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Originally published in Revolutionary World, vol. 33 (1979): 71–84.
MECHANISM AND THE UNITY OF MATTER, SPACE, AND
TIME IN CLASSICAL MECHANICS
When Newton formulated the laws of motion which form the basis of what is known as classical mechanics, he assumed the existence of absolute space and time. Although these notions were retained in physics essentially up to the time of Einstein, alternate views were being put forth on the basis of analyses of the interconnections of the philosophical categories (or concepts) of matter, motion, space, and time. Thus we find Hegel writing early in the 19th century:
Its essence [of motion] is to be the immediate unity of space and time.... Thus we know that space and time belong to motion; the velocity, the quantum of motion is space in relation to a definite time that has elapsed.... [l]t is the concept of space itself that creates its existence in matter…. Often a beginning has been made with matter, and then space and time regarded as forms of matter…. Just as there is no motion without matter, so also there is no matter without motion.t
These views of space, time, matter and motion were adapted by Engels to the fundamental materialist viewpoint that matter is primary to our ideas about it. He could then argue in 1887–78 against attempts to turn space and time into mere sense impressions. “The two forms of existence of matter [i.e., space and time—E.M,] are naturally nothing without matter, empty concepts, abstractions which exist only in our minds.”2
In the sphere of mathematics, the discovery of non-Euclidean geometry in the first half of the 19th century led to the questioning of Newtonian concepts of absolute space. Thus long before the theory of relativity was propounded by Einstein, the mathematician Riemann conjectured the dependence of the geometry of physical space on the action of matter.3
Classical physics of the 19th century remained largely unaffected by these views, probably because the dialectical concepts reflected in them are contrary to the spirit of the mechanistic philosophy which dominated the physics of that period. Nevertheless, it could have been possible to work these ideas into the basic framework of Newton’s laws of motion by re-interpreting some of the content of the laws. It is still important that this be done, since Newton’s laws retain their usefulness as the basis of nonrelativistic mechanics today with only minor modification in the way they are formulated. We can then look at the laws as an approximate, partial, and incomplete reflection of the objective reality they describe, rather than as a model of the world which must now be discarded in favor of the better model provided by relativistic mechanics. It will turn out, however, that the laws of motion still retain their basic mechanistic stamp.
It will therefore be the purpose of this paper to reassess the foundations of classical mechanics on the basis of the interrelationship of matter, space, and time. For this purpose it will be sufficient to examine Newton’s first and second laws of motion. But it will not be necessary here to look at the laws through the very eyes of Newton, although it will be useful at times to refer to Newton’s own words.
Newton’s laws stand at the threshold between philosophy and physics. We will approach Newton’s laws by considering the properties and interrelations of the general categories (or fundamental philosophical concepts) of materialist dialectics, such as matter, motion, space, time, quantity, quality, and measure, and certain specialized categories of physics associated with Newton’s laws, such as velocity and force. It is possible to pursue this investigation with less stress on the philosophical content, but the widespread influence of the operationalist viewpoint, with its emphasis on the quantitative side of the physical world, makes it desirable for the sake of clarity to draw attention to the underlying philosophical framework employed here. Let us therefore begin with a brief summary of the content of some of the most relevant categories. A more detailed discussion can be found in the book of Konstantinov et al.4
“Matter is a philosophical category denoting the objective reality which is given to man by ... and reflected by our sensations, while existing independently of them.”5 This reflection “is not a simple, not an immediate, not a complete reflection, but the process of a series of abstractions, the formation and development of concepts, laws, etc.”6
As a philosophical category, motion is the universal attribute and mode of existence of matter and embraces all changes and processes.7
Space and time are objectively real forms of the existence of matter in motion. “The concept of space expresses the coexistence and separateness of things, their extent and order of disposition in relation to one another.”8 The concept of time “characterizes the sequence of the occurrence of material processes, the separateness of the various stages of these processes, their duration and their development.”9
Matter, motion, space, and time form an inseparable unity which can he summed up as: “There is nothing in the world but matter in motion and matter in motion cannot move otherwise than in space and time.”10
The content of the other general categories will become clear as we use them. Unlike other philosophical systems (for example, those of Plato, Aristotle, Kant), there are no limits on the number of categories of materialist dialectics; their number and content develop along with the development of our knowledge of the world. Moreover, each specialized field will make use of specialized categories such as element in topology, point in geometry, material point in mechanics.
Categories in Newton’s First Law
The first law reads:
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.11
The most important categories contained in this law are: body, rest, uniform motion, right line, forces. Before discussing the dynamical aspects of the law, that is, the changes produced by the forces, let us consider the kinematical side.
Kinematics deals with the motion of material bodies (as distinct, say, from the motion of material fields). Material bodies are distinguished from fields in that they have limited extension in space. When the scale of motion is large compared with the extent of the body, it is useful to introduce the abstraction known as the material point in which the effects of the structure, including size or extent, are neglected in the motion under consideration.
In the Scholium preceding the laws of motion, but following his definitions, Newton described motion as the translation of a body from one place to another.12 Since he believed in the existence of absolute space (and time) independent of matter, he allowed for both absolute and relative motion, (In his second law he used the term “motion” in the same sense that momentum is used today.) The concept of rest was described as the continuance of the body in the “cavity [relative or absolute— E.M.] which the body fills.” Uniform motion will be discussed in detail below.
In what follows we will accept without discussion the categories of the various branches of mathematics, so that right line in Newton’s usage is simply a straight line in Euclidean geometry, but at this point of the discussion, we have not yet associated it with the geometry of real space.
Finally, we shall put off a discussion of the concept of forces until we have had more opportunity to consider the question of uniform motion.
Motion of a Material Point and the Measure of Time
Since we are ignoring any change or motion as regards the internal development of a material point, the simplest motion will be displacement (= translation) in space. In the case of Newton’s absolute space and time, one could consider the motion of a single material point. But in the case of displacement, the inseparability of matter, space, and time requires the presence of at least one other material body (or point), since spatial and temporal concepts cannot be formed from a material point, which itself has no extent and undergoes no process of change. We will call this other material body (or point) a reference body (or reference point). We can now proceed to develop the concept of the position of one material point relative to the reference body in a Euclidean space. We fix a coordinate system to the reference body and designate the position of a material point by the coordinates. It should at once be apparent that we still have to establish quantitative association with our qualitative concepts of space as an expression of the coexistence and separateness of things, and order of their disposition in relation to one another. Any property of matter displays both quantitative and qualitative characteristics, and these two categories are united through the philosophical category that Hegel called measure (similar to the role played by the term metric in topology). To establish the measure for our space we can make use of the extension of the reference body, say, by using idealized meter rods (of arbitrary number) for the reference body.
Here we come upon the problem of how to ascertain that the meter rods transported to form our coordinate system remain congruent with one another. The problem of congruence was, of course, considered by Riemann, who, as we indicated earlier, considered it possible that physical space was not Euclidean. As d’Abro noted, the type of geometry we obtain is dependent on our definition of congruence: “If we define as congruent the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry.”13 For Einstein it was not merely space that was associated with matter, but space-time. For Riemann, on the other hand, it was simply space that would be subject to the action of matter. As an assumption for our interpretation of classical mechanics, we can take the empirical fact that Euclidean geometry gives a good approximation of the spatial properties of coordinate systems constructed from physically real meter rods, which we then consider to be ideally rigid, one-dimensional, rectilinear segments. In doing so, we then impart the following modified “classical” characteristic to our physical description: the geometry of space is Euclidean, independent of mass and time, but not independent of matter.
What about time?
First of all, we need some sequence of events to establish duration or time intervals. With his assumption of absolute time, Newton never really established a theoretical basis fur the measure of time. His ultimate reference for time was empirical, that is astronomical observation (apparent tune) corrected by an astronomical equation.14 But to justify the use of astronomical observations, it is necessary to show that the physical motions observed (even if they were properly corrected by the astronomical equation) represent the “equable” flow of time “without relation to anything external” in accordance with Newton’s own definition of time.15 To do so requires the use of his three laws of motion and the law of gravitation. While Newton was able to obtain results indicating the consistency (within the limits of classical mechanics) of his assumptions with his conclusions, this measure of time is not suitable for an examination of the fundamental principles underlying classical mechanics.
Fock suggested the use of electromagnetic radiation for specifying the position of a body in space at a given instant in a fixed reference frame, as well as to establish straight lines.16 Light can be used because of the interconnections of its properties with space and time. But we should try to approach classical mechanics on the broadest possible base of general philosophical considerations and make use of as few specialized postulates of physics as possible. Therefore we are not particularly anxious to introduce processes associated with electromagnetic, strong, or weak interactions. (We already know that classical mechanics is an expression of the gravitational interaction, although we will have no need to make use of this information here.)
Thus the sequence of events that allows us to fix a concept of time must entail the simplest motion of a material point, a sequence of changes in position. The association of changes in position with changes in time for a material point leads us immediately to the specialized category of physics, velocity. But the quantitative definition of velocity as the ratio of change in position to change in time requires a quantitative measure of time. Neither velocity nor time can be elaborated quantitatively without the other. Thus the two sides of the simplest mechanical motion of a material point, that is, changes in space and time expressed through the category of velocity, constitute a dialectical unity that cannot be reduced to a simple combination of two factors, each with its own properties, as Newton had sought to do, even within the framework of nonrelativistic mechanics.
Suppose we wished to examine the consequences of using two different kinds of clocks for the investigation of the properties of velocity of a material point. Let one of these clocks be based on the frequency of the electromagnetic radiation of cesium atoms (atomic clock). For the other clock let us take the time intervals between successive beta decays from some (initially) definite amount of the phosphorus isotope P32. Our physical system thus consists of: two devices emitting time signals, a Euclidean-space coordinate system fixed to an arbitrary material body (for example, rigidly joined meter rods can serve as both the reference body and the coordinate system if we wished to consider one-dimensional motion); and a material point, the velocity of which we wish to investigate. Let us take as the definition of the quantitative side of (average) velocity the magnitude of the displacement between successive time signals. For simplicity we can assume one-dimensional motion. For a material point undergoing continuous displacement of an unspecified character, we would obtain two different sets of velocity determinations when the two clocks are used concurrently. We can define uniform motion to be that motion which gives equal displacement between time signals. This definition, of course, does not establish the possibility of such motion actually occurring with the use of either clock.
If the material point were undergoing random motion, then both clocks would give us sets of random magnitudes for the velocities. In both cases the probability of obtaining uniform motion in either set would tend to zero as the number of time intervals increase (assuming arbitrary accuracy of the position determination). But Newton’s first law postulates uniform motion under certain idealized conditions. Under these conditions, uniform motion is not just one of an infinite number of quantitatively different, but qualitatively similar, possible sets of velocities determined for the moving material point over some number of time intervals. In other words, Newton’s first law can be regarded, in effect, as positing the negation of randomness of motion through the idealization of uniform motion. The idealization involves the absence of any other material bodies (or fields), or at least, any interconnection (or interaction) between them and the material point in uniform motion or the reference body. On the other hand, the nature of the interconnection between the reference body and the material point has been established by the way the concept of motion of a material point was introduced.
The qualitative character of uniform motion enables us to make a choice among natural clocks in the absence of other material bodies or forces arising out of the interactions of the material point with other matter. The clocks that can serve as a correct measure of time in classical mechanics are those that result in intervals corresponding to equal displacements. In fact, Newton’s first law establishes our isolated system of a noninteracting material point and noninteracting reference body as precisely such a natural clock. In other words, Newton’s first law provides the measure for time in classical mechanics.
With the aid of the second and third laws and the law of gravitation, it then becomes possible to establish theoretically the connection of time intervals associated with uniform motion to those associated, say, with astronomical time.
The preceding discussion somewhat parallels that of d’Abro, who wrote:
Now both Galileo and Newton recognised as a result of clock measurements that approximately free bodies moving in an approximately Galilean frame described approximately straight lines with approximately constant speeds. Newton then elevates this approximate empirical discovery to the position of a rigorous principle, the principle of inertia, and states that absolutely free bodies will move with absolutely constant speeds along perfectly straight lines, hence will cover equal distances in equal times. When expressed in this way as a rigorous principle, the space and time referred to are the absolute space and time of Newtonian science. In other words, it is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time.17
It should, however, be noted that Newton’s space and time were absolute in the sense that both were independent of matter and of each other. The concept of space which we have employed can be considered absolute only in the sense that the geometry does not depend on the distribution of mass, and, although independent of time, it is still a reflection, albeit approximate, of the properties of matter. The concept of time is now related to that of space through Newton’s first law, also as a property of matter, but again it is absolute in the sense that it does not depend on the distribution of mass.
Our analysis above started by building our spatial coordinate system from congruent meter rods. Would it not be possible to show, similarly, that the first law can be regarded as establishing a measure for space if we were to choose congruent time intervals as our starting point?
One is tempted to try to answer this question by considering which concepts of congruence arc primary: spatial or temporal. But how should we seek the answer to this question?
We would suggest that the answer lies in the logical movement of scientific theory, that is, the ascent from the abstract to the concrete.18 In this framework, the points of departure of any theory are the most general categories of philosophy. The expression of physical theory also requires the categories of mathematics. It would then appear that mathematics occupies an intermediate position between philosophy and physics, the latter being a more concrete expression of reality. And it is the geometry of space (rather than time or space-time) that enters classical mechanics as an extensively elaborated branch of mathematics.
We would therefore be inclined to regard the measure of time to be dependent on the measure of space, with the first law providing this connection.
Nevertheless, despite our attempt to introduce dialectical elements into the conceptual framework, the conceptions of space and time associated with the first law retain an absolute character. This absolute character is rooted in the assumptions that the properties of space and time, although considered to be properties of matter, are unchanging properties of matter and are unaffected by changes to which matter may be subject.
Dynamical Content of the First Law
The dynamical content of the first law arises from a second negation, a negation of some specific state of uniform motion (which can also include rest). Before continuing our discussion, however, we will have to consider a difficulty in the Motte-Cajori version of the law.
In its Latin original, the first law reads:
Corpus omne perseverare in statu suo quiescendi vel movendi unifomtiter in directum, nisi quatenus a virabus impressis cogitur statum ilium mutare.19
The two words nisi quatenus are more properly translated except in so far as rather than by the single word unless as we find in the Motte-Cajori translation given earlier in this paper. The qualification except in so far as it is compelled to change that state envisages the possibility of a second negation, the negation of a given uniform motion, however, not one that takes us back to random motion, but to a qualitatively and quantitatively determinate changed state of motion, the qualitative and quantitative aspects of the change being directly associated with the forces impressed upon the body; but the relation-ship between the change and the forces is not yet specified explicitly, which a complete expression of law-governed causality would require. The expression nisi quatenus does not go far enough to meet this requirement, but certainly goes further than the single English word unless. It will take the second law to complete this expression of causality.
In discussing Newton’s concept of inertia, Shapere makes an observation which conveys some of the sense of the moments just made about causality:
(I)t is a logical property of the notion of “cause” (or “causal explanation”), or at least of the most usual usage of that expression, that a variation in an effect requires a corresponding variation in the cause. Thus it is no accident that inertia—the inertial mass—enters into calculations in Newtonian physics not as a “force” causing or explaining the rest or uniform rectilinear motion of a body; rather it enters into other sorts of calculations entirely: fundamentally, into calculations of the impressed forces acting on the body (or, when the forces are known, into calculations of the body’s response thereto)....20
In his initial attempts to arrive at the laws of motion, Newton tried to work through a Descartes-like causality principle which also lacks a complete specification of the relationship of the cause to the effect. In axiom 100 of the Waste Book, Newton expresses what Herivel described as a general philosophical principle:21
100. Everything doth naturally persevere in that state in which it is unless it be interrupted by some external cause, hence axiom 1st and 2nd ....22
Thus Newton used this principle as justification for his first expression of the principle of inertia which he had written at some earlier time and had designated as axioms 1 and 2:
1. If a quantity once moves it will never rest unless hindered by some external cause.
2. A quantity will always move on in the same straight line not changing the determination nor the celerity of its motion) unless some external cause divert it.23
It should be noted, however, that Newton, even in the Principia, used the term “force” both as an external cause as well as an internal one. As Dijksterhuis commented, Newton had not completely abandoned Aristotle’s view that every motion required a motor,24 and still retained that concept in modified form through the vis insita or innate force of matter, which he called “inertia” or “force of inactivity.”25 This matter is of importance in connection with the discussion on the externality of causation later in this paper.
There has been much discussion about the redundancy of the first law of motion relative to the second.26 For the purposes of this discussion it is not necessary to go into this question in great detail, but a few comments would, nevertheless, be worthwhile. Let us first state the second law.
The change in motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed.27
Dijksterhuis, for example, points out that the second law, by stating that the change in motion (here, actually momentum) is proportional to the force (impulse), is only establishing a sufficient condition, but not a necessary one. The first law rules out the possibility of the momentum changing of its own accord.28 Although this observation is valid, it could lead to the conclusion that the first law would not be necessary if the second law were formulated somewhat after the fashion:
An acceleration a of a body of mass rn occurs if and only if there is impressed upon it a force F such that F = ma.
Such a viewpoint reflects a Machian predilection for equations as expressions of causality, that is, an approach which reduces causality to correlations between states.29 The problem of establishing the concept of uniform rectilinear motion, the measure of space and time, and other questions raised earlier would not be resolved by a simple quantitative relationship.
I. Bernard Cohen discussed several historical bases for Newton having stated the first and second laws of motion separately.30 Of particular relevance to the present discussion is the following basis offered by Cohen: the first law “embodies a radical departure from traditional physics to the extent that it declares motion ... to be a ‘state’ and not a ‘process’—in the Aristotelian sense”—and equates the “state” of resting with the “state” of uniform rectilinear motion. It does this, according to Cohen, without the complicating factors of the quantitative relation of force to change of motion (see the more extensive discussion by Koyré).31
By asserting that a particular form of motion of a body is a state, rather than a process, the first law postulates the theoretical possibility of an unchanging state of matter. We have pointed out that the concept of uniform rectilinear motion already involves properties of space and time that are independent of the changes of matter. To this, one may add another “unchanging essence” reflected in Newton’s laws of motion, namely the mass of the moving body.
In both Hegelian and Marxian dialectics, the term “metaphysical” is applied to philosophical outlooks which see things as potentially isolatable and with an unchanging essence (or with unchanging properties), in contradistinction to the dialectical view that all things are universally interconnected and constantly changing. As we have seen above, the direction of development of the science of mechanics which culminated in Newton’s laws clearly has the first character. Therefore, in the literature written from a dialectical viewpoint the terms “metaphysical” and “mechanistic” have become interchangeable.
In the case of an unchanging essence, no internal development or “self-determination” occurs. The sources of all change or motion must come from the outside, that is, they must be external and produce only unessential changes in the relationship of one thing to another. We will now see how this mechanistic basis of classical mechanics is reflected in a causality principle.
Mechanism and the Causality Principle
As we have seen from the entire preceding discussion, there is no way of eliminating the mechanistic character from Newton’s laws of motion, despite our ability to introduce certain dialectical elements into their interpretation today. The principal source of this mechanism appears to lie in the general philosophical principle expressed in axiom 100 of Newton’s Waste Book, to which Herivel drew attention and which we reproduced above. Herivel32 (11) and Koyré33 13] both commented extensively on similarity of language in the laws of inertia of both Newton and Descartes although the differences between them were by no means unessential. But the general philosophical principle in Newton’s axiom 100 is so close to Descartes’ formulation that it is worthwhile looking at the latter more closely, for it is an effort to express causality as a reflection of the immutability of God (whom Descartes regards as the first cause):
The first law of nature: that each single thing, by its own nature, always perseveres as far as possible in the same state, and so whatever is once moved always continues to move.
Now from this same immutability of God can be known some rules or laws of nature, which are the secondary and particular causes of the diverse motions we observe in individual bodies. The first of these is that each single thing, in so far as it is simple and undivided, and by its own nature, remains as far as possible in always the same state, and never changes except through external causes. Thus if some part of matter is square, we convince ourselves easily that it will constantly remain square, unless something arrives from elsewhere to change its shape.34
Descartes’ use of change of shape as an illustration indicates that he meant more than displacement, say, of a material point. By his qualification “in so far as it is simple and undivided,” Descartes excluded from consideration any internal causes. But Newton, according to his later writings, believed that matter was formed “in solid, massy, hard, impenetrable, moveable particles ... so very hard, as never to wear or break in pieces: no ordinary power being able to divide what God himself made one in the first creation”.35 So he was able to dispense with Descartes’ qualification and the only causes were therefore clearly the external ones. We note, however, that this general philosophical principle expressed in Newton’s axiom 100 is still an incomplete expression of a causal principle in the sense of law-governed causality, as we shall now show.
In seeking an expression of the modern causal principle, Bunge arrived at the following:
If C happens, then (and only then) E is always produced by it.36
According to Bunge, this formulation reflects the concepts usually regarded as essential components of the category of causality: conditionalness, uniqueness of the bond, one-sided dependence of the effect upon the cause, invariability of the connection, and the productivity or genetic nature of the link, As Binge further demonstrates, externality of the cause is an essential feature of the causality principle.37 The quantitative as well as the qualitative specification of changes in state requires not only the first law, but also the second. These two laws of Newton taken together constitute an expression of this causal principle. The description of processes unfolding in accordance with such a causal principle is what has come to be known as the mechanistic viewpoint. Its chief features are the absence of internal development, externality of the cause, and a tearing away of the bonds of universal interconnection for a static description of the states.
It is important here to distinguish between the mechanistic concept of causality as expressed in Bunge’s formulation and the general philosophical category of causality which is an expression of necessary interconnections arising during the transformation of one form of motion of matter into another. All expressions of causality entail some degree of severance of universal interconnections. The dialectical nature of all processes of change involve the “all-sidedness and all-embracing character of the interconnections of the world, which is only one-sidedly, fragmentarily and incompletely expressed by causality.38
We can now sum up the major points made above.
We have seen, as Mach did long ago, that Newton’s laws do not require a concept of space and time that are absolute in the sense that they are entirely independent of matter. The Euclidean geometry used to describe the spatial relationships between material bodies is an approximation to the properties of the space associated with the bodies. Newton’s first law is then an (almost complete) expression of causality in mechanics and serves to establish a measure of time. The time so established is likewise an approximation, not only because it is based on the use of Euclidean geometry, but also because it is independent of the mass and disposition of the material bodies with which it is associated. Thus space and time still retain an absolute character, although different from that imparted to them by Newton.
The first law of Newton has a philosophical and theoretical content distinct from the second law and cannot be considered redundant in relation to the second. Taken together, the two laws constitute a complete reflection of the causality principle and impart an indelible mechanistic character to classical mechanics.
I wish to express my gratitude to. Professors John Earman and Alan F. Shapiro for helpful discussions.
School of Physics and Astronomy
University of Minnesota
1 . G. W. F. Hegel, Naturphilosophie. Quoted in F. Engels, Dialectics of Nature, ed. and trans. Clemens Dutt (New York: International Publishers, 1940), 343
2. Engels, Dialectics of Nature, 327.
3. A. Grunbaum, Philosophical Problems of Space and Time (New York: Knopf, 1963), chaps. 1 and 14.
4. F. V. Konstantinov et al., The Fundamentals of Marxist-Leninist Philosophy (Moscow: Progress Publishers, 1974).
5. V. I. Lenin, Materialism and Empirio-Criticism, vol. 14 of Collected Works (Moscow: Progress Publishers, 1972), 130.
6. V. I. Lenin, Philosophical Notebooks, vol. 38 of Collected Works, (Moscow. Progress Publishers), 182.
7. Konstantinov et al., The Fundamentals of Marxist-Leninist Philosophy, 80–85.
8. Ibid., 85.
10. Lenin, Materialism and Empirio-Criticism, 175.
11. “The Motion of Bodies,” in vol. 1, book 1, of Sir Isaac Newton’s Mathematical Principles, of Natural Philosophy, and His System of the World, ed. F. Cajori (Berkeley: University of California Press, 1946), 13.
12. Ibid., 7.
13. A. d’Abro, The Evolution of Scientific Thought from Newton to Einstein (New York: Dover, 1927), 60 (cf. also 48).
14. Newton, “The Motion of Bodies,” 7–8.
15. Ibid., 6.
16. V. Fock, The Theory of Space, Time and Gravitation. 2nd rev. ed. (New York: Macmillan, 1964), 9–10.
17. d’Abro, Evolution of Scientific Thought, 75.
18. In his discussion of the method of political economy, Marx wrote: “It seems to be correct to begin with the real and the concrete, with the real precondition.... However, on closer examination this proves false.... The concrete is concrete because it is the concentration of many determinations, hence unity of the diverse. It appears in the process of thinking, therefore, as a process of concentration, at a result, not as a point of departure, even though it is the point of departure in reality and hence also the point of departure for observation and conception. Along the first path the full conception was evaporated to yield an abstract determination, along the second, the abstract determinations lead toward a reproduction of the concrete by way of thought. In this way Hegel fell into the illusion of conceiving the real as the product of thought concentrating itself, probing its own depths, and unfolding itself out of itself by itself, whereas the method of rising from the abstract to the concrete is only the way in which though appropriates the concrete, reproduces it as the concrete in the mind.” “Introduction to the Critique of Political Economy,” in Grundrisse (New York: Vintage, 1973), 100–101.
19. “An Historical and Explanatory Appendix,” in vol. 2 of Sir Isaac Newton’s Mathematical Principles, of Natural Philosophy, and his System of the World, ed. F. Cajori (Berkeley: University of California Press, 1946), 644.
20. D. Shapere, “The Philosophical Significance of Newton’s Science.” in The Annus Mirabilis of Sir Isaac Newton, 1666–1966, ed. R. Palter (M.I.T. Press, 1970), 288. Cf. also Texas Quarterly 10, no. 3 (1967).
21. J. Herivel. The Background to Newton’s “Principia” (Oxford: Oxford University Press, 1965), 45.
22. Ibid., 153.
23. Ibid., 141.
24. E. J. Dijksterhuis, The Mechanisation of the World Picture (Oxford: Oxford University Press, 1969), 466. Cf. also J. E. McGuire, “Comment” on contribution by I. Bernard Cohen, in The Annus Mirabilis of Sir Isaac Newton, 1666–1966, 186.
25. Newton, “The Motion of Bodies,” 2
26. See, e.g., J. Earman and M. Friedman, “The Meaning and Status of Newton’s Law of Inertia and the Nature of Gravitational Forces.” Philosophy of Science 40 (1974): 329–59.
27. Newton, “The Motion of Bodies, p.13.
28. Dijksterhuis, The Mechanisation of the World Picture, 474.
29. For a detailed critique of this viewpoint see, e.g., M. Bunge, Causality. The Place of the Causal Principle in Modern Science (Cambridge: MA: Harvard University Press, 1959).
30. I. Bernard Cohen. “Newton’s Second Law and the Concept of Force in the Principia,” In The Annus Mirabilis of Sir Isaac Newton, 1666–1966.
31. A. Koyré, Newtonian Studies (Harvard University Press, 1965).
32. Herival, The Background to Newton’s “Principia,” 11.
33. Koyré, Newtonian Studies, 13
34. R. Descartes. Principia Philosophiae, Pars Secunda. Quoted in A. Gabbey, “Force and Inertia in Seventeenth Century Dynamics,” Studies in History and Philosophy of Science 2, no. 1, (1971): 56–57n134.
35. Newton, “An Historical and Explanatory Appendix,” 638.
36. Bunge, Causality, 47.
37. Cf. Bunge, Causality, chap. 7.
38. Lenin, Philosophical Notebooks, 159. Cf. also E. Marquit, Revolutionary World 23–25 (1977): 171–79.