it was a mere dispenser of light and heat […] in contrast, Kepler, as we have seen, accorded the sun an indispensable role as motor of the planetary motions. Matter is quite vitally involved in physically determining the motion of other matter. This indeed is the core of the Keplerian revolution, […] It was this conviction which led Kepler to suspect that the planetary motions were governed by simple mathematical laws in which the position of sun was the key to everything else. Without this conviction the area law, and with it the ellipse too, could never have been found.’[128] In my opinion, this is also a plea for plenism! A plea for place and matter against space and the void (a position I advocate for in this website following Aristotle, Kepler, Descartes and Mach, at least). In fact, as Barbour says: ‘He was not at all enamoured of the idea of space existing on its own as a physical entity independent of tangible and perceptible matter. To quote from the Epitome: “if you are speaking of void space, that is, of what is nothing, what neither is, nor is created, and cannot oppose a resistance to anything being there, you are dealing with quite another question. It is clear that [this void space], which is obviously nothing, cannot have an actual existence”.’[129] Barbour continues: ‘it is true that in Chap. 2 of the Astronomia Nova Kepler grants that a body might be able to follow a straight line through the empty ethereal air. But at the back of his mind he always had the stars to define such a motion.’[130] The striking similarity between Kepler and Mach is now evident, and ‘although in Kepler’s world the planets find their way by looking to markers that he believed to be fixed – the sun and stars – his conceptual framework is in fact only one short step from the solution to which Mach was led when he confronted the fact of universal motion of all matter in the universe. The natural progression from Kepler’s scheme is not one in which all the bodies in the universe look to invisible space ‘to see where they should go’ in their motion but rather the fully Machian one in which ‘all look to all’ and perform coordinated motions relative to each other.’[131]  Indeed, there is a striking affinity between Kepler and Mach with respect to the way the ultimate phenomena of reality should be understood. Barbour goes on with the similarities of approach between the two men across different epochs: ‘Kepler’s dedication to the essential objects in which he believed the divine text to be written is paralleled by Mach’s dedication to the phenomena – for Mach had a true devotion to sights, sounds, and colours and wanted to show that they, and not conjectural atoms devoid of all phenomenal accidents, were the true bricks of the world.’[132]  And, in the same pages, Barbour closes this parallel with the following observation that I cannot help but subscribe, given the nature of my research on concepts of place and space: ‘Because Kepler and Mach share a certain childlike primitivity, we should not underestimate the vitality of their ideas’.

So we are now approaching the end of the Chapter that Barbour has dedicated to Kepler, and we are going to assist to a real turning point (if not a turn upside down, properly) in the forthcoming chapters: ‘about 40 years after Kepler published the Epitome of Copernican Astronomy, the concept of motion had passed from being completely matter-based to completely space-based.’ [133] And, to come back to a frequent refrain of mine, this implicated a shift of focus and interest from the concept of place (which, by tradition, has an inextricable conjunction with matter) to the concept of space (which is intrinsically alien to matter, severed from it, as in the ancient atomistic tradition where the void – the ancestor of space – was the counterpart of matter). It is was a period where at first Galileo and then Descartes – one more inclined to practice, observations and experiments, the other more inclined to philosophical theorization – contributed to creating the necessary conditions to the emergence of the modern world, before the final systematization operated by Newton. It is Galileo, properly, the subject of Barbour’s next chapter.

7. Galileo: the geometrization of motion

If Kepler definitely dismantled the traditional thinking regarding celestial motions, the Italian scientist Galileo Galilei (Pisa, Italy, 1564; Arcetri, Italy, 1642) dismissed traditional ideas concerning terrestrial motions.  Galileo is often considered the father of modern science – Barbour says; this is ‘due to the enthusiasm and vigour with which he applied mathematics, above all Euclidean geometry, to the problem of describing and understanding the world.’ [134]

With respect to the questions we are particularly interested in in this website – questions of space and place – Galileo has as prominent position with respect to the diffusion of a concept of spatiality that will have in Descartes and Newton two focal points: I’m speaking of the formalization of the concepts of geometric space and physical space. It is my conviction that Galileo possessed both concepts in nuce, that is in embryonic form, in the sense that they were there – he used them as operative concepts (see Image/Table 20) -,  but he did not formalize them explicitly, as Descartes and  Newton did.  One of the main points of my research is that concepts of place and space should be analyzed and judged on a historical basis; therefore, the shift of interest from one concept to the other is the result of converging sociocultural and intellectual – or symbolic – factors, that is, it is the result of many interlaced interests regarding the work and the spirit of many different people (philosophers, physicists, astronomers, mathematicians, artists, architects, poets…) across different epochs. Nonetheless, should I be forced to make a very short list of people who, more than others, were responsible for the definite shift of vision – a true ‘revolution’ of thinking – regarding the nature of the world, from ancient to modern (a shift which I have symbolized by way of the aforementioned turn from place to space) I would definitely point my finger at the triad Galileo, Descartes and Newton (Kepler is the first left out from the list, since, as we have seen, while his attitude is definitely modern, he still clings to old concepts – that’s why we have said, with Barbour and Koestler, that Kepler is a ‘watershed’). In no one better than Galileo is evident the conceptual shift of interest from place to space. In a certain sense, I believe Barbour summarizes this important ‘passage’ (and belief of mine) when, at a certain point in the text, he says: ‘I have emphasized […] Galileo’s mathematization of motion not only for its intrinsic importance in the development of the methods of modern science but also because it is characteristic of the move away from the contingent world [which I interpret as that which is concrete and ingrained in the concept of place/matter] into the perfect world of mathematics [which I interpret as that which is abstract: the proper domain, or breeding ground, for the development of the concept of space]. This movement gained great strength from the success of Galileo’s work and was taken still further by Descartes. These two men can be regarded as the founders of modern rationalism. Although the empirical content was much more pronounced in Galileo’s thinking than in Descartes’, they were united in seeing the clarity of the concept as all-important. Such a concept was space. We shall see in the next chapters how space gradually acquired all the attributes of a perfectly clear concept. For Copernicus and Kepler, space was truly nothing; after Galileo and Descartes, it became almost palpable. The overthrow of Aristotle was complete, [by ‘the overthrow’ we should also – if not especially – refer to the abandonment of the traditional conceptualization of the extensive continuum as place] and space became such an ingrained part of our thought that it remains, I believe, the biggest single obstacle to a Machian, i.e., relational, conception of motion.’[135]  I couldn’t find a better allied than these words pronounced by Barbour to sustain my thesis concerning questions of space and place! One of the axioms of my research is that the concept of space became a fully-fledged concept, or ‘a perfectly clear concept’ to use Barbour’s terminology (I mean space understood as a concrete three-dimensional extent –  the actual background-scene of all events), only after the decisive formal steps made by Galileo, Descartes and Newton were done: at first Descartes geometrized space (on the base of the myriad of past investigations and debates that culminated in Galileo’s work), then Newton reified that geometrical entity and turned it into a physical entity. It seems to me this view of mine is consonant with the aforementioned statement made by Barbour.

Galileo is especially remembered for having mathematized motion: ‘Galileo applied essentially the same approach [of the astronomers] on the earth: he mathematized terrestrial motions – Barbour says.[136] One of the first argumentation that will lead Galileo so far apart from Aristotle’s theory of motion is because ‘he noted that hailstones of very different sizes landed more or less simultaneously. Since it was reasonable to assume they had all started to fall simultaneously, the observation was in conflict with the Aristotelian teaching that larger and heavier bodies fall faster than lighter and smaller ones.’[137]  This kind of mundane observations guided his studies until he drew far-reaching conclusions on the motion of free-falling bodies, on the motion of projectiles, on the mathematization of the concept of inertia, and the subdivision of motion into parts. In spite of Galileo’s specifically modern attitude towards experiments and the use of mathematics to describe experiments, in the initial stage of his investigations into the nature of motion, Galileo still clung to Ptolemaic and Aristotelian models and conceptions: this is traceable to his early unpublished book De Motu (c. 1590) which was ‘an essentially Aristotelian study of motion’ – Barbour says.[138]  That’s also the reason why the analysis of Galileo’s works is particularly important to understand the passage from old conceptions where motion was understood with respect to matter and place, to a modern conception of motion understood against the backdrop of space. For sure, the conversion to the Copernican system – which happened around 1595 (we know that from a letter to Kepler written in 1597) – was an important step.

As regards Galileo’s studies on terrestrial motions – the studies that took him to the discovery of the law of free fall, a restricted form of the law of inertia, and the parabolic motion of projectiles -, they covered a period from 1602 to 1608. However, his interest for terrestrial motions was set aside for a brief ‘astronomical interlude‘, concluded with the publication of the Siderues NunciusThe Starry Messenger – written in Latin (1610). Very briefly, this is the course of the events that contributed to ‘the destruction of the Aristotelian division of the world into the perfect heavens and the corruptible earth.’[139] After having heard about the invention of the telescope – an invention licensed by the Dutch government in 1608 -, he suspended his work on terrestrial motion and began to perfect that invention. After he magnified the optic of the instrument to a great extent, he began to observe the heaven and he found what others before him couldn’t find or observe: among other things, he discovered the Jovian moons, he discovered the phases of Venus, and he described the mountain surface on the Moon other than noticed spots on the sun. These apparently minor facts were symbolically important for the strengthening of a new vision of the cosmos, which completely demolished/replaced the ideal perfection of the ancient cosmology and astronomy. Galileo’s observations were a confirmation of Kepler’s arguments on celestial motions: especially, the Jovian moons were a confirmation of the fact that celestial bodies circle around material bodies, and not around void points determined after mathematical reasons; this was a very significant physical fact.

It is in his opus magnum  Dialogo Sopra i Due Massimi Sistemi del Mondo, (1632) – Dialogue Concerning the Two Chief World Systems – that Galileo espoused his thinking and theories about motion. The book is written in Italian, in the form of a dialogue between three symbolic characters – the intelligent layman Sagredo, the clever intellectual Salviati, and the ‘dim-wit’ Aristotelian Simplicio -, in four different days: in Day 1 Galileo, by means of the three characters, confronts the heliocentric and the geocentric systems; in Day 2, where he speaks about the daily rotation of the earth on its axis, Galileo ‘formulates, albeit somewhat imperfectly and in a restricted form, the law of inertia and the principle of Galilean invariance [while in Day 3, Galileo] shows how the dynamical arguments against the diurnal rotation of the earth can be overcome by means of his theory of motion, and gives astronomical arguments for the annual motion of the earth around the sun… [the dialogue ends on Day 4, where Galileo takes] the supreme proof of the earth’s mobility – his erroneous theory of the tides.’ [140]

What is peculiar with Galileo – Barbour says – is that he was interested in the description of motion rather than in the physical causes of it; that’s why his work can be ‘more appropriately referred to as motionics than as dynamics’.[141] Barbour explains that Galileo, at the beginning of his studies on motion, had an Aristotelian and Copernican conception of motion; he made real advancements only when he began to apply mathematics to Aristotle’s qualitative investigations. The Apology of Mathematical language – a ‘credo in geometria’ – as a way to decipher the book of Nature is explicitly traceable to a memorable passage published in the book ‘Il Saggiatore’ (The Assayer), in 1623; here, Galileo says: ‘Philosophy is written in this immense book that stands ever open before our eyes (I speak of the Universe), but it cannot be read if one does not first learn the language and recognize the characters in which it is written. It is written in mathematical language, and the characters are triangles, circles, and other geometrical figures, without the means of which it is humanly impossible to understand a word; without these, philosophy is a confused wandering in a dark labyrinth.’ [142]

Aristotelian and Copernican traditional conceptions of motion and place (rather than space) were important inputs for Galileo: in fact starting from the distinction between the different types of motion devised by Aristotle, Galileo begun to think at ‘primordial motions‘ so that he could almost propose ‘an atomic theory of motion’ where the traditional Aristotelian distinction between rectilinear and circular motions could be maintained. However, we also apprehend from Barbour that ‘although Galileo strikes one as more Aristotelian than not […] he followed Copernicus’s lead in extending the class of allowed natural motions, which […] he tells us are of three kinds: straight, circular, and mixed circular-straight.’[143] This was Galileo’s preliminary theoretical situation when he began to make experiments aiming at the mathematical treatment of motion. We apprehend from Barbour, ‘the entire world […] was turned upside down by the mere rolling of a “perfectly hard ball” down an “exquisitely polished plane”. That is what happened sometime around 1603 […]. He discovered and, very important, correctly analyzed the law of free fall. At much the same time he clarified his thoughts about what one may call a law of “circular inertia[Galileo – as Barbour explains – did not use the word ‘inertia‘ but he spoke of ‘persistence of motion’] and how it could be combined with the law of free fall to give the law of the motion of projectiles [parabolic motions] on the surface of the earth’. [144]

RSAP_008_22_GalileoInclinedPlane_96dpi
Image 22: Galileo’s inclined plane used for experiments on the nature of motion (photograph: Museo Galileo).

But before we briefly analyze the steps Galileo made to formulate such laws, there is an important precondition we should emphasize: it regards the modes of thinking the medium in which motions happen, which, in turn, is a question that is inextricably linked to one of the main arguments I have argued for in this website, so far: the development of the concept of space and the almost correlate ‘abandonment’ of the concept of place. At this regard, we also read from Barbour, Galileo had an almost opposite view with respect to Aristotle: while the Stagirite believed the medium was an active agent with respect to the possibilities of a body to move, Galileo believed that such active agent-as-medium was only an ‘annoying perturbation‘ or, in Barbour’s own words: ‘for Galileo the effect of the medium was nothing but an annoying perturbation of the perfect mathematical law that the body would follow in its absence.’[145] While with respect to Aristotle I would qualify that medium – or ‘active agent’ – as place (this is a positive agency played by place with respect to the motions of bodies – suffice to remind Aristotle’s concepts of proper place and natural motions), conversely I would qualify that ‘nothing but an annoying perturbation…’ as a negative agency that leads to the concept of as space. This means that there is a sort of ambivalent conceptualization in the mind of the Italian scientist when he thinks about the motion of bodies: ‘something’ is physically present – the medium understood as annoying perturbation, properly; but it is as if the almost impalpable effect and presence of such medium can be neglected once we shift the focus from the impalpable perturbation to the very motion of bodies, or, better, to the ideal status of motion with which bodies are naturally endowed: linear motion, circular motion or mixed motions. Given the negligible perturbation exerted by that medium – indeed a perturbation impossible to be detected with the lack of proper technologies – I believe the step was short before that ‘annoying perturbation’ could be considered as physically irrelevant and forgotten once it was gone out of focus. This is in effect what the concept of space means according to Newton: something which is physically and diffusely present but has no physical effects on the motion of bodies (the aforementioned block of glass without glass…). Such impalpable presence of the medium may certainly have been affected by the modes Galileo conceived his experiments where the actual sometimes conflates with the ideal. We read from De Motu: ‘But this proof must be understood on the assumption that there is no accidental resistance (occasioned by roughness of the moving body or of the inclined plane, or by the shape of the body). We must assume that the plane is, so to speak, incorporeal or, at least, that it is very carefully smoothed and perfectly hard, so that, as the body exerts its pressure on the plane, it may not cause a bending of the plane and somehow come to rest on it, as in a trap. And the moving body must be [assumed to be] perfectly smooth, of a shape that does not resist motion, e.g., a perfectly spherical shape, and of the hardest material or else a fluid like water.’[146] This is indeed the description of an idealized experiment rather than the description of an actual experiment: an abstract world which is the proper breeding ground for the development of an abstract/ideal concept like space, which is juxtaposed to the actual forces exerted on bodies by the more traditional, Aristotelian, place.

Given this important premise regarding the different modes of thinking about the relation between matter and the where of motion (place/space), by following Barbour’s exposition we now come back to the steps Galileo made to identify the Laws pertaining the terrestrial motion of bodies. The first step was the substitution of the concept of ‘natural motions’ (a concept of Aristotelian tradition) with the concept of  ‘neutral motions’ (I believe this substitution operated by Galileo can also be read as an anticipation of the incoming substitution between the concept of place and the more ‘neutral’ space as the medium where motion occurs). Another important step was the abandonment of the Ptolemaic system in favour of the Copernican model, a conversion that happened sometime around 1595, and which obliged Galileo to consider motion – and its dynamics – under a new perspective. The following step he made was to determine the speed of free-falling bodies whence he finally arrived at the Law of free fall. This was possible because of the accurate experiments he made with falling bodies, in the form of a sphere moving down on an inclined plane (image 22, above).

What Galileo discovered was a law of unsurpassed simplicity and beauty. Namely, that if the distance traversed in the first unit of time is taken as unity, then the distance traversed in the second unit of time is equal to three, that in next to five, and so on.’ This is mathematically expressed by the so-called ‘odd-numbers law […]. For Galileo, this was a law of miraculous Pythagorean harmony […]. At a stroke, the whole direction of thinking about motion was changed: Geometry is concerned with laws not causes […]. The first important result that Galileo deduced from [the odd-numbers law] was what is now called his law of free fall […] which states that the distance fallen is proportional to the square of the time of descent. […] only later deduced the correct result that the speed is proportional to the time t, i.e. v=at.’ [147]   What is the real consequence of this law? The real meaning – Barbour explains – is the fact that for the first time motion was made amenable to mathematization, and it opened the way to the possibility of applying the same scientific method (observation, experiments, mathematization, verification) to other phenomena that occur in nature. Of course, the other phenomena that immediately interested Galileo were the other types of ‘atomic motions’: circular inertia and compound motions to begin with. Galileo’s law of free fall is the first out of the six insights pertaining terrestrial motions which were necessary for the discovery of dynamics (up to now, the eighth of the baker’s dozen to use Barbour’s expression).

The analysis of circular motion – and with it the discovery of the Law of circular inertia – is what comes next in Galileo’s agenda, we learn from Barbour. Galileo believed that a quantity of motion was intrinsic within the nature of any physical body; he called that quantity ‘persistence of motion‘. From ruminations on his experiments with the inclined planes he undoubtedly must have made ‘the assumption that on a horizontal plane the motion would persist forever’ – Barbour says; and given that – Barbour continues – ‘in Galileo’s mind a horizontal surface meant the spherical surface of the earth […] this means that for Galileo the persistence of motion was persistence in circular motion – he had a concept of circular inertia.[148]  These facts, coupled to his ideas of atomic motions (in this case circular motion would be another type of atomic motion amenable to mathematization) and to the traditional idea of perpetual, uniform, circular motion, led Galileo to the discovery of  the second law of terrestrial motions – Galileo’s Law of inertia, properly; indeed, ‘imperfectly recognized as it was’ – Barbour says –  and given that ‘he failed to raise it to the status of the first law of nature he shares the honour for its recognition with Descartes.’[149]  The law of inertia is the second insight of a baker’s dozen (concerning terrestrial motions) that were necessary for the discovery of dynamics, Barbour notes.

After having analyzed linear motions and circular motions, and after having found the correspondent Laws, Galileo focalized his attention on mixed or compound motions; more precisely, he focalized on the study of the parabolic motion of projectiles. What he did was a ‘great service’ for the discovery of dynamics itself; Barbour says: ‘the development of the technique of composition of primordial atomic motions.’[150] Again, the discovery of this law regarding the parabolic trajectory of projectiles was the result of Galileo’s careful experiments with the inclined plane. This discovery represents the third insight of the baker’s dozen associated with terrestrial motions awarded to Galileo.  We read from Barbour: ‘It is quite clear from the amount of space that Galileo devotes to the question of the composition of motions in both the Dialogo and the Discorsi that he regarded this as one of the most important of his insights (…). Before Galileo, there was no clear conception of such superposition of motions, each conceived to have an independent physical existence.’ [151]

Barbour concludes the analysis of Galileo’s work, by introducing the fourth and final great service that he offered for the discovery of dynamics: the formulation of the principle of Galilean invariance (one of the famous experiments that Galileo espouses in the Dialogue, is that of a ship travelling at constant velocity on a smooth sea: the observer within the ship, without any visual contact with the external world, is not able to judge whether the ship moves or if it is stationary – this is, in the end, what the principle of Galilean invariance consists of), which is ‘the fourth of the baker’s dozen associated with the study of terrestrial motions, though in this case Galileo must share the honour for it with Huygens.’ [152]

Now, I want to touch upon the argument that Barbour deals with, in the paragraph ‘Galileo and absolute motion’. This paragraph is important with respect to Barbour’s aims – the absolute/relative debate -, as well as to the main concern of this site: the relation between concepts of place and space. Barbour, by explaining Galileo’s attitude towards motion, tries to get into Galileo’s mind to determine whether Galileo’s understanding of motion was with respect to space or to matter (actually, I believe we’d better say ‘with respect to the place of the fixed stars‘ rather than simply ‘with respect to matter’, since this implies the recognition of the intimate correlation between place and matter, which is something that takes us closer to the Aristotelian tradition and the way of thinking about matter and place). I will take some distance from the overall argumentation that Barbour offers in this paragraph. Let’s see. The incipit of the new section is quite remarkable; Barbour says: ‘certain ideas about the nature of things can become so ingrained in the mind as to defy almost completely any possibility of being dislodged.’[153] This is exactly the same belief that I have with respect to the possibility of speaking about a reformed meaning of the concepts of place and space, explicitly: the traditional meaning of those concepts is so ingrained in our minds that it is very difficult to avoid deep-rooted prejudices in the mind of people since at least three centuries. Now, we get closer to the main point of the discussion: the ‘belief in the independent and real existence of space, can be discerned in Galileo’s writings‘ – Barbour says.[154]  Again, I cannot help but subscribe to the quite evident fact that Galileo has clearly developed or, at least, he was developing a certain notion of space, were it not for the fact that, unlike almost all of his predecessors, in his writings he openly used the term ‘space’ (he used the Italian word ‘spazio’, actually) along with ‘place’ (‘luogo’).[155] However, and this is where my position diverges from Barbour, it is all another question to say that what Galileo had in mind when he was thinking about the motions of bodies was definitely absolute space (even if Barbour wisely avoids to explicitly use the terminology ‘absolute space‘ when he speaks of Galileo, what is ‘independent and real existence of space‘ if not absolute space, precisely?). I believe the question is not as crystal clear as Barbour seems to depict it. The issue that Barbour is directly addressing for the circumstance is: ‘How is motion to be described? To what is motion to be referred: space or other bodies? […] Galileo’s beliefs, perhaps unconscious, have to be deduced from the manner in which he describes motion’, Barbour says.[156]  And Barbour’s answer is quite direct: according to him, in the text there are pieces of evidence that Galileo thought the motion of bodies to be referred to space. And he brings what he believes are pieces of evidence. He says: ‘Right at the start of the Dialogo […] comes a passage which suggests that Galileo conceived of motion as taking place relative to space.’[157]  I have reported the extended passage Barbour refers to in the note above, but, to me, is everything but clear the way Barbour deduces the existence or attribution of (absolute) space in the mind of Galileo. As it is unclear to me the fact that saying with Galileo that the fixed stars are at rest instead that they define the state of rest – like Copernicus and Kepler have said explicitly -[158]  implies that they are at rest in (absolute) space as Barbour wants to convince the reader; at this regard, I would not put so much emphasis, or draw such an important conclusion, by relying on the literal distinction between to be at rest and to define a state of rest; moreover, to me, the fixed stars could be at rest in their own place – that is the place of the fixed stars, properly, without this fact implying the question of the where of place (and if I am obliged to answer about ‘the where of place’ – I would reply that it is matter which is/defines the place of its own existence, whence we could deduce the primordial correlation between place and matter). Later in the text, Barbour adds: ‘The clearest evidence for Galileo’s belief in the reality of motion with respect to space is to be found in his theory of the tides […] where it is particularly interesting that Galileo actually uses the words absolute motion several times.’[159]  Now, as Barbour himself notes, the point is that Galileo does not clearly express with respect to which kind of ‘entity’ absolute motion takes place, and he adds: Galileo ‘does not seem to feel that it is necessary to explain precisely what it [absolute motion] means.’[160]  Notwithstanding, Barbour excludes another possibility, for which I instinctively feel very close to Ernst Mach. ‘With respect to what is this absolute motion?Barbour asks, and he quotes a sentence expressed by Mach himself: ‘It is noteworthy that Galileo in his theory of the tides treats the first dynamic problem of space without troubling himself about the new system of coordinates. In the most naive manner he considers the fixed stars as the new system of reference.’ [161]  How far the position of Mach is with respect to the one argued by Barbour. I deem this possibility suggested by Mach very plausible: after all, as Barbour himself said few pages before, that possibility (motion understood with respect to the fixed stars) had been already anticipated by Kepler. At this specific regard, it is not my point to take part in the different views between Mach and Barbour: I suggest the interested reader to read the book to have a personal opinion; my idea is that in Galileo’s mind there was an ambivalent presence of notions of place and space, that is, he could rely on a traditional understanding of place (as a notion intimately connected to matter), other than on what I have called ‘a certain notion of space’ – space as an operative concept in between the physical and the mental domain; a concept that will be developed, clarified and formalized by Newton as ‘absolute space’ (not without the important theoretical/mathematical contribution of Descartes). Again I redirect the reader to the synoptic table above (image 20). I believe this question is epistemologically relevant and, independently of my different view on this subject, it is the explicit merit of Barbour’s book to have attempted to address the historical question concerning notions of motion and, more or less explicitly, space, place and related concepts from a scientific perspective; a contribution that can have effects beyond the scientific boundaries. We are here in between mind – the mind of the father of modern science, Galileo – and reality. As a concluding remark concerning such an important question, Barbour says: ‘Laws of such apparent geometrical perfection point to a transcendent world – beyond direct sense perception – governed by mathematics and geometry. Science in this view is not the ordering of sensual empirical facts that in and by themselves constitute the totality of reality. It is rather a journey by which the soul finds its way back laboriously, through the Socratic process of recollection, to its antenatal disembodied state.’ [162]

Barbour’s final words express the same conviction that I have, a conviction which moved me to the present journey through the pages of this website: ‘The debate about the absolute or relative nature of motion is at root a debate about reality’, Barbour says.[163] Since the debate on motion is ingrained in the more encompassing question regarding the notions of place and space, we now understand why space and place are such important concepts: their discussion ‘is at root a debate about reality’ to paraphrase Barbour’s concept. The mode we understand concepts of space and place is definitely related to the way we understand reality and the phenomena around us, not just from a scientific perspective, but also and foremost, from a plurality of perspectives: the perspective of the scientist, of the philosopher, the social scientist, the artist, the poet, the politician, the architect, etc.

8. Descartes and the new world

As Barbour says in the introduction to the work of the French philosopher, mathematician and scientist René Descartes (La Haye en Touraine, France 1596; Stockholm, Sweden, 1650), while Galileo was the last tenant of the old world, that is, the last tenant of ‘the grandiose edifice of Peripatetic physics’, René Descartes ‘was at work on plans for a new edifice.’[164] A new world, literally. Barbour mainly ascribes Descartes philosophical merits and, concerning physics, the idea of inertial motion in a form quite close to that adopted by Newton. Barbour especially takes into consideration two works written by Descartes, which have important consequences for the future of the absolute/relative debate: (i) Le Monde – The World – published posthumously in 1664, despite having been elaborated many years before and more or less complete by 1633,[165] and containing the key ideas of Cartesian Physics; (ii) the Latin book Principia Philosophiae – Principles of Philosophy, published in 1644. Some words are also spent by Barbour on Descartes’s (iii) Discours de la method – Discourse on Method (1637) in which Descartes’s credo in the primacy of the mind – cogito ergo sum – is espoused.

(i) The World. This is Barbour’s incipit about Descartes’s The World: ‘his aim was no more and no less than to give a rational explanation for all the phenomena of the material world. The work is written as a charming fable: Descartes says he will not attempt to explain the real world but will instead describe an imaginary world, a ‘new world’, about which the most important thing is that it is completely determined and described by an absolute minimum of properties. In fact, he puts into this imagined world nothing but matter, whose sole property is that it possesses extension, and, vitally important, motion. His assertion is that, provided this matter and motion satisfy (by God’s ordinance) certain almost self-evident laws, such a world, whatever its initial condition, would of necessity evolve into a world indistinguishable from the one we observe around us. Hence his famous assertions: “Give me extension and movement and I will reconstruct the world” and “the entire universe is a machine in which everything is made by figure and movement”.’[166] Fundamentally, by continuing Barbour’s initial metaphor, Descartes is giving a definite form to the new edifice for the coming centuries: a new vision of the world, – an extreme rationalist programme, Barbour also says – shaped on a strong determinism and mechanism: a new Weltanschauung, indeed.

I make a brief digression by saying that Descartes’s literary device – the work written as a charming fable – is an interesting way to address the relationship between the actual and the ideal as the two correlate parts of the Real World beyond any supposed dualism. I want to highlight this argument since the distinction actual/ideal, understood as a correlation of two different but contiguous moments is critical to understand concepts of place and space. Dualism is the hallmark of Descartes’s philosophy, according to the usual interpretations. In spite of that, I believe this literary device, as well as certain passages contained in the Principles of Philosophy (passages where the complete union – or correlation – between the mind and the body prevails over their separation),[167] could shift our focus from Descartes’s dualism understood as the solution to the human way of knowledge (this is a traditional and accepted interpretation which cast a shadow on Cartesian philosophy), to Descartes’s dualism understood as the putting on the table of the question  regarding  human knowledge in modern terms: a starting point rather than a conclusion.[168] After all, this is exactly the reason why ‘modern philosophy is usually considered to have begun with Descartes’, as Heidegger said.[169] I believe the ambiguity of Descartes’s conceptions of place, space, and matter – whose differences should be read on the base of a conceptual distinction rather than on their actual difference -[170]  is an exemplification of Descartes’s difficult search to overcome the dualism that he posed as a condition for knowledge.

In line with the Aristotelian tradition, Descartes understood the world as a plenum: from a physical point of view, he needed such a plenum because of his theory of light and for certain other physical phenomena, Barbour also hypothesized. However, – Barbour adds – what is truly remarkable in Descartes’ treatment of physical phenomena is his insistence on the concept of motion. As we anticipated in the chapter dedicated to Galileo, the Italian and the French scientists share an important insight that was decisive for the discovery of dynamics: I’m speaking of the Law of inertia. Specifically, Descartes believed ‘that any piece of matter, once set in motion, would continue to move forever with the speed initially imparted to it if it were not for the intervention of other matter and […] that this motion would be along a straight line in space, again if it were not for the intervention of other matter […]. These ideas […] were fused by Huygens and Newton into the modern law of inertia.’[171]  Descartes – as Barbour has shown – arrived at those two insights, which were fused into a single law, by studying the phenomenon of centrifugal force. From the way he studied that phenomenon we also understand the difference between Galileo’s and Descartes’ attitudes toward motion: while Galileo was interested ‘in motion in its own right, as a deeply interesting subject of empirical study’– that is, Galileo was interested in ‘the how’ of motion, what Barbour called ‘motionics -, Descartes understood motion as a means of explanation for physical phenomena; that is, he was interested in ‘the why’ of motion, what we actually call dynamics. Coming back to Descartes’ understanding of the world as a plenum, it is interesting to understand the mechanisms of motion devised by the scientist; these mechanisms – especially the reference to circular of motion – remind us the traditional understanding of motion and place and the correlate rejection of the notion of the void as the background for matter. We read from Barbour: ‘One of the most characteristic features of Descartes’ physics is his circle of motion. Since the world is a plenum, matter can only leave one place if other matter comes in to fill the void. Depending on the disposition of the pieces of matter, this may involve quite an extended circular readjustment of matter. The circle must however always close because volume of matter is exactly conserved in Cartesian physics. […] The notion of a circle of motion […] is central to one of the most distinctive features of the Cartesian cosmology – the theory of vortices. According to Descartes’ “laws of nature” […] God created the world with a certain initial amount of motion of the individual pieces of matter and then, through the laws of collision, ensured that this motion always remained in the world in an unchanged amount, being merely passed from certain pieces of matter to others […]. Descartes posited that as a result of the combined influence of these factors – the collisions and the constancy of the total amount of motion – the world would, on a large scale, settle down into a system of huge vortices, in which the various particles swarmed around centres at which he placed the sun and the other stars. Thus, each vortex was assumed to be at least as large as the solar system […]. The planets […] are carried around the sun in the circles…’ [172]

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Image 23: Descartes’ understood the cosmos as a plenum; he devised a Theory of Vortices to explain the positions and motions of celestial bodies (detail of Descartes’s drawing included in Le Monde, 1664).
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Image 24: Interstellar Space as a plenum. Photograph: M. Kaufman; B. Saxton (NRAO/AUI/NSF); ALMA (ESO/NAOJ/NRAO); NASA/ESA Hubble Space Telescope.
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Image 25: In the wake of the Aristotelian tradition, Descartes understands the cosmos as a material plenum. No void really exists between bodies, and space exists as an abstract (conceptual) characterization of matter, which coincides with its internal place. That’s why Descartes says: ‘space or internal place and the corporeal substance which is contained in it are not different…’. Figuratively, while this actual red chair I’m sitting on at this moment can be really identified with its own place, the idea I have of this red chair is not different from the real red chair I’m sitting on, in the sense that I’m really thinking of this same red chair and not another red chair (that’s why these entities ‘are not different’) but, at the same time – and this is the omission that eventually caused so much confused interpretations – while this red chair-as concrete place exists in actuality – res extensa –  my idea of the same red chair exists in an ideal domain only, it is a conceptual abstraction – res cogitans -, just like space is, ultimately (photography: vancouvermom.ca)

According to Barbour, even if Descartes never offered a detailed mathematical explanation of the physical phenomena he spoke about, some of his ideas were amenable to mathematical treatment. That’s why his work was quite remarkable: it was a question concerning the method, which will become the hallmark of modern science. Among the physical phenomena that Descartes took into account in The World, and which Barbour briefly exposed, we find an account on hardness and liquidity, and the phenomenon of light and of weight. However, we learn from Barbour, it is Descartes’ emphasis on motion and his anticipation of Newton’s law of inertia, which was so relevant for physical and dynamical questions. In the paragraph The Stone that put the stars to flight Barbour looks into the origins of Descartes’ important contribution to the discovery of dynamics with more detail; here, Barbour puts some specific emphasis on two main points: Descartes’  considerations regarding the possibility to find a theory of motion through which explaining all of the phenomena of nature by means of motion alone; and  Descartes’ important contribution regarding the formalization of the notion of laws of nature – ‘the rules according to which these changes take place I call ‘laws of nature’Descartes says when he is speaking about the changes that pertain the states of material bodies. [173]

Descartes’ important contribution to the discovery of dynamics is especially evident in the three laws of nature that he formulated; as Barbour says, ‘Descartes actually set up three laws (or rules) of nature in The World. The first can be called a generalized law of inertia, according to which things have an innate tendency to stay in the same state […]. The essence of his first law is that the quantity of motion that a body has will remain constant unless it interacts with other bodies.’[174] The second law (or rule) is the law of collisions and regards the idea that ‘the quantity of motion is conserved in collisions.’[175] The third law is the conservation law, which regards the rectilinear persistence of motion of the parts of a body subjected to motion along a curved line.[176]  Then, Barbour summarizes the key ideas of the scientific revolution provided by Descartes’ The World: materialism – ‘all the variety of the visible world is to be explained by matter in motion’ -,[177]   determinism – ‘the whole world unfolds from an initial state in accordance with immutable laws’-,[178] and – most important for our interests on space and place – the idea of ‘isotropy and homogeneity of space and time – the laws of nature are exactly the same at all times and in all places’, Barbour says. [179]

With respect to the specific question regarding space I will take some distance from Barbour’s interpretation; in fact, according to Barbour, the question of Descartes’ interpretation of the concept of space ‘brings us now to the most surprising twist of all in the history of the discovery of dynamics’; a ‘twist’ I do not agree with. Let’s see why.

(ii) Principles of Philosophy. According to Barbour’s opinion, in the interval of time from the completion of The World, in 1633, to the publication of Principles of Philosophy (Principia Philosophiae), in 1644, Descartes passed from an absolute to a relative conception of space because of the intervention of the Inquisition (with respect to the different concepts of space in Descartes, Barbour also speaks of pre-Inquisition concept of space). This is the subject of the second part of the story regarding Descartes’: the analysis of the (ii) Principles of Philosophy. In 1633 Descartes was going to finish and publish The World when he heard the sad news concerning Galileo’s condemnation. Then, he decided to withhold the book from publication and begun to work on the Principles of Philosophy. The reason why he withheld the book we directly apprehend from the words of Descartes in a letter sent to a friend: ‘since I would not wish, for anything in the world, to write a discourse containing the slightest word which the Church might disapprove, I would, therefore, prefer to suppress it, rather than publish it in a mutilated version.’[180] It seems to me that Descartes didn’t want to make compromises – that is, to make changes or to cut some parts – to please the Church: instead of it and instead of going against the church, ‘better to be silent’ is Descartes’s response. This is the way I read Descartes’ behaviour in that specific circumstance. It seems to me Barbour’s view is quite opposite: according to the British author, Descartes began to work on the Principles of Philosophy, and because of the intervention of the Inquisition against Galileo, he made the aforementioned ‘surprising twist’ in his interpretation of physical concepts: he turned from an almost absolute concept of space and motion that he expressed in The World – ‘he seems to have had an intuitive concept of motion as taking place in space, this space being to all intents and purposes much the same as Newton’s absolute space’, Barbour says – [181] to a relative concept of space and motion. This fact takes to Barbour’s conviction that Descartes ‘was simultaneously the effective founder of the two diametrically opposed concepts of motion that are the subject of this book – both the absolute and the relative![182]  I do not agree with Barbour’s interpretation about that twist; moreover, and most significant, I have some doubts concerning the possibility of interpreting ‘absolute space’ as a concept that Descartes already possessed. Let’s see this question more in detail. Barbour starts explaining his thesis about ‘the turn’ by describing Descartes’ early concept of motion in The World. This is the incipit of the paragraph: ‘It is quite clear that at the time Descartes wrote The World he instinctively conceived space as something that exists independently of matter. He has the concept of space as the container of matter. He describes the world of his fable as “a wholly new one [now it is Descartes who is speaking] which I shall cause to unfold … in imaginary spaces. The philosophers tell us that these spaces are infinite, and they should very well be believed, since it is they themselves who have made the spaces so”. There is obvious irony here, – Barbour comments – but we have no reason to doubt that Descartes did at this stage actually believe in space.’[183]  First of all, I would pay attention to the fact that we’d better discern between imaginary spaces – this is what Descartes refers to, properly – and space as (absolute) container of concrete matter – this is what Barbour attributes to Descartes. I would say that the two kinds of spaces are not quite the same entity – there is an ontological, other than epistemological, divide between the two concepts – and given that Descartes is particularly sensitive to the separation between res extensa (the concrete) and res cogitans (the abstract), this is not a secondary question. Then, Descartes’ reference to space – infinite spaces, actually – as an invention of philosophers, makes this question about the nature of space everything but ‘quite clear’ to use Barbour’s own words. At least, from this introduction, I would deduce that Descartes may have believed in a certain idea of space belonging to the realm of the mind rather than to the realm of physical matter, as Barbour wants us to believe. Then, Barbour continues: ‘He talks of stopping in it at “some fixed place”. He says of the matter which he conceives that “each of its parts always occupies a part of that space”.’[184] Here, by, relying on a literary device – the fable… – Descartes is making recourse to his imagination: it is ‘hypothetical matter’ occupying an ‘imaginary space’. The possibility to understand space as an imaginary entity – hence, space belonging to the realm of res-cogitans – also opens the way to the metaphorical use of the term space. Again, this is Barbour speaking: ‘Significantly, in view of Descartes’ later passionate denial of the vacuum , in The World he grants that there could be void in some parts of the world: “if there can be a void anywhere, it ought to be in hard bodies”.‘’[185]  However, Descartes’ proposition is a hypothetical proposition – ‘if there can be…’– , which does not express a categorical conviction, and which is counter-balanced by the following statement, always included in The World, where Descartes openly denies the hypothesis of the void within material bodies: ‘recall that among the qualities of matter, we have supposed that its parts have had various different motions from the moment they were created, and furthermore that they are all in contact with each other on all sides without there being any void between any two of them…’ [186]

Barbour cites some other quotations from The World to support his absolutist/relativist thesis regarding Descartes’s turn; in the overall, I do not find those quotations convincing and I redirect the reader to the book to have an opinion. What I want to point out is that Descartes had certainly developed a certain notion of space in the book The World; however, I’m everything but sure that such notion was close to the absolute notion of space Barbour attributed him. I would dare to say that, at the time Descartes was writing The World (completed by 1633), probably, he hadn’t conceived a fully-fledged notion of space yet, but he was still pondering on a notion which could be in the middle ground between a geometrical and a physical notion (a notion suspended between abstraction and concreteness). Probably, in the end, a more abstract notion than a concrete notion, given that Descartes could be considered the inventor of the geometric concept of space (an abstract concept indeed) as Einstein said in the article ‘The Problem Of Space, Ether, and The Field in Physics’.[187]  Space – I mean space as a clear cut modern concept, the three-dimensional continuum – was an innovative concept and there is no guarantee at all that this innovative concept could be equated to (or be the direct extrapolation of) any sort of intuitive (absolute) container of physical bodies as many people say by forgetting the millenary debate around that concept. That’s why, for me, Descartes ruminated for so many years on such a difficult and ambiguous concept after he finished the World, before delivering his final understanding of space (place and matter) in the Principles of Philosophy (published in 1644) and after having almost certainly grasped the notion of geometric space as a three-dimensional continuum, as also Einstein suggested (Descartes’s work on Geometry dates back to 1637, which was published as an annex to his Discourse on Method).

RSAP_008_26_CartesianSpace_96dpi
Image 26:Let AB, AD, EF, GH… be any number of straight lines given in position…’: in Descartes’s Geometry, 1637, ‘lines given in position but not in length… becomes lines of reference or coordinate axes’;[188]  this is the first step necessary to define an abstract space of mathematical origin intended as an immersive continuum with respect to which geometrical elements can be located.

Then, I do not believe he made any ‘surprising twist’ as a direct response to the Inquisition, as suggested by Barbour. Ultimately, the definitions Descartes formalized and delivered us (space, internal/external place and matter) are anyway ambiguous, and this is probably the reason why they are so open to different interpretations and so difficult to be grasped without a deep commitment to his work and to the elucidation of the concepts of space and place, from the ancient Greek meanings to the modern interpretations. For instance, concerning the ambiguity of interpretations I have just mentioned, Barbour says that in The Principles Descartes ‘denies the existence of space’ but, again, I believe this interpretation is too simplistic and should be rejected, as I’m going to show in a few lines.[189]  Concerning Descartes’ thinking, I believe we assist to a refinement of concepts in epistemological sense and not to a revision in the sense of the surprising twist – from absolute to relative – proposed by Barbour. At this regard, I believe that Descartes’ great contribution concerning geometry and mathematics is critical to understand the ‘refinement’ of the concept of space in the (abstract) sense I’ve just mentioned. The unprecedented introduction of the idea of a spatial continuum within Euclidean geometry certainly contributed to the focalization and subsequent elucidation of the concept of space, but a new obstacle for its interpretation arose (and I think my divergence with respect to Barbour is due to that obstacle): is geometrical space – the space that contains figures and numbers Einstein speaks of – a possible reduction of physical space or is it another kind of space? I believe Descartes kept the two conceptualizations apart (the concrete and the abstract), that’s why he still needed the concept of place: place, just like in the tradition, is the concept that relates actual bodies (this is external place for Descartes, a concept very similar to the place of Aristotle); for Descartes, place is physical, conversely space is geometric, that is abstract (it was Newton who abolished the difference between actual physical place and geometrical space: space became physical and place was reduced to a part of space). As concerns space, the ambiguous epistemological relation between abstraction and concreteness, between reality and imagination is also traceable to the following words written by Barbour:  ‘it should be emphasized that the ubiquitous use of Cartesian coordinates in problems of dynamics does much to disguise the problem of the ultimate invisibility of space. Because the conceptual axes are drawn on paper, the actual absence of reference marks in space can be conveniently forgotten.’[190] I believe the concept of absolute space and its astonishing diffusion thrived on that risky omission, which is the omission of (actual) place, ultimately. Then space thrived on a fallacy:  the fallacy of misplaced concreteness – the belief that an idea could be reified (geometrical space – or anyway an abstract idea like space – reified in the form of absolute space by Newton) fulfilling the concrete task of a frame of reference, which, before reification, was demanded to a couple of conceptual axis. With respect to the Cartesian coordinates and the invention of geometrical space, I believe it is interesting to quote this passage from Barbour, whence we learn the correlated contribution of Leibniz in the formalization of geometric space: ‘by introducing the idea that the distance of a point from a line can be represented algebraically by a number, Descartes made geometry amenable to algebraic treatment and thereby made a most important contribution to the development of mathematics in the seventeenth century. The value of his method was greatly enhanced by Leibniz, to whom the name Cartesian coordinates is due, when he pointed out the convenience of making the coordinate axes orthogonal. Leibniz thereby introduced for the plane orthogonal coordinate systems…’.[191] See Image 26, above.

The ambiguous concept of spatiality that we find in The World, (what Barbour interprets as absolute space, I interpret as a tentative concept, or a working concept, in between geometry and physics, maybe, as I’ve already said, a conceptualization more geometrical than physical, more abstract than the concrete absolute container of physical bodies suggested by Barbour) is opposed to the conception of spatiality founded on the relational basis that we find in the Principles. Again, here, my view diverges a bit from that of Barbour: ‘the Principles pose the question of the referential basis of motion as a matter of prime importance. Space is said categorically not to exist and all motion is declared to be purely relative’, Barbour says.[192] While, as Barbour rightly notes, in the Principles there is evidence throughout the text that Descartes has a relational position with respect to motion – Aristotelian relativism still holds -, there is also evidence that space is not categorically said not to exist: suffice to read Principles X, XI, XII, XIII or XIV. I believe Barbour’s affirmation about the non-existence of space should have been complemented by the proposition ‘as an actual entity‘ (then the resulting phrase would sound like this: ‘…non-existence of space as an actual entity’); this fact does not exclude the existence of space as an abstract entity – this is the way Descartes’ conception of space should be interpreted according to me. The fact that space is not considered by Descartes as a notion standing on its own, without the necessity to speak about place or matter to elucidate the very sense of space itself, is another question, which cannot be rendered through the words ‘space is said categorically not to exist’ or ‘is anyway superfluous’. It seems to me that Barbour developed a certain idiosyncrasy with respect to the ambiguous concepts devised by Descartes (we have to admit a certain difficulty to disentangle the ambiguous identity between space, matter and place – internal place, properly); such ‘idiosyncrasy’ towards Descartes’ conceptions reaches his climax when Barbour admits that ‘there is not much point in following him through all his contortions, some of which do him little credit, and are often hard to follow…’.[193]  So he concludes the section on Descartes with the following words: ‘ this, then, is the confused state in which Descartes’ Principles left the theory of motion in the middle of the seventeenth century.’[194] Where Barbour speaks of a ‘confused state’, I would prefer to speak of an ‘ambiguous state’; this is after all the state in which Descartes also left concepts of place, space and matter at the end of his journey. It is out of such ambiguity that Christiaan Huygens and Isaac Newton offered their different attempts to cast some light on the question: one, Huygens, directing toward a relational understanding of the world and concepts; the other, Newton, putting a definite word on the debate and clearing those concepts out of any ambiguity. Then, even if my overall consideration concerning Descartes’ work and concepts diverges a bit from that of Barbour, I rejoin him in his fascinating journey towards the discovery of dynamics – which is also a journey towards the discovery of the meanings of place, space, and matter.

9. Huygens: relativity and centrifugal force

Christiaan Huygens (The Hague, the Netherlands, 1629; The Hague 1695) was a Dutch mathematician, physicist, astronomer and a brilliant inventor: he invented the pendulum clock, which transformed time-keeping and had many important scientific implications; he also made some important contributions in dynamics and optics. During his lifetime, we learn from Barbour, Science became a ‘major intellectual activity if not a business’; in fact, different scientific Institutions were given birth: among them of the Royal Society in England (1660) and the French Academy of Sciences in French (1666), of which Huygens was a founding member. Huygens was deeply impressed by Descartes – of whom he was a friend of the family; from Descartes, he inherited a mechanical and deterministic vision of the world. We read from Barbour: ‘Huygens was carried away by the idea of a mechanical explanation of the phenomena of nature along the lines proposed by Descartes. In accordance with this programme, all phenomena must be given a microscopic interpretation in terms of invisible particles of matter, the only elements allowed in this explicatory enterprise being, as we have seen, the size, shape, and motion of the particles […]. His main difference from Descartes […] was in following Gassendi’s lead in reviving the ancient atomistic theory of particles moving in a void […]. By his introduction of mathematics and at least some sound physical principles he transformed atomism from qualitative philosophy into a genuine science.’ [195]

As concerns the discovery of dynamics and the overall absolute/relative debate, Huygens made important contributions proposing a theory of collisions, an accurate analysis of centrifugal force, and, most of all, he anticipated of the theory of relativity. Huygens’ works taken into consideration by Barbour are: (i) De Motu Corporum ex Percussione (On the Motion of Bodies in Collisions) finished in 1656, but published posthumously in 1703; (ii) De Vi Centrifuga (On Centrifugal Force), again published posthumously. We learn from Barbour that both books contained anticipations of the key ideas and techniques Einstein adopted for his restricted and general principle of relativity. According to Barbour, Huygens’s works were an exemplification of the ‘rigour of mathematical discipline coupled with accurate observation’ that determined the success of the modern scientific method. [196]

In the opening part of De Motu we find two important assumptions: the first pertaining the Law of inertia, expressed with an unprecedented clear exposition – ‘once a body has been set in motion, it will, if nothing opposes it, continue that motion with the same speed in a straight line’;[197]  the second regards a ground-breaking statement of the Principle of relativity: ‘the motion of bodies – Huygens says – and their speeds, uniform or nonuniform, must be understood as relative to other bodies that are regarded as being at rest even if these together with the others partake in a further common motion…’.[198]  In the overall, the formulation of the principle of relativity enunciated by Huygens is a bit vague: in fact, Barbour comments: ‘the principle of relativity is formulated with words that seem to tie it in to the real contingent world, but in a curiously vague way. […] the central point is in clear focus, but the picture shades off gradually into nothing. There is no supporting background, the picture hangs literally in the air. We see a boat and there is talk of a river bank, but there it ends […]. The opening statement seems like an acceptance of Cartesian relativism (The motion of bodies… must be understood as relative to other bodies that are regarded as being at rest) but some other more stable background is immediately implied by the further common uniform motion.’ [199]  The reference to the ‘contingent world’, which Huygens’ principle of relativity is tied to, redirects our attention from a conceptualization of space understood as absolute – which, at that time, was a concept in the air as we have seen in the past chapter – to the correlation between place and matter, which is a conceptualization very close to the interests of this website (a correlation that comes from Descartes and, earlier, from Aristotle; a correlation that looks forward to Mach and Einstein rather than to Newton). However, in spite of the first mathematical attempts that tries to come to term with a basically relational understanding of certain physical phenomena, the enigma of relativity was far from being resolved (and still it is) even if, Barbour says that ‘one fact […] comes before all theorizing and detailed observation: any observation of the world is relative […]. Relativity (in the wide sense of the word) is a fact of life.’[200] I cannot help but subscribe to this observation. I would add that place – and matter which is its correlate – is a fact of life; while space (not just absolute space, but any kinds of space) is an ingenious concept, the product of the rational mind: it is the rational mind that has reduced the actual variability of place to a neutral space in the attempt to come to terms with the complex phenomena of reality. This is basically my point of view concerning the concepts of space and place.

Apart from relativity, there are other important physical implications that accompany Huygens’ work – Barbour says; these are three of the most fundamental laws of nature: (i) the conservation of momentum, (ii) the conservation of (kinetic) energy in collisions, and (iii) the centre-of-gravity (centre-of-mass) law.[201] As regards the science of dynamics, the first enunciation of the principle of relativity and the conservation of momentum law (which was rediscovered by Newton) are the two final insights belonging to the ‘baker’s dozen’, which, according to Barbour’s scheme, were necessary for the discovery of dynamics. If the principle of relativity is ‘a fact of life’ – as Barbour says – there are two other important observational facts that were specific subjects of investigation for Huygens:  inertial motion and centrifugal force (the latter is ‘the tendency of bodies swung in a circle to recede from the centre’). [202] The concept of centrifugal force has been a critical concept for physics and specifically for dynamics; as Barbour says, centrifugal force ‘played an extremely important role in the arguments about the rotation of the earth, in the discovery of the law of inertia, and in the discovery of the law of universal gravitation; it played a key role in the emergence of the precise concept of force and, thus, of fully-fledged dynamics; and it continues to play a central role in the discussion about the absolute or relative nature of motion.’[203] That concept was due to Huygens (vis centrifuga), but ‘the credit for having put the problem of its accurate mathematical description at the centre of attention in the study of motion is undoubtedly Descartes.’[204] Huygens’ merit, we know from Barbour, was to understand that ‘a precise definition of force could not be obtained unless Cartesian dynamics was complemented with an awareness of something that it completely lacked: acceleration. He realized that through acceleration it was possible to form a precise concept of force.’[205] These steps – the correct definition of the concept of force and the introduction of the notion of acceleration – were necessary steps to pass from an embryonic to a fully-fledged dynamics.

After having celebrated Huygens’s scientific records the time is ripe for Barbour to introduce the work of the greatest scientific genius of all time – Sir Isaac Newton – and to complete what, in the author’s plan, should have been the first of two volumes; Barbour’s overall plan has remained incomplete and the second volume, which ought to have been a development of the Machian hypothesis concerning the relational approach to dynamics, was never published.

10. Newton I: the discovery of dynamics

Barbour dedicates the last three chapters of the book to the prominent figure of Sir Isaac Newton (Woolsthorpe-by-Colsterworth, United Kingdom, 1642 – London, United Kingdom, 1727). Chapter 10 is a survey on the development of Newton’s ideas on dynamics culminating in the publication of his opus magnum – Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), in 1687. Chapter 11 covers the issue of absolutism against relationism, by considering Newton’s concepts of absolute space and time, as he devised them in the famous Scholium. The main issue of relationism versus absolutism is also considered in the final chapter, where Barbour considers the conceptual clarification of Newtonian dynamical concepts in the post-Newtonian period.

As Barbour rightly points out, Newton’s work was remarkable since it marked the conclusion of this great enterprise that began more than two and a half millennia ago in ancient Greece and prepared the ground to the reactions that took to the General Theory of Relativity. Such a ‘great enterprise‘ ultimately regards the rational inquiry into physical reality and, more specifically, from Barbour’s privileged point of view, it regarded the clarification of motion, as well as the clarification of other concepts necessary to inquire into the nature of motion: space, place, and time to begin with.

As concerns Chapter 10, Barbour divided the analysis of Newton’s work into two periods: the early writings, and the period culminating in the publication of the Mathematical Principles of Natural Philosophy. In the overall, we learn from Barbour, there was a certain inevitable progression between the early works – in which the influence of Kepler, Galileo and Descartes could be still noticed – and the concepts Newton developed in the Principia, even if no relevant turn can be found.

With the following words concerning the importance of the concept of force, Barbour introduces the work of Newton: ‘The most distinctive feature of Newton’s dynamics is the concept of force, or, rather, forces, since Newton actually employed three related but distinct concepts of force.’[206] These three concepts of force were: (i) inherent force, (ii) motive force and (iii) centrifugal force. The first two types of force were the fruitful result of Newton’s theory of collisions, which he developed in the Waste Book, a sort of collection containing mathematical papers and theories;  the third concept he elaborated in a work on the centrifugal force‘conatus a centro‘ (endeavour from the centre). Newton’s framework for the theory of collision derived from Descartes: as Barbour observes, ‘everything is based upon the idea that the natural state for a body is to continue in a straight line with uniform motion (or else to remain at rest). Collisions are events that deflect a body from one such state into another.’[207]  In order to quantify a force, some preliminary conceptual steps were necessary; Newton’s insight was to understand that the measure of a force was possible ‘through change in inertial motion’, and since change of motion was related to the size of the body, Newton identified ‘force with change in speed multiplied by the bulk of the body’; thus, Barbour continues, ‘he arrived at a concept very close to Descartes’ quantity of motion.’[208]  It is also important to point out that, concerning the concept of force, Newton recognized the importance of the direction of motion of a force; Newton was the first to have a clear understanding of the notion of force as a vectorial quantity. Then Barbour starts explaining the different concepts of force adopted by Newton, which is a passage I omit, if not to say that Newton always believed inherent force had a definite and unique value, this fact meaning that such condition could only be realized if a definitely fixed framework of reference existed with respect to which defining that force. At a distance of centuries, we still don’t know if this fixed frame of reference – absolute space – existed only in the mind of Newton, or if it also exists as a matter of fact (this is, after all, a core issue both for Barbour and for the arguments that I deal with, in this website). As Barbour says: ‘Newton never wavered in his belief in what he later came to call absolute space. Initially, it was no doubt purely instinctive; later it hardened into a formalized conviction […] at this stage, for Newton, objects moved through a space that was as real for him as the green felt of a snooker table.’ [209]

The critical relation between motion and force was clear in the mind of Newton and it was synthesized in the Preface of the Principia, revealing his great plan: ‘the whole burden of philosophy seems to consist in this – from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.’[210]  Between words, we understand that the barrier between terrestrial and celestial phenomena is finally broken: what for Descartes was only a declaration of intents, for Newton became law (the three laws of motion and the law of universal gravitation, precisely).[211]

Before introducing the different steps and circumstances that took Newton to the publication of the Principia, Barbour deals with the question about the influence that others scientists had with respect to the possibility for Newton to crack nature’s secrets (Galileo and Descartes, especially, but also Robert Hook, with respect to the groundbreaking idea of Universal Gravitation, and Kepler, for his area law which permitted Newton to concentrate his attention on what he would have called centripetal force). Barbour sums up the key elements of Newton’s contribution to dynamics: ‘In what does the essence of this work consist? It consists in the adaptation of the key elements of Galileo’s work on projectiles […] to the much more difficult situation in which the motions are not orthogonal and the acceleration is not uniform. Newton achieved the generalization by ‘infinitesimalizing’ Galileo’s procedure, that is, he broke the motion up into a succession of very small stretches, in each of which he applied the Galilean technique.’ [212]

We have finally reached the climax of a long process that started two millennia before in the territories of Greece and its Mediterranean colonies. Barbour enumerates a sequence of exciting chronological steps that culminated in the printing of the masterpiece; in April 1686 a first version of the work with the complete title – Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) – was presented to the Royal Society and after a possible question of plagiarism was resolved with a patch into the body of the Principia, ‘the printing of the great work could proceed and was completed on 5 July 1687.’[213]  This is Barbour’s overview of the Principia: ‘The main body of the Principia is divided into three Books. Books I and II are basically of a formal mathematical nature with Book I treating the case of motion when there is no resistance while Book II treats motion in resisting media. These two books are essentially the mathematical theory of the motion of bodies with mass moved in accordance with Newton’s three laws and interacting in accordance with definite forces […]. The overall logic of his book is to posit formally the concept of force in conjunction with his laws and the definition of mass, derive the mathematical consequences that flow from them, and then demonstrate that the existence of certain forces with mathematically well-defined properties, above all forces of gravity, is proved by the phenomena of nature […]. The climax of the work is thus in Book III, in which Newton, as he announces in the Preface, gives “an example of this [the finding of the forces and the demonstration of the other phenomena] in the explication of the System of the World; for by the propositions mathematically demonstrated in the former Books [I and II], in the third I derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea”.’ [214]

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Image 27: Newton’s own copy of his Philosophiae Naturalis Principia Mathematica, 1687 (photograph: from Wikipedia, Andrew Dunn).

I will only briefly deal with the contents of the three books, giving a more extended treatment of those parts that specifically meet our interests in the question of space and place.

Book I begins with the definitions of the main physical concepts; in sequence: the mass concept, the momentum, the concept of inherent force, the concept of impressed force, and that of centripetal force. Then, there is the famous Scholium on absolute and relative space, time, and motion, to which, as we anticipated, Barbour dedicates an entire Chapter (Chapter 11). After the Scholium, the remaining parts of Book I regard the formulation of the laws of motion and some consequences deducted from them. I directly refer to what is enounced in the Principia with respect to the three fundamental laws of motion: in  Law I Newton says that ‘every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it’; Law II says that ‘the change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed’; Law III says that ‘to every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.’ [215]

Barbour’s attention is especially given to those arguments of the Principia that are directly related to ‘the discovery of dynamics and the problem of whether motion is absolute or relative’. In turn, I will just mention those specific arguments that I believe were important to determine the modes of thought of an epoch and the shift of vision from a relational to an absolute understanding of place and space (and motion as consequence). At this respect, when Barbour deals with the influence that the Keplerian area-law had on Newton, he made an interesting analogy between the two astronomers; Barbour says: ‘just as Kepler learnt to move freely in his mind’s eye through the solar system once he had firmly established the halving of the eccentricity of the earth’s orbit, so too did Newton learn to treat the most general of orbital problems once he had grasped the dynamical significance of the area law. And thus the second great synthesis (the creation of dynamics) was added to the first (Kepler’s discovery of the laws of planetary motions.’ [216]  As a comment, I would say that there is a significant increment of abstraction in the passage from Kepler to Newton: while Kepler is still bounded to the Aristotelian place of the cosmos and therefore he can only hypothesize mental journeys in a space of trigonometric relations acting as a sort of correlate – an alternative image – of the actual place of the cosmos, Newton, being totally detached from the influence of the Aristotelian boundaries of place, makes no ontological distinction between space and place; for him the distinction is merely quantitative: they are both concrete entities, one – place – smaller than the other – space -, and contained in it – precisely, ‘a part of space’  to appeal to Newton’s own definition. Then, the ethereal, trigonometric and mental extent (or expanse) that supports Kepler’s mind’s eye corresponds to Newton’s ethereal extent (or expanse) that defines the very nature of absolute space. In the end, those two kinds of extents (or expanses) offered the same support to the ability of the mind to roam freely through ideal or supposedly actual domains. In the passage, I believe we have assisted to the reduced ontological significance of place (and of its practical function), and we have assisted to the reification of an abstract entity like space, which was intrinsically amenable to mathematization. With absolute space, not only the barrier between the terrestrial and the celestial was broken; a more subtle and elusive barrier was removed as well: the barrier between abstract and concrete domains (I believe a limit between the two domains exists; the only way to remove it or to bypass it, to arrive at a more comprehensive understanding of reality, is to elucidate the correlation between them and not to act as if that limit does not exist, or as if such limit is insurmountable. Correlation is the arm against dualism and, as an architect, my first step to avoid such dualism is to show the difference and the correlation between place and space, which are two of the most representative concepts of those two correlate realms or domains: the concrete and the abstract). The mathematization of nature – and with it modern science – begins to thrive only after such operation of removal was definitely accomplished. Newton, more than the pragmatic Galileo or the theoretician Descartes, was the real beginner of such modern tendency: ‘he uses mathematics to study nature in a manner of which Descartes never even dreamed’, Barbour says.[217]  Two of the most important topics that Newton has dealt with in Book I of the Principia – the orbital problem and the solution of dynamical problems with arbitrary initial conditions – could only be accounted for with highly sophisticated mathematical approaches that ‘provide the basis of dynamics and with it most of modern science’;[218] approaches that also established ‘the concept of physical determinism […] so firmly […] in the scientific mind’. [219]  In all of the cases and the problems considered in the Principia, the concept of absolute space underpins Newton’s mathematical approach. The scientific mind is ingrained in the concept of (absolute) space and vice versa. Following Barbour exposition, among the other important results contained in Book I of the Principia, there is Newton’s proof of potential theory and the question regarding the mutual interaction of bodies. [220]

Barbour’s engagement with Book II, which treats motion in resisting media, is quite brief; in this section of the Principia, ‘the main topics that Newton considers are the motion of bodies that are resisted in accordance with two basic laws (both suggested to him by experiment): in proportion to their velocity through the medium and in proportion to the square of their velocity. He reports extensive experiments made with pendula to establish the actual resistance of air. He considers circular motion of bodies in resisting media. Then follow sections on the density and compression of fluids and hydrostatics. He considers the motion of fluids and the resistance they offer to bodies moving through them. Then comes the section on the propagation of motion (pulses) through a fluid in which he attempted to derive the speed of sound. Finally, he devotes a section to the circular motion of fluids…’.[221] This is anyway an important section since, from Barbour’s impressions, we learn that by trying to establish the nature of the medium within which or through which bodies move Newton tries ‘to create a framework of theoretical continuum mechanics by means of which Newton could conclusively prove his own conviction that the planets move through an ether completely free of resistance and simultaneously demolish comprehensively Descartes’ vortex theory.’[222]  What is that ether completely free of resistance if not absolute space?

Now, as for Book III, this is the culmination of Newton’s great plan: to explain the System of the World, that is to explain the mechanisms of natural phenomena on the base of the principles and concepts Newton enunciated in the first two books; ‘It remains that, from the same principles, I now demonstrate the frame of the System of the World’, Newton says.[223] From the analysis of several natural phenomena, Newton is able to show that in Nature exists one specific and universal force acting on all bodies: the force of gravity. Then, universal gravitation regards ‘the notion that each piece of matter attracts with a force that is proportional to its mass and equally is attracted in a given gravitational field with a force that again is proportional to its mass.’ [224] The ultimate barrier between celestial and terrestrial physics has been removed: in fact, ‘by his pendulum experiments Newton showed that all terrestrial matter is attracted to the earth by a force proportional to its mass. He then points out that the celestial motions prove that this property must also hold for the material of which the planets and satellites are made.[225] As Barbour notes, in the Principia there were many hints that the gravitational force elucidated by Newton was not the only universal force that existed in nature: Newton was conscious about that, however, since he was ‘very loathe to speculate in public about things he could not demonstrate mathematically’,[226] he never risked hypotheses about those forces or, better, about the reasons behind those forces (gravity in primis), leaving open the possibility to metaphysical interpretations, which he (apparently) never embarked on. This is the context in which Newton’s famous saying ‘hypotheses non fingo‘ should be interpreted (that famous statement was added in the second edition of the Principia, in 1713). Even if in the beginning that motto was about the explanation of gravitation, it became Newton’s way to eliminate the occult metaphysical or transcendental religious entities from science; curiously, with just one relevant exception: his theory of absolute space. [227]

After having celebrated the methodological importance of the Principia for the development and success of modern science (above all the methods of observation and of projecting numbers into nature, which Newton increased dramatically), Barbour’s concluding remark for this section regards Descartes’ silent presence behind many passages of the book (the very title is a proof for that:  Mathematical Principles of Natural Philosophy is Newton’s counterpoint to Descartes’ Principles of Philosophy). Barbour concludes: ‘The extent to which the Principia is a polemic against Descartes should never be forgotten, least of all in the Scholium on absolute and relative motion. Which brings us now to the central conflict.’ [228]

11. Newton II: absolute or relative motion?

This is a crucial chapter for the development, evolution, and understanding of the concepts of motion, space, place and time. Barbour introduces the argument with a brief historical overview: ‘broadly speaking, the concept of motion employed by the really major figures in the history of science was relational up to Galileo. The main reasons for this appear to have been, first, the intellectual dominance of Aristotle, second, the fact that, alone among the disciplines, astronomy developed as a quantitative empirical science and third, related to the second, the unchanging aspect of the heavens. The need to relate all motions to the distant sphere of the stars, which are assumed to be relatively fixed, is most pronounced in both Copernicus and Kepler.’[229]  Motion was understood relative since that with respect to which motion occurred – this is ‘place’ – was relative (to matter). Place was a commanding concept in Aristotle (place as the immediate relational container of matter), as well as in Copernicus and Kepler. Descartes saved the Aristotelian concept of place when he introduced the difference between external and internal place. Up to Kepler, for astronomical purposes, the ultimate place was nothing other than the place of the fixed stars (and place was before all the place of matter). But, since the ancient times, against this vision, there was an antagonist vision: the vision of the atomists and of those who believed Aristotle’s finite concept of place (topos) was not a convenient concept; a belief often expressed through ‘the problem of what exists outside its outermost sphere‘… Those alternative visions contributed to the ascendancy of a ‘space-based concept of position and motion [which] seems to have gathered strength in the second half of the sixteenth century’ Barbour says.[230] Then, Barbour asks for the reasons of that transition from the relational to the absolute standpoint, (a transition that concerned motion as well as place and/or space), and he enumerates a host of factors. First, he mentions the psychological need for a stable foundation for the world when the material frame started to fall apart: for that reason, the concept of space (absolute space) had to be ‘more or less invented‘.[231] Then the study of motion (which started at the time of the astronomical revolution) and the thrust to the geometrization of motion possibly contributed to the emergence of space. Another possible reason was ‘the revival of the ancient doctrine of atomism and the idea that material particles moved through a void’;[232] at this specific respect, Barbour mentions the role that Gassendi might have had on that question: ‘It is quite possible that his ideas about space and time – which Gassendi taught existed independently of their content and provided the general frame of any knowledge of reality – influenced Newton’s views.’ [233]  Barbour concludes: ‘thus, the desire for a solid foundation went hand in hand with the search for a geometrized container of the world. Absolute space could be seen as the reified and simultaneously Platonized (i.e., geometrized) void.’[234]  In the end, I’m very close to Barbour’s position about the fact that space could be seen as an act of reification. Then the British author mentions the role of Galileo – a transitional figure Barbour says – where the Aristotelian, relational context (whence he derived his concept of inertial motion and his ideas of a well-ordered cosmos) coexists with the absolutist bias that he showed in ‘his theory of the tides and numerous revealing tell-tale passages in his writings.’ [235]

Before turning his attention to Newton’s concepts, once again Barbour deals with Descartes ambivalent/ambiguous conceptualizations in the middle between relationalist and absolutist positions. Here, again, I take the occasion to say that where Barbour sees absolute space in Descartes’ The World, I only see an attempt to focus on a concept – space – which hasn’t been elucidated yet beyond any doubt; where Barbour reads a turn from an absolute to a relational space, I just see Descartes’ attempt to make his thinking clearer and clearer in between ontological and epistemological considerations. I believe the difficulty, or ambiguity, in reading Descartes derives from the fact that epistemology, as a modern theory of knowledge, starts with the Cartesian ‘I’ – I doubt, I am. A proper language or a proper vocabulary was not developed by Descartes to address these questions for the very first time, therefore causing many confused interpretations. So, I believe Descartes was actually in balance between an abstract space (which can indeed be easily misplaced for ‘absolute space‘ – I suspect this is also what Barbour calls ‘intuitive space’ and interprets as an antecedent state to ‘absolute space’) and actual place/matter: one has intellectual or cognitive character – res cogitans (this is space after all) -, the other has extensive character – res extensa, this is place as the bearer/holder of matter. It seems to me Barbour casts on Descartes what is actually his own belief or interpretation: that an abstract space – such is the space Descartes refers to in The World, evidently (the space invented by philosophers Descartes said) -, ‘intuitive space’ and ‘absolute space’  are fundamentally the same – or very similar – entities, which is an interpretation I reject.

Finally, we come to Newton’s spatial concepts and their ‘linear’ development from the early writings to the publication of the Principia. This is Barbour’s overall introduction of the argument: ‘I believe that throughout his life Newton believed instinctively in the reality of motion through space.’[236] The first text that Barbour takes into consideration is the collection of papers, under the title Waste Book: here, there is no explicit hint at the concept of ‘absolute space‘, however, – Barbour says – ‘he clearly worked instinctively in an intuitive (absolute) space, just like Galileo and the pre-Inquisition Descartes.’[237] The first explicit pronouncement on space, place and motion made by Newton is traceable to the paper Laws of Motion, which is later than the Waste Book. Newton says: ‘There is an uniform extension space or expansion continued every way with out bounds: in which all bodys are each in severall parts of it; which parts of space possessed and adequately filled by them are their places. And their passing out of one place or part of space into another, through all the intermediate space is their motion.’[238]  According to Barbour, Newton elaborated his concept of (absolute) space starting from Descartes’ idea of the uniform rectilinear motion of undisturbed bodies; this is the question that Newton probably asked and answered, according to Barbour hypotheses: ‘uniform rectilinear motion of undisturbed bodies with respect to what? […] Galileo, the pre-Inquisition Descartes, and Newton all had the same instinctive response – it is with respect to space: absolute space.’[239] Now you know:  needless to say, I only partially agree with Barbour’s opinion; we cannot say that intuitive space is the same of absolute space. In my opinion, that would be an anachronism in the end, other than a possible epistemological fallacy. I believe we can have an intuitive understanding of place (and of the matter related to that place): this is for me the name of the extensive as well as intensive domain that surrounds and compounds physical bodies. Space is a concocted concept that comes after place and after different related concepts and conceptualizations through which, in the course of time, man engaged with the where of physical realm: I’m thinking of the Greek terms for extension, interval, distance, void, place, infinite… (diastema, kenon, topos, chora, apeiron, etc.). We cannot simply argue that the Greeks, the Latins, or, even back in time, the primitive man had an intuitive notion of space the way the modern man has (a three-dimensional before than tetra-dimensional notion) since before the concept of space could be invented  man had to pass through a gradient of terms – from more concrete to more abstract – indicating different modal approaches to reality: space is a relatively late-term that, among other domains, includes the geometrical domain. I would say that we could only speak of space in modern terms after Descartes (geometrical space as a three-dimensional abstract continuum could be devised after the mathematical intuitions of the French scientist and philosopher) the same way we can speak of absolute space after Newton (physical space as a three-dimensional continuum). Before them, there were numerous attempts to delineate the properties of a novel spatial conceptualization: from the ancient Greeks, down to Descartes and Newton.

The next text that Barbour considers is the paper De gravitatione et aequipondio fluidorum, a text associated with the Principia, and dated around 1680. The paper was an attack against Descartes’ relational concepts. These are some of the introductory definitions given by Newton: ‘the terms quantity, duration and space are too well known to be susceptible of definition by other words. Def. 1. Place is a part of space which something fills evenly. Def. 2. Body is that which fills place. Def. 3. Rest is remaining in the same place. Def. 4. Motion is change of place.’[240]  The argument at issue in the paper is that of how to determine the motion of bodies; Newton rejects ‘the idea that immediately contiguous bodies play significant role in defining or determining the motion of a body’,[241] as well as he rejects the idea that distant bodies may be used as a reference for motion.[242] In brief, his aim is to show ‘once and for all, that relativism and a meaningful statement of the law of inertia as formulated by Descartes himself are simply incompatible.’[243] Newton’s conclusion is inescapable: ‘it is necessary that the definition of places, and hence of local motion, be referred to some motionless thing such as extension alone or space in so far as it is seen to be truly distinct from bodies.’[244] Then space (absolute space) is the conditio sine qua nonto lay truer foundations of the mechanical sciences.’[245] Newton’s dynamics is founded on absolute space. At this point, Newton, in a remarkable passage that anticipates the Scholium, directly analyzes the critical concepts of extension and space: ‘we can clearly conceive extension existing without any subject, as when we may imagine spaces outside the world or places empty of body, and we believe [extension] to exist wherever we imagine there are no bodies, and we cannot believe that it would perish with the body if God should annihilate a body, it follows that [extension] does not exist as an accident inherent in some subject. And hence it is not an accident. And much less may it be said to be nothing, since it is rather something, than an accident, and approaches more nearly to the nature of substance. There is no idea of nothing, nor has nothing any properties, but we have an exceptionally dear idea of extension, abstracting the dispositions and properties of a body so that there remains only the uniform and unlimited stretching out of space in length, breadth and depth.’[246] And this is Barbour’s illuminating comment before he starts enumerating the properties that Newton attributes to space: ‘the properties that Newton lists as belonging to extension show how remarkably “substantial” it appeared to him. It really is a perfectly uniform and translucent block of glass extending from infinity to infinity and has all the properties of such a block of glass except the glass!’ [247]

RSAP_008_28_AbsoluteSpace_BlockOfGlass_96dpi
Image 28: Newton’s conceptualization of absolute space is like a transparent block of glass extending in every direction, Barbour says (a block of glass without glass; then – I ask – what does remain? I believe only the idea remains. That’s what space is, after all: an idea, an abstract conceptualization. There is no such thing as ‘physical space’).

I have already said it: what is a block of glass without glass? Just the idea remains. That’s what space is: a remarkable, ingenious idea. Then, this is Newton’s list of six points about the properties of space: ‘1. In all directions, space can be distinguished into parts whose common limits we usually call surfaces […] 2. Space extends infinitely in all directions […] 3. The parts of space are motionless […] 4. Space is a disposition of being qua being. No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated. And the same may be asserted of duration: for certainly both are dispositions of being or attributes according to which we denominate quantitatively the presence and duration of any existing individual thing. So the quantity of the existence of God was eternal, in relation to duration, and infinite in relation to the space in which he is present; and the quantity of the existence of a created thing was as great, in relation to duration, as the duration since the beginning of its existence, and in relation to the size of its presence as great as the space belonging to it […] 5. The positions, distances and local motions of bodies are to be referred to the parts of space […] 6. Lastly, space is eternal in duration and immutable in nature, and this because it is the emanent effect of an eternal and immutable being. If ever space had not existed, God at that time would have been nowhere…’.[248]  Interestingly enough, at point four and six, Newton engages with the metaphysical foundation of space associating the qualities of God to the qualities of space – eternal in duration and infinite in extension. Contrarily to the conviction, he will explicitly sustain in the second edition of the Principia (remember the motto ‘hypothesis non fingo‘), here, it seems he does not care to the mix between religious, or transcendental entities and science. Then, as also Barbour noted, it is all the way remarkable Newton’s nonchalance in combining the extremely realistic, sober and positivistic point of view regarding his conception of matter, with his aprioristic (and transcendental) position regarding his ideas of space and time.[249] According to me, this is Newton’s greatest limit, given that the same dualist attitude permeates the concepts that he will ultimately deliver to the readers of the Principia.

We have reached the climax of a long process that started in the Mediterranean regions more than two millennia before the publication of the Principia: the Scholium on absolute space, time and motion. Given the historical importance of this moment for the modern science, and for dynamics, in particular, Barbour decided to state in full Newton’s Scholium. He left his comments to a subsequent paragraph. I will merely report Newton’s own definitions and give a synthesis of the individual paragraph after item IV. Then I will return to Barbour’s interpretations expressing my point of view on the subject, where opportune.

This is Newton speaking:

I do not define time, space, place, and motion, as being well-known to all.

Newton continues: ‘Only I must observe that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external […]

II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.

Newton continues: ‘Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies, and which is commonly taken for immovable space […]

III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative […]

IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another…’.[250]  After these fundamental definitions Newton introduces a list of individual paragraphs. Paragraph (a) is a discussion on time, absolute time and the role of astronomy; (b) is about space and time as orders of situation and succession; (c) is an attack against those who believe in the relativity of motion; in (d) Newton comments on the impossibility of determining absolute motion (or absolute rest) ‘from the position of bodies in our regions’;[251]  paragraph (e) is an attack against ‘the notion that contiguous bodies are to be used to define motion.’[252] Paragraph (f) is about the impossibility of determining motion accurately, with respect to distant objects. Barbour notes, this is the only place in the Principia were we find a direct reference to absolute, immovable space as ‘infinite’ (‘Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute immovable space’).[253]  Paragraph (g) is about true motion and relative motion: once again, Newton specifies that true motion does not consist in the relations between contiguous bodies so that this is another implicit attack on Descartes’ relativism. Paragraphs (h), (i) and (j) are crucial paragraphs: we find the famous bucket experiment regarding a vessel containing water and twisting around its axis and the experiment regarding the two globes connected by a cord and revolving around their common centre of gravity. According to Newton’s hypothesis, these are ‘arguments for the absolute nature of circular motion’ and, therefore, for the possibility to determine the existence of ‘immovable space’ from the nature of true motions and the forces impressed upon bodies. Barbour says he will dedicate another section to the evaluation of this argument, but, most important, he anticipates that Newton’s attempt to determine the instantaneous direction of the motion of bodies in absolute space (as in the famous experiments) is faulty for the following reasons: ‘1) Newton has only shown (within the framework of his theory) that the distant bodies do not rotate; he cannot from that deduce that they are in a complete state of rest (the system as a whole may have a uniform translational motion in absolute space). Therefore, his claim falls to the ground. (2) To complete the determination of motion, Newton seems to need distant bodies such as the stars, which are assumed to be at rest. So the final conclusion of the Scholium appears in truth to be an admission of defeat: it is not possible to determine the direction of motion without reference to other bodies!’ [254] Then, it seems the case for absolute space is desperate; at least, it is a desperate case if we rely on the physical ground only, the ground through which Newton has tried to probe the nature and existence of absolute space. In the end, Barbour has shown that Newton ‘worked out’ his theory of motion (in Book I) under the (a-priori) assumption of absolute space; when he had to work out his ambitious program ‘to demonstrate the frame of the System of the world’, in Book III, ‘we find that, de facto, Newton refers all motions to the centre of mass of the solar system, assumed to be at rest, and makes in addition the assumption that the distant stars are at rest. It is this assumption which enables Newton to fix directions.’[255]  These assumptions – I say – reveal the incongruence of a theory worked out on the concept of absolute space and absolute motion, but which cannot dispense with the distant stars (that is, it cannot dispense with the intrinsic relation between matter and its place).

Following Barbour’s comments on the Scholium, there are distinguished merits and some flaws in the argumentation taken by Newton to support his absolutist versus relationalist thesis. Barbour comments, ‘seen in terms of his polemic with Descartes (whom we note that Newton does not now even mention), the Scholium was an almost complete success… Newtonian dynamics conquered the world. Men came to accept his concepts of absolute space and time – and they worked brilliantly. Descartes’ confused notions were completely forgotten.’[256] In spite of this clear success achieved by Newton, there are at least three flaws in the Scholium, Barbour notes: first, it was too specifically directed against Descartes and ‘although purified of the extreme polemicism of De gravitatione’, it was nevertheless distorted by such polemic vein.[257] Coming to more technical and probing arguments, I will directly quote Barbour’s own words about the other two points: the second flaw regards the fact that ‘the physical arguments that Newton invokes to prove his point are not taken from the full generality of the dynamics […]. Instead he reverts to the restricted dynamics of his early work; the only dynamical problem he considers is that of perfectly circular motion. This gives the impression that the all-important distinction is between rotation and absence of rotation, whereas in reality the decisive distinction is between unaccelerated and accelerated motion. [Third] Newton persistently fails to acknowledge the existence of one of the most important results of his own dynamics, the famous Corollary V to his Laws of Motion. It is this corollary that, in the Newtonian scheme, gives expression to the Galileo-Huygens relativity principle [which] demolishes the basis of any claim that the unique speed and direction of absolute motion can be determined from the phenomena, quite counter to what is implied in the Scholium.’ [258]

Then, by espousing the limits of Newton’s theory, Barbour takes his final step to introduce the argument of the last chapter: the amendment to the Newtonian system. Barbour notes: ‘The Scholium was Newton’s attempt to interpret the content of his dynamics by identifying the referential basis of motion. It was his attempt to explain visible motion in terms of invisible space and time. The final clarification of the immense achievement of Newtonian dynamics only came about two centuries later when Neumann, Mach and Lange showed that the true understanding of what Galileo and Newton had achieved required one to interpret visible motion, not in terms of invisible space, but in terms of visible matter.’[259]

Here we are, right to the point: this statement of Barbour concerning the interpretation of Newton’s dynamics and its provisional amendment in order to have a more appropriate understanding of the physical world, is in tune with my overall conception of reality understood as the place where matter appears after certain processes occur – physicochemical, to begin with; then, biological, social and symbolic processes, respectively (as for space, it is a concept necessary to cope with reality at a symbolic level). That’s why I say that place – which is inextricably correlated to matter – is a founding concept at an ontological level, while with space we enter an epistemological domain of abstract interpretations and correlations with the actual reality of place, or places. Barbour continues: ‘They [Neumann, Mach and Lange] achieved the final conceptual clarification which put Newtonian dynamics in its true perspective. It was not, as Newton believed, a distant intimation of the relationship between the material world and God, but something rather closer to home, though not, I think, any the less wonderful for that: the intimate and unbelievably delicate and precise bond of matter to matter, the fine and subtle net that permeates the palpable created world.’[260] More precisely, reconnecting the sense of those words to the metaphysical foundations of reality through the way I see concepts of place, matter, time and space, I would say that it is this net, properly ( this ‘fine and subtle net that permeates the palpable created world’  is what I have also called the fabric of reality or physical continuum – see the article Space, Place and the Fabric of Reality), which constitutes the real concrete structure of any place as the concretization of processes (place-as-matter) on which we can build abstract spaces independently of their peculiar nature (be it geometrical, mathematical, literary, artistic, perceptual, architectural… space). As an architect, it was that comprehensive net (the concretization of a myriad of processes that unify objects and subjects, the concrete and the abstract) which I have tried to render at the scale of architectural phenomena, by means of what I have called ‘archi-textures’ – the preliminary placial and spatial state of reality out of which architecture emerges and in which it exists as the reification/outcome of physicochemical (geological, hydrological, morphological, meteorological…), biological (physiological, psychological), ecological, sociocultural, and symbolic – or intellectual – processes through which objects and subjects are also correlated (figures below).

RSAP_008_29_Archi-textures 01_CalviRollinoArchitetto_96dpi
Image 29: Archi-textures and ‘the fine and subtle net that permeates the palpable created world’. By means of the series of studies that I have called ‘archi-textures’, I have tried to represent the foundational and unifying net of placial and spatial processes that are necessary for architecture to emerge as one of the most distinguishing aspects of the palpable created world. Photograph: study model for The Green Library, Legnano, ITA, 2009; author: Alessandro Calvi Rollino Architetto.
RSAP_008_30_Archi-textures 02_CalviRollinoArchitetto_96dpi
Image 30: Archi-textures can be seen as a metaphysical breeding ground for architecture: in and out of place, matter, time and space architecture emerges as the concretization of a concatenation of processes: physicochemical, biological, social and symbolic processes, to begin with. Photograph: study model for The Green Library, Legnano, ITA, 2009; author: Alessandro Calvi Rollino Architetto.

I close this section anticipating a remarkable sentence made by Barbour in the following chapter, which is perfectly apt to the tone of the last few lines, where we have argued for the priority of matter (I would say matter-as-place, given the particular interpretation of place I’m maintaining in this website): ‘the reality of dynamics is that it relates matter to matter, not matter to space and time. What the whole debate is really about is not whether motion is relative to matter or to space but the clarification of the precise nature of the connection which dynamics establishes between matter and matter’.[261] Matter is where everything begins. That ‘where’ – I say – is logically (and etymologically) a place – the place of matter – and not space, which is an abstract concept of mathematical origin.

12. Post-Newtonian conceptual clarification of Newtonian dynamics

So we have already anticipated the argument of the final chapter of Barbour’s brilliant book: the clarification of the foundations of Newtonian dynamics, which especially occurred in the second part of the XIX century. Among others, the relevant aspects concerning this clarification were three: the fact that from the empirical analysis of phenomena, the essential features of dynamics could be identified (in particular forces); the fact that, from empirical analysis and observations, ‘a priori’ elements were banished from dynamics (and more generally from science). However, there is a point which is particularly meaningful and which Barbour treated extensively in this chapter: the implementation of certain empirical definitions which ‘provided paradigms of the operational definition of the basic concepts of dynamics in terms of observable objects and processes.’[262] These operational definitions regarded the conceptualizations of ‘inertial system’ by Ludwig Lange, ‘equality of time intervals’ by Carl Neumann, and ‘mass’ by Ernst Mach. As Barbour noted, these definitions ‘were of academic and epistemological rather than practical value; they did not significantly change our knowledge or understanding of the real contingent world…’.[263] Therefore, Barbour takes under closer inspection the contributions of those three scientists. Of course, even if from different perspectives, their contributions had some relevance in the overall debate between relativist and absolutist positions. I believe the epistemological value of their work can also contribute to having a more appropriate understanding of the concepts of space, place and matter.

As regards the work of the German mathematician Carl Gottfried Neumann (Königsberg, now Kaliningrad, Russia 1832; Lipsia, Germany,1925), Barbour takes into consideration two different concepts: apart from the aforementioned ‘equality of time intervals‘, he also presents us Neumann’s ‘Body Alpha’ concept. This is an interesting concept, which has the function of replacing Newton’s absolute space without falling into the epistemological traps Newton fell into. Let’s see it: ‘There must be a special body in the universe which serves us as the basis of our judgement, with respect to which all actual or even conceivable motions in the universe are to be referred…’ – Neumann says;[264]  this ‘special body‘ is the Body Alpha, and Neumann proposes that ‘the motion of any material point left to itself (i.e., subject to no disturbance) is rectilinear with respect to Body Alpha.’[265] Evidently, this ‘body‘ had the analogous function that absolute space had for dynamical purposes; yet, where Newton reified an abstract concept (space) which – I say – was a complex synthesis of, at least, cognitive, geometrical and metaphysical factors, Neumann – more prosaically and with a subtle epistemological strategy – hypothesized the existence of an actual entity somewhere in the universe with respect to which the motion of any material point left to itself was rectilinear; therefore, he gave a different interpretation of Newton’s first law, but saved Newton’s hypothesis of absolute motion and his entire dynamics. In short, with Neumann, we assist at the real materialization of the frame of reference, as a genuine material embodiment of a central point, the Body Alpha. This was Neumann’s way to leave the ambiguous absolute space concept out of dynamics.

As we have anticipated, there is another important contribution offered by Neumann to the clarification of dynamics: this regards his concept of time with particular reference to the modes of establishing the equality of time intervals, which are indeed necessary for the study of motion and, in particular, are necessary to determine the uniformity of motion of material bodies. ‘The key element in Neumann’s concept is that of a material point subject to no forces […]. For brevity, we can call such an object ‘force-free’. The essence of Neumann’s definition of equal intervals of time, which Lange modified to define an inertial system, is that motion and time are to be defined relative to force-free bodies.’[266]  In short, to materialize a clock we need to observe two or more force-free bodies ‘for in this case the motion of one of them can, by convention, be taken as defining equal units of time (the times required to pass through equal distances). [267]

Within the revision climate of the second part of the XIX century, the history and the work of Neumann, as anticipated by the aforementioned sentence, intersects that of the German physicists Ludwig Lange (Gießen, Germany, 1863; Weinsberg, Germany, 1936) who, starting from Neumann’s ideas of time and free-force bodies, formulated the concept of inertial frame of reference which has since replaced any direct reference to absolute space.

Finally, to complete this section regarding the conceptual clarification of dynamics, Barbour introduces the work of the Austrian physicist and philosopher Ernst Mach (Brno, now the Czech Republic, 1838; Munich, Germany, 1916) and his concept of mass. To introduce the argument, Barbour briefly analyzed the historical development of that concept and the difficulties behind it, in between geometrical origins and physical (dynamical) argumentations. He started mentioning Kepler – whose search for a cause of motions beyond geometrical determinations led to the concept of force -, and continued with Newton – who shifted his vision from Kepler’s belief in laziness of bodies with respect to motion to laziness with respect to change in motion – this is ‘precisely Newton’s concept of inertial mass.’[268] Finally, Mach changed somewhat the interpretation of the concept of mass given by Newton, shifting the attention from resistance to acceleration: ‘the essence of the mass phenomenon is not resistance per se but mutuality of acceleration […] of at least two bodies’,[269] which underlies the intrinsic proximity between this operational understanding of the mass concept and the content of Newton’s Third Law (which, under the perspective of Mach’s concept of mass, becomes redundant). Mach’s operational definition of the mass concept, other than physically relevant was epistemologically relevant since it reveals Mach’s attitude to the analysis of phenomena: ‘the object of natural science is the connection of phenomena’ [270] – Mach says in the first decade of the XX century, revealing an attitude which anticipates by many decades our contemporary and now-diffused mode of thinking about the interconnection of the phenomena in nature. This also gives a more precise determination of the essence of dynamics. Barbour asks: ‘What then is the essence of dynamics? It is in the recognition of universal correlations in the observed behaviour of bodies. The basis of it all, the ground on which the correlations are observed, is the possibility of measuring distances and times and the existence of bodies that can be recognized at different times […]. Using these basic possibilities of observation, we establish the characteristic way in which bodies accelerate themselves from one inertial motion into another. The very specific and completely universal manner in which the mutual accelerations always occur permits the introduction of the mass concept‘. [271]

Given the importance of Mach’s contribution to the history of dynamics (and to physics in general), at the end of Vol. I, Barbour gives some introductory description of Mach’s work and intents, which I only briefly mention here: ‘Mach set out to clear out all the metaphysical cobwebs from the kitchen of physics. Totally committed to empiricism […], more clearly than any of his contemporaries, Mach realized that any successfully functioning scientific theory or discipline must in the last resort rest on experimentally observable phenomena […]. Mach’s idea was to build up the concepts of dynamics systematically, starting with the most basic geometrical elements and progressing from them to the more sophisticated dynamical concepts. […] He accepted distance measurement as given and also, for the purposes of the clarification of the concepts of mass and force, that clocks exist and that the distant stars define a frame of reference with respect to which all motions are to be defined. Then the first observational fact on which dynamics rests is the law of inertia: in the frame of reference defined for practical purposes by the stars, bodies are usually observed to travel in straight lines with uniform speed. So far, there is little difference here from Newton; it is on the transition from velocities to accelerations that Mach parts company from Newton, who, made force the primary concept of dynamics. […] Developing his systematic approach, Mach insists that the observable accelerations should come first. When this is done, it transpires that the definition of mass (and force) cannot be separated from the content of Newton’s Third Law. For the observed facts are as follows: bodies normally travel on straight lines relative to the stars but, under suitable conditions, two or more bodies can mutually accelerate each other. Mach’s great insight was that it is the very special law which governs the manner in which this mutual acceleration takes place which makes possible the meaningful definition of mass.’ [272]

We close this very long journey that started two and a half millennia ago with Barbour’s synoptic overview of the discovery of dynamics and an important observation (concerning the concepts of place and/or space) we can derive from it. Barbour qualifies the state the concept of motion was left after this long journey as ‘semi-interactive‘: the first laws of motion devised by the first astronomers were completely ‘non-interactive‘ (Barbour speaks of geometrokinetic laws), and we have to wait for Kepler to instil force into matter. As a matter of fact, ‘Kepler transformed the glorious lantern [the sun] that God had hung at an advantageous point to illuminate the mysterious cavity of the world, into the motor which drove the planets.’ [273]  Galileo, with his terrestrial ‘motionics’, pulled out of the sky the basic elements of motion and after him the possibility to close the gap between celestial and terrestrial motion was foreshadowed: while Descartes realized the philosophical possibility of unifying those different laws, and – as Barbour says – initiated the real debate about the ultimate nature of motion (remember the following question that the Cartesian concept of inertia aroused: ‘uniform rectilinear motion of undisturbed bodies with respect to what?’),  we had to wait for Newton to realize concretely (or should we say abstractly?) that unification by means of the laws of motion and universal gravitation. Yet, Barbour says, ‘Newton’s solution to the problem, absolute space and time, left dynamics in a semi-interactive state. It divided the original geometrokinetic law of perfectly uniform circular celestial motions into two: the law of inertia, which remained geometrokinetic (though space-based instead of matter-based), and the law of universal gravitation, which epitomized the totally new interactive concept of motion.’[274] The new gulf between the invisible space and the visible matter, generative of concrete forces, had to be bridged: and so it was that ‘under the pressure of applications, which of necessity are matter-based and not space-based (you cannot make an observation of how something observable moves relative to something that is unobservable) the factual nature of absolute space and time were identified: a family of immaterial inertial systems whose location, orientation, and motion must ultimately be deduced from the observed motion of matter and, as it were, “painted in” on that matter.’[275] Yet, an incongruous disjunction between the invisible inertial systems and the observable matter still existed;[276] to bridge that gulf between space and matter, or between inertial systems and matter or, again, as Barbour says, ‘between the seen and the unseen, Mach proposed the elimination of the cut between matter and the inertial systems’[277] through the conceptualization of interactive gravity, according to which the sum of all the masses of the universe constitute the dynamical frame of reference for motion. In a few words, Mach completed what Kepler initiated (the introduction of the idea of interaction between bodies) and Newton partially implemented: in the end Mach rendered dynamics completely interactive (in the intentions of Barbour, the implementation of this fundamental idea ought to have been the core issue of Volume II, which he never published).

Then, by following Barbour’s footsteps, we have come to the partial end of this journey that started two and a half millennia ago, and, with Barbour, we can now draw the provisional conclusion: [278]

the dynamical frame of reference, and not the ground under our feet or a sphere of fixed stars in the heavens, is the true backbone of the world.

To come back to the main theme of my research on space and place, I believe we can derive the following conclusion. Time passes by, but there is a continuous thread that links different, yet epistemologically-similar modes of thinking about the world; either the common man (or even the primitive man) thinks at the ground under his feet, or the first philosophers and astronomers think at the sphere of the fixed stars in the heavens, or the modern scientists think at the dynamical frame of reference for determining the motions of bodies, in any case, one thing seems to persist: the necessity to rely on concrete matter to have a (frame of) reference with respect to which understanding the phenomena of nature. And since – as I’m maintaining in the pages of this website – we can understand concrete matter as the place where processes concretize, we can reassign the concept of place a prominent ontological and epistemological position in determining and expressing our modes of thinking about the world. Given that I believe place and matter are only conceptually-separated entities behind which there is unity, I believe we can close the first part of this discussion regarding the scientific perspective on concepts of place and space, pointing out a fundamental correspondence between the philosophical and the scientific perspective that we have analyzed through the works of Casey and Barbour.

It seems to me, the fundamental scheme… remained unchanged

It seems to me, the fundamental scheme we have apprehended by reading the article based on Casey’s work remain unchanged: at first, the placial instances of Aristotle determined the nature of the debate and the triumph of the concept of place (topos) as a fundamental conceptual instrument to explain the basic phenomena of the world. Then a long phase begins, which, to use Casey’s terminology, determined at first the ascendancy of space and then its supremacy, with the triumph of the Newtonian concept of absolute space. But this is only the partial result of an ongoing process as both Casey and Barbour maintained; in fact, after Newton – we have just seen it with Barbour – Mach envisaged new epistemological possibilities to understand and represent the phenomena of nature: the return to the concreteness of matter is a way to come back to, or, better, to go forward to the concreteness of place, a ‘new’ place, which extends the limits of the old Aristotelian notion, preserving its foundational ontological character; a turning back to place, which is, actually, a going forward, given the extension of meaning of the traditional concept of place (as Casey showed in his book), and which many scholars are turning their interests on for many decades now.

As we know, Barbour didn’t finish his work as in his intentions, and we now have to rely on the works of other scientists to continue this journey regarding the scientific perspective on questions of place, space and the related concepts, in more recent times. For this, I redirect you to the next item: Space and Place: A Scientific History – Part II.

Notes

[1] From Barbour’s perspective ‘the debate about the absolute or relative nature of motion is at root a debate about reality’, in Julian B. Barbour, The Discovery of Dynamics (New York: Oxford University Press Inc., 2001), 402. Since speaking about motion means speaking about place and/or space, I cannot help but subscribe to such an important pronunciation. To elucidate the concepts of space and place is to elucidate the fundamental nature of reality.

[2] Ibid., 13.

[3] Inertial motion is the characteristic, primary natural state of a body which moves uniformly with a purely rectilinear motion unless it is affected by some external body or force. It took a very long time before this characteristic state of bodies was understood and isolated; it was finally incorporated into Newton’s First Law, which states that ‘every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it’, 21.

[4] Ibid., 1-2.

[5] Ibid., 3.

[6] Ibid., 19.

[7] Ibid.,  21.

[8] Ibid., 21-27.

[9] Ibid., 47.

[10] Ibid., 56.

[11] Ibid., 56-57.

[12] Ibid., 41.

[13] Ibid., 44.

[14] Barbour’s scientific bias is traceable to the presentation of the Pythagoreans before the Milesians and the Atomists before Anaxagoras; in this brief account on the pre-Socratic philosophers, I will keep the usual order of presentation.

[15] G.S. Kirk and J.E. Raven, The Presocratic Philosophers (Cambridge: Cambridge University Press, 1984), 73.

[16] Julian B. Barbour, The Discovery of Dynamics (New York: Oxford University Press Inc., 2001), 61.

[17] Guthrie W. K. C. The earlier Presocratics and the Pythagoreans (Cambridge: Cambridge University Press, 1985),  173-174.

[18] J. Barbour, The Discovery of Dynamics, 60.

[19] Ibid., 61.

[20] Ibid., 61.

[21] Ibid., 61.

[22] Ibid., 62.

[23] Anaxagoras’ detour into Metaphysics, in between the material and the immaterial, may have influenced Plato’s thinking: ‘…Proprio l’impatto di Anassagora segnerà una svolta decisiva nel pensiero di Platone, il quale ci dice espressamente, per bocca di Socrate, di aver imboccato la nuova strada della Metafisica per sollecitazione e insieme per delusione provocata dalla lettura del libro di Anassagora.’ In Reale-Antiseri, Il Pensiero Occidentale dalle Origini ad Oggi, Vol. I, p.45.

[24] J. Barbour, The Discovery of Dynamics,  61-62.

[25] Ibid., 62.

[26] Ibid., 63.

[27] Ibid., 64, 65.

[28] Aristotle’s Metaphysics, Book XII, Chap. 8.

[29] J. Barbour, The Discovery of Dynamics, 67.

[30] Ibid., 87.

[31] ‘Why Aristotle put such emphasis on motion is not clear to me from his work’ Barbour says at a certain point in the text, page 74.

[32] For instance see pages 64, 65, 67, 89, 90, 106, 109, 224 or 247.

[33] Edward S Casey, The Fate of Place: A Philosophical History (Berkeley: University of California Press, 1997), 51, 133-134.

[34] Pierre Duhem in: The Concepts of Space and Time, ed. Milic Capek (Dordrecht:Reidel, 1976), p. 29.

[35] The philosopher Martin Heidegger and the mathematician Salomon Bochner were quite explicit concerning the impossibility to attribute the concept of space to the ancient Greeks: see Note 5 of the article Back to the Origins of Space and Place

[36] Ibid., 75.

[37] Ibid., 75.

[38] Ibid., 78.

[39] Ibid., 89, 91.

[40] Ibid., 101.

[41] Ibid., 103.

[42] Ibid., 109.

[43] Ibid., 107.

[44] Ibid., 109-110.

[45] See Jean Piaget and Barbel Inhelder, The Child’s Conception of Space (London, Routledge & K. Paul, 1956). In this book, the two authors deal with the development of the notion of space in the child, passing from topological to metrical considerations; from the concretion of the two modalities that happen through different phases – or stages – of the cognitive development of the child, the spatial character of objects and the concept of space are finally acknowledged. Then, what Barbour roughly call ‘intuitive space’ is actually the result of different and complex stages of cognitive development and adaptation to the environment and regarding physiological, psychological, logical and symbolic processes.

[46] J. Barbour, The Discovery of Dynamics, 110-111.

[47] See chapter two – The Emergence of Space – in Edward Casey’s The Fate of Place; see also my article Place and Space – A Philosophical History.

[48] J. Barbour, The Discovery of Dynamics, 112.

[49] From an observer on the earth ‘The sun is observed to move around the heavens on a great circle (the plane in which the earth moves of necessity contains the earth and, since it is a plane, it must cut the celestial sphere in a great circle), and this circle keeps a fixed position in geoastral space. Since eclipses of the sun or moon can only occur when the moon, which normally is not situated in the plane defined by the motion of the earth around the sun, is actually in this plane, the great circle of the sun’s apparent motion is called the ecliptic’, 114.

[50] Ibid., 117.

[51] Ibid., 113.

[52] Ibid., 115.

[53] Ibid., 118.

[54] Ibid., 153.

[55] Ibid., 153.

[56] Ibid., 155.

[57] Ibid., 168.

[58] I have already spoken about those two fallacies (one of which – the fallacy of misplaced concreteness – became notorious after the work of Alfred N. Whitehead) in my previous items. For reference, see the items: Preliminary Notes and The Τόπος of a Thing.

[59] Ibid., 175.

[60] The difference between sidereal time and solar time is about 4 minutes: while solar time is the passage of time we are all accustomed to, the 24-hour rotation of the earth measured relative to the sun, the same rotation relative to the distant stars takes 23 hours, 56 minutes and 4 seconds…

[61] Ibid., 185.

[62] Ibid., 189.

[63] ‘The collapse of Peripatetic physics did not occur suddenly; the construction of modern physics was not accomplished on an empty terrain where nothing was standing. The passage from one state to the other was made by a long series of partial transformations, each one pretending merely to retouch or to enlarge some part of the edifice without changing the whole. But when all these minor modifications were accomplished, man, encompassing at one glance the result of his lengthy labor, recognized with surprise that nothing remained of the old palace, and that a new palace stood in its place.’ are the words expressed by Pierre Duhem and reported by Barbour (page 191).

[64] This is the passage taken from Whitehead: ‘I do not think, however, that I have even yet brought out the greatest contribution of medievalism to the formation of the scientific movement. I mean the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. Without this belief the incredible labours of scientists would be without hope. It is this instinctive conviction, vividly poised before the imagination, which is the motive power of research: – that there is a secret, a secret which can be unveiled. How has this conviction been so vividly implanted on the European mind? . . . My explanation is that the faith in the possibility of science, generated antecedently to the development of modern scientific theory, is an unconscious derivative from medieval theology’, page 192.

[65] Ibid., 194.

[66] Ibid., 195.

[67] Ibid., 195.

[68] Ibid., 196.

[69] Ibid., 197.

[70] Ibid., 197.

[71] Ibid., 197.

[72] Ibid., 199.

[73] At this regard, these are the words of Buridan reported by Barbour: ‘it seems to me that it ought to be said that the motor in moving a moving body impresses (imprimit) in it a certain impetus (impetus) or a certain motive force (vis motiva) of the moving body, [which impetus acts] in the direction toward which the mover was moving the moving body, either up or down, or laterally, or circularly… And by the amount the motor moves that moving body more swiftly, by the same amount it will impress in it a stronger impetus [this is the momentum]. It is by that impetus that the stone is moved after the projector ceases to move. But that impetus is continually decreased (remittur) by the resisting air and by the gravity of the stone…Thus the movement of the stone continually becomes slower, and finally that impetus is so diminished or corrupted that the gravity of the stone wins out over it and moves the stone down to its natural place.’ To a certain extent, Barbour says, ‘Buridan anticipated not only Newton’s First Law but also Newton’s identification of momentum as the most fundamental dynamical concept’. Ibid., p. 200. In the following pages Barbour explains why Buridan cannot be actually credited for that: ‘there is no clear statement that, in the absence of resistance and gravity, impetus would carry a body on a straight line at a uniform speed for ever’; moreover, ‘there is no attempt by Buridan to give a precise and quantitative mathematical treatment.’ And finally, ‘just as in the case of the Mertonians, there still seems to be a complete lack of any idea of tying the physical concepts he has developed to experimental measurements.’

[74] Ibid., 203

[75] Ibid., 208.

[76] Ibid., 215.

[77] Ibid., 216.

[78] Ibid., 217.

[79] Ibid., 221.

[80] Ibid., 221.

[81] Ibid., 222.

[82] Ibid., 217.

[83] Ibid., 218.

[84] Ibid., 219.

[85] Ibid., 223.

[86] Ibid., 223.

[87] Ibid., 224.

[88] Ibid., 224.

[89] Ibid., 227.

[90] ‘Although Copernicus sensed the importance of the sun and, equally important, made the earth a planet like the others, his final scheme remained to a remarkable degree geocentric. First, the motion of the earth was quite different from that of the planets. Alone among them it moved uniformly in helioastral space. Second, the motions of the five planets were not coordinated on the sun but on the centre of the earth’s orbit, the orbis magnus. Moreover, they were strongly correlated to the earth’s orbital period…‘, Ibid., 242.

[91] Ibid., 247.

[92] Ibid., 247-248.

[93] from a quick digital query on a .pdf format of De Rivolutionibus, I just found few explicit references to the Latin spacium – the medieval Latin term for the English space – this fact meaning that if there is a conceptualization of something comparable to our modern concept of space in the mind of Copernicus (and of his contemporaries), this conceptualization is still very far from being a fully-fledged concept  expressed by a precise term – spacium or spatium – with a definite three-dimensional meaning, as it had for Newton; in fact the spatial sense of the vocabulary used by Copernicus is still very often conveyed by the term locus which, in general, is more close to the concept of place (wich is connected with matter) rather than to the modern concept of (three dimensional) space (which is disconnected from matter).

[94] Ibid., 249.

[95] Ibid., 253.

[96] Ibid., 253-254.

[97] Ibid., 254.

[98] Ibid., 259.

[99] Ibid., 261.

[100] Ibid., 255.

[101] Ibid., 256.

[102] Ibid., 255

[103] In the introduction of De Rivolutionibus  Copernicus gives an almost explicit definition of what an organic system is.

[104] Ibid., 258.

[105] Ibid., 264.

[106] Ibid., 266.

[107] Arthur Koestler, The Sleepwalkers. A History of Man’s Changing Vision of the Universe (New York: The MacMillan Company, 1959).

[108] Ibid., 270.

[109] Ibid., 313.

[110] J. Barbour, The Discovery of Dynamics, 351.

[111] A. Koestler, The Sleepwalkers,  314.

[112] J. Barbour, The Discovery of Dynamics, 272.

[113] Ibid.,  273.

[114] Ibid., 275.

[115] Ibid., 275.

[116] Ibid., 275.

[117] Ibid., 277; see also:  A. Koestler, The Sleepwalkers, 318.

[118] A. Koestler, The Sleepwalkers, 324.

[119] Ibid., 324-325.

[120] J. Barbour, The Discovery of Dynamics, 294.

[121] Ibid., 297-298.

[122] Ibid., 312.

[123] Ibid., 338.

[124] Ibid., 338.

[125] Ibid., 339.

[126] Ibid., 339.

[127] I often use the following notation ‘place/matter’ since matter  is for me a field of localization – that is a place – of certain processes (physicochemical, biological, social and symbolic processes): or, otherwise said, everything that appears to our senses under the guise of matter is a place – a material place of actualized processes.

[128] Ibid., 340.

[129] Ibid., 341.

[130] Ibid., 341.

[131] Ibid., 341.

[132] Ibid., 341.

[133] Ibid., 344.

[134] Ibid., 356.

[135] Ibid., 376-377.

[136] Ibid., 353.

[137] Ibid., 353.

[138] Ibid., 353.

[139] Ibid., 355.

[140] Ibid., 355.

[141] Ibid., 356.

[142] Ibid., 356.

[143] Ibid., 365.

[144] Ibid., 365.

[145] Ibid., 368.

[146] Ibid., 368.

[147] Ibid., 372-373.

[148] Ibid., 377.

[149] Ibid., 377.

[150] Ibid., 378.

[151] Ibid., 381.

[152] Ibid., 395.

[153] Ibid., 396.

[154] Ibid., 396.

[155] this fact – the explicit use of spatial and placial terms in the original works of authors of the past epochs and the way those terms were translated by scholars in the following epochs – could be an important subject for a specific research, something I’m currently working on, in bits and pieces, something that immediately interested me, as soon as I began to deal with spatial and placial questions reading ancient texts or pieces extracted from ancient texts (I’m speaking of Plato, Aristotle, Simplicius, Lucretius, Vitruvius, Ptolemy, Copernicus, Bruno, Alberti, etc.); such an important philological argumentation is often completely underestimated by those scholars who deal with concepts of place and space.

[156] Ibid., 396-397.

[157] Ibid., 397. Barbour refers to the following statement (which I report entirely) made by Salviati, when he was talking about the creation of order from chaos: ‘primordial chaos, where vague substances wandered confusedly in disorder, to regulate which nature would very properly have used straight motions . . . But after their optimum distribution and arrangement it is impossible that there should remain in them natural inclinations to move any more in straight motions’. Then Barbour says that ‘straight motions’ can be logically defined operationally only with respect to space.

[158] Ibid., 398.

[159] Ibid., 399, 400.

[160] Ibid., 400.

[161] Ibid., 400.

[162] Ibid., 401.

[163] Ibid., 402.

[164] Ibid., 406.

[165] The book was withheld from publication by Descartes himself after he became acquainted with Galileo’s condemnation by the Inquisition; in a letter to a friend he wrote: ‘…since I would not wish, for anything in the world, to write a discourse containing the slightest word which the Church might disapprove; I would, therefore, prefer to suppress it, rather than publish it in a mutilated version’,  436.

[166] Ibid., 409.

[167] In Principle 48, Part 1, Descartes says: ‘I do not recognize more than two principal kinds of things: one is intellectual or cogitative things, that is, things pertaining to the mind or to thinking substance; and the other, material things, or things pertaining to extended substance or body. Perception, volition, and all modes of perceiving and willing pertain to thinking substance; while size (or extension in length, width; and depth), figure, motion, situation, divisibility of its parts, and such, pertain to extended substance. However, we also experience in ourselves certain other things which should be attributed neither solely to the mind nor solely to the body, and which, as I shall show later in the proper place* originate from the close and profound union of our mind with the body: specifically, the appetites of hunger, thirst, etc.; and similarly the emotions or passions of the soul (which do not consist solely in thought), for example, the emotions of anger, merriment, sadness, love, etc. ; and finally all sensations, such as pain, pleasure, light and color, sounds, odors, tastes, heat, hardness, and the other tactile qualities’, in René Descartes, Principles of Philosophy (Dordrecht: Kluwer Academic Publishers, 1982), 21, 22. *See Principles 189-91, Part 4.

[168] With respect to the possibility of an alternative benign interpretation of Descartes’s dualism I read these words by Alfred North Whitehead: ‘In one sense the abstraction [that is the separation of body and mind] has been a happy one, in that it has allowed the simples things to be considered first’, in Alfred N. Whitehead,  Mind and Nature (Cambridge: Cambridge University Press, 1934), 66.

[169] See: Martin Heidegger,  What is a Thing? (South Bend: Gateway Editions Ltd., 1976), 98.

[170] See: Heidegger, What is a thing?Descartes identifies space, or internal place with the body which occupies it: ‘for, in truth, the same extension in length, breadth, and depth, which constitutes space, constitutes body. The distinction we make is only conceptual’, 17.

[171] J. Barbour, The Discovery of Dynamics, 411, 412.

[172] Ibid., 412, 414.

[173] Ibid., 428.

[174] Ibid., 428.

[175] Ibid., 429.

[176] Ibid., 429.

[177] Ibid., 431.

[178] Ibid., 431.

[179] Ibid., 431.

[180] Ibid., 436.

[181] Ibid., 436.

[182] Ibid., 437.

[183] Ibid., 437.

[184] Ibid., 437.

[185] Ibid., 437.

[186] The Philosophical Writings of Descartes Vol. I (Cambridge: Cambridge University Press, 1985), 93.

[187] Einstein wrote: ‘Euclid’s mathematics, however, knew nothing of this concept [space] as such; it confined itself to the concepts of the object, and the spatial relations between objects. The point, the plane, the straight line, the segment are solid objects idealized. All spatial relations are reduced to those of contact (the intersection of straight lines and planes, points lying on straight lines, etc.). Space as a continuum does not figure in the conceptual system at all. This concept was first introduced by Descartes, when he described the point-in-space by its coordinates. Here for the first time geometrical figures appear, in a way, as parts of infinite space, which is conceived as a three-dimensional continuum’. See Einstein’s 1934 article: ‘The Problem Of Space, Ether, and The Field in Physics’. In Ideas and Opinions (New York: Crown Publishers, Inc., 1954), 279.

[188] René Descartes, La Geometrie – The Geometry of René Descartes, translated by D. E. Smith and M. L. Latham, (New York: Dover Publications, Inc., 1954), p. 26, 27.

[189] J. Barbour, The Discovery of Dynamics, 439.

[190] Ibid., 423.

[191] Ibid., 423.

[192] Ibid., 436.

[193] Ibid., 447.

[194] Ibid., 450.

[195] Ibid., 452, 453.

[196] Ibid., 457.

[197] Ibid., 462.

[198] Ibid., 462.

[199] Ibid., 463.

[200] Ibid., 476.

[201] Ibid., 472.

[202] Ibid., 478.

[203] Ibid., 478.

[204] Ibid., 482.

[205] Ibid., 483.

[206] Ibid., 503.

[207] Ibid., 504, 505.

[208] Ibid., 505.

[209] Ibid., 505-506.

[210] Ibid., 527.

[211] At this regards, Barbour spoke about ‘Descartes’ dream of explaining the entire material world by a single primitive substance differentiated solely by figure, magnitude, and motion. At a stroke Newton replaced this Cartesian world – which was a pure figment of the imagination – with a scheme that did as much, but of the three Cartesian pillars, retained only one (motion), the other two (figure and magnitude) being replaced by mass and gravitational force’, 564.

[212] Ibid., 555-556.

[213] As to the question of plagiarism, it was about some anticipations Hooke had made to Newton in the course of their correspondence; it especially regarded the inverse square law and the suggestion of a central force of attraction in the determination of planetary motions. That question was ‘patched’ with a citation, see pages 566-567.

[214] Ibid., 578-579.

[215] Ibid., 575.

[216] Ibid., 580.

[217] Ibid., 580.

[218] Ibid., 584.

[219] Ibid., 589.

[220] Ibid., 590-591.

[221] Ibid., 592.

[222] Ibid., 591.

[223] Ibid., 592.

[224] Ibid., 593.

[225] Ibid., 594.

[226] Ibid., 595.

[227] Max Jammer, Concepts of Space (New York: Dover Publications Inc., 1993), 98.

[228] J. Barbour, The Discovery of Dynamics, 597.

[229] Ibid., 598.

[230] Ibid., 599.

[231] Ibid., 599.

[232] Ibid., 599.

[233] Ibid., 453.

[234] Ibid., 600.

[235] Ibid., 600.

[236] Ibid., 608.

[237] Ibid., 609.

[238] Ibid., 609.

[239] Ibid., 610.

[240] Ibid., 612.

[241] Ibid., 613.

[242] Ibid., 614.

[243] Ibid., 617.

[244] Ibid., 617.

[245] Ibid., 618.

[246] Ibid., 618.

[247] Ibid., 618.

[248] Ibid., 618-620.

[249] Ibid., 622, 623.

[250] Ibid., 623-624.

[251] Ibid., 625.

[252] Ibid., 636.

[253] Ibid., 626.

[254] Ibid., 638.

[255] Ibid., 641.

[256] Ibid., 629.

[257] Ibid., 630.

[258] Ibid., 630.

[259] Ibid., 631.

[260] Ibid., 631.

[261] Ibid., 657.

[262] Ibid., 645.

[263] Ibid., 645.

[264] Ibid., 647.

[265] Ibid., 648.

[266] Ibid., 655.

[267] Ibid., 655.

[268] Ibid., 678.

[269] Ibid., 678.

[270] Ibid., 688.

[271] Ibid., 687.

[272] Ibid., 684, 685.

[273] Ibid., 690.

[274] Ibid., 691.

[275] Ibid., 691.

[276] ‘There is a nice way of summarizing the Copernican Newtonian revolution. It eliminated the Aristotelian gulf between the heavens and the earth but created a gulf between the seen (matter) and the unseen (inertial systems)’, 691.

[277] Ibid., 691.

[278] Ibid., 691.

 

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