Whitehead and Newton on Space and Time Structure
by Robert R. Llewellyn
Robert R. Llewellyn is Assistant Professor of Philosophy at Southwestern at Memphis, Memphis, Tennessee. The following article appeared in Process Studies, pp. 239-258, Vol. 3, Number 4, Winter, 1973. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.
The construction of a theory of space-time structure is clearly a fundamental concern of Alfred North Whitehead in his early writings in the philosophy of natural science (see, for example, the "Prefaces" to PNK, CN, and R). There are important modifications in Whitehead’s theory in his later, more metaphysical, writings; but these modifications only serve to emphasize that the development of such a theory remains a major task in his attempts at philosophical analysis (see especially chapters IV and VII in SMW and part IV in PR).1 In general, Whitehead constructs a theory that is reactionary in its analysis when compared with the theories of space-time structure in the special theory of relativity (STR) and in the general theory of relativity (GTR),2 and that is in opposition to the theory of absolute space and absolute time in the Newtonian cosmology (see PNK 1-8; and PB part II, chapters II, III, and IV). In this article, the extent of ‘Whitehead’s opposition to the Newtonian theory is investigated. The focus of the investigation is on Whitehead’s analysis in Process and Reality, although important features of this analysis are derived from his earlier analysis in The Principle of Relativity.
Whitehead makes quite explicit the fact that his theory of space-time structure differs in two major respects from the Newtonian theory. First, the theory of space-time structure in the Whiteheadian cosmology is a relational theory as opposed to the "receptacle-container" theory in the Newtonian cosmology (PR 108f, 441). Space-time structure concerns relations between and sustained by the actual occasions of the universe; it is not an actual thing in which the real events of the world occur. Second, the extensive continuum, of which spatiotemporal extensiveness is a more specific determination, is a "real potential" factor of thc universe in the Whiteheadian cosmology as opposed to absolute space and absolute time continua as real and actual things comprising the universe in the Newtonian cosmology (PR 113f; cf. 101-06). The coming-to-be of present actual occasions actualizes -- specifically, spatializes and temporalizes -- an extensive order for the universe. These two points will be referred to again as this investigation concludes.
In as equally clear a manner Whitehead acknowledges that he incorporates in his cosmology certain of Newton’s notions about space and time structure. Three notions are mentioned. First, the quanta of spatiotemporal extensiveness, or the "regions," correlative to actual occasions and unique to each actual occasion in the Whiteheadian cosmology are to be identified with the absolute places of material objects at absolute instants of time in the Newtonian cosmology (PR 113; cf. 108f). Second, the nature of an actual occasion is such that the actual occasion is constituted by its relations, including extensive relations, with other actual entities; it, therefore, cannot be the actual occasion that it is and be "somewhere else" spatially and/or temporally. An actual occasion never moves, according to the Whiteheadian cosmology. This parallels the notion that absolute places and absolute times are immutable and immobile, according to the Newtonian cosmology. Newton’s absolute places and absolute times are places and times for themselves as well as for material objects, and thus it is not meaningful to say of them that they are moved out of themselves (PR 113; cf. 109f). And third, for each actual occasion there are other actual occasions which, because they occur in causal independence of that actual occasion, are its contemporaries. Within this nexus of contemporary actual occasions are multiple "durations," defined by the characteristic that any two actual occasions comprising it are contemporaries. A duration is thus a temporal cross-section of the universe, according to the Whiteheadian cosmology. This is analogous to the notion of the immediate present state of the universe, according to the Newtonian cosmology; with the exception that whereas in the Whiteheadian cosmology there are multiple durations associated with an actual occasion one of which -- the "presented" duration -- may have special significance, in the Newtonian cosmology there is one and only one immediate present state of the universe (PR 188-92; cf. 486-89).
The investigation below into certain features of the Newtonian and the Whiteheadian theories of space and time structure is an argument that other important and essentially Newtonian notions about space and time structure are fundamental to Whitehead’s theory. Specifically: the Whiteheadian cosmology embraces the notion of a uniform metric structure for the space-time continuum, that is independent of the material objects, commonly said to be "in" space-time, and especially that is independent of the material objects appropriated as standards of spatiotemporal measurement (see, for example, JR 58-59). The Newtonian cosmology sets forth the potions of absolute space "remaining always similar and immovable in its own nature" and of absolute time "flowing equably of itself and from its own nature," and the claim that the measurement of lengths in absolute space and of durations in absolute time is independent of any "sensible and external measures’ of them (PNP 6-8). There seems to be a clear and definite parallel here between the two cosmologies.
Simply noting this parallel is not sufficient to establish Whitehead’s theory as fundamentally Newtonian in important and essential aspects. However, it can be argued that Whitehead’s theory is fundamentally Newtonian if, in addition to noting the above parallel, it can be shown that Whitehead and Newton share a common understanding of the status and function of such a uniform and independent space and time structure in scientific inquiry. In the investigation to follow it is shown that both Whitehead and Newton maintain that such a space and time frame of reference is presupposed by any satisfactory scientific analysis of the physical nature of the universe. This common understanding of the status of space and time structure is reflected in similar treatments by Newton of gravitational forces and by Whitehead of what he terms "impetus" (a concept having gravitational and electromagnetic significance); both treat these as real physical phenomena against a framework of a uniform and independent space and time structure.
Before this investigation begins, it is to be noted that the immediate point of interest is not the soundness of the respective arguments that either Whitehead or Newton offers in defense of his position;3 the objective is to delineate within Whitehead’s theory of space-time structure what frequently is an overlooked but fundamentally Newtonian position.4
The argument that is presented here is not that Whitehead’s theory is simply a return to Newton’s theory. A clarification of a third difference between Whitehead’s theory and Newton’s theory, alluded to above, may serve to bring out the contrast of the two theories and yet show the fundamentally Newtonian position in Whitehead’s theory with which this investigation is concerned. In the Whiteheadian cosmology there are multiple durations associated with a particular actual occasion. Every other actual occasion included in these durations is a contemporary of that actual occasion; however, it is not necessarily the case that any two other actual occasions in these durations are themselves contemporaries (see PR 188-92). In this way the Whiteheadian cosmology incorporates the principle of the relativity of simultaneity, actually the principle of the causal independence of contemporary events, characteristic of modern theories of relativity. The Newtonian cosmology, in view of its affirmation of a unique time continuum, posits one and only one immediate present state of the universe, any two points of which are to represent places where contemporary events happen. The contrast of the two theories is clear on this point.
Whitehead’s theory is further articulated by the notion of the presented duration. Perception in the mode of presentational immediacy defines for a particular actual occasion one of the multiple durations associated with it, and this is its "presented" duration (PR 191f, 488-90).5 The actual occasions that might be said to actualize the presented duration are not perceived; however, the extensive relations within the presented duration are perceived (PR 95-97, 188). And, it is part of Whitehead’s theory that these extensive relations are such that a "systematic" geometry is sustained by them (PR 194-96, 497-99). Whitehead seems to mean by a systematic geometry that it is a geometry representing a structure of uniform curvature yet not a geometry identified as one of the particular uniform metric geometries.6 Further, this systematic geometry, exemplified within the presented duration, is taken to be a defining characteristic of the "geometrical" society, a basic member of a hierarchy of societies comprising the present cosmic epoch (PR 148-59K Thus Whitehead explains that the perceptive mode of presentational immediacy is of importance, for it exhibits a ". . . complex of systematic mathematical relations which participate in all the nexuses of our cosmic epoch, in the widest meaning of that term." And, "it is by reason of this disclosure of ultimate system that an intellectual comprehension of the physical universe is possible. There is a systematic framework permeating all relevant fact" (PR 498f). It is argued, as a result of the present investigation, that this is a fundamentally Newtonian position: there is a uniform and independent metric structure of the space and time continuum.
II. Space and Time Structure and Physical Analysis
Newton: It is evident that the "axioms," or "laws of motion," as stated by Newton, presuppose some determination of a spatial frame of reference and a temporal frame of reference prior to the use of those laws to analyze the physical motion of material objects:
LAW I: Every body continues in a state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
LAW II: The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. (PNP 13. Italics mine.)
It is not that Euclidean geometry fixes the meaning of "right line" for Newton that is important for present purposes; rather, it is that the notion of a "state of rest" or a "state of motion in a right line" presupposes some spatial reference, and "uniform motion" and "change of motion" presuppose some temporal reference.
More to the point, Newton’s "Scholium" which introduces the notions of "absolute, true, mathematical" space and time, and "relative, apparent, common" space and time (PNP 6-12), makes clear that absolute space and absolute time continua are thought to be necessary for a satisfactory theory of dynamics, that is, a theory of the forces which determine •the motion of material objects.7 The main idea in Newton’s position is that not all physical frames of reference are suitable for satisfactory analysis of the motion of material objects; in fact, no physical frame of reference is completely suitable for this purpose. A review of Newton’s position in the "Scholium" clarifies the status and function of absolute space and its relation to a theory of dynamics.
Newton defines "absolute motion" and "relative motion" using his previously introduced notions of absolute space and absolute time (PNP 7-8). All relative motions are said to presuppose absolute motions (PNP 9). Relative motions are readily perceived since such motion involves the translation of a material object relative to some perceived system of material objects. Absolute motions, however, present a major problem; they are not readily perceived since absolute space, relative to which such motion takes place, is not accessible to the senses (PNP 8). It is by reference to the causes and the effects of motions that absolute and relative motions can be distinguished and consequently that the notion of absolute space can be determined: ". . . we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes, and effects" (PNP 8).
Newton’s position is detailed in two essential steps:
The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body; but relative motion does not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be preserved, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Thus, true motion by no means consists in such relations. (PNP 10)
Absolute motion is not generated or altered except by a force that is exerted on the material object in motion. However, it is not possible to determine when a force is acting on a material object by simply noting the change in motion of that material object relative to a system of other material objects. Uniform rectilinear motion relative to some physical system is the classic example; such motion may be altered or terminated altogether by imparting motion not to the material object said to be in motion but to the physical system that is used as a frame of reference. The status of the material object relative to absolute space -- whether at rest or in motion -- cannot be determined. Rotational or circular motion (and accelerated motion in general), in contrast, does present the possibility, according to Newton, of determining absolute motion:
The effects which distinguish absolute from relative motion are the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. (PNP 10)
Rotational or circular motion gives rise to, for example, the familiar centrifugal force, the presence of which may be recognized without reference to changes in motion relative to any surrounding system of material objects.8 Newton illustrates the significance of this step in his position with his famous "bucket experiment" (PNP 10f).9
The details of the "bucket experiment" need not be specified for present purposes; the experiment uses a bucket filled with water, and the bucket is suspended from a rope so that when the rope is twisted, the rope is the axis of rotation for the bucket and eventually for the water. The point of this "experiment," as Newton understands it, is as follows :10 The shape of the surface of the water in the bucket -- either flat or concave -- is independent of the motion of the water relative to the sides of the bucket -- whether at rest or rotating. The concave shape of the surface of the water, considered by Newton as a deformation of its normal shape, is the effect of a force that is exerted on the water. The effect of a force exerted on a material object is to give it an accelerated motion; thus, the force that is exerted on the water gives rise to an accelerated motion of the water. Since the shape of the surface of the water -- the observed phenomenon showing the presence of a force -- is independent of the motion of the water relative to the sides of the bucket, the accelerated motion must also be independent of the relative motion of the water and the bucket. It is a motion that is accelerated relative to an absolute frame of reference, namely absolute space.
Recapitulating the crucial point: if all motions were relative motions, then the sides of the bucket would be a perfectly suitable frame of reference in terms of which to analyze the motion of the water. However, the concave surface of the water cannot be accounted for simply in terms of the relative motion of the water and the bucket. Newton maintains that the deformation in the shape of the surface of the water must be due to forces that give rise to accelerated motion relative to absolute space.
Now, given any system of material objects that might be considered as a frame of reference in a theory of dynamics, that system is either at rest, or in motion with uniform velocity, or in accelerated motion relative to absolute space. It is impossible to determine mechanically whether any such system is at rest or in motion with uniform velocity relative to absolute space; there are no forces present the effects of which reveal the status of the system relative to absolute space.11 Frames of reference at rest or in motion with uniform velocity relative to absolute space (and hence in motion with uniform velocity relative one to another) are the "inertial frames" of classical Newtonian dynamics. Any system of material objects in accelerated motion relative to absolute space manifests forces -- the "inertial forces" -- that vary in accordance with the degree of acceleration; the centrifugal force discussed above is an example. Therefore, no system of material objects may serve as a frame of reference and be completely suitable for the purpose of analyzing in terms of laws the motions of material objects; for such laws of motion should be stated in such a manner that they are unaffected by the peculiar absence or presence of inertial forces in particular physical frames of reference. The laws of motion are to be stated, then, in terms of the one privileged, or "absolute," frame of reference, that is, absolute space. The uniformity of motion along a path in absolute space, or the lack of it, would require reference in like manner to the one privileged, or "absolute," frame of reference, that is, absolute time.
Thus, for Newton, absolute space and absolute time are presupposed by a theory of the dynamics of material objects in motion. It is only in reference to such continua that the motion of material objects can be analyzed without reference to the peculiar features of particular systems of material objects that might be taken a frames of reference.12
Newton himself acknowledges that he is not completely satisfied of material objects that might be taken as frames of reference.13
This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes use to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances. . . . But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I feign no hypotheses . . . . to us it is enough that gravity does really exist, and act according to the laws which we have explained. (PNP 546f) 14
Newton does not believe that gravity is an essential property of matter, but he also denies that it can be a force that acts at a distance.15 Newton is clear on one point with regard to his analysis of gravitation; he is concerned with the mathematical treatment of gravitation, not with the nature or causes of gravitation or the manner in which gravitation acts but with the conditions under which gravitation acts (PNP 5f, 192). Thus, for Newton, the gravitational force due to the presence of some material object is analyzed as superimposed on an inertial frame of reference. Whenever a second material object is considered, then relative to an inertial frame of reference the two material objects attract each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them. Though the presence of a gravitational force manifests itself by imparting an acceleration to a material object, the theories of inertial and gravitational forces are treated as distinct due primarily to the belief that the gravitational force is a special datum apart from the motion of material objects. By analyzing the action of gravitational forces against the background of an inertial frame of reference Newton is able to measure and to describe that action in terms of coordinates assigned to an inertial frame. Recalling that an inertial frame of reference is either at rest or in motion with uniform velocity relative to absolute space, and that the addition of a velocity to an acceleration does not affect the acceleration and hence would not affect the action of gravitational forces (accelerations due to gravity), Newton’s law of gravitation is, in effect, formulated relative to absolute space.
Newton’s theory of space and time structure is more detailed than is indicated in the preceding analysis; for instance, there is a theological dimension to Newton’s theory.16 For present purposes, however, only one further notion concerning the nature of absolute space and absolute time in Newton’s theory is of interest. Absolute space and absolute time have natures of their own, that is, extensional features which are independent of any external means of measurement as for example by measuring rods and pendulums:
I. Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.
II. Absolute 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 spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. (PNP 6)
What Newton wants to claim here is that absolute space and absolute time have intrinsic structures in terms of which measures such as equal lengths and equal durations may be determined. Physical measuring standards, such as measuring rods and pendulums, simply would function to reveal these structures of absolute space and absolute time which contain those standards. The difficulty, which Newton recognizes, is that there may be no physical motion whereby absolute time is accurately measured (PNP 8); and there may be no material object at rest in absolute space in reference to which absolute positions are determined (PNP 8). Nevertheless, Newton maintains that, for example, absolute time may be distinguished from relative time "by the equation or correction of the apparent time" (PNP 7f). His point is this:17 If the times marked out by (1) a carefully constructed pendulum and (2) the motion of the sun across the sky (the solar day) were both to be used as temporal frames of reference for scientific inquiry, then it would be discovered that certain physical phenomena would manifest unsuspected and unusual seasonal variations, for example in the motions of Jupiter’s satellites (and in the motion of the pendulum), if the solar day is taken as the standard but not if the pendulum is taken as the standard. Thus, the motion of the pendulum is taken as the standard, as a more suitable "approximation" to the uniform time flow of absolute time, and some physical explanation is then sought to account for the seasonal variation of the solar day relative to this standard.18 Were standards other than the pendulum chosen as operating "more uniformly," then the pendulum would be abandoned in favor of a "more uniform approximation" to absolute time.
Whitehead: The presence in Whitehead’s position of important notions about space and time structure which are essentially Newtonian becomes clear as one investigates the meaning of such representative claims as the following:
The structure [of the continuum of events] is uniform because of the necessity for knowledge that there be a system of uniform relatedness, in terms of which the contingent relations of natural factors can be expressed. Otherwise we can know nothing until we know everything. (H 29)
Unless we start with some knowledge of a systematically related structure of space-time we are dependent upon the contingent relations of bodies which we have not examined and cannot prejudge. (R 59)
[The perceptive mode of presentational immediacy] exhibits that complex of systematic mathematical relations which participate in all the nexuses of our cosmic epoch, in the widest meaning of that term. . . .
It is by reason of this disclosure of ultimate system that an intellectual comprehension of the physical universe is possible. There is a systematic framework permeating all relevant fact. By reference to this framework the variant, various, vagrant, evanescent details of the abundant world can have their mutual relations exhibited by their correlation to the common terms of a universal system. . . . The discovery of the tnie relevance of the mathematical relations disclosed in presentational immediacy was the first step in the Intellectual conquest of nature. Accurate science was then born. Apart from these relations as facts in nature, such science is meaningless. (PR 498f)
The immediate issue, for Whitehead, concerns a point of opposition to the Einsteinian formulation of GTR. Whitehead offers an alternative formulation and claims on behalf of his formulation that it can account for all experimental results accounted for by the Einsteinian formulation but that it represents a different interpretation of these results in terms of a more adequate concept of nature (PNK vi; CN vii, 182; IS 125-35).18The major theoretical difference between the two formulations is that whereas in the Einsteinian formulation the metric structure of the space-time continuum is variable from point to point and in differing directions (that is, heterogeneous and nonisotropic), in the Whiteheadian formulation the metric structure of the space-time continuum is uniform from point to point and in differing directions (that is, homogeneous and isotropic). The metric structure is dependent on, according to the Einsteinian formulation, and is independent of, according to the Whiteheadian formulation, the distribution of matter in the universe. It is this uniform and independent metric structure of the space-time continuum that Whitehead refers to as the "system of uniform relatedness" or the "systematic framework" in the quotations above. The uniformity of metric structure does not limit the geometrical description of this structure to a unique geometry. Whitehead did hold in his writings in the philosophy of natural science that Euclidean geometry provided the simplest analysis for the purposes of scientific inquiry:
our experience requires and exhibits a basis of uniformity, and in the case of nature this basis exhibits itself as the uniformity of spatio-temporal relations. This conclusion entirely cuts away the casual heterogeneity of these relations which is the essential of Einstein’s later theory. It is this uniformity which is essential to my outlook, and not the Euclidean geometry which I adopt as lending itself to the simplest exposition of the facts of nature. I should be very willing to believe that each permanent space is either uniformly elliptic or uniformly hyperbolic, if any observations are more simply explained by such a hypothesis. (R v; see IS 134)20
"[O]ur experience requires and exhibits a basis of uniformity, and in the case of nature this basis exhibits itself as the uniformity of spatio-temporal relations." Whitehead’s "method of extensive abstraction" is used not only in his early writings in the philosophy of natural science but also in his later, more metaphysical, writings to abstract from the complexity of the relations which comprise the datum of sense-perception and to isolate by a conceptual analysis those relations which express a uniform metric structure, that is, to "exhibit" a basis of uniformity in nature.21 It is the sense in which this uniformity is "required" that is the crucial point for further investigation.
Whitehead considers this uniformity a requirement for measurement in the space-time continuum;22 and thus, as a presupposition of any physical analysis that would be considered scientific.
Whitehead’s position involves the following considerations. All experiments that might serve as bases for the construction of a physical theory or that might serve as tests for the confirmation of a physical theory are subject to the demand that standard conditions prevail or that suitable correction factors be introduced to ensure the consistency and the comparison of the experimental results. Otherwise, the experimental results would be one-time reports with no significance beyond isolated experiments, certainly not beyond the domain of the peculiar conditions that do prevail in the experiments Also, were there not an assumption of standard conditions, it would follow that theories would be constructed and confirmed with reference only to peculiar conditions prevailing in particular areas where the experimentation takes place (PR 194f, 499, 502f).
The establishment of standard conditions, or the introduction of correction factors, is particularly important to ensure the uniform operation of measuring standards (for example, measuring rods and pendulums) for it is generally in terms of measured results that the various experiments are evaluated for consistency and are compared. A presupposition of practical measurement is that a specified measuring rod, for example, is rigid; that is, that the rod does not undergo deformation in its length as it is transported from point to point or in differing directions during the process of measurement. In selecting a measuring rod, care is exercised to eliminate or to correct for thermal, electric, electromagnetic, and other physical influences which would result in a dependence of the rod’s length on its particular physical composition; that is, the influence of the so-called "differential" forces is eliminated. This may be accomplished by establishing standard conditions under which measurement is to take place or by introducing correction factors in the results obtained with the measuring rod.
Standard conditions may be shown to prevail in the space-time continuum, or the need for correction factors indicated, by a simple test. Select a measuring rod and construct a second measuring rod of a material differing from the first but otherwise identical; place the two rods together so that their endpoints coincide; and move the pair of rods about in the space-time continuum. If they coincide at every point and in every direction, standard conditions prevail, and the first measuring rod is not subject to deformation in its length due to its particular physical composition. If they fail to coincide, correction factors must be introduced to correct for deformation in its length due to its particular physical composition. The measuring rod is thus either free of the effects of the "differential" forces, or may be corrected to counterbalance the effect of such forces.23
The question remains, even after "differential" forces have been taken into consideration, whether measuring standards remain self-congruent under transport from point to point and through a lapse of time. Whitehead disavows the thesis of geochronometric conventionalism, which maintains that this self-congruence is a matter of stipulation and affirms that this self-congruence must be established by referring the measuring standard to a uniform metric structure:
Measurement depends upon counting and upon permanence. The question is, what is counted, and what is permanent? The things that are counted are the inches on a straight metal rod, a yard-measure. Also the thing that is permanent is this yard-measure in respect both to its internal relations and in respect to some of its extensive relations to the geometry of the world. (PR 499f, italics mine; see 501f.)24
The uniform metric structure would serve as a framework in terms of which the self-congruence, or the lack of it, of a measuring standard could be determined. Once this self-congruence is determined, the measuring rod simply makes evident in a convenient manner the metric structure of the events with which it is compared (R 58).
The importance of the uniform metric structure to Whitehead is underscored by contrasting it briefly with the variable metric structure in the Einsteinian formulation of GTR. Suppose that a measuring rod is selected, and that this definition of "the shortest distance," or "a straight line" is given: "that path along which the measuring rod, under standard conditions or properly corrected, is laid down the least number of times." Utilizing the measuring rod, two rectilinear triangles are constructed in diverse regions with corresponding sides having the same lengths. In a space-time continuum of variable metric structure, the angle-sum of the interior angles of those two rectilinear triangles will generally not be equal, contrary to expectations derived from fundamental congruence theorems in the Euclidean geometry of a plane surface for example.25 The failure of the interior angle-sums of the two rectilinear triangles to be equal suggests that the straight lines that arc the respective sides of the two triangles are different. In light of the definition of "a straight line," however, this would mean that in diverse regions of the space-time continuum under consideration the measuring rod, despite the prevailing standard conditions or the introduction of correction factors, undergoes deformation in its length as it is moved from point to point or in differing directions in the space-time continuum.26 The extent of the deformation, according to the Einsteinian formulation, is dependent on the distribution of matter in the vicinity of the measuring rod. In a space-time continuum of uniform metric structure, the angle-sum of the interior angles of these two rectilinear triangles will be equal. The equality of the lengths of the respective sides and the equality of the interior angle-sums suggests that it would be possible to "move" one triangle out of its region and to transport it, without disturbing its spatial configuration, and superimpose it on the second triangle or bring the two triangles into coincidence. In light of the definition of "a straight line," however, this would mean that in diverse regions of the space-time continuum under consideration the measuring rod, under prevailing standard conditions or the introduction of correction factors, is not deformed in its length as it is moved from point to point or in differing directions in the space-time continuum. The measuring rod may be "moved freely" in the space-time continuum without deformation. This feature of a space-time continuum of uniform metric structure, which is familiar because of its association with fundamental congruence theorems such as in the Euclidean geometry of a plane surface, is not unique to Euclidean geometry. Two rectilinear triangles constructed on the surface of a sphere so that their corresponding sides have the same length represent a two-dimensional analogue of a space-time continuum of uniform metric structure having a Riemannian geometric structure. 27 Geometries describing the uniform metric structure of space-time continua are frequently designated "congruence geometries" since they retain the traditional concept of congruence as involving superimposition, or embody the axiom of "free mobility."
Now, if measurement can be significant only if the measuring rod is free of all deforming influences, not simply those attributed to the influence of "differential" forces, then given a space-time continuum of variable metric structure, no set of conditions can be specified to ensure that the measuring rod does not undergo deformation in its length as it is transported from point to point during the process of measurement. And this is one of Whitehead’s objections to the Einsteinian formulation of GTR:
it must be remembered that measurement is essentially the comparison of operations which are performed under the same set of assigned conditions. If there is no possibility of assigned conditions applicable to different circumstances, there can be no measurement. . . . For this reason I doubt the possibility of measurement in space which is heterogeneous as to its properties in different parts. I do not understand how the fixed conditions for measurement are to be obtained . . . . But Einstein’s interpretation of his procedure postulates measurement in heterogeneous physical space, and I am very skeptical as to whether any real meaning can be attached to such a concept. (IS 134)
In the absence of a uniform frame of reference in terms of which the self-congruence of the measuring rod can be determined for all regions in the space-time continuum, Whitehead holds that the peculiar congruence behavior of that rod must be known for all regions in the space-time continuum, and this seems not to be possible.28
The basis of uniformity that Whitehead maintains is required by scientific inquiry -- particularly for spatiotemporal measurement -- is not only a uniform system of relations but also is an independent system of relations (R 81; cf. PR 192-94). The sense in which it is independent will be specified below; it is sufficient to note here that it is not a substantial independence. The reason why Whitehead believes that it must be independent is that only if it is a system of relations independent of the changing, variable, physical characters of material objects can it serve as a frame of reference in terms of which the self-congruence of material objects under transport -- which makes them suitable as measuring standards -- can be determined.
The use to which Whitehead puts this uniform and independent metric structure is exemplified in an analysis of what he calls "impetus":
I start from Einstein’s great discovery that the physical field in the neighborhood of an event-particle should be defined in terms of ten elements... According to Einstein such elements merely define the properties of space and time in the neighborhood. I interpret them as defining in Euclidean space a definite physical property of the field which I call the "impetus." (IS 135, italics mine; see also CN 181f; R 71; PR 506f)
The "physical field" in the neighborhood of an event-particle, an event approximating to a point without spatial or temporal extension, is a means of expressing the complete sphere of influence due to the presence of that event and its specific character; in other words, the actual occasion that occurs at that point as it were. For both Einstein and Whitehead, this physical field is expressed by a mathematical function which represents physical magnitudes such as the density of matter at a point, the internal stresses of matter at that point, etc. According to the Einsteinian formulation of GTR, the physical magnitudes so represented define the metric structure of the space-time continuum, and fields of gravitational force are interpreted as inextricably intertwined with this structure.29 According to the Whiteheadian formulation of 0TH, the physical magnitudes so represented define a physical characteristic, the "impetus," against the background of a uniform metric structure.30 In effect, Einstein incorporates fields of gravitational force into a theory about the metric structure of the space-time continuum, whereas Whitehead retains such fields as special physical factors constitutive of the physical field associated with an event.
Whitehead’s position, as investigated above, is reflected in his explication in Process and Reality of the physical and geometrical order of nature in terms of "a hierarchy of societies" (PR 147-50, 506-08). Basically, a "society" is a grouping of events which manifest a common characteristic, the presence of that characteristic being guaranteed by the relations which the events sustain. The physical and geometrical order of nature is constituted by at least three societies, "the society of pure extension," "the geometric society," and "the electromagnetic society." The point to be noted is the relationship of the geometrical society and the electromagnetic society. The latter is embedded, so to speak, in the former, so that a determination of the variable physical quantities which characterize the electromagnetic society is obtained against a background of relationships which comprise a uniform metric structure:
The whole theory of the physical field is the interweaving of the individual peculiarities of actual occasions upon the background of systematic geometry. (PR 507)
[T] hese diversities and identities are correlated according to a systematic law expressible in terms of the systematic measurements derived from the geometric nexus. (PR 150)
The preceding investigation into the Newtonian and Whiteheadian theories of space and time structure suggests that there is a clear and definite parallel between the two theories and argues that the parallel is significant because Newton and Whitehead share a common understanding of the status and function of a uniform and independent space and time structure in scientific inquiry. This investigation supports the contention that for both Newton and Whitehead some privileged space and time structure is presupposed by any physical analysis that is to be considered scientific and, specifically, shows that in Newton’s analysis of gravitation and in Whitehead’s analysis of impetus this privileged space and time structure functions as the framework, or background, in terms of which definite physical characteristics are then analyzed. For Newton, absolute space and absolute time are presupposed by a theory of the dynamics of moving bodies and in particular are necessitated by the fact that no available physical frame of reference seems suitable for a satisfactory analysis of the accelerated motions of material objects. For Whitehead, a uniform and independent metric structure is presupposed by measurement in the space-time continuum and in particular is necessitated by any attempt, so it seems to Whitehead, to establish standard conditions or to introduce correction factors ensuring the self-congruence of measuring devices under transport.
It might be argued that all scientific inquiry, whether classical or contemporary, presupposes, perhaps in the sense that it makes some assumption with regard to, a theory of space and time structure, and that it obviously may be either an absolutist or a relational position. The various arguments for one or the other may be evaluated and the assumed theory may be modified. There would be nothing surprising about this and especially nothing of interest in the fact that two particular cosmologies might share this common presupposition. However. the crux of the argument in this investigation is not simply the common presupposition of a theory of space and time structure by scientific inquiry; rather, it is the common presupposition of a privileged space and time structure, and the use of this privileged space and time structure in subsequent physical analysis in a strikingly similar manner. Further, it is not simply an arbitrary decision to conduct physical analysis against the background of such a space and time structure; rather, both Newton and Whitehead maintain that this structure is a real factor within the universe -- it is there to be discerned, either through the analysis of forces as for Newton or through a method of abstraction as for Whitehead:
how we are to obtain the true motion from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise [i.e., the Principia]. For to this end it was that I composed it. (PNP 12)
[T] he point of the definition [of straight lines by the method of extensive abstraction] is to demonstrate that the extensive continuum, apart from the particular actualities into which it is atomized, includes in its systematic structure the relationships of regions expressed by straight lines. These relationships are there for perception. (PR 496)
What this investigation means in terms of more comprehensive metaphysical issues is evident. Neither Newton nor Whitehead ascribes to a purely relative theory of space and time structure, that is, a theory which would maintain that any selected system of material objects may serve as a suitable framework in terms of which to analyze physical reality. Newton and Whitehead maintain, on the contrary, that there are important considerations presupposed by scientific inquiry which result in the selection of certain specific frames of reference, perhaps even a unique frame of reference, in terms of which the spatial and temporal relations of material objects are to be determined.
There is a difference in the considerations that influence Newton and Whitehead to assert the theories of space and time structure that they do. And this difference leads to a difference in approaches to questions about the content of physical geometry, that is, about the metric structure of space and time.31 The Newtonian approach is accurately described as "holistic"; it is an approach that envisions the fixing of geometrical structure by and within the confirmation of some satisfactory physical theory. In support of this, it is to be recalled that Newton’s argument for absolute space involves certain definite ideas about forces, their effects, and what constitutes a deformation of a normal situation. Consequently, for Newton there is no prior demarcation between physical geometry and physics. Supposedly then, Newton’s theory of absolute space and absolute time and his theory of dynamics would be confirmed, or disconfirmed, as a whole. The Whiteheadian approach may be characterized as affirming the "autonomy" of physical geometry; it is an approach that envisions the fixing of a uniform and independent metric structure regardless of the type of physical analysis that is conducted. Any and all measurement presupposes such a structure. Consequently then, for Whitehead physical geometry is the science of the uniform relatedness of events, and a physical theory is the science of their contingent relatedness (R v-vi).32
This investigation certainly does not claim any innovative or original interpretation of the Newtonian theory; it is intended to be an accurate description of certain essential features of the Newtonian position. It is a result of this investigation, however, that one should be quite cautious in assuming an all-embracing endorsement by Whitehead of the concepts that define twentieth-century relativity science; Whitehead himself continually warns his reader of this fact. This investigation argues that indeed one may find that Whitehead retains certain important and essentially classical notions of seventeenth- and eighteenth-century science.
A final note puts the Whiteheadian position, and this investigation of it, into proper perspective. Although both Newton and Whitehead claim that there is a uniform and independent metric structure of the space and time continuum, Whitehead’s claim is on behalf of a uniform and independent system of relations, not a substantial entity in its own right. This system of relations may be discerned within nature and is sustained by -- or actualized by -- the events or actual occasions and their relations that are the fundamental constituents of the universe. In Science and the Modern World, such a system is termed "an ideally isolated system":
This conception lithe concept of an ideally isolated system] embodies a fundamental character of things, without which science, or indeed any knowledge on the part of finite intellects, would be impossible. The ‘isolated’ system is not a solipsist system, apart from which there would be nonentity. It is isolated as within the universe. This means that there are truths respecting this system which require reference only to the remainder of things by way of a uniform systematic scheme of relationships. Thus the conception of an isolated system is not the conception of substantial independence from the remainder of things, but of freedom from casual contingent dependence upon detailed items within the rest of the universe. Further, this freedom from casual dependence is required only in respect to certain abstract characteristics which attach to the isolated system, and not in respect to the system in its full concreteness. (SMW 46)
It is an "ideal" system in that it is discerned by thought through abstraction, not through immediate sense-perception; it is an "isolated" system in that it has a completeness in its own right guaranteed by the sameness of the relations that it exhibits; but it is not a self-subsistent entity. In this respect the Whiteheadian theory of space-time structure is unmistakably different from that of Newton. Nevertheless, it is interesting that Whitehead credits Newton with recognizing the importance of the concept of an ideally isolated system to scientific inquiry (SMW 46).
Thus, Whitehead’s theory of space-time structure represents an interesting intermediary position in the controversy between the positions traditionally labeled "relational" and "absolutist"33 Whitehead’s theory is "relational" with regard to the fundamental nature of space-time and is "absolutist" with regard to a structure exhibited within and sustained by the extensional relations of events.
PNP -- Sir Isaac Newton, Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World (1686) trans. Andrew Motte (1729). Translation revised and annotated by Florian Cajori. Berkeley: University of California Press, 1934
1Some of the modifications of importance are: (1) the substitution of the relation of extensive connection for the relation of extension as the fundamental extensional relation; (2) the analysis of metric structure in terms of the geometric society instead of the relations of extension and cogredience; (3) the emphasis on the potential character of extensional relations.
2Whitehead acknowledges this himself; see CN vii, and R v.
3Ernst Mach’s criticism of Newton’s theory is well-known; see his The Science of Mechanics: A Critical and Historical Account of Its Development, trans. by Thomas J. McConnack (6th ed.; La Salle, Illinois; Open Court Publishing Company. 1960), chapter II, section VI, para. 2-6. See also Stephen Toulmin, "Criticism in the History of Science: Newton on Absolute Space, Time, and Motion, I and II," Philosophical Renew, 68 (January and April, 1959), 1-29 and 203-27, for a renewed evaluation of Newton’s theory. Adolf Grünbaum, especially in his Philosophical Problems of Space end Time, Borzi Books in the Philosophy of Science (New York: Alfred A. Knopf, 1963) chap. 15, and Robert M. Falter, in his Whitehead’s Philosophy of Science (Chicago: University of Chicago Press, 1960), chapters V and VI, are contemporary critics of Whitehead’s theory.
4There is an evaluative task to be undertaken. Whitehead’s retention of certain fundamentally Newtonian notions about space and time structure puts him at variance with the theory of space-time structure in the Einsteinian formulation of GTR. The basic issue in question is philosophical; however, there is reason to believe that Whitehead’s reading of the Einsteinian formulation is not necessarily the most satisfactory. If an alternative interpretation is adopted, and one which seems more reasonable, then it is possible that Whitehead’s process metaphysics could be consistent with the Einsteinian formulation of GTR. It would involve the abandoning of the Newtonian notions about space and time structure investigated here. This will be presented and argued in a supplementary article.
5Actually, it is the "presented locus" that is defined by perception in the mode of presentational immediacy (PR 195f, 492), and Whitehead notes in several places that in keeping with the results of physical science the presented duration and the presented locus are not to be identified (PR 192-96, 492f). Another reason for this distinction is to emphasize that a duration signifies actual occasions in unison of becoming whereas presentational immediacy only reveals extensive relations of some contemporary actual occasions (PR 491-93). The exceedingly difficult discussions of "strains," "strain-feelings," and "strain-loci" are relevant (PR part IV, chapter IV), but comprehension of these notions does not seem to be crucial to the argument here.
A footnote to PR 196 indicates that this distinction between the presented duration and the presented locus is a change from the position in CN (also PNK and R, for that matter). This change is of tremendous significance as one attempts to understand the development of Whitehead’s theory of space and time structure (see the important "Notes’ to PNK). At issue is Whitehead’s account of the real individuality of events.
6Grünbaum, in his Philosophical Problems of Space and Time, points out that to assert uniform curvature is meaningless without tacit reference to some particular metric geometry (p. 427). He finds Whitehead’s position untenable for this and other reasons. This issue is discussed briefly in the supplementary article mentioned above.
7Newton concludes this "Scholium," PNP 12: "But how we are to obtain the true motions from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise. For to this end it was that I composed it."
8The forces involved are the familiar centrifugal force and Coriolis force, which on a rotating disk are the forces on a mass-particle pulling it towards the edge of the disk and pulling it sideways away from a simple straight line from the center to the edge. Also included is the force experienced by a mass-particle when a frame of reference is accelerated along a straight line -- a force holding the mass-particle back -- and is decelerated along a straight line -- a force pulling the mass-particle forward. The term "inertial forces" refers to all such forces associated with accelerated motions, whether rotational or rectilinear.
9Toulmin, in "Newton on Absolute Space," pp. 25-27, argues that this experiment is designed to "illustrate" the distinction between absolute and relative motion which is essential to concepts relating to Newton’s theory of dynamics, and not to give evidence of the objective existence of absolute space as it is frequently interpreted.
10See the very clear explication in Ernest Nagel, The Structure of Science: Problems in the Logic of Scientific Explanation (New York: Harcourt, Brace & World, Inc, 1961), pp. 207-09.
11Newton assumed that appropriate optical and electromagnetic experiments might determine the state of uniform velocity, or of rest, of a mass-particle relative to absolute space. This was shown not to be the case, and this insight was incorporated in STR.
12Ernest Nagel, in providing a contemporary evaluation of Newton’s theory of mechanics, emphasizes that there is an element of truth that can still be attributed to Newton’s position, an element closely related to the analysis given above. Newton’s second law of motion states that if a force is impressed on a body, then that body is accelerated with the magnitude of the force and the direction of the acceleration in the direction of the force. The forces referred to cannot in general be measured independently of the acceleration involved; consequently, if the fact that a mass-particle is accelerated must first be determined in order to know that a force is operative, then in those cases some frame of reference must be adopted relative to which the motion of that mass-particle may he analyzed. This procedure means, in effect, the assigning of a logical priority to the specification of a frame of reference for the purposes of subsequent physical analysis. Nagel, in The Structure of Science, (p. 214), comments on the significance of this procedure as follows: ‘Newton’s procedure in assigning logical priority to the selection of a frame of reference, with respect to which motions are to be analyzed in terms of his axioms, was thus entirely cogent, however faulty may have been his argument for absolute space.
13This has been a point of emphasis in recent studies of Newton’s writings. See Alexandre Koyré, Newtonian Studies, A Phoenix Book (Chicago: The University of Chicago Press, 1965), pp. 149-63.
14Koyré’s suggested reading of "feign" instead of "frame" is used in the claim -- " . . . I frame no hypotheses."
15See Koyré’s discussions in Newtonian Studies.
16See the "General Scholium" to PNP 543-47. There may be points of similarity in Newton’s and Whitehead’s treatments of the relation of God to space and time, that is, God’s Omnipresence and Eternality. For a discussion of issues concerning Whitehead’s treatment see Lewis S. Ford, "Boethius and Whitehead on Time and Eternity," International Philosophical Quarterly, 8 (March, 1968), 38-67; and Lewis S. Ford, "Whitehead’s Conception of Divine Spatiality," Southern Journal of Philosophy, 6 (Spring, 1968), 1-13. Professor Ford has brought to my attention the recent discussion of Newton’s theory in Ivor Leclerc’s book, The Nature of Physical Existence (New York. Humanities Press, 1972), in which Leclerc argues that for Newton place and space are grounded in God’s activity. As Professor Ford noted, this is significantly different from the "Newtonian view of space and time that Whitehead criticizes in PR. I am not able to assess what directions an investigation of the possible similarities in Newton’s and Whitehead’s treatments of this topic might take.
17Newton’s example is elaborated by Toulmm in" Newton on Absolute Space," pp. 15-17.
18Toulmin, "Newton on Absolute Space," pp. 17, 21, uses the words "approximate" and "approximations" to describe Newton’s understanding of the distinction between "absolute time" and "apparent time."
Toulmin’s analysis is valuable and convincing to a certain extent. Toulmin argues that absolute space and absolute time are to be interpreted in Newton’s Principia as "theoretical ideals" or "mathematical ideals" in an axiomatic system. Consequently, it is incorrect to interpret Newton as maintaining that absolute space and absolute time can be measured without reference to some material objects, that absolute space and absolute time are real existents apart from all material objects, and that absolute space and absolute time are founded on essentially metaphysical considerations (see part I of Toulmin’s two-part essay).
Nevertheless, Toulmin himself must acknowledge that Newton does not consistently reflect such a view in the language he uses in the Principia, or in late additions to the Principia (for example, the "General Scholium"), or in other writings (for example, the Opticks) (see part II of Toulmin’s two-part essay). In view of the qualifications that Toulmin makes it would seem that Toulmin is arguing for a more satisfactory reformulation of Newton’s theory rather than for a more accurate reinterpretation. See Howard Stein, "Newtonian Space-Time," The Texas Quarterly, 10 (Autumn, 1967), 174-200. for another and perhaps more balanced interpretation of Newton’s absolute space and absolute time.
The theory attributed to Newton in the present investigation is believed to be a justifiable account of the comprehensive Newtonian position. But in spite of the above criticism of Toulmin’s analysis, the present investigation embodies his insistence on the relation in Newton’s Principia between his theory of dynamics and absolute space and absolute time. This is also a point of emphasis in Stein’s article.
19Palter, in Whitehead’s Philosophy of Science, chapters VIII, IX, and appendix IV, discusses the specific scientific and mathematical differences between the Einsteinian and the Whiteheadian formulations.
20Cf. PR 503, where Whitehead claims that the choice of a geometrical theory to describe the uniform metric structure is . . . to he found by comparing rival theories in respect to their power of elucidating observed facts."
21The issues here are complex and detailed; nearly every commentator on Whitehead’s philosophy of natural science discusses and criticizes this method. The success of Whitehead’s method in exhibiting a uniform metric structure is a basic question but not an issue in this investigation. This question is raised in the supplementary article referred to above. It need only be remarked here that the method is not a deductive method, nor is it designed to isolate for immediate sense-perception the geometrical relations defined by the method.
22Associated with both of the quotations, B 29, 59 and PR 498f, given on pages 246f above are examples involving the measurement of spatial distance and times of revolution; see R 58f and PR 499.
23This is but one of several tests for the presence of differential forces, and the test itself is described in very simplistic terms; nevertheless, the point illustrated is correct, namely that the presence of "differential" forces may be ascertained without the use of instruments which themselves involve established measuring scales.
24In fairness to Whitehead’s position it is to be noted that he disavows the "operationalist" understanding of self-congruence; yet, his position is also decidedly anti-conventionalist.
25To simplify matters the measurement of the angle-sums can be performed with the specified measuring rod and expressed in "radians." One radian is the measure of the angle at the center of a circle, subtending an arc of the circle equal in length to the radius (approximately equal to 57.3_).
26These deformations would be true of any measuring rod chosen regardless of its physical composition, for as is noted the deformation occurs despite prevailing standard conditions or the introduction of correction factors. It might be said that the deformation is the result of so-called "universal" forces, forces which influence every physical body in the same manner and are not dependent on their physical composition. The designation of forces as "differential" and "universal" is utilized extensively by Hans Reichenbach, The Philosophy of Space & Time, translated by Maria Reichenbach and John Freund, A Dover Paperback (New York: Dover Publications, Inc., 1958), pp. 10-28.
27A two-dimensional analogue of a space-time continuum of uniform metric structure having a Lobachewskian geometric structure may be constructed as well; see Nagel, The Structure of Science, pp. 237-40.
28It may seem plausible to introduce additional correction factors to correct for the deformation of the measuring rod by the distribution of matter. However, in order to avoid a vicious circle in this procedure, the extent of deformation must be determined without invoking additional measurements. It is difficult to see how this could be accomplished inasmuch as the extent of the deformation is dependent on the distribution of matter, a determination of which seems necessarily to involve the measurement of spatial intervals. See PR 506f.
The possibility of the circularity involved in practical measurement in a space-time continuum of variable metric structure is emphasized by C. I. Lewis, "The Categories of Natural knowledge," The Philosophy of Alfred North Whitehead, ed. by Paul Arthur Schilpp (2nd ed.; New York. Tudor Publishing Company, 1951), pp. 710-13.
It is to be recalled that the soundness of Whitehead’s argument (or of Lewis’s commentary on this point) is not the immediate point of interest in this investigation. A critical evaluation is contained in the supplementary article mentioned above, n. 4.
29There is reason to believe that it is not altogether accurate to interpret fields of gravitational force as consequences of the metric structure, although this is the understanding attributed to the Einsteinian formulation of CTR; see Reichenbach, The Philosophy of Space & Time, p. 256f. This issue is an immediate point of interest in the supplementary article mentioned in n. 4.
30The use of the term "impetus" is probably intended to suggest analogies with a fourteenth-century theory of motion, formulated in opposition to the Aristotelian theory of motion (see Herbert Butterfield, The Origins of Modern Science, 1300-1800, A Free Press Paperback [rev. ed.; New York. The Free Press l957], chapter 1).
The Aristotelian theory maintained that a mass-particle continued in motion only so long as a moving-agent was in contact with the mass-particle and was imparting motion to it at every moment. The continued motion of a projectile was explained by the rush of air, which was compressed in front of the projectile, to the rear of the projectile, where the air then pushed the projectile. This rush of air was brought into effect by the initial motion of the projectile as it was put into motion. In contrast, the fourteenth-century theory of impetus maintained that the motion of a mass-particle was due to an "impetus" which it acquired either from the moving-agent or from the mere fact of being in motion. The impetus was a physical character attributed to the mass-particle itself, and the measure of mass was used as n measure of the impetus corresponding to a given velocity. The essential point was that given the theory of impetus, the continued motion of a projectile could be explained after contact with the moving-agent was terminated without appealing to a rush of air around the projectile, a phenomenon not easily made compatible with other empirical and theoretical aspects of the Aristotelian theory.
To the extent that an analogy is intended between the fourteenth-century theory of impetus and Whitehead’s formulation of GTR, the following observation may be made: the concept of impetus in both cases designates a physical characteristic attributed either to the mass-particle or to the physical field which it causes and in both cases is developed in opposition to theories which would attribute this physical characteristic to the surrounding air or to the spatiotemporal frame of reference and its metric structure.
31The designations "holistic" and "autonomous" below are derived from Arthur Fine, "Reflections on a Relational Theory of Space," Synthese, 22 (1971), 450. Fine describes the Newtonian position as "holistic."
32The specific task of physics for Whitehead is the analysis of the relationships of events with the goal ". . . to contrast the sphere of contingency by discovering adjectives of events such that the history of the apparent world in the future shall be the outcome of the apparent world in the past" (B 29, cf. PR 150).
33Alexandre Koyré, From the Closed World to the Infinite Universe, The Cloister Library, Harper Torchbooks (New York: Harper & Row, Publishers, 1957). p. 252, characterizes the Leibnizian -- or relational -- theory of space as "a lattice of quantitative relations" and the Newtonian -- or absolutist -- theory of space as "a unity which precedes and makes possible all relations that can be discovered in it." See Fine, "Reflections on a Relational Theory of Space."