It is a widespread opinion among historians of science and philosophy, and also among scientists and epistemologists, that the concept of “absolute space” constitutes, in Newton’s scientific theory, a “metaphysical vestige” from which the successive evolution of physics has rightly been freed.
In our opinion the underestimation of the scientific and epistemological importance of this Newtonian concept derives from the fact that the role played by the process of idealization in the construction of scientific theories has been neglected [cf. Nowak, 1980]. We should like in this essay to show that absolute space is not a “metaphysical” concept - concept that derives from the ambiguous way in which the term “metaphysical” is used, this term being still coloured too much by the dismissive judgements on it which have been expressed by a whole epistemological tradition of a neo-positivistic character - but that on the contrary, in the light of the idealizational conception of science, it is without doubt an indispensable concept for the overall economy of Newton’s physical theory, although its role is complex and is situated at different levels of conceptualization.
More precisely - to summarize what we shall try to maintain later - it is possible to say that
1) the need to introduce the concept of absolute space derives from an overall “metaphysical” and “philosophical” conception of nature that Newton always has in mind and which constitutes the conceptual background which justifies its admissibility: this is the philosophical level of his inquiry;
2) absolute space is an idealizing hypothesis, and a mathematical construct indispensable to the conceptual construction of the theory: this is the mathematical level of the Newtonian theory in which one notices the idealizational nature of his scientiflc construction;
3) absolute space is a physical hypothesis: it is, in fact, the empty space of matter with its dynamic properties which generates the forces of inertia: this is the physical, or to be precise, experimental level of Newtonian science;
4) finally, absolute space is a theological hypothesis in that it is the “sensorium dei”, and this is the theological or “fideistic” level which does not represent, we maintain, an indispensable aspect of Newton’s construction but which is, on the contrary, a concession both to his adversaries and to his own Christian conscience, and which does not play any role in the overall economy of his scientific construction.
1. Is Newton’s absolute space a metaphysical principle?
Absolutc space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute space; which our senses determine by its position relative to bodies; and which is commonly taken for immovable space [Newton, 1686, p. 6].
And further on:
All things are placed in time as to order of succession; and in spacc as to order of situation. But because the parts of space cannot be seen, or distinguished from one other by our senses, therefore in their stead we use sensible measures of them. For from the position and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs [Newton, 1686, p. 8].
So wrote Newton.
Many historians of science have interpreted these famous passages as concessions that Newton is supposed to have made within his physical theory to his metaphysical conceptions. For example, Jammer asserts that:
It is well known that Newton, himself a religious man, never denied the existence of beings and entities that transcend human experience; he contended only that their existence had no relevance to scientific explanation: in its mundus discorsi, science has no place for them. Intimately acquainted with the problems of religion and metaphysics, Newton managed to keep them in a separate compartment of his mind, but for one exception, namely his theory of space [Jammer, 1964, p. 96] .
However, elsewhere the same author writes on the contrary that “to Newton, absolute space is a logical and ontological necessity” [Jammer, 1954, p. 99] .
From another point of view Koyré affirms that:
Si la philosophie l’occupe, ce n’est donc pas ex professo, mais seulement dans la mesure où il en a besoin pour poser les bases de son investigation mathématique de la nature [Koyré, 1962],
and in his most important book dedicated to Newton:
Metaphysical hypotheses, so Newton told us, “have no place in experimental philosophy”. Yet it seems quite clear that metaphysical convictions play, or at least have played, an important part in the philosophy of Sir Isaac Newton. It is his acceptance of two absolutes — space and time — that enabled him to formulate his fundamental three laws of motion [Koyré, 1965, p. 113].
In addition, a historian of philosophy, as shrewd as Cassirer often is, writes:
It is in fact in having vigorously prevented the entry of religious and metaphysical problems into the field of concrete physical investigation that Newton’s philosophical greatness is revealed, and this in spite of his strong personal interest in these problems. Only in one place does Newton abandon his critical reserve: his scientific doctrine of space and time ends up by appearing to us as a description of two divine attributes, which plunges us into the questions of speculative theology of the period [Cassirer, 19622, p. 485] .
From these positions, which subsequently became widely consolidated in the historiographical tradition, there would seem to emerge a basic ambiguity in Newton, on the one hand a “coherently” empirical scientist, on the other a “metaphysical” one tending to introduce the notion of absolute space into purely physical questions and so in contradiction with himself. This supposed ambivalence has also been interpreted as the symptom of an irreconcilable conflict in Newton between his mathematical rationalism and his equally present experimental empiricism [cf. Burtt, 1927; Randall, 1942]. As Strong affirms, when Newton speaks of space, time and motion as “absolute real and mathematical” he seems to be a realist mathematician, while when on the contrary he refers to his experimental work and makes explicit declarations about the method followed he shows a positivistic mentality [Strong, 1971, p. 427].
But either absolute space has a meaning of its own in the context of the Newtonian theory (“a logical and ontological necessity”, for Jammer; it is useful “pour poser les bases de son investigation” and “enabled him to formulate his fundamental three laws of motion”, for Koyré) and then it is not clear in what sense it is considered a metaphysical vestige, giving this last term a pejorative meaning; or it has no meaning — in the sense that his physical theory could have omitted it without for this reason losing its generality and validity — and in that case it is not clear what is meant by the assertion that it is necessary for the formulation of the three laws of motion. And moreover, if Newton is a realist mathematical rationalist it is hard to understand his repeated affirmations about the necessity for the experimental and inductive method; if, on the other hand, he is an inductivist and a positivist, absolute space becomes a foreign body in bis theory and only the symptom of an incorrigible metaphysical vocation deriving from his personal theological and alchemic predilections, about which much has been said recently.
In reality such dilemmas are, in our opinion, none other than the result of erroneous epistemological tenets that have given rise to a wrong interpretation of Newton’s scientific work by trying to fit it into a framework of insufficient conceptual categories: it is neopositivist empiricism which, more in Jammer, less in Koyré, has up to now been the dominant paradigm of interpretation. And it is not an accident that it was Mach, one of the putative fathers of contemporary neopositivism, who was one of the most severe critics of the metaphysical character of absolute space in his attempt to construct a physical science without it (as we shall subsequently see). We are in the presence of a characteristic distortion, as has happened several times in the history of science: a good “scientific practice” is linked to a “bad philosophy” which is often the remnant of pre-existent epistemological doctrines.
The above-mentioned epistemological tradition of a neopositivist derivation began to decline in the middle of the 1960’s and this led to a sort of “explosion” of the philosophics of science and to phenomena, in some ways abnormal, such as the epistemological anarchism of Feyerabend with the consequent bankruptcy of any possibility of a “rational reconstruction” of a history of science. However, in the last years, thanks to the work done by the idealizational methodology, there have emerged new patterns of interpretation of scientific research which allow us, in our opinion, to have a better understanding of the concept of absolute space and so enable us to free ourselves of the above mentioned contradictions and ambiguities.
2. Idealization and mathematicization in Newton’s physical theory
By examining in general the way in which Newton constructs his own physical theory one can have an idea of the role played therein by the idealization procedures which, as I. B. Cohen has well written, pervade what he calls “the Newtonian style”:
The Newtonian style has three phases. Phase one usually begins with nature simplified and idealized, leading to an imaginative construct in the mathematical domain [...] Consequences are deduced by means of mathematical techhiques and are then transferred to the observed world of physical nature, where in phase two a comparison and contrast is made with experimental data and the laws or rules derived from such data. This usually results in an alteration of the original mathematical construct or system, or a new phase one, which in turn leads to a new phase two [Cohen 1980, p. XII-XIII].
In fact, in the last decade various writers have increasingly emphasized the importance in the science of construction of ideal models which do not give an immediate description of physical reality but which refer to it by means of counterfactual statements that must then be concretized in order that they may be ever closer to phenomena. Thus scientific statements are considered more fertile the more they grasp the “essence”, in other words, the inner regularity of the phenomena that in their immediate manifestation are disturbed by a multiplicity of secondary factors. Therefore, in order that it may be studied scientifically reality is first simplified, while the range of the phenomena that are considered to make up the field of study is selected from it (this is the phase defined as the “categorization of the world” [Nowak, 1980, pp. 112 ff.]) then it is necessary to define both the magnitudes and the type of relation which within such a field of study are to be considered most essential and finally it is possible to construct a theoretical model within which one may hypothesize some laws which describe the essential link binding the objects belonging to the above-mentioned range.
Such a way of proceeding, however, presupposes a conception of science and its role in the process of knowledge that breaks with what was the pre-Galileian pattern or “ideal of science”1. In short, science had first to cross what can be called its “threshold of maturity” and which was marked by the abandonment of a whole series of philosophical assumptions and by the adoption of an authentically new “vision of the world”. This break was made by the work of Galileo and completed by a point of view that was more strictly scientific and mathematical by Newton.
This general philosophical reorientation has seen the transition from a conception of science as a theoretical contemplation to an “active and operative science”, to quote Bacon (who was the principal exponent of this new mode of understanding); it has led to the rise of a new conception of experience which is no longer understood as the immediate world of the senses but rather as the point of arrival of a preceeding theoretical modelization (and it is by now historiographically well known that medieval Aristotelism did not err on the side of abstractness but rather on the side of empiricism). And it is precisely this new concept of experience that Galileo has in mind when he speaks of the great book of Nature written in mathematical language: it is not here so much a question of a supposed “Platonism” as of the fact that Galileo considers that it is first necessary to simplify Nature, eliminating from the consideration of science all those aspects that cannot be quantified, and then to idealize it by speaking of perfectly or absolutely rigid bodies and only apply the mathematical or geometrical instrument, without which it is impossible to make Nature “speak”. But the mathematical and experimental method of investigation, by the very fact that it refers to ideal entities which cannot be perceived immediately and permits the use of instruments which are not the simple extension of Man’s faculties, ends up by referring to realities and laws which go beyond human senses. In such a way “from the world of objects directly accessible to the senses, science moves progressively into a world of abstract objects (but here it would be better to say ‘ideal’ objects) which move in an abstract geometrical space governed by universal laws” [Amsterdamski, 1981, p. 548].
This is the most general metaphysical outlook that Newton, more or less consciously, had in mind and which he further elaborated by enunciating explicitly in his “Rules of Reasoning in Philosophy” those ontological presuppositions (the simplicity and uniformity of Nature) which, already clearly present in Galileo, now became the metaphysical horizon within which every scientist carries out his own investigation and which Newton explicitly returns to (as we shall see) to support his scientific theses: this is the plane that we indicated before as the philosophical level of his enquiry. In this general context one can understand why the introduction of the concept of absolute space did not meet any basic objection. The postulation of not immediately perceptible entities is compatible with an “essentialist” attitude of scientific research, in that Newton did not aim at the generalization of experimental facts but at the construction of a theoretical edifice by means of which he could interpret every kind of physical experiment, which always came after the construction of a coherent conceptual apparatus. This is what Newton explicitly indicates when he affirms that:
[...] in philosophical disquisitions, we ought to abstract from our senses, and
consider things themselves, distinct from what are only sensible measures of them [Newton, 1686, p. 9].
In this distinction between “things themselves” and “sensible measures” there is the whole difference between a science understood as a pure generalization of sensible aspects and a theoretical science which distinguishes between “surface” and “essence”, the former being understood as the starting-point in order to reach, by means of the measures, the latter.
What has been said can be easily shown by an overview at the way in which Newton constructs his physical theory.
As early as the “Preface” to the first edition of the Principia ... of 1686 Newton clearly asserts:
For 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 [Newton, 1686, p. XVII].
It is clear from this passage that for Newton it is necessary to start from the complexity of phenomena in order to arrive at the ideal concept of force with which later to reconquer reality, now no longer present to the senses, but ideally thought.
And in fact Newton in Books I and II of Principia... proceeds by treating of forces, laws and bodies in the abstract, considering then as pure mathematical entities constructed by means of a conscious work of idealization, which in Book III are concretized and interpreted physically. In doing so Newton warns the reader that in his definitions as well in his axioms and propositions, he is considering purely ideal and mathematical entities (such as forces, motions, material bodies etc.).
Thus, for example, in Book I Newton, after having idealized bodies and forces, utilises the definitions given and the idealized laws of motion to prove:
Proposition I. Theorem I. The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described [Newton, 1686, p. 40]
and afterwards its inverse2. The two theorems are therefore interpreted as stating a pure ideal regularity. Later, from the phenomenological law obtained from the generalization of astronomical observations described in “Phenomenon I” (Book 3) and thanks to the utilization of theorems I and 2 (book 3) and Corollary 6 of the 4th theorem (Book 1) Newton can deduce Proposition 1 of Book 3:
That the force: by which the circumjovial planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to Jupiter’s centre; and are inversely as the squares of the distances of the places of those planets from that centre [Newton, 1686, p. 406].
The centripetal force is thus concretely interpreted as the force produced by the planet Jupiter, and the material points which revolve around the centre (identified with the planet Jupiter) become the satellites. Later, by means of the “Rules of Reasoning in Philosophy”, Newton can still concretely interpret the ideal centripetal force (Proposition 5)3 and so identify it with the force of gravity and extend it to all the planets as a result of rules 1, 2 and 4.
Newton’s way of proceeding will subsequently show us how he passes from the most abstract model to ever more concrete ones in which account is taken of previously neglected variables in such a way as to reconstruct the various levels of complexity of the physical phenomena under examination. In fact if in the first approximated model the reciprocal interactions of the planets were neglected, in a successive phase it is necessary to take them into account:
It is true, that the action of Jupiter upon Saturm is not to be neglected [...] As the planet is differently situated in these conjunctions, its eccentricity is sometimes augmented sometimes diminished; its aphelion Is sometimes carried forwards, and its mean motion is by turns accelerated and retarded [Newton 1686, p. 421].
If this is Newton’s way of proceeding it becomes clearer why he also wanted to include in his Principia the 2nd Book which has been seen as an interruption of the continuity between the general theory (Book 1) and its application (Book 3) [cf. Hall, 1976, p. 252]. In fact in this book, in which the mechanics of fluids is discussed, Newton prepares the instruments which are necessary for the successive concretizations of the general theory elaborated in Book 1, by taking into account hitherto neglected factors. So, for example, he introduces the concept of resistance opposed to a body in general in order to concretize this concept by applying it to the means in which the resistance first hypothesized as depending on velocity, is proportional to the square of the velocity; and then again, in the concrete practice of experimentation, he successively considers the resistance of the wire, the imperfect sphericity of the bodies and so on.
Having, thus, summarily indicated the role played by idealizational procedures in general in Newtonian science, we can go into detail and analyze the place occupied by the concept of absolute space in Newton’s physical and mathematical theory.
3. Idealization and mathematization in the process of construction of absolute space
Newton was led to the concept of absolute space by his axioms on motion which constitute the foundation of his mechanics. It is from these, therefore, and in particular from the first axiom that we rnust start.
First, one assumes the existence of an empty space (not to be confused with absolute space4) and of bodies which move freely in it. That already implies the first law of motion seeing that, in virtue of the principle of non-sufficient reason, space must be homogeneous, isotrope and infinite. It is therefore supposed that the interactions with other bodies present in the universe are annihilated and now, equivalently, it is possibie to study the behaviour of only one body. The body itseif is reduced to a material point moving in an ideal empty space. It is now possible for Newton to formulate, in relation to a hypothetical system on inertial reference, the famous axiom of motion:
Every body continues in its state of umiform motion in a right line, unless it is compelled to change that state by forces impressed upon it [Newton, 1686, p. 13].
It is clear that so far Newton has taken a series of steps of an idealizational nature.
He is, moreover, well aware that the law of inertia makes sense only when one specifies exactly the system of reference with respect to which motion is uniformly rectilinear and so he looks for the system in which such a law is unconditionally valid. He will no doubt have asked himself: “Is the Earth a good system of reference?”. The answer must of course be negative in that the Earth has a movement of rotation and acceleration. To an observer rotating with the Earth the movement of the body would have seemed to be curved. Nor would the situation change if the Sun were chosen as the system of reference as it might not have an uniform motion. It is thus clear that a concrete system of reference made up of material bodies does not help us to arrive at the concept of inertia. The need for the introduction of an inertial system is the first step towards the idea of absolute space and time which are, as Newton declares, “true and mathematical”.
It is significant that Newton refers to absolute space and time as “true” and “mathematical”. These adjectives must be linked to the declaration already quoted in which Newton speaks of the need, in “Phiiosophical disquisitions”, to abstract from the senses. In fact, if it is necessary when we speak about absolute places and motions to use sensible measures, as it is impossible directly to see and distinguish such places, it is also necessary for a theoretical science which wants to go beyond the sensible surface to postulate the existence of ideal entities which allow one to derive universally valid laws which will only later be concretized to bring them closer to the surface, that is to that concrete reality which is present to our senses. “Truth” is, in this case, the corelative of “essence” which alone admits of mathematical treatment. Therefore one could schematically describe Newton’s way of proceeding in the following way: on the basis of a general metaphysical conception (the philosophical level of his inquiry), or rather thanks to this, ideal entities are introduced and idealizing assumptions are made which allow a mathematical derivation of certain laws (this is the mathematical and idealizational level) which in their turn, when they are concretized and referred to more specific physical systems, lead to the need to postulate the physical existence of absolute space (this is the physical or experimental level). Let us see how one arrives at the final result.
Thanks to the above-mentioned simplifications and idealizations, Newton is able to reach a more strictly mathematical level and write the equation of natural motion of the material point with reference to an inertial system. In Corollary 5 of the laws of motion we read:
The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion [Newton, 1686, p. 20].
This implies that the laws of motion must remain valid for every system at rest or in uniform rectilinear motion in absolute space. Let us suppose an ideal situation in which a body C moves in uniform rectilinear motion in relation to a system K at rest in absolute space with O as the origin. If K is at rest and the “impressed force” is Fx = 0, the equation of motion wili be:
The result of equation  will be that vx is constant; and conversely if vx constant then Fx = 0.
The material point C in system K, as there is no impressed force, will move in uniform rectilinear motion because of inertia. In this case, K is an inertial system and Newton’s law is valid. The same will apply to a system K’ that moves in uniform rectilinear motion with respect to K. The new coordinates of point C wilI be:
x’ = x - v0 t
y’ = y
z’ = z
v0 is the relative speed between K and K’. The new speed of C will be
v’ = vx - v0
and from this it follows that v’x = v’0 and vx = v0 , both constants. Thus in K’ the converse is valid: if v’x = v’0 , then F’x = Fx = 0.
Let us now consider a system K” which is accelerated in relation to K; we shall have:
x”= x - v0 t - ½ a0 t2
vx”= vx - v0 - a0 t
There thus appears an inertial additional force equal and opposite to acceleration multiplied by the mass of the body. Absolute space is no longer, now, merely an ideal system of reference with respect to which the laws of motion are valid, nor is it more simply the vacuum of Democritus, but it is also a dynamic structure which explains the origin of inertial forces. It is in this subtle passage that the concept of absolute space becomes complicated, giving rise to serious difficulties (in fact for many it seems absurd that absolute space can generate forces that produce verificable effects without being influenced in its turn: would this not violate the principle of the reciprocal dependence of physical phenomena?). These difficulties will be faced first by Mach and then by Einstein, the latter trying to solve them by means of his own theory of general relativity. Let us see briefly how these attempts are made.
4. Absolute space in Mach and Einstein
The person who led the most decisive attack against the concept of absolute space in Newton was the scientist and epistemologist Ernst Mach who, analysing in his famous work The science of mechanics the basis of Newtonian physics, writes in connection with absolute space:
It would appear as though Newton in the remarks here cited still stood under the influence of the mediaeval philosophy, as though he had grown unfaithful to his resolves to investigate only actual facts [Mach, 1960, p. 272].
Absolute space, like absolute time, is for Mach an “idle metaphysical conception” [Mach, 1960, p. 273] and in the preface to the 7th edition he writes thus:
With respect to the monstrous conceptions of absolute space and absolute time I can retract nothing. Here I have only shown more clearly than hitherto that Newton indeed spoke much about these things, but throughout made not serious application of them. His fifth corollary contains the only practically usable (probably approximate) inertial system [Mach, 1960, p. XVIII] 5 .
In such a way Mach, coherently with his fundamentally epistemological outlook which later had a great influence on the formation of the Vienna Circle and then of neopositivism, considers that scientific knowledge is founded only on the observation of facts, on induction and the construction of scientific laws understood as a generalization of duly observed factual links and on this basis he criticizes Newton’s concept of absolute space precisely because it is not susceptible to observation and experimentation and is therefore metaphysical.
However, Mach did not merely expose what in his opinion was the scientific and epistemological weakness of the concept of absolute space, he also proposed a bold physical interpretation. While in Newton’s theory absolute space has the function of justifying the origin of inertial or fictitious forces, for Mach on the contrary the inertial forces associated with the motion of a body are generated by all the other masses of the universe. Both the inertial and gravitational properties of matter are seen as a consequence of the mutual relationship between bodies.
Einstein too, strongly influenced by his reading of Mach, later exposed the epistemological defect inherent not only in Newton’s mechanics but also in his own theory of special relativity. In both, in fact, the cause of the origin of inertial forces is presumed to be absolute space, which for Einstein is not susceptible to experimentation either. Instead, the cause of the different shape of two distant masses which do not interact in space (a sphere and an elliptical body) must be sought, according to Einstein, in the other rotating masses, which are the only ones to generate the diversity between the two bodies.
There cannot be privileged space, such as that of Galileo or absolute one to which to refer the laws of physics. This, according to Einstein, must be of such a nature as to be able to be applied to systems having any kind of motion. Thus a system K” which is uniformly accelerated with respect to K can equally be considered to be at rest and so could be chosen as a system of reference for the laws of physics, provided that the spatial and temporal region possesses a gravitation field. Einstein would like in this way to incorporate in his physical theory Mach’s idea which consists in reducing the inertia of bodies to their mutual relationship and so tends to extend the validity of the laws of mechanics also to no-inertial systems of reference by means of his theory of general relativity (this last need is foreign to Mach’s thought). The gravitational field, as expressed in the geodetic equation, takes the piace of Newton’s absolute space. But the gravitational field, in Einstein’s equations, is linked to the distribution of masses: field and matter replace Newton’s absolute space and matter. For Einstein not only is there no empty space, there is no absolute space either in the sense of an ideal system of reference with an intrinsic metric not depending on matter. Reformulating Mach’s principle and incorporating it in his own theory of general relativity Einstein tried to eliminate the epistemological anomaly which is absolute space, consisting in the fact, to use the author’s own words, that:
it is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon [Einstein, 1956, p. 54].
For Einstein, on the contrary, both the inertial and metric properties of the space-time continuum are entirely deterrnined by the tensor Tik. In that consists Mach’s principle in the meaning given to it by Einstein6. In it there is a tendency towards an organic fusion of the ideas developed by Riemann with those of Mach. The former, starting from the theory of the three-dimensional surfaces of Gauss, had elaborated an abstract theory of space extended to n-dimensional manyfolds. The latter, more interested in the problems of physics, had, as already mentioned, brought the inertial properties of matter back to the mutual relationship of bodies. In Einstein, Mach’s conviction is blended with the ideas developed by Riemann to construct thereby the principle of general relativity (as already enunciated).
The distribution of matter, expressed by the energy tensor Tik, not only determines the inertial and gravitational properties of a body but also its metric fieid gik. The equation of the gravitational field
Rik - ½ gik R = -k Tik.
links the metric tensor gik to the energy tensor Tik, geometry to physics: the metric properties of space are no longer independent from matter but are determined by it.
It could thus seem that Einstein has resolved the epistemological anomaly of absolute space by means of his equations of the gravitational field. However, several physicists and epistemologists have drawn attention to the fact that with the solutions that have been found for Einstein’s equations absolute space has not been eliminated, rather it has reappeared unchanged. So, for example, Grünbaum, facing the problem of the solutions of Einstein’s equations, has written that “far from having been exorcised by GTR (General Theory of Relativity), the ghost of Newton’s absolute space is nothing less than a haunting incubus” [Grünbaum, 1968, p. 420]. In fact, with Schwarzschild’s solution Mach’s principle is violated in a double sense:
First the boundary conditions at infinity then assume the role of Newton’s absolute space, since it is not the influence of matter that determines what coordinate systems at infinity are the Galilean ones of special relativity and second, instead of being the source of the total structure of space-time, matter then merely modifies the latter’s otherwise autonomously flat structure (Grünbaum, 1968, p. 420].
We believe that such considerations can in a certain sense be ambiguous and misleading. Do they mean that it is thought that Newton’s metaphysical hypothesis has reappeared in contemporary physics, attributing once more to absolute space a negative connotation in line with the positions of Mach and other historians of science quoted by us?
We do not consider that this is so and we think that in this case too, as in the case of Newton’s absolute space, we are in the presence of an idealizational and mathematical construct which is fully legitimate from a scientific point of view and so is far from being a metaphysical hypothesis (in the negative sense of this term). In fact in Schwarzschild’s solution of the equations of the field of Einstein, the metric
when r tends to infinity, tends to Minkowski’s metric:
Thus in this solution of Schwarzschild’s it is assumed as an idealizational hypothesis a spherically symmetrical field generated by only one mass-point. By formulating this hypothesis the effect of every other mass that could disturb the spherical symmetry has been eliminated. Minkowski’s space, being a freely invented ideal entity within a more complex mathematical theory, cannot be reduced to a metaphysical principle. Thus, we may conclude, Schwarzschild’s solution does not reintroduce “the ghost of Newton’s absolute space”, as Grünbaum believes, but clearly shows the character typical of an idealizational and mathematical construct not only in the case of the solution given of the equations of Einstein’s gravitational field, but also as far as Newton’s absolute space is concerned.
We have seen that absolute space originates in the process of idealization of a system of non-interacting bodies which can consequently be considered in isolation from one another. The first law of motion derives its origin from this idealization. At this stage it is not yet possible to distinguish empirically between absolute space and relative space. Nor does the second law permit us such a distinction, in spite of the fact that it gives rise to inertial forces in accelerated systems in that it is simply the definition of the cause or force which causes variations in the state of rest or motion with respect to absolute space and, in this sense, it can be derived from the first (obviously only from a conceptual point of view, mathematically, on the contrary, it is the first law that can be derived from the second). In fact, it is with the third law that Newton carries out a concretization taking into account hitherto neglected interactions between bodies and so attributing the cause of the forces to the very matter of which the bodies are made. But the inertial forces seem to go beyond this interaction and if there is motion in a straight line they are not operationally ascertainable, it is being possible to think that they are found in matter itself; when there is motion along a curve, the situation is different. Thus, the centrifugal forces that are generated, for example, in the experiment with the bucket can no longer be attributed to matter: the possibility emerges of distinguishing between absolute space and relative space. The latter now appears as a mathematical fiction while the former takes on a measurable, physical reality which cannot be otherwise explained. To eliminate the scandalous concept of absolute space Mach hypothesized the interaction with other masses of the universe, and Einstein tried to formalize this idea but did not succeed. Even if with the geometricization of inertial forces the problem of attributing forces to matter seems to be solved, nevertheless it remains unchanged in the choice of the boundary conditions to be given to the field equations.
What has been said so far accounts for the first three levels (the philosophical and metaphysical one, the idealizational and mathematical one and the physical and experimental one), that we mentioned in the Introduction and have developed in this paper. We must still clarify in what sense it is also possible to say that in Newton absolute space is also a theological hypothesis. In fact the definition of absolute space as the “sensorium dei” is well known as are also the clearly fideistic conclusions of the General Scolium added, be it noted, to the second edition of his Principia ... where the existence of God is postulated and defined with all the characters of traditional theology to explain the origin and the working of the solar system and the machine of the universe. Well, this part of Newtonian philosophy not only does not enter into the construction of his physical science but can be considered completely superfluous; his physics loses nothing without this theological conclusion. So Newton’s train of thought can be summarized in the foliowing way: from a general philosophical conception of the world, inherited from Galileo, he is led to postulate absolute space as an ideal concept which allows him to elaborate his physics and which, in the successive development of his theory, ends by taking on physical reality. At this point he makes a concession to theistic conceptions of the time, which he accepted, and presents the concept of absolute space, and in general the image of the world at which he has arrived, as the expression of a provident divinity.
From what has been said we have a clear vision of the central role of the idealizational procedures in the construction of the physical theory on Newton’s part which, if correctly understood, liberate the concept of absolute space from the condemnation which historically it has been the object of. The latter resulted from insufficient epistemological instruments and from the consequent confusion of levels in which, as we have tried to show, Newton’s scientific work is elaborated.
Translated by Andrew Brayley, Istituto Universitario di Magistero Catania, Italy
* Thanks are due to S. Notarrigo and F. Coniglione for illuminating discussions, to S. Agostino for an valuable help with the bibliography, and to A. Brayley for the accurate English translation of the manuscript.
1 This expression is used by S. Amsterdamski .
2 Every body that moves in any curved line described in a plane, and by a radius drawn to a point either immovable, or moving forwards with an uniform rectilinear motion, describes about that point areas proportional to the times, and is urged by a centripetal force directed to that point” [Newton, 1686, p. 42].
3 That the circumjovial planets gravitate towards Jupiter; that circumsaturnal towards Saturn; the circumsolar towards the Sun; and by the forces of their gravity are drawn off from rectilinear motions and retained in curvilinear orbits” [Newton, 1686, p. 410].
4 In philosophical historiography and epistemological reflection empty space has been confused with absolute space [cf. e.g. Grünbaum, 1968, p. 6) while on the contrary the former is a physical concept — in the sense of absence of bodies — and the latter is an idealizing assumption. The two concepts are, moreover, independent of each other: Descartes believed in absolute space and full space, Newton in empty space and absolute space, Mach (who elaborated a physical science of interacting material bodies and not a field physics) was a bitter critic of the concept of absolute space but accepted the existence of an empty space. It is clear that the two concepts of empty space and absolute space can co-exist in the same author but can also be mutually exclusive.
5 In the following passage, however, Mach seems to recognize the ideal character of absolute space and time: “No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience” [Mach, 1960, p. 250].
6 0n the different interpretation of Mach’s principle cf. [Dicke, 1964].
Amsterdamski, S. . Scienza..In Enciclopedia Einaudi, vol. XII. Torino.
Burtt, B. A. . The metaphysical foundations of modern physical science. New York.
Cassirer, E. . Storia della filosofia moderna. Milano.
Cohen, I. B. . The Newtonian revolution. Cambridge.
Dicke, R. H. . The many faces of Mach. In Chin & Hoffman (Eds.) Gravitation and relativity. New York.
Einstein, A. . The meaning of relativity. London.
Grünbaum, A. . Philosophical problems of Space and time. Dordrecht.
Hall, A. R. . La rivoluzione scientifica 1500/1800. Milano.
Jammer, M. . Concepts of space. Cambridge.
Koyré, A. . Du monde dos à l’univers e infini. Paris.
Koyré, A. . Newtonian studies. London.
Mach, E. . The science of mechanics: A critical and historical account of its development. La Salle, Ill.: Illinois.
Newton, I. . Mathematical principles of natural philosophy (ed. by Cajori). Berkeley 1960.
Nowak, L. . The structure of idealization. Towards a systematic interpretatìon of the Manrxian idea of science. Dordrecht.
Randall, J. H. jr. . Newton’s natural philosophy: its problems and consequences. In Clarke & Nahm (Eds.). Philosophical essays in honor of Edgar Arthur Singer jr. Philadelphia.
Strong, E. W. . Il punto di vista matematico di Newton. In Wiener & Noland (Eds.). Le radici del pensiero scientifico. Milano.