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The Concept of Mass in Process Theory

by Granville C. Henry and Robert J. Valenza

Granville C. Henry is Professor of Mathematics and Philosophy at Claremont McKenna College, Claremont, CA, 91711. He is the author of Forms of Concrescence: Alfred North Whitehead’s Philosophy and Computer Programming Structures (Bucknell University Press, 1993).

Robert J. Valenza is W. M. Keck Professor of Mathematics and Computer Science at Claremont McKenna College. He is the author of Linear Algebra: An Introduction to Abstract Mathematics (Springer-Verlag, 1993). The following article appeared in Process Studies, pp. 292-307, Vol. 27/3-4, Fall-Winter, 1998. 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.

I. Introduction

The four fundamental abstract dimensions of physics are mass [M], length [L], time [T], and charge [Q]; all physical parameters are built out of products of these and their inverses. Indeed, measurement systems derive their names from the choices of corresponding units. Thus the MKSQ system fixes upon meters, kilograms, seconds, and coulombs as its fundamental units. Forces, for instance, are then measured in units of kilogram-meters per second squared, called newtons, whose abstract dimensions accordingly are [M] [L]2[T]. Note that this dimensional characterization is independent of any measurement system; forces always take this form. Similarly, Planck’s constant h intrinsically has abstract dimensions [M] [L]2/[T]. Clearly then, mass is key to physics. Like the other fundamental dimensions, it seems to quantify an. essential attribute of matter and substance. Moreover, entities that exhibit mass seem to do so in a way that is constant over time -- a state of affairs ostensibly supporting a traditional ontology where substance is understood to exist through space and time. In this sense, the concept of mass would itself seem to lie in opposition to process theory. Why should short-lived Whiteheadian actual entitles or their collective societies attract each other or resist changes in motion?

In this essay we examine the connection between mass and substance, from both the traditional and process perspectives. We begin by highlighting the most important historical developments, including the notion of mass as it appeared prior to and in Newton’s work, the formal characterization of Mach, and its eventual role in relativity and quantum theory. We next deploy a superstructure which frames the rest of the discussion: the complementary notions of preprojective and postprojective discourse. Within this framework we prepare for a characterization of mass in process terms by a refined appeal to elements of Quine’s metaphysics, and we hope here to have made a new point of contact and reconciliation between Whitehead and the more traditional metaphysical theories. Moving into a fully process-theoretic context, we use one of the Categoreal Obligations to deduce the very idea of mass and then speculate on why the body has mass, but the soul does not, without a simple, pluralistic dismissal of the idea as a mere category error.

Finally, we address a problem posed by Charles Hartshorne concerning a process view of God and relativity’s denial of absolute simultaneity. We suggest that only entities or societies of a technically restricted type are subject to relativity’s constraints.

Two aspects of this brief program require some early advisories. First, we do not claim to establish some tight correspondence between the rigid formalisms of current-day physics and the more colloquial terms of a less adamantine metaphysics. One cannot hope to give a metaphysical definition of mass. We seek only to make plausibly consistent our understanding of the world as mediated so effectively by physics and certain general metaphysical notions, including fundamental aspects. of process thought. Second, our attempt at some reconciliation between Quine and Whitehead, which many might judge to be doomed from the outset, is not an attempt to reconcile their ontologies. Rather, we are applying certain methodological features of Quine’s analysis of things in the world to Whiteheadian actual entities in order to recover aspects of the less radical Quinian world view. In the same sense that experience underdetermines science (TDE 42 or PMN, Part Three), this approach is hardly peculiar.

II. A Brief History

Allow us first to make a loose but useful distinction, which we promise to make taut in the following section. The term mass has two senses in ordinary discourse. First, to say that this pear has mass might be to say that within the framework of physics and systems of physical measurement and associated units, there is some number m greater than zero, which, to some unspecified tolerance, functions as a parameter in a mechanical analysis of the pear’s behavior. Second, outside this elaborate technical framework, but nonetheless within the even more elaborate framework of ordinary language, to ascribe mass to this pear is to acknowledge a host of fundamental properties and behaviors that need no formalization. In this latter sense, the pear has mass, but not its shadow. At the heart of the matter we find a kind of exclusion principle: a pear resists sharing the spacetime it occupies with any other material object.1 Extending this informal usage, we attribute more mass to a bigger pear that requires more effort to lift. Thus things with mass seem to partake of a quantity of continuous material, of substance. We learn to speak of objects or societies with these familiar properties as having mass long before we ever appreciate mass as a physical parameter. We recognize, moreover, that this property is largely independent of physical form.

To appeal to the Quine’s metaphor from his famous essay "Two Dogmas of Empiricism" (TDE 42-46), if the synthetic and the analytic are not dichotomous, but rather the respective periphery and center of a web of knowledge and beliefs, then our casual use of the word mass lies at the periphery, while our technical meaning lies near the center. With this in mind, let us now make some remarks about the evolution of the term as it developed analytically, i.e., as it moved toward the center of the Quinian web.

Max Jammer, in Concepts of Mass (CM 37-73), recounts the emergence of mass as a working concept in both metaphysics and physics. He observes that the Aristotelian notion of substance, encompassing both prime matter and corporeal form, with the latter contributing the property of extension, led directly to the problem of characterizing corporeal form, and eventually to our technical notion of mass.

To summarize Jammer’s development with merciless brevity, the evolution of the concept is punctuated principally by Avicenna (980-1037), Averroës (c. 1126-1298), Aegidius2 (c. 1247-1316), Kepler (1571-1630), and Newton (1642-1727). Avicenna characterized corporeal form as the tendency of prime matter to manifest dimensionality, which is to say spatial extension. Averroës proposed to correct Avicenna in two ways: first, that the essence of corporeal form lies in indeterminate dimensions, thus asserting the priority of spatiality in the abstract over the particular incidental form of a body, its so-called determinate dimensions; second, a characterization of corporeal form as "merely the capacity of prime matter for natural motion and as merely the tendency to move to its natural place" (CM 40), a harbinger of the formalities of dynamics to come. Aegidius then took the crucial step: he refined Averrës thesis by identifying indeterminate dimensions with quantitas materiae (quantity of matter) and explicitly detaching this attribute from the determinate dimensions. This move begins to resolve what we now understand to be a long-standing entanglement among the concepts of mass, density, and extension (volume).

If quantitas materiae is somehow the proper concept of mass -- as much for what it discards as irrelevant as for what it asserts definitively -- it is left to Kepler and Newton to systematize this notion and to incorporate it into the technical panoply of working physics. The process begins when Kepler submits to the ellipticity of the planetary trajectories. For an ellipse, an apparent distortion of the putatively more natural and ideal circle, implies that inertial mass, as it lies in opposition to force, must be attributed to the planets, and that more generally the motions of the planets require dynamic, as opposed to merely geometric or kinematic, analysis3 Here indeed may lie the kernel of a Kuhnian paradigm shift. Moreover, Kepler speaks decisively when he says, "Inertia or opposition to motion is a characteristic of matter; it is stronger, the greater the quantity of matter in a given volume4 because in this statement he essentially identifies Aegidius’s raw concept with a soon-to-be measurable parameter in a full-blown theory of dynamics. In developing this theory, however, Kepler still suffered from a confusion, initiated by Aristotle, between force and momentum. This confusion was ultimately resolved by Isaac Newton.

Newton had difficulties of his own in defining mass, and seems guilty of a double circularity. First, his definition -- independent of dynamics -- of mass as the product of volume and density begs the question insofar as density (mass per unit volume) must then be introduced into physics as an independent fundamental dimension. Indeed, it then becomes exactly as difficult to define density as to define mass. Second, to the extent that one can derive mass from the relation

Force = Mass x Acceleration

his famous second law of motion, mass again depends on a prior fundamental dimension: in this case force. However, here one could postulate the possibility of equal forces (say, a spring compressed to a fixed limit), from which relative mass could be measured inferentially from the ensuing kinematics. But perhaps a better way to frame these perplexities lies in the linguistic philosophy of our own century. Rather than looking for an unambiguously grounded formal theory in Newton, we should perhaps look instead for simple coherence. It does assert something about the world to say that two parameters such as force and mass can be deployed in such a way as to predict or describe a large class of mechanical phenomena. This accords very well with Newton’s third law of motion: we take it starkly as conservation of momentum; Newton more likely saw it as reciprocity between mass and force. Again, coherence.

The Austrian physicist and philosopher Ernst Mach (1838- 1916) did succeed, in a most straightforward way, to give a theoretical definition of mass. Consider an isolated system consisting of a reference body held at some fixed distance from test body B1. Then (by gravitation) our reference body experiences an acceleration a1 in the direction B1. If we replace B1 in our system by another body B2, our reference mass will experience an acceleration a2. Now arbitrarily taking the mass of B1 to be unity, we can define the mass of B2 (or any other test body) as the ratio a2 /a1. In other words, the measured mass simply is proportional to the observed acceleration, and here we have our definition. Of course, this is highly idealized, but nonetheless, it does yield a definition without reference to force and is certainly consistent with the law of universal gravitation.

While our historical sketch to this point is sufficient background for our subsequent discussion, some remarks on the role of mass in postclassical physics will be illuminating.

In the special theory of relativity, one shows that the mass of an object is no longer constant, but varies with velocity according to the relationship

m = m0 divided by the square root of ([1-v2] divided by c2)

where m0 measures rest mass, v is velocity, and c is the speed of light. From this one deduces -- as an exercise in first-year calculus, incidentally – the famous Einstein mass-energy equivalence, E= mc2. At first glance, it would seem that any interpretation of mass as a measure of quantitas materiae is obliterated by its dependency on velocity. But, in fact, since we can always compute rest mass from observed mass, it still makes sense to compare two objects moving at different velocities, and to that extent quantitas materiae survives. Perhaps more significantly from our point of view, however, is that the equation above and the Einstein mass-energy equivalence in some sense redefine the concept of mass away from any intuitive interpretation. For instance, now light and heat have mass, too. To revert to Quine’s metaphor, this illustrates the extraordinary measures science, as a holistic enterprise, will take to protect the center of the web.

In quantum theory we find a similar migration of meaning by formalism. To measure mass is to measure energy, and the uncertainty principle, which embodies the essential complementarity of wave and particle descriptions of physical objects, tells us that

D E× D t£ h


where _E is the uncertainty in energy, At is the duration of the measurement, and h once more is Planck’s constant. This is to say that any measurement of energy made over a finite time is fundamentally uncertain, thus relegating both mass and energy to probability distributions, rather than ordinary numbers. Again quantitas materiae might be said to survive weakly to the extent that one might compare the means of such distributions, and again one observes how privileged is the center of the web.

III. The Preprojective and the Postprojective: A Synopsis

In a previous article (PP), we have attempted to characterize physics, and perhaps science more generally, as proceeding via a class of correspondences, suggestively termed projections,

E ® p(E)

which to each event E associate a point E(E) in some mathematical space. The key aspect is the explicit separation of an event from its projection: they are held as distinct members of disjoint classes. To illustrate briefly, we consider classical dynamics. Here, for a given event E, we might take (E) to lie in the 5-dimensional representation space

R3 x R x R

The first factor represents position (requiring three coordinates, each a member of the set of real numbers R), the second time, and the third mass. (R denotes the set of nonnegative real numbers).5 In this model we recognize a persistent identity as a connected set of images in the codomain. In fact, in Newtonian physics we would expect that the image of a single substantial object would be identifiable not only by the continuity of its first three factors as an implicit function of the fourth, but also by the constancy of the fifth. In relativistic dynamics, the final factor is no longer constant, but still varies continuously.

To extend our example, note that in dealing with electrodynamic phenomena the representation space cited above is insufficient. Physicists are then wont to refine the projection, enlarging the representation space to:

R3 x R x R+ x R

introducing a new factor of R to represent electrical charge. A mathematician says that the projection has been made more nearly faithful. In this augmented space, one might now better analyze systems of charged particles in motion among their mutual gravitational and electromagnetic effects.

Note that in this second example all of the fundamental dimensions as given in the introduction are now represented. One should not feel, however, that this somehow exhausts the possibilities: the species of representation spaces are limitless. (Further examples are given in PP.) In particular, discrete mathematical spaces are also available. These include the kinds of spaces used to describe the behavior of computers and even the space of ordinary written language.

One of our main points in setting out this framework is the claim that in some deep sense, to do physics is to find laws that relate (E) to (E1) for separable events E and E1 (including of course the special case that E and E1 pertain to the same substance or object in the naive sense of those words. These laws do not necessarily constitute predictions of future values of these projections, but are sometimes merely constraints on associated quantities or their relations. So, for instance, one might explicitly derive the ballistic trajectory of an object in a vacuum, or one might be content to assert that the sum of the potential and kinetic energy of said object is constant. Or one might declare that (E) and (E1) are necessarily independent if the distance between their spatial coordinates divided by the difference between their temporal coordinates, in some common frame, exceeds the speed of light.

IV. Mass and Matter

It is clear from the preceding discussion that the second, more technical sense of mass that we introduced in our historical synopsis is indeed the postprojective one. That various particular projections may have held sway historically, with more or less precision, is inessential. The point is that the concept simply never gels without projection, and the key characteristics of the projection are these:

The mass attributed to an ordinary object varies at worst continuously; in many contexts it suffices to assume that it is constant.

The mass attributed to ordinary objects is required as an independent parameter in predicting the motions that result from the interactions among a system of physical objects.

Thus while the idea of the predicate has mass is sensible preprojectively as an attribution to be made at the periphery of the Quinian web, it seems to have no deep meaning -- this is to say, no embedding into a network of concepts and actions that allows us to describe and circumscribe the phenomenal world -- without recourse to mathematical abstraction. Indeed, the imprecise notion of quantitas materiae is already a groping in that direction: it anticipates a representation space without making explicit a projective mechanism; that is, a kind of measurement.6

We approach a natural inference here reminiscent of Russell’s attack on Kantian metaphysics. Just as Russell asked how could relations among phenomena not reflect relations among noumena, we can ask the following question: if a key feature of mass is continuity, then does this not imply some underlying substrate that must persist without change from measurement to measurement? More generally, does not the effectiveness of continuous projections in describing the physical world suggest an underlying substance metaphysics, certainly at variance with the discontinuities of process thought? We have then in our usual objects and substances not misplaced concreteness, but simply inferred concreteness. This seems to us to be a deep and difficult question, to which we now turn.

V. Ontology, Projection, and Continuity

We shall now argue that the metaphysical implication from projection to substance of the preceding section is escapable: the continuity or connectedness of a projection such as mass as it appears in the mathematical model that is physics does not imply a preprojective substance ontology insofar as an examination of terms -- or more properly the origin of particular terms -- reveals a circularity of argument. This leaves the way clear for a process metaphysics that is consistent with both physics and contemporary theories of entification.

In laying out his metaphysics, Quine explicitly connects entification with continuously varying properties. He says, for instance, "Most of our general terms individuate by continuity considerations, because continuity favors causal connections" (TTPT 12). Implicit in this, of course, is the recognition that the causal connection that looms largest in the world is stability: the most likely precondition for the existence of this pear at a given location at any given time is its existence at a nearby location at a proximate prior time. In our framework and language, we would say that what entifies is that which tends to admit a continuous projection into generalized mathematical spaces; moreover, the particular projections that we choose are those that are useful in the derivation of regular mathematical features. Now we do not mean to suggest here that the everyday ontology of the non-physicist is somehow driven by the high abstractions of mathematics -- although to some extent it is of course driven by current scientific models. Indeed, the river almost always flows in the opposite direction: our informal, intuitive notions of cardinality, time, space, and causality in fact drive the evolution of abstractions such as functions, geometric spaces, real numbers, and coordinate systems. That these abstractions appear natural and ineluctable -- and at times even have attained the lofty status of Kantian transcendence -- is the result of effective habits of thought that favor a certain class of proto-abstractions.7 (See MFF for detailed and compelling development of this thesis.) With this caveat taken to heart, we may assert that the thrown ball as it crosses my field of vision is viewed as a single object to the extent that its geometric features and trajectory vary continuously, and similarly for a moving picture of that same ball.8

Mass, as we have seen, is one element of a composite projection appropriate to physics. Thus together with spatiotemporal coordinates and other postprojective features, mass helps to define that to which it is later attributed. So, for example, if I turn away from my desk for a minute and see a brown pear where before had stood a green one, this violation of continuity leads me to believe that the same pear no longer sits before me. Similarly, if my pear suddenly seems to suffer a gross attenuation of density, I again doubt its sameness. In other words, the various projections that we carry out intuitively or with the refined instrumentations of physics are indeed the facilitators of entification. In this sense, the common objects of the world are the posits of our projections, so that the continuity of these projections for such objects is in this respect tautological.9

The implications of this line of reasoning for process metaphysics are huge. One may now assert that a physical element of a process-driven reality does not have mass because it is a substance, but rather that it appears as a substance because it has mass -- in the postprojective sense. Thus the attribution of substance should in fact be viewed as postprojective. The key point is that projection and entification constitute a kind of looping negotiation with reality: we entify largely by the continuity of some projections, are guided in the choice of subsequent projections by what has previously entified, and the process iterates. The pattern is similar in its gross dynamics to the story of how astronomers and physicists deduced the existence of the outer planets and subatomic particles.

VI. A Process View of Mass

In our general view of process thought we follow Charles Hartshorne’s interpretations and revisions of Whitehead, especially on the issue of the nature of eternal objects. For us, Whitehead’s eternal objects are too abstract, too Platonic, too flat, and, as defined as absolutely pure potentials, seem completely vacuous. We seek to reduce the polarity of the "certain extreme finality" (PR 22) of the contrast between actual entities and eternal objects. Our "eternal objects" are more or less abstract and more or less potentials.10 We believe Whitehead’s eternal objects cannot include the vast, incredible variety of kinds of abstract feelings that are part of actual entities -- the only place that so called eternal objects can occur. We have come to this conclusion reluctantly from a study of Whitehead’s attitude towards mathematics. In our judgment, Whitehead was led astray by a typical posture towards mathematics at the time. We believe that this error on his part was the primary reason for his quick slip into professional mathematical irrelevance (see WEPM).

Any revised interpretation of the basic polarity of actual entities and eternal objects must ripple through the rest of Whitehead’s philosophy. Although we see minimal distortion of orthodox process thought by our change, we do notice that it has informed our position on the nature of mass primarily through the interplay of the "more" and "less" abstract prehensions during the process of concrescence. The standard interpretation of concrescence pictures a transition from the physical pole to the consummate satisfaction associated with the mental pole. However, Whitehead began to allow the physical pole to have aspects of mentality through his hybrid physical prehensions, which as a type of physical prehension occur at the earliest stages of concrescence. If one relaxes the rigidity of Whiteheadian eternal objects, and interprets them as varying degrees of abstract feelings, one can have the full range of such feelings, both physical and mental, as subjective forms of prehensions as well as their objective data. This means that one can more easily describe the transition in concrescence from the preprojective physical feeling of mass to the postprojective mathematical description that occurs as an idea in the mentality of the concrescence. More importantly, one can interpret how the postprojective mathematical description of mass, prehended at an early level of concrescence, can guide a concrescing entity to a physical feeling of mass in a satisfaction. This is important to us, for we believe – à la Kuhn’s theory of paradigms -- that the feeling of mass in objects by professional scientists is significantly conditioned by their understanding of the theoretical description of mass. This theoretical description is influenced by the culture of science, which must have its grounding as science in experimental evidence. Again, a looping negotiation.

Although Whitehead does not define mass in Process and Reality, he presents the philosophical background for its preprojective origins in Chapter III, "The Order of Nature," Sections IV through VII. He first introduces the top level of his hierarchical description with what he calls the general presumption of the whole and part, which is characterized as pure extension. This has a nice mathematical analog in the idea of a topological space, a set with no a priori notion of distance that nonetheless formalizes the ideas of separation and proximity. His second level, the geometrical society, is a particular kind of topological space, one that does admit some sort of metric -- a measure of distance -- although not a unique one. He characterizes this in terms of straight lines, by which he means geodesics relative to a differential notion of length.

The third level, the level of so-called electromagnetic societies, is of most importance to our present discussion. Here Whitehead posits the universe of the current epoch: individuation and the laws of physics. This is a beautiful idea, although his terminology is puzzling. Consider that electromagnetics constitute only one of the four fundamental forces as currently understood. That Whitehead excludes the weak and strong nuclear forces is understandable -- these were not identified until the 1930s -- but why is gravity omitted? One might conjecture that perhaps as a thorough student of relativity, Whitehead felt that gravity, in accord with the general relativistic notion that it is a property of space, should more properly be folded into the second level of his hierarchy. But still there is something most curious about the level of electromagnetic societies: mass is ignored, as though perhaps its only manifestation were gravity -- which is simply false, even in our current understanding of the universe. Our point is that Whitehead, within the limits of his knowledge, might more properly have spoken of material electromagnetic societies, or have used some other term to acknowledge mass explicitly. Possibly he did not mean to exclude mass at this level but to emphasize the nongravitational forces between masses, so that the concept of mass is not absent by design, but merely left implicit. One might be persuaded to this position, but nonetheless the terminology is revealing and we are left to ourselves to derive the notion of mass from the more basic elements of his metaphysics.

Whitehead’s argument in preparation for our characterization of the preprojective kernel of mass is as follows. Every society of actual entities exists within a larger and changing environment of actual entities in which it "may be more or less ‘stabilized’" in that environment, which is to say, persistent in some group of characteristics despite environmental changes (PR 100). Continuing in this vein, Whitehead then argues that for a society to maintain itself through a changing environment, it must be "unspecialized" in the sense of being resilient and flexible in its responses to the world, with the consequence that an unspecialized society is also likely lacking in the strong structural characteristics that would lead to an intensity of satisfaction. "Thus the problem for Nature is the production of societies which are structured with a high ‘complexity,’ and which are at the same time ‘unspecialized.’ In this way, intensity is mated with survival" (PR 101).

One solution to the problem of complex resilience, says Whitehead, occurs by appeal to the Category of Transmutation. The society in question takes a characteristic approach to the world, in particular to its objectification of the nexus that surround it. The approach maintains the stability of the society by "eliciting a massive average objectification of a nexus, while eliminating the detailed diversities of the various members of the nexus..." (PR 101). In other words, "It ignores the diversity of detail by overwhelming the nexus by means of some congenial uniformity which pervades it. The environment may then change indefinitely so far as concerns the ignored details so long as they can be ignored" (PR 101). It is important to realize that this objectification reflects a process of abstraction, occurring in a concrescing entity, that determines a common form of a society. For an entity that is part of the society, the form is internalized in such a way as to maintain that formal aspect. An entity not a part of the objectified society, but of some other, may perceive the form of the society as an objectified property of the society.

So what then is mass? We hold that mass, in its preprojective sense, is an objectified form of certain physical societies, a form that acts as a universal leveler of relations in societies, and one that, in a distant echo of Averroës, completely ignores elements of internal structure. Thus in this objectification, the constituents of a society with mass are so flattened that what remains is a kind of counting measure, or its continuous analog, a density integral11 And thus at least informally mass as quantitas materiae can be recovered for process thought.

To the extent that one feels that something has been achieved in a process-theoretic characterization of mass that seems to answer to the first, historic gropings for the meaning of this idea, one is tempted to ask to what extent our thesis answers to the later, more sophisticated, formal, and patently postprojective notions. We proceed here with two observations in response to two fundamental questions that prepare the ground for the more radical analysis of the following section.

First, we ask, to what kind of societies is mass attributable? The obvious proposal here is to restrict this attribution to societies that admit a serial order (see PR 34). The point is that the very idea of mass -- either preprojectively, with time regarded as mere serialization, or postprojectively, with time as a topologically sophisticated abstract coordinate system -- is insupportable without temporal context. Preprojectively, we see from Whitehead’s analysis of the survival of complex resilient systems that the Category of Transmutation must apply across the serial evolution of a nexus. This is consonant with essential features of the postprojective formalization of mass: we may speak of a mass m at an instant t, but

(i) in classical or relativistic mechanics, mass is only assessed in the context of possible accelerations, which in turn require the context of a continuous time interval;

(ii) in quantum theory, the instantaneous measurement of mass is completely indefinite.

To summarize, the upshot of our first observation is this: the notion of mass apart from time is nonsense. Therefore in process thought, mass must be a certain kind of objectification of a determinate form of a nexus that persists over time. This entails that it be a society with serial order.

Second, we ask, what role does the attribution of mass play in our successful negotiation of reality? For this, let us begin with the postprojective as a metaphor for the preprojective. Consider again Newton’s second law of motion, rewritten thus:

Acceleration = Force1/Mass

From this we see that in an environment of bounded forces, the reciprocal of mass bounds the observed accelerations. Thus when we ascribe mass to an entity, we are asserting a kind of limit on its dynamics.12 Bringing this back to the preprojective, we are saying that insofar as a high-grade society objectifies a given nexus by virtue of the Category of Transmutation in the flattened form that is mass, the dynamics of that nexus are bounded, at least with regard to its extensive relations with the world. What is most striking here is that in the preprojective discourse the connection between mass as quantitas materiae and inertia is not accidental: the value and the goal of the transmutation is to slow things down; to facilitate a universe that is negotiable.

VII. Why the Body Has Mass but the Soul Does Not

Whitehead speaks of a structured society as one that "provides a favorable environment for the subordinate societies which it harbors within itself" (PR 99). These societies, called "inorganic" (PR 102) by Whitehead, can exist outside the structured society with only slight differences resulting from the change in their environment. A molecule is an example of an inorganic subordinate society within the structured society of a living cell (PR 99). Some general features of the molecule, including its mass obtained by the abstractive levelings described above, are independent of the environment of the molecule. Speaking postprojectively, we can claim that the mass of the molecule is the same inside or outside the cell, subject to relativistic considerations.

However, there is much more to the living cell than just its societal components that have mass. There are other actual entities that cannot sustain themselves outside the environment of the cell, for example, "living occasions" (PR 102). In the society of a live animal body there are such occasions, together called an "entirely living nexus," (PR 103) that may assume dominance over the body. Whitehead conjectures that this entirely living nexus is not a society for two reasons. First, it requires the protection of the structured society of the body (PR 103). Second, he understood such a nexus to be deficit of any "defining characteristic," which is that required common element of form that describes and is inherited by each actual entity of the nexus of a society. In particular, he asserts that the entirely living nexus does not have the form of "life."

It follows from these considerations that in abstraction from its animal body an "entirely living" nexus is not properly a society at all, since "life" cannot be a defining characteristic. It is the name for originality, and not for tradition. (PR 104)

Whitehead is claiming here that by its very nature, the essential creativity of a living occasion is too evanescent to be subject to the Category of Transmutation. By analogy, we may look at the surface of the ocean on a calm day and see waves, and even, if we so choose, assess their amplitude and frequency. But this mode of description fails entirely in the random turbulence of a cataclysmic storm, so that concepts such as amplitude and frequency may no longer apply. In the same way, the entirely living nexus does not have that form whose abstraction gives mass because of the extent and nature of its originality. It is so creative that it cannot be subject to that common form that is the general leveler of all physical societies. Mass captures only the inherited, noncreative aspects of societies. The entirely living nexus has no such characteristics. Indeed, when Whitehead speaks of entirely living occasions, he removes all that is hereditary and leaves only raw originality, which in essence can have no inertia.

While Whitehead’s denial of any persistence of form in purely living occasions leads to a satisfying explanation of why the soul does not have mass, in another sense it seems to fly in the face of experience: our living cohorts in the world do not seem so psychologically indeterminate. The existence of the most elementary social structures and the sciences of psychology and ethology suggest strongly that living occasions may indeed exhibit the structure of enduring objects that have personal order. Hence we question whether there exists any living nexus that is as purely creative as Whitehead suggests; that is, an entirely living one. To the contrary we have asserted previously (PAWM 39) that there is good evidence in Process and Reality that the physical pole is part of the unified satisfaction of any entity. Under this interpretation, the dominant occasions that constitute the soul of a personal society could have some aspect of physicality. If so, why cannot that inheritance of physical form in the living nexus of the human soul be leveled to give at least some mass? Our highly speculative answer: it might do so, but not normally.

We think that there are two reasons in addition to the explanation given above for denying mass to living nexos in ordinary occasions. The first is the highly emotive nature of the soul. This complexity of emotion that is the core of the soul and the essence of creativity of life as described above may so overwhelm the forms of physicality of the soul that they cannot be leveled to get mass. The second is the normal nature of the physicality of the soul, which explains why it can be overwhelmed by emotion and creativity. Concrescence begins in physical experience, the prehension of past actual entities, which is a necessary condition for the development of mental experience. But as the process of concrescence unfolds, that which was physical normally becomes more mental, that is, abstract. We acknowledge the firm influence of the physical body on the soul through efficient causes. Further, we have attempted to describe how the soul controls the body via final causes through abstractions of physical processes derived from the body (PAWM 41). The point is that the soul does have a structure that includes forms of physicality inherited from the physical pole. However, like emotions, these forms as higher abstractions do not have mass. Whitehead’s enduring objects internalize a common form as part of their actual or physical being. Soul events internalize physical forms for purposes other than being physical; for example, in order to influence physical events. Thus, whatever physical existence, if any, is attributable to soul events is easily obscured by creativity and emotion.

VIII. Some Speculations on Knowledge and God

Our process-theoretic analysis of mass and earlier discussion of ontology suggest that science is generally limited to the study of only a subset of the features recognized by Whiteheadian metaphysics. Perhaps the three most salient strictures are these:

(i) Science restricts its attention to societies that admit serial order. This is because science deals in continuity, and hence science apart from time is nonsense.

(ii) Science restricts itself to abstractions that depend not on the full structure of an actual entity, but on those elements of structure that an entity inherits through its objectified past. In other words, immediate creativity, in the Whiteheadian sense, is beyond its reach.13

(iii) In describing the world, science can at best associate a cluster of eternal objects (high abstractions) with an event. Indeed, Richard Rorty (in ME) has asserted that one of the distinguishing characteristics of process theory is the denial that an event is nothing more than the sum of its ingressed eternal objects. It is precisely these eternal objects that lend the accustomed intersubjectivity to scientific measurement and analysis.14

We raise these limitations because they bear directly on Charles Hartshorne’s question of the reconciliation of special relativity’s denial of absolute simultaneity with the process view of God.15 How indeed can God participate both as possible subject and object in every actual occasion in a universe subject to a principle of locality? In a universe for which action at a distance is impossible and the transmission of signals is limited by the speed of light? Our speculative answer is this: the principle of localization pertains only to causal relations mediated by the restricted class of abstractions admissible to science. In particular, this restricts the application of localization to only those abstractions (such as mass) that depend only on the ingression of the objectified past. This point of view accords well with Bell’s theorem, which provides for the instantaneous correlation of separated events, in this case what appears to be the simultaneous choice of orientations. Thus we hold that creative synthesis, as such, is then not subject to localization, and thus without contradicting the current laws of physics God is free to participate in the creative process across the entire universe, because God’s contact with the world is not mediated by scientific abstraction and is therefore not subject to the restrictions of a local, postprojective theory.

IX. Conclusion

Broadly speaking, this paper has presented a rationale for materialism in a process-driven universe, in the sense that we have tried to account for the origins and effectiveness of materialist-substance ontology in a Whiteheadian world. Following Whitehead, we see the appearance of mass in the world as a fundamental element in the way that a complex, resilient society might approach reality and survive. (The same might be said for the notions of duration, extension, and even electrical charge.) That this approach is successful is undeniable: after all, we are here to affirm it. But a metaphysical penalty is assessed in connection with the consequent inferred and misplaced concreteness, which drains reality of its creative richness. By asserting the process-theoretic foundations of our world, we can maintain both science and God and thus escape the materialist malaise-perhaps never better expressed than in this brief excerpt from a work held by many to be the greatest novel ever written:

From the moment Levin saw his beloved brother dying and for the first time looked at the problems of life and death in the light of what he called the new convictions that between the ages of twenty and thirty-four had imperceptibly taken the place of the beliefs of his childhood and youth, he was horrified not so much by death as by a life without the slightest knowledge of where it came from, what it was for, and why, and what it was. The organism, its dissolution, the indestructibility of matter, the law of conservation of energy, evolution -- these were the words that had replaced his former faith. These words and the concepts associated with them were very useful for intellectual purposes, but they made no contribution to life, and Levin suddenly felt he was in the position of a man who had exchanged a warm fur coat for a muslin blouse, and who the first time he finds himself in the frost is persuaded beyond question, not by arguments but by the whole of his being, that he’s no better than naked and is inevitably bound to perish miserably.16

Process metaphysics requires no such mutually exclusive exchange: we have our muslin blouse and our warm coat, and an abundant, nonexistentialist universe in which to enjoy both.



1. We acknowledge the eventual technical insufficiency of this naive exclusion principle, which sounds rather like the essential distinction between waves and particles: waves superimpose; particles do not. Insofar as this distinction is blurred by quantum theory, so that ordinary light has particle aspects and ordinary baseballs have wave aspects, much more needs to be said. But we only propose this feature as a starting point in acquiring the informal use of the predicate has mass. One learns quickly, indeed, that one can walk through the shadow of a lamppost, but not through the lamppost itself.

2. Perhaps more commonly known as Giles of Rome.

3. This is no small feat, as Bronowski and Mazlish point out (WIT 111), in that the four-element tradition did not recognize the heavenly bodies as material. Indeed, their lofty perch in the upper reaches of the world was not a place held by things material. See also CM 55.

4. Quoted from CM 56.

5. One must not confuse our idea of a representation space for an event with that of a state space for a dynamic system, which in the case of a one-particle system would also require a momentum vector.

6. At this point, one might well ask with some urgency to what extent philosophers are enjoined to accept the definitions of science in their own analyses. Some would maintain that philosophy is entirely free to dismiss scientific formalisms. Others might declare that the goal of philosophy is not the explication of science (as Russell and Whitehead once tried to explicate mathematics), and hence the question is irrelevant. We hinted at our own view in the introduction: philosophy can sometimes profitably illuminate the phenomenological and semantic path from experience to science, and, in so doing, both the beginning and the end of the journey are prime data.

7. The very notion of continuity, which ultimately requires the context of an abstract mathematical domain, is a case in point. The idea of continuous change presupposes some measure of nearness, an abstraction that distills itself into the mathematical formalism of a topological space. While the attendant technicalities may be necessary to systematize this idea, we are hardly without it naively.

8. One might say that everyday entification occurs somewhere between the periphery and the center of the web. What is critical for us here is only that it does not occur at the periphery. While one might assert that both Whitehead and the logical positivists mandate that we always begin at the edge, Kuhn (SSR), Quine, and Rorty have shown that one cannot remain there for long and still make sense of the world.

9. Physicists often suffer the same circularity in defining a closed system.

10. Abstractions have a complex ordering, which is not adequately symbolized by Whitehead’s abstractive hierarchies. We have attempted to show a more appropriate ordering of abstractions as forms of concrescence in PAWM.

11. We are being blatantly, but we hope suggestively, metaphorical here; what we intend by this characterization of mass is entirely preprojective. Nonetheless, that the metaphor recalls Newton is most satisfying.

12. Recall that Kepler used the converse of this implication to deduce the substance of the planets.

13. This is a generalization of the conclusion we reached in PP about measurement.

14. A measurement is intersubjective, not in the sense that everyone would record the same numbers, but in the sense that everyone reading the records agrees on what they are seeing, and in particular on when two measurements agree. Thus a recorded value of 3.1416 centimeters may have whatever experimental uncertainty, but as a record it is unambiguous and eminently distinguishable from 2.7183 or even 3.1417.

15. In a personal letter to Henry dated October 20, 1969, Hartshorne states, "My own worst problem is how to reconcile a process view of God with relativity’s denial of absolute simultaneity." His solution to the problem came in response to a modified Whiteheadian theory of events proposed by physicist Henry Pierce Stapp in "Quantum Mechanics, Local Causality, and Process Philosophy" (PS 7 [1977]: 173-182). Hartshorne’s article followed Stapp’s article in the same issue and is titled "Bell’s Theorem and Stapp’s Revised View of Space-Time."

16. From Anna Karenina (1876) by Leo Tolstoy, Part Eight, Chapter Eight; translation by Joel Carmichael (1960).



CM Max Jammer. Concepts of Mass in Classical and Modern Physics. Cambridge: Harvard University Press, 1961.

ME Richard Rorty. "Matter and Event," Explorations in Whitehead Philosophy. New York: Fordham University Press, 1983.

MFF Saunders Mac Lane. Mathematics: Form and Function. New York: Springer-Verlag, 1986, 455-456.

PAWM Granville C. Henry and Robert J. Valenza. "The Principle of Affinity in Whiteheadian Metaphysics," Process Studies 23(1994), 50-49.

PP Granville C. Henry and Robert J. Valenza. "The Preprojective and the Postprojective: A New Perspective on Causal Efficacy and Presentational Immediacy," Process Studies 27(1998), 33-55.

PMN Richard Rorty. Philosophy and the Mirror of Nature. Princeton: Princeton University Press, 1979.

SSR Thomas S. Kuhn. The Structure of Scientific Revolutions. Chicago: The University of Chicago Press, Second Edition, 1970.

TDE Willard Van Orman Quine. "Two Dogmas of Empiricism," From a Logical Point of View. Cambridge: Harvard University Press (second, revised edition, 1961); New York: Harper and Row, 1963.

TIPT Willard Van Orman Quine. "Things and Their Place in Theories," Theories and Things. Cambridge: The Belknap Press of Harvard University Press, 1981.

WEPM Granville C. Henry and Robert J. Valenza. "Whitehead’s Early Philosophy of Mathematics," Process Studies 22 (1993), 21-36.

WIT J. Bronowski and Bruce Mazlish. The Western Intellectual Tradition: From Leonardo to Hegel. New York: Harper and Row, 1960; Harper Colophon edition, 1975.

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