Completing the Picture:

What Do You Give a Theory

That Has Everything?

Jonah Bossewitch

PHI 327

Professor Bas van Fraassen

Preceptor Chad Mohler

May, 1996

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have a knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements. Clearly, then, systematic knowledge of nature must start with an attempt to settle questions about principles."

Aristotle, Physics I.I.184a

As Aristotle wisely reminds us, a meaningful discussion of whether a theory is complete, must begin with an attempt to settle our criterion for completeness. The criterion for the completeness of physics must be particularly demanding, since physics is the study of the World, a relentless search for the principled cause(s) which unite the totality of its elements. Systematic knowledge of physics entails systematic knowledge of the world, and a claim that physics is complete, is a claim that we have found a complete theory of the world, which provides us with a complete understanding of it. What would a complete theory of the world be capable of telling us about the world? Is there any knowable knowledge that a complete theory of the world might be exempt from expressing? What types of objects are included the world described by physics? Can we ever know when a theory is complete?

Some physicists have a two word answer to all of these questions: Quantum Mechanics. They believe that Quantum Mechanics (QM) is the final step for physics, capable of expressing all that can ever be expressed about the universe, since it is the empirical bottom line. The trouble is, not only do physicists disagree over how to relate the empirical bottom line to the theoretical bottom line, they also disagree over whether such a project is possible or even meaningful. QM has brought many ancient philosophical dilemmas into contemporary focus, and formerly paranoid skeptical positions, such as denial of an external absolute reality, or the admission of "true" indeterminism, have become integral components of many well accepted interpretations. This outcome is certainly surprising, as most people expected that a complete theory would leave no questions unanswered, and would settle all controversy once and for all. QM has certainly not provided us with this utopian state of knowledge, and some might point to this fact alone to prove that QM is incomplete. But, is there anything else about the world that any theory could ever possibly tell us, and if not, is QM truly lacking?

This strange incarnation of completeness does not fit the intuitive notion of completeness, and in order to get a better sense of the concept, it is helpful to examine various rigorous formulations of completeness that have been proposed by philosophers of science (some of whom were also physicists on the side). By comparing crude attempts to define completeness, we will get a better feel for the task at hand, and hopefully learn how to evaluate and refine a more sophisticated criterion.

The Deterministic Completeness Criterion: "We have a complete scientific account of a type of phenomena when we have exhibited it as a part of a deterministic process [a process in a deterministic system] and the task of science is not achieved until such a complete account has been given."

van Fraassen lecture notes, 2/15 [1].

Despite the ambiguity in the concept of determinism, this completeness criterion is a first stab at what we are searching for, and provides us with a model of what a criterion might look like. We can bracket the difficulties associated with defining determinism, and focus on what this criterion implies.

This criterion reminds us that it is important to distinguish between necessary and sufficient conditions of completeness. All criterion must explicitly provide us with sufficient conditions, such that if these conditions are met, the theory in question is complete. However, technically speaking, these conditions need not be necessary, and some criterion might not specify the only conditions that determine if a theory is complete. In other words, completeness might be satisfied by some other conditions, which also imply completeness. Thus, there need not be a unique formulation of completeness, and in practice, different criterion may describe identical sets of theories.

The deterministic criterion asserts that a theory is complete if and only if the phenomena is described by a deterministic process, enforcing the necessity of determinism in any good theory. This is a glaring assumption, since it implies that all phenomena are a part of a deterministic process, and until we have uncovered that process, our theory is incomplete. Ironically, if some phenomena are truly indeterministic, our theory is doomed to be considered incomplete, even if it is capable of expressing all that there is to say about a phenomena.

Under this criterion, different interpretations of QM do not receive equal judgment, despite the fact that they are all empirically adequate, and effectively say exactly the same things (i.e. make the same predictions) about the world's observables. The Modal interpretation and some collapse interpretations, make liberal use of nondeterministic processes, while Bohm's interpretation and other collapse interpretations, preserve determinism by sacrificing other properties. Thus, the deterministic criterion would judge the former interpretations incomplete, while the latter interpretations, which produce identical assertions, would be judged complete. In the final analysis, this "objective criterion" makes a substantive claim about the nature of reality, which is grounded in blind faith. The deterministic criterion has more to do with dogma than completeness, and it teaches us to be wary of hidden agendas disguised as completeness criterion.

Even if the deterministic criterion did not suffer from faulty assumptions concerning the nature of the universe, it would still fail to serve as a reliable criterion for completeness. Simply put, the deterministic criterion permits us to be lazy, and settles for explanations that are far from complete. Consider Kepler's Laws of planetary motion. They provide a deterministic model for the phenomena they describe, and, since Kepler exhibited the motions of the planets as part of a deterministic process, the deterministic criterion would judge this to be a complete scientific account of the phenomena. Yet, almost every physicist would agree, that any scientific account of the motions of planets which does not include gravity, is fundamentally incomplete. The deterministic criterion allows us to become arrogant and complacent, claiming to have a complete understanding of a phenomena while there is still much to learn about it.

Finally, the deterministic criterion does not discriminate between "correct" accounts of reality and fabricated accounts which happen to match the observables. Should a criterion for completeness assume that there is a way that things "really" are, or is this another dogma that we ought to discard when evaluating the merits of a theory? This criterion is not the only which faces this problem, and as we shall see, this question strikes a deep epistemological chord, which we will soon grapple with.

The Common Cause Completeness Criterion: "We have a complete scientific account of a type of phenomena when we have exhibited it as part of a 'causal' process [a process in a system in which all positive correlations between factors derive from 'common causes' in the common past of those factors] and the task of science is not achieved until such a complete account has been given."

van Fraassen lecture notes, 2/15 [3].

The common cause criterion is very similar to the deterministic criterion, but differs in a few key respects. Their similarity should come as no surprise, since early formulations of determinism often invoked the concept of causality, and the two concepts are fundamentally related. A common cause is an event C, that always precedes two correlated events A and B.1 The common cause criterion states that given any two correlated events (A,B) there must exist a third event C which is their common cause. Under this criterion, failure to find C indicates an incomplete scientific theory that is simply ignorant of the true common cause.

The common cause criterion suffers from virtually the same flaw that the deterministic criterion does. It assumes that all events have a common cause, and if we should happen to live in a universe where events can not be correlated in such a manner, our theory will be judged incomplete, whether or not there exists a "true" common cause to be found. Furthermore, from an epistemological standpoint, we can never be sure that we have identified the "real" common cause. At best, all the theorizing in the world can only produce a common cause model that accurately describes and predicts the behavior of the system.

However, in contrast to the deterministic criterion, the common cause criterion demands an unending, relentless, pursuit of knowledge, and will never allow us to retire to a tropical island with a complete theory in hand. Even if we manage to find an event C which is the common cause of A and B, we will then be forced to account for the common cause of all events like C, and this process leads to an infinite regress. Once again, the criterion demands that we find a common cause, whether or not there is one to be found. For all we know, the world is composed of infinite chains of causes, since there is no a priori reason to assume otherwise.2

This contrast between the two criterion we have examined so far, teaches us to differentiate between a theory's completeness, and the possibility of our knowledge of its completeness. This distinction hinges on our ability to know whether a theory has satisfied the criterion, and we need to keep a careful watch for criterion which judge incomplete theories to be complete, or complete theories to be incomplete. In most cases, we would rather err on the side of ignorance than of arrogance, and would prefer a more stringent evaluation which filters out false positives (judges T complete when T is not complete) at the expense of false negatives (judges T not complete when T is complete). Depending on your attitudes towards scientific progress, a standard which can never be satisfied, may not be problematic, since it can be regarded as an unattainable ideal, which ought to be strived for, but can never be achieved.

In 19353, Einstein, Podolsky, and Rosen, proposed a rigorous formulation of the completeness criterion that is hauntingly similar to Aristotle's criterion cited at the top of this paper. EPR argue that correctness, (e.g. "the degree of agreement between the conclusions of the theory and human experience") is not identical with completeness, in opposition to the proponents of the deterministic and the common cause criterion. Beyond correctness, "every element of the physical reality must have a counterpart in the physical theory." EPR recognized the philosophical quagmire that they had entered by introducing "elements of physical reality" into a physical theory. They deftly avoided having to provide a comprehensive definition of reality by proposing a sufficient condition for element of reality, which equates a physical element of reality with the ability to predict its value with certainty: "If, without in any way disturbing the system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." Thus, a theory might be correct, but not complete in two ways: (1) There are elements of the theory that do not correspond to elements of reality, or (2) there are elements of reality that are not represented by the theory. In either case, the theory may accurately predict and describe every observable, but it does this the "wrong" way, by fabricating imaginary processes which do not happen to correspond to reality, or by neglecting to inform us of something that is knowable with absolute certainty.

The EPR condition for completeness captures the gist of what we've been after all along. As Redhead puts it, "The idea behind it is very simple. The language in which the physical theory is formulated must be sufficiently rich that every proposition signifying putative relations between the 'elements of reality' can at least be expressed in the language of the theory."4 EPR dodge the issue of absolute reality by defining reality in way that admits possible realities as elements of reality. This is crucial when considering the various interpretations of QM, since by the EPR criterion, they all either pass or fail completeness as a group, since all elements of reality are represented by theoretical elements (albeit not the same ones) in each of the respective interpretations.

EPR did not consider explanations that go beyond "brute facts" to be idle metaphysical baggage. However, they recognized the futility in demanding that the elements of the theory which correspond to the unobservable elements of reality (a description of the system when it is not in an eigenstate), also correspond to the "correct" elements of reality. EPR would have disagreed with the positivists, the instrumentalists, and the anti-realists who believe that interpretations which go beyond the observables should be dispensed with. Instead, EPR would have been sympathetic with Gribbon, who believes that "all such interpretations are myths, crutches to help us imagine what is going on at the quantum level and to make testable predictions. They are not, any of them, uniquely 'the truth'; rather they are all 'real', even where they disagree with one another."5 Although EPR might have believed in a unique truth, they conceded that any theory which could possibly correspond to reality, ought to be considered complete.

The difficulty applying the EPR criterion enters when we attempt to specify exactly what constitutes a "physical quantity". What this question amounts to is a question about the ontological status of elements of reality. Do counterfactuals qualify as elements of reality? EPR argued that we can predict with certainty what the value of an observable would be if we would measure it. According to their criterion, this observable must have a real value, even though we have not yet measured it. They use this reasoning to show that, in certain cases, QM tells us nothing about these elements of reality, and consequently, it must be incomplete.6

Counterfactuals are not the only controversial elements of reality. Consider the case of identical particles? Should they be considered collectively as a single element of reality, or does our theory need to describe them individually? This example can be illustrated vividly by an analogous parable proposed by David Lewis:

"Two gods live respectively on a flat mountaintop and a sharp mountain peak; one throws down manna, the other lightning bolts. Each has godlike omniscience, that is, for every true eternal sentence A, each knows that A—but neither knows anything else. Is there anything else to know?"

van Fraassen, lecture notes 2/27 [4].

Neither god knows which god he is, and it seems that this is a significant piece of knowledge. However, if the gods are truly identical, and differ only with respect to their position (which determines their behavior), then there might not be any content to an assertion which distinguishes between the two. While this might sound strange, this is precisely the Quantum scenario proposed by Bohm and is also intimately related to the Bose-Einstein statistics. Is there anything more to know about particles in identical states that QM doesn't tell us? This is a slippery question which many Quantum mechanics are still grappling with.

Finally, we need to wonder about higher order relationships, such as abstract machines or intentional states? If we can predict with certainty that, "If I jab you with a pin, you will become angry", should we consider the intentional state of anger to be an element of physical reality that physics ought to describe? Can someone truly understand the operations of a computer from the atomic level, without an appreciation for the emergent patterns of information processing described by theories of computation? Answers to this line of questioning seem to be further off than answers concerning counterfactuals or identity, but they also seem more urgent. Roger Penrose has argued that consciousness is proof that physics is incomplete, and has called for the development of a science of consciousness. Should any such project prove successful, it would unequivocally demonstrate that QM is incomplete.

Until some of these questions are settled, a theory's completeness will often be matter of opinion. Technically speaking, a theory can only be said to be complete within a given domain. While we may be able to assert that a theory is incomplete, the above uncertainties show that it is impossible to prove that a theory is complete according to the EPR criterion, without specifying what constitutes an element of reality. Perhaps that's the way that EPR would have wanted it, sending us back to the laboratory and the blackboard to continue the never ending task ahead of us. In the end, we may never be able to claim that our theory of completeness is complete, but this should not stop us from continuing the discussion.

To the extent that I can express in the languge of QM, I pledge my complete honor that this paper was written in accordance with the University's real elements of conduct.

Jonah Bossewitch