ABSOLUTE SPACE AND IDEALIZATION IN NEWTON *
Giuseppe Boscarino
Pre-print a
cura dell’autore. Originale pubblicato su Poznan
Studies in the Philosophy of the Sciences and the Humanities 1989, Vol. 17,
131, 149 and IDEALIZATION II: FORMS AND APPLICATIONS, edited by J. Brzezinski,
F. Coniglione, T.A.F. Kuipers, L. Nowak, Rodopi, Amsterdam-Atlanta, GA 1990
0. Introduction
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:
[1] 
The result of equation [1] 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 is
so that
[2] 
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
y”= y
z”= z
So that:
vx”= vx
- v0 - a0 t
i.e.
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.
5. Conclusions
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
NOTES
* 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
[1981].
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].
REFERENCES
Amsterdamski, S.
[1981]. Scienza..In Enciclopedia Einaudi,
vol. XII. Torino.
Burtt, B. A. [1927]. The
metaphysical foundations of modern physical science. New York.
Cassirer, E. [19622].
Storia della filosofia moderna. Milano.
Cohen, I. B. [1980]. The
Newtonian revolution. Cambridge.
Dicke, R. H. [1964]. The many faces of Mach. In Chin & Hoffman
(Eds.) Gravitation and
relativity. New York.
Einstein, A. [1956]. The meaning of relativity. London.
Grünbaum, A. [1968]. Philosophical problems of Space and time. Dordrecht.
Hall, A. R. [1976]. La rivoluzione scientifica
1500/1800. Milano.
Jammer, M. [1954]. Concepts of space. Cambridge.
Koyré, A. [1962]. Du monde dos à l’univers e infini. Paris.
Koyré, A. [1965]. Newtonian studies. London.
Mach, E. [1960]. The science of mechanics: A critical and
historical account of its development. La Salle, Ill.:
Illinois.
Newton, I. [1686]. Mathematical principles of natural philosophy (ed. by
Cajori). Berkeley 1960.
Nowak, L. [1980]. The structure of idealization. Towards a
systematic interpretatìon of the Manrxian idea of science. Dordrecht.
Randall, J. H. jr.
[1942]. 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. [1971].
Il punto di vista matematico di
Newton. In Wiener & Noland (Eds.). Le radici del pensiero scientifico. Milano.