Bell’s Theorem, H. P. Stapp, and Process Theism

by William B. Jones

William B. Jones teaches philosophy at Old Dominion University, Norfolk, Virginia. He holds a Ph.D. in physics (University of Virginia, 1970) and a Ph.D. in philosophy (Vanderbilt University).

The following article appeared in Process Studies, pp. 250-261, Vol. 7, Number 4, Winter, 1977. 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.


An analysis of a concept in process thought dealing with one aspect of quantum mechanics which theorizes that the earlier of two events cannot affect the other if the distance between them is so great that a light signal cannot traverse it during the time interval separating the two events.

In a series of papers (3:1-10, 4:1303-20, 5:270-76, 6:313-23, and 7) Henry Pierce Stapp has argued that quantum mechanics is incompatible with the principle of local causes. According to this principle, the earlier of two events cannot affect the other if the distance between them is so great that a light signal cannot traverse it during the time interval separating the two events. Stapp arrives at this conclusion through a quantum mechanical analysis of the possible results of a double spin-measuring experiment, i.e., an experiment in which one measures the spins of each of two slow neutrons, say, which have just collided. (The "spin" of a particle like a neutron is roughly analogous to the spin of a top or a billiard ball.) He shows that a contradiction ensues if it is assumed that the possible results of measuring the spin of one of two such particles in various directions are independent of the direction chosen for measuring the spin of the other member of the pair of particles.

Stapp’s way of handling this difficulty is to postulate (1) that the fundamental events of which the universe is composed are well-ordered as regards their coming into being (which, according to Stapp, is not the same as their being well-ordered with respect to measurable time) and (2) that the character of all events prior to any given event is available to it regardless of whether or not a light signal could traverse the distance between them in the time interval separating them. It probably should be emphasized that these conditions apply only at the level of fundamental events and in no way contradict the limitations imposed by relativity theory upon the transmission of signals accessible to human beings. The application of these principles to the double spin-measuring experiment leads to the conclusion that one or the other of the measuring events came into being first and that the other is, indeed, not independent of it.

Charles Hartshorne (2) finds Stapp’s proposal noteworthy because making the assumption that the fundamental events of the universe are well ordered effectively disposes of a recalcitrant problem that has plagued theistic process philosophers for some time. If one accepts the relativistic view that all events are not temporally well-ordered (i.e., that in some cases the question of which of two events precedes the other has no unique answer), then it becomes very difficult, if not impossible, to work out a consistent account in which God has a temporal aspect of the sort Hartshorne envisions, i.e., an awareness and enjoyment of the successive satisfactions attained by the actual occasions. According to relativity theory the question of which of two events precedes the other can only be answered (or asked) with respect to some frame of reference, all of which are on a par and have no claim to special status. Consequently, one is left with no clear answer as to the order in which events present themselves to God, and the problem is compounded by the stipulation that each actual occasion prehends God and thus has access to his awareness of other actual occasions (God’s "consequent nature"). Thus the different time-orderings are brought into confrontation with each other.

All these problems evaporate if the fundamental events of the universe are indeed well-ordered, even if relativistic considerations do prevent us human beings from determining what some of these orderings are. Of course, the crucial question is: what reasons are there for thinking that the fundamental events really are well-ordered? Stapp’s argument for this conclusion has two parts. He first presents a mathematical proof, inspired by Bell’s theorem, that a contradiction results if one assumes: (a) the principle of local causes, (b) an elementary theorem of quantum mechanics, and (c) what Stapp calls the "assumption of contrafactual definiteness" (roughly the assumption that measuring procedures which were not carried out would have yielded definite results had they been carried out and that these possible but unrealized results are restricted by the same laws that apply to the results of actual measurements). The second part of his argument is that the proper, if not the only, way to remedy this situation is to reject the principle of local causes.

The intent of this paper is twofold: (i) to present Stapp’s proof in a nontechnical form so that nonphysicists can be apprised of just what it is that he has proved and (ii) to draw out and examine the assumptions of the proof so that its significance for philosophy, especially process philosophy, can be assessed. Part I below addresses the first of these concerns; part II, the second.

I. Stapp’s Generalized Version of Bell’s Theorem

Stapp’s proof is a modified version of what is generally referred to as "Bell’s theorem." In its original form, this theorem is addressed to the question of whether it is theoretically possible for the statistical conclusions of quantum mechanics to be explained by a hypothetical, experimentally inaccessible realm of microevents characterized by a set of "hidden variables." Bell’s conclusion is that such an explanation is not possible as long as the principle of local causes is assumed. According to this rather common-sensical principle, if two physical systems (particles, for example) have ceased interacting or have never interacted, neither is affected by changes induced in the other. Thus, two widely separated systems which have interacted in the past would each be unaffected by measurements performed upon the other. The system considered by Bell and by Stapp is made up of two particles, neutrons, for example, whose "spin" directions have become correlated through some collision process -- in somewhat the same way that the axes of rotation of two spinning billiard balls might be correlated after they have sideswiped each other (with each spinning in a direction opposite to that of the other). According to the principles of quantum mechanics, two slow" neutrons which have brushed past each other would tend to have their spins in roughly opposite directions.

Stapp, unlike Bell himself, is not concerned with "hidden variables." He rather develops a version of "Bell’s Theorem" which is intended to show that, given certain very general assumptions, quantum mechanics itself, as it manifests itself in a simple theorem, is incompatible with the principle of local causes. Stapp’s argument rests directly upon a theorem in quantum mechanics concerning the behavior of a certain class of interacting pairs of particles, among which are neutrons. He focuses his attention upon the correlation that is established between the "spins" of two "slow" neutrons when they scatter off each other (collide with or brush past one another). It happens that the spins of neutrons are quantized, i.e., allowed to assume only certain discrete values. Indeed, the magnitude of the spin of a neutron is the same in every case, being absolutely fixed and unchangeable. However, the direction of the spin of a neutron can change, so that one can ask whether a neutron has "spin-down" or "spin-up" with respect to a given coordinate axis in much the same sense that one can ask whether a spinning top is rotating in a clockwise or a counterclockwise direction when viewed vertically downward. If it is ascertained that one member of a pair of neutrons, which have interacted in the above manner, has its spin up with respect to a given axis, then it is reasonably certain that the other member of the pair, if measured, will be found to have its spin down with respect to this same axis.

Stapp’s proof, however, requires that the effects of measuring the spins of the two neutrons along different axes be considered. It is easier to grasp what this involves if one employs the following model of the experiment. The model is not Stapp’s, but its use makes it easier to understand his arguments. The model is the following: think of the neutrons, after the scattering event, as traveling in opposite directions (away from each other) down a hollow axle or pipe at the ends of which spin-measuring devices are attached like thin wheels. These "wheels" thus lie in planes perpendicular to the line along which the neutrons are moving. When a neutron passes through the center of one of these devices, it measures the spin of the neutron along an axis coinciding with a radius of the "wheel." More specifically, it determines whether the neutron has spin-up or spin-down along this axis, which is, of course, perpendicular to the neutron’s line of flight. For convenience, let it be assumed that the tube through which the neutrons are traveling is lying horizontally and that initially both of the "wheels" are so turned that they will determine whether a passing neutron has spin-up or spin-down with respect to the vertical. Note that since it is with respect to a radius of a wheel that the spin is measured (and not with respect to a diameter), the "up" direction for any orientation is always well-defined: up is "outward" along the radius that is taken as a coordinate axis. In effect Stapp considers the results of employing two different orientations for each of the "wheel" spin-measuring devices for a total of four distinct configurations. In the first two cases, one of the wheels is left as just described (i.e., making spin-determinations -- "up" or "down" -- with respect to the vertical) while the second "wheel" is initially given the same orientation, to form the first configuration (1), and is then turned through an angle of 90_ to form the second configuration (2), in which the two axes along which spin-determinations are to be made are perpendicular. In the third configuration (3), the second "wheel" is returned to its initial orientation (so that it will make spin-determinations with respect to the vertical) while the first "wheel" is turned through an angle of 135_ -- in the same direction as the second "wheel" was turned in case (2). Leaving the first "wheel" in this position and also turning the second "wheel" through an angle of 900 results in a fourth configuration (4). The angle between the two axes (one in each "wheel") along which the spin-projections are measured is, in these four cases: (1) 0_, (2) 90_, (3) 135_, and (4) 45_

The convention followed here will be to assign a spin value of + 1 to a neutron if it is found to have spin-up with respect to a given axis and to assign a spin value of -1 if it is found to have spin-down with respect to that axis. For a given configuration of the two axes and for a given scattering event between two neutrons, the "wheel" spin-measuring devices will record a value + 1 or -1 for each neutron. From each such pair of numbers, one can then form a product, which itself can assume only the values + 1 and -1. The theorem from quantum mechanics which Stapp employs states that if one records the values of these products for a large number of neutron-scattering events (for the same configuration of the "wheels") and averages them, then the result will equal the negative of the cosine of the angle between the axes along which the spins are being measured. This is a statistical conclusion, and a large number of cases must be considered for it to be accurate. It is by applying this theorem to each of the four configurations described in the preceding paragraph that Stapp arrives at the conclusions which he calls the generalized Bell’s theorem.

In his analysis, Stapp considers a set of neutron-scattering events, N in number, and the possible results of measuring the spins of the neutrons involved along the four different sets of axes just delineated. It is easier to understand Stapp’s argument if one thinks of a set of N apparatuses like that described, in each of which a neutron-neutron scattering event has just occurred at the center of the horizontal tube and the two neutrons have ceased to interact and are on their respective ways down the opposite halves of the tube. Let it be supposed that the N apparatuses are all lined up with their long, horizontal tubes parallel and that they have been fitted with a control device that enables an experimenter stationed at either end of this array of instruments to control the orientation of the spin-measuring devices at his/her end of the tubes. Suppose one of the experimenters has the choice of selecting either the vertical orientation for all of the devices under his/her control or an orientation that is 135° away from the vertical. Suppose further that the experimenter at the opposite end of the array of horizontal tubes is allowed to choose either a vertical orientation for all of his/her spin-measuring devices or an orientation that is 90° away from the vertical. Finally, suppose that each experimenter has sufficient time after the neutron pairs in each apparatus have ceased interacting to choose between the two orientations allowed. Thus in each of the N scattering processes there is a period of time after the neutrons have ceased to interact during which it is possible that their spin-projections will be measured along any one of the four pairs on axes described above. Furthermore, according to the principle of local causes, the selection of an axis along which to measure the spin-projection of one of the neutrons should have no effect on the other neutron if the choice is made after the two particles have ceased to interact. Thus the set of N spin-measurements obtained by means of one of the spin-measuring devices should be the same no matter how the second spin-measuring device is oriented when it measures the spin of the other member of each of the neutron pairs.

There are thus two possible sets of spin-values for the neutrons coming down the tubes toward the first experimenter: one set A, (with a spin value of + 1 or -1 for each neutron) that will result if he/she chooses the vertical orientation for the spin-measuring devices under his control and another set B that will result if he/she chooses the orientation differing from the first by 135°. Similarly, there are two sets of possible spin-values for the neutrons approaching the other ends of the tubes: one set C that will result if the experimenter stationed at that end of the tubes chooses the vertical orientation for the spin-measuring devices under his/her control and another set D that will result if he/she chooses the orientation achieved by a 90° rotation from the vertical. Since by this time the neutrons have ceased interacting, the choice made by either of the experimenters should have no effect upon the values obtained by the other. Consequently, since there are two possible sets of spin measurements for the N neutrons that come down each end of the horizontal tube, there are four sets of the products that are formed by multiplying a possible spin value for one neutron by a possible spin value for the other member of the pair, one set of such products for each of the pairs of sets, A-C, A-D, B-C, and B-D. Obviously, one could form four such sets of products of possible spin-values -- not by considering N simultaneous double spin-measuring experiments, but by considering the possible outcomes of a set of N serial experiments yet to be performed, in each of which it is possible to decide upon the orientation of the spin-measuring devices after the two neutrons have ceased to interact. Either approach may be utilized in developing Stapp’s argument.

The next step in the proof is to note that each of the four sets of spin-value products must satisfy the quantum mechanical theorem referred to above. Since each set of possible spin-values is possible, it represents a set of results of measurements which may, indeed, be realized. If they are realized, these spin measurements would have to satisfy the quantum mechanical theorem, else the theorem would be false -- and there is good evidence available for the correctness of the theorem.

In the case of the first set of products of possible spin-values, the orientation of the axes of the spin-measuring devices is the same; the angle between them is zero; and the cosine of this angle is one (1.0). Consequently, according to the quantum-mechanical theorem in question, if one adds up the N products and divides the result by N, the result should be negative one (-1.0). In the case of the second set of such products, the angle between the relevant axes is 90° which has a cosine of zero (0.0). So averaging the second set of products should yield zero (0.0). And in the third case, since the cosine of 135° (the angle between the axes) is the negative reciprocal of the square root of two, the average of this set of products of spin-values must equal the positive reciprocal of the square root of two. Finally, in the fourth case, the angle between the axes is 45°, of which the cosine is the reciprocal of the square root of two. Consequently, the average of the last set of products must equal the negative reciprocal of the square root of two.

Now each statement to the effect that one of the averages just discussed equals the number given may be regarded as a mathematical equation relating the spin-values that are averaged. There would be four such equations. What Stapp shows mathematically is that these four equations lead to a contradiction. Proving this is a straight-forward exercise in algebraic manipulation. It cannot be reviewed here. The interested reader is advised to consult Stapp’s own treatment (4:1306-08). The vital question that must be addressed here is what is to be made of the fact of this contradiction. What is to be made of the fact that the four statements in the preceding paragraph about the four product averages cannot all be true?

II. The Assessment of the Proof

The immediate inference to be made from the fact that the four equations described just above lead to a contradiction is either (a) that one or more of the equations is false or (b) that it is somehow illegitimate to combine them algebraically in the way Stapp does in arriving at the contradiction. One can put aside, with reasonable confidence, the suggestion that all four of the equations are false. Given an experimental set-up like that described above, it is always possible actually to carry out the measurements for one of the possible configurations of the spin-measuring devices. Furthermore, experiments of this sort have been performed, and, as Stapp notes (1:938-41), their results obey the mathematical equations which are appropriate according to the quantum mechanical theorem Stapp uses in his proof. Thus, the theorem is experimentally well confirmed. But if one accepts the correctness of the theorem, is one not also obliged to accept the correctness of all four of the equations? They are all instances of the same general relationship, which is expressed by the theorem, and are thus on the same footing. Their common status is also shown by one’s being free to choose any one of the four for experimental test. As was explained in the previous section of the paper, under the right conditions, one can even make this choice after the pair of particles being considered have ceased to interact.

The unavoidable conclusion would seem to be that the second of the two alternatives mentioned at the beginning of the section must be correct: some error must have been committed in combining the four equations in the way required to reach the contradiction. This is the line of argument followed by Stapp. According to him, the error lies in assuming that one is dealing with the same set of possible spin-measurement results for the particles coming out one side of the apparatus described no matter what orientation one considers for the spin-measuring device (s) on the other side of the apparatus. Put more formally, this is the assumption that in averaging the first two sets of spin-value products one is dealing with the same set of possible spin-values A for the particles emerging from one side of the apparatus whether one combines them with the set Gin the first spin-averaging process or with the set D in the second averaging process. Stapp’s claim is that the set of numbers labeled by A in the first averaging process is not really the same as the set labeled by A in the second averaging process. In other words, the possible spin values (with respect to a given axis) for one member of a pair of until-recently interacting particles are not the same in case the spin of the second member of the pair is to be measured along one axis as they would be if the spin of the second particle were to be measured along another axis -- even if the selection of the axis for the second particle can be made after the two particles have ceased interacting.

Stapp’s thesis about possible spin-values must not be confused with similar claims about the results of actual measurements. Stapp is not suggesting that the actual results of spin measurements for one member of a pair of until-recently interacting particles will be affected by the choice of an axis for measuring the spin of the other member of the pair -- even if this choice is made after the particles have ceased interacting. Stapp’s thesis is quite compatible with its being determined experimentally that changes in the orientation of the spin-measuring device applied to one member of such a pair of particles have no significant effect upon the statistical make-up of spin-measurement results for the second member of such particle pairs.

In order to understand this seemingly paradoxical situation, one must keep in mind that, for a given collision event, it is theoretically impossible to carry out more than one of the four experiments corresponding to the four possible configurations of the spin-measuring devices described above. Of course, it is manifestly impossible to give either of these devices more than one orientation at a time, but the problem goes deeper than that. Given, as noted above, that the overall magnitude of the spin of the neutron is fixed, simultaneously measuring its spin in two different directions, like the two available to each of the spin-measuring devices, would violate the Heisenberg indeterminacy principle. Thus only one of the four possible sets of spin-value products discussed in part I could ever be formed from spin-values actually obtained from experiment. The four sets of spin values, A, B, C, and D, from which these products are formed are all possible, but they are not all simultaneously possible. Performing the measurements that yield the spin values making up sets A and C precludes in principle performing measurements that yield the spin values making up sets B and D. Furthermore, the principles of quantum mechanics also forbid postulating that a neutron really "has" definite spin values in two such different directions whether they can both be measured or not -- if these postulated spin values are taken as (possible) parts of a theoretical account of the neutron which has experimentally testable implications. Indeed, Stapp’s proof itself may be regarded as a demonstration of the incompatibility of quantum mechanics, as embodied in the theorem used in the proof, and the assignment of definite spin values for different directions to the neutron.

Accordingly, Stapp is careful to distinguish between (a) attributing definite spin values in more than one direction to a particle like the neutron and (b) asserting that if the spins of certain pairs of such particles are or were to be measured in this or that direction, a specific mathematical relation will or would be found to hold, on a statistical basis, between the spin values of the members of the pairs. Stapp argues that while the uncertainty principle does forbid ascribing such spin values to particles like the neutron, it does not rule out as meaningless or unphysical talk about the outcome of various possible measuring procedures, even if it is theoretically impossible for more than one of them actually to be carried out.

The argument here is subtle. It is clear enough that the spin values dealt with in Stapp’s proof are not intended as hypothetical characteristics of particles; but, taken as a class, they are not all possible experimental spin values either. Each corresponding pair of members of the sets A and B (as well as C and D) are alternative possible experimental spin values. One may choose to measure the spin of a given particle in either of the two relevant directions, but cannot choose to measure it in both. One may choose to determine the membership of set A or set B, but cannot choose to determine them both. The same may be said of sets C and D. Now, Stapp’s proof deals with the joint class made up of A, B, C, and D. It is upon the properties of this joint class that his conclusion rests. But from what has just been said it is clear that half of the quantities which make up this class are unknowable in principle. Stapp’s conclusion, then, is not about the results of actual measurements nor about a class of class of measurements that are all possible. It is rather about a class of measurements that are possible individually but not jointly: no more than half of them can ever be performed.

The assumption that it is meaningful and proper to employ symbols designating the results of unperformed and, in half the cases, unperformable measurements Stapp calls the "assumption of countrafactual definiteness," which he has defended in one of a series of lectures given at the University of Texas in the spring of 1977. In these lectures Stapp counters the suggestion that symbols which designate quantities that are unknowable in principle are meaningless by pointing out that philosophers of science have come to recognize the need for theories that have reference to such unknowable quantities. He cites as an example Maxwell’s electromagnetic theory with its postulated electric and magnetic fields propagating in a vacuum. The magnitude of these fields in a vacuum cannot be measured because any attempt to do so will destroy the vacuum. They are thus unknowable in principle. Stapp concludes that being unknowable in principle is not a fatal defect. He further notes that the consideration of alternative possibilities is surely meaningful and of considerable theoretical and practical usefulness.

Stapp is certainly correct in arguing that talk about quantities that are unknowable in principle is not necessarily meaningless, and the example he cites surely supports this claim. However, there appears to be an important difference between the unknowable quantities found in Stapp’s proof and the kind cited in his defense of the meaningfulness of such discourse. Consequently, this argument contributes very little either toward defending the cognitive significance of Stapp’s proof or toward clarifying the status of the unknowable spin values with which the proof deals; the same can be said of his argument defending the general significance of talk about possibilities. Stapp’s use of Maxwell’s electromagnetic theory as an example to illustrate the propriety of having reference to unknowable quantities is particularly unfortunate in that it beclouds the central issue of the proper status of the spin values dealt with in the proof. The notion of an electromagnetic field propagating in a vacuum is meaningful precisely because it is part of an elaborate physical theory which has numerous well-confirmed observable implications. Clearly, the basis for the meaningfulness of Stapp’s unknowable spin values must be of a different character. He goes to considerable length to distinguish his treatment of possible spin values from such a unified, theoretical account of the behavior of particles like the neutron. He does this in order to escape the strictures of the uncertainty principle, as explained above. Furthermore, he makes it quite clear that the view that he is advancing has no real implications as regards the outcome of actual experiments.

So the meaningfulness of talk about the joint class, A, B, C, and D, cannot be defended as being the same kind of theoretical talk as is found in such as Maxwell’s electromagnetic theory. But can it be defended as nothing more than talk about alternative possibilities such as everyone engages in almost daily? Certainly, there is nothing particularly problematic about asserting all of the following: (a) if two classes of measurements A’ and C’ are performed, their results will be related in a particular way; (b) if the two classes of measurements A’and D’ are performed, their results will be related in a certain way; (c) if the two classes B’ and C’ are performed, their results will be related in a certain way; and (d) if the two classes B’ and D’ are performed, their results will be related in a particular way. Furthermore, no special problems would seem to be posed by the fact that only one of these four possibilities can be realized even in principle. (Only one member of the pair A‘, B’ and one member of the pair C’, D’ can be realized.) And one can certainly represent the relationship between the two classes of (possible) measurements in each of the four cases by a mathematical equation. Four equations would thus result.

But it does not follow from all this that one can then treat these four equations as a set of simultaneous equations expressing various relationships between the members of the joint class A, B, C, and D. Certainly, the four statements (a), (b), (c), and (d) do not in themselves provide a basis for treating the equations in this way. It is easy to see that this is so if one contrasts (1) merely affirming these hypothetical statements with (2) entertaining the possibility that the antecedents of two or more of them might be satisfied simultaneously. Now, each of the statements says that if certain conditions are met, certain measurements made, then the results of these measurements will satisfy a particular mathematical relation, one of the four equations in question. If it were possible simultaneously to satisfy the conditions mentioned in the antecedents of, say, the first two statements, (a) and (b), then one could form a compound statement of the following form: If conditions "a" and conditions "b" are satisfied, then equation "one" and equation "two" will hold between the results obtained. And one could then combine equations "one" and "two" by addition, subtraction, substitution -- by any of the ways that are appropriate for simultaneous equations.

This same point can be put in terms of possible spin-measurements: If the three classes of spin measurements A’, C’, and D’ are all performed, then (a) the average of the spin-value products formed from sets of spin- values A and C will equal negative one (-1.0) and (b) the average of the spin-value products formed from sets A and D will equal to zero (see the discussion near the end of part I). In such a case, one could combine the resulting equations by addition, subtraction, substitution, etc. But, of course, it is absolutely impossible, because of the indeterminacy principle, simultaneously to determine experimentally the spin values making up both set C and set D. Consequently, it is impossible to use this method to provide a basis for treating the two equations as simultaneous equations over the quantities contained in the joint class A, C, and D.

It should be clear from the above discussion that while entertaining the possibility that the antecedents of the hypothetical statements (a), (b), (c), and (d) might all be satisfied at once would permit the four equations under discussion to be treated as simultaneous equations, merely affirming the four statements themselves does not. The meaningfulness of talk about alternative possible measurements does not in itself ensure the propriety of conjunctively combining equations which each express relationships among the results of one of the four possible measuring procedures. And since Stapp has provided no further arguments for the meaningfulness of the joint class A, B, C, and D or for the propriety of treating the four equations relating the four sets of spin-value products as simultaneous equations, one can only conclude that both of these matters stand in need of considerable clarification and that any philosophical claims which depend upon the conclusion reached in Stapp’s proof are in jeopardy.

Quite obviously, the crucial question for those who favor a Whiteheadian type of framework is precisely what status do alternative possibilities like the two sets of spin-measurement results C and D have in such a scheme. This question is of great moment not only to those who, as Hartshorne, are interested in Stapp’s proof and the issues upon which it bears but also to all who recognize the importance of fitting quantum mechanics, one of the two absolutely fundamental theories of modern physics, into Whitehead’s scheme. Stapp has certainly demonstrated that this task is no trivial exercise but calls for some first class philosophical work. Indeed, his proof shows quite forcefully that quantum mechanics poses special problems for any view that, instead of regarding real (as opposed to pure or purely logical) possibilities as subjective fictions, accords them objective status and makes them independent of the observer. But Whiteheadians may very well wish to raise the following question: does granting that certain possible (but unperformed) experiments would have yielded definite results (had they been performed) commit one to admitting the propriety of treating these "possible" results as definite, but unknown, quantities governed by a set of simultaneous linear equations? Some Whiteheadians may want to reject talk about sets of spin-values that would have been obtained had certain experiments been carried out on the grounds that, in a Whiteheadian context, one can properly talk only about sets of such values that might have resulted. Thus some may wish to argue that one cannot assume that a certain set of spin-values would have been obtained no matter which of conditions a and conditions b are met -- on the grounds that the provisions for novelty in Whitehead’s system preclude such an assumption.

The very possibility of this line of argument calls attention to the fundamental question of this entire matter: even if the contradiction arrived at in Stapp’s proof does constitute a genuine problem, must one accept his method of disposing of it? Is Stapp’s postulation of a well-ordered set of fundamental events with effectively instantaneous communication or causation between them the only or even the best solution to the problem posed by his modified Bell’s theorem? This is a question which will surely need to be considered by philosophers who have an interest in the very significant issues raised by Stapp’s work.



1. Stuart J. Freedman and John F. Clauser, "Experimental Test of Local Hidden-Variable Theories," Physical Review Letters 23/14 (April 3, 1972), 938-41.

2. Charles Hartshorne, "Bell’s Theorem and Stapp’s Revised View of Space Time," Process Studies 7/3 (Fall, 1977), 183-91.

3. Henry Pierce Stapp, "Correlation Experiments and the Nonvalidity of Ordinary Ideas About the Physical World," LBL-5333, Berkeley, California, 1968.

4. Henry Pierce Stapp, "S-Matrix Interpretation of Quantum Theory," Physical Review D 3/6 (March 15, 1971), 1303-20.

5. Henry Pierce Stapp, "Bell’s Theorem and World Process," II Nuovo Cimento 29B/2 (October 11, 1975), 270-76.

6. Henry Pierce Stapp, "Theory of Reality," Foundations of Physics 7/5-6 (1977), 313-23.

7. Henry Pierce Stapp, "Quantum Mechanics, Local Causality, and Process Philosophy," edited by William B. Jones, Process Studies 7/3 (Fall, 1977), 173-82.

Support from the Old Dominion University Research Foundation is gratefully acknowledged by the author.