Science in Christian Perspective

Logical Indeterminacy, 
Levels of Meaning and Mystery 
DENNIS FEUCHT
Tektronix, Inc.
Portland, Oregon 97077

From: JASA 29 (December 1977): 150-153

An understanding of the nature of mysteries involves a consideration of their logical status and an analysis of the paradigms and analogies used to describe them. Several powerful models for dealing with important mysteries have arisen, and their application to theological problems is considered here.

A central mystery of the Christian faith-the relation of God's sovereignty, man's freedom, and evil-is uncannily similar to the antinomies that arise in science¹ mathematiCS², and philosophy^3.

To begin, let us examine the nature of a simple antinomy. Consider the following two statements, both of which we assume are true:

If statement (1) is true, then statement (2) is also true. However, if statement (2) is true, then by its assertion, statement (1) is false. But if statement (1) is false, then it follows that (2) is also false and thus statement (1) is true. But if statement ( 1) is true, this leaves us where we started and the argument could go around in circles indefinitely. Notice that these two statements cannot be said to be contradictory since we cannot start with one of them and reason consistently to the negation of the other. Instead of converging, the logic "oscillates" without ever reaching a logical solution.

Sometimes the word paradox is used to describe this condition. Formally speaking, however, a paradox does not necessarily mean that a logically indeterminate state has been reached as happens in an antinomy. No amount of further clarification or careful attention to details will change the logical status of an antinomy; a paradox could possibly be resolved with further investigation. One question we will want to ask of a mystery is whether an antinomy is actually occurring or whether we could logically resolve the mystery with more information.

A good example of a mystery in science is the waveparticle dualism of light. Under certain conditions light is best modeled as a particle, such as when it reflects off a mirror. Other behavior, such as interference patterns, causes us to consider it to be like waves in a water pool. Mathematical models of light from quantum physics give more depth to the picture as the particle can be viewed as a constructive superposition of a number of waveforms that cancel out everywhere else. A total model of light requires a consideration of both the particle and wave behavior in a way which does justice to both. Presently, by manipulating the mathematics, a model of light can "fade" from one view into the other in what has been called a complementary relationship between the two views.4,6 As an electron, considered a particle, goes from rest to the speed of light, it changes from a particle to a wave in its observed properties, and by doing so, displays its complementary nature. This does not mean that some transformation of what it is has occurred; what changes is what we observe of it. The scientific question about light is open to further research (a search for "hidden variables") and does not necessarily present an antinomy.5

This idea of light being both a wave and particle in essence but only presenting itself in one manifestation when we observe it-the idea of complementarity has been used to try to explain bow the brain embodies the mind. The mind is somehow represented in its physical embodiment, the brain, and does not exist apart from it, yet is not the same as it. If this phenomenon is viewed from a physical point of view, only the physical brain is seen, but if the meaning of the information which the brain upholds is considered, the mind is seen instead.

The same is true of a computer, since one could completely explain the physical operation in terms of Maxwell's equations, but in that explanation, never allude to any programming that has been done to effect its operation. From the mind-like view, a programmer may never know circuit theory but be able to explain the operating system software precisely. A part of the physical description would be that of the physics and chemistry of the ink and paper of the printout. However, the programmer views the physical embodiment with only a subsidiary awareness of its existence as he pays focal attention to the meaning contained in that physical embodiment.⁷

The meaning of a physical embodiment and that embodiment itself can therefore be different manifestations to us, depending on the viewpoint from which we observe. The entity in question is not a brain plus a mind, but a singular entity. The distinction between brain and mind is made by us. This does not mean that we can make what we observe into anything we want, but that our subjective understanding of what something is depends on our objective observation of it. In the example we consider here, it happens that we observe more than one manifestation which should ultimately be related in a singular view, since we presuppose that what we observe is a singular entity and not a separate one for each manifestation. Because we are confronted with different manifestations, or as they are often called, different levels of meaning, we face a mystery which is similar to the scientific wave/particle problem.

We are not dealing here with the existence of spiritual beings, but only with the problem we have of reconciling different levels of meaning for those things which have a physical embodiment. How God, who is a spirit, actively relates to the physical system, is a different mystery. Also, to say that man's spirit is only a different level of meaning is unwarranted.

Another mystery arises from a related but distinct topic which has a mathematical counterpart in Gbdel's Incompleteness Theorem. This Theorem essentially states that if a mathematical system is developed from given postulates and a given set of rules of logic, then it is possible that true statements can be made which lie within the framework of the system but cannot be proven by the system.2 Hence, all logical systems are incomplete in themselves. From this mathematical foundation, the concept of logical incompleteness has permeated the developing disciplines of information theory, cybernetics, and artificial intelligence. Along with it comes a basic antinomy which will be presented as a purely philosophical thought-experiment, though it is couched in a scientific-like setting. It is based on the idea that no system (whether physical or purely mathematical) can contain a complete representation of itself, for the representation which it contains would also need to include that representation. Now it needs another representation of this representation, ad infinitum. (A similar effect occurs when two mirrors face each

Several basic mysteries of the Christian faith and the philosophy of religion have analogous counterparts in science and mathematics, which may be applied to theological problems.

other.) The only way the system could contain a complete representation of itself would be to have an infinite number of representations of representations. ⁸ This is known as infinite regression and is a default solution to the problem of incompleteness.

This argument was presented by Donald MacKay⁹ to show that even if we were physically determined in our willful actions and decisions, that we would still be correct to regard ourselves as free; that is, that no prediction about our future actions would be true regardless of who knew the predictions. Let us say that some non-interacting observer could completely explain our physical operation by knowing enough about how we function physically. (Whether this could be done is highly questionable from a Christian view since it does not take into account God's immanent action in the physical system, but for the sake of the argument we assume it to show its inadequacy even if it were so.) This observer would then be able to predict, in a deterministic fashion, all of our future actions, taking into account not only our physical operation, but also including the effects on us of interaction with our environment. Now suppose that he were to offer these predictions to us. If we were to disbelieve them (i.e., discount them so that they have no effect on us) we would go along, and eventually they would come true as predicted, and we would have been wrong in our disbelief. If instead we believe them, they affect us, and that effect was not taken into account when the predictions were made, so the predictions are wrong because they do not adequately account for the effect they will have on us when we believe them. Thus, we are wrong to believe them. We are faced with an antinomy. The predictions are true, but if we accept them as true, then they are false; but if we regard them as false, they are then true.

To try to get around this, the observer may modify the prediction by including in it the effect it will have on us when we believe it so that it no longer will be false for us to believe it. This will not do, however, for we would be correct. to disbelieve it, since in disbelieving it, we do not fulfill the conditions that it is necessarily based upon. Another attempt to foil the antinomy is to "close the loop" so that when a true prediction is believed by us and affects us, the observer recomputes and offers us another prediction which we believe. This new prediction reaffects us so he again computes, and this goes on and on. Does it ever converge to a logical solution? No, for this is just the infinite regress solution. Since the observer is constantly offering us corrected predictions, he is no longer in isolation from us as before, and now the system is not just "us" but us and him, and the Incompleteness Theorem applies.

Another rebuttal to this argument is that we may go ahead and do what is predicted whether we believe the prediction or not, so its affect upon us is then negligible, and we are correct in believing it. For these cases, we may arrive at the same outcome only approximately, and we may not be surprised to be told what we are going to do-it's something we probably would have done anyway, like get out of bed in the morning. The point is that we did not necessarily have to decide to do as the prediction specified just because it was offered to us. All that is needed is one exception to invalidate the claim that the predictions are binding upon us (i.e., determinate for us regardless of our acceptance of them). If we were told that tomorrow we are going to decide to go swimming and as a consequence of our decision, drown-if we really believed this, we would decide not to go swimming at all, and we would not drown as a result of our future decision to go swimming. Therefore, predictions concerning our future decisions are invalid once they are disclosed to us, since future decisions we decide upon now are already made.

If predictions are given to us over which we have no control, then they are binding upon us regardless of our belief in them; but again, only one exception is needed to invalidate this as necessarily true. The observer would be correct in saying we make no decisions because for him everything we do has a causal explanation. But for us to say we make no decisions is incorrect since we are not the observer, and if we try to become the observer, our decisions interfere with our observations. The Incompleteness Theorem does not allow us to make accurate predictions about ourselves, so we never know whether we are acting deterministically or not; our knowledge of ourselves is never sufficient to decide that.

It is because we view ourselves as free that this argument is even relevant. In thinking of ourselves as capable of making choices which will affect the future, we wonder whether our choices are real, and it is because of this presupposition of individual freedom that we see ourselves in the place of the one being observed. Therefore, this argument brings across a very important conclusion: it is not possible to view the knowledge of a non-interacting observer in the same perspective (or logical framework) as the knowledge of the subject under observation. In doing so, in our thinking we would attempt to combine the knowledge of both into a single system (namely, our thought structures about this) and result in an incomplete view.

Because what is true for the observer is not true for the subject seems to imply that truth is relative. If it can be shown that both of "us" know something that for one is true but concurrently is false for the other, then the relativity of truth has been established. This thought-experiment shows just the opposite. The observer is compelled to say that from his reference frame we are determined and that from our reference frame we are free. We are compelled to say the same concurrently. Our relation to the observer's prediction is that we must believe it is true though we do not yet .know it, just as he must believe it is true. Either of us would be wrong to believe it was not true while it remained private with him and not disclosed to us. After offering it to us, we both would be wrong to believe it.

Furthermore, if more than one observer makes predictions about the subject, their predictions must necessarily agree and be true to what the subject is. The basis for truth is the determinate nature of the subject to which they all must appeal for correct predictions.

This conclusion may be helpful in dealing with some tough theological problems. Specifically, it may offer a working relation between the doctrines of God's sovereignty and man's responsibility for his willful actions, including the problem of evil in light of God's sovereignty. If we begin with the biblical teaching that all that God does is good in His sight, then if God is truly sovereign, we come to an impasse-God creates evil, yet it is good in His eyes (c.f. Prov. 16:4). From our point of view, good and evil are real and distinct as Scripture clearly bears out. If God is sovereign and views the world as determined by the counsel of His will, then from His point of view, all that He does is good, and only from our point of view are good and evil distinct (though God knows our point of view as well), as God has communicated this to us. ("From our point of view" does not mean " in our opinion" but positionally our knowledge as the subject before God who, in this regard, is like the observe) Since God's ways are past our finding out and His thoughts beyond ours (Isa. 55:10, 11), His communication to us is not like the observer offering his predictions to us, but is already spoken in compatibility with our frame of reference. At the judgment, God would not judge us from His reference frame (knowable only to Him) but on the basis of the truth He has revealed to us in our reference frame. In this way, God does not judge His own sovereign actions that He predestined us to follow, but judges us from our point of view as free agents, responsible to what His communication to us was. In thinking about this, there is a natural tendency toward integration of the two points of view, but as we saw, this results in a contradiction where an antinomy should be instead. The Scriptures distinguish between the human and divine reference frames (c.f. Phil. 2:12, 13; Prov. 16:1, 9).

The value of this model applied to theological problems lies in its ability to provide operational solutions, but not complete and final ones. The thought-experiment does not account for how God acts into the physical world (in determining it) and this inimanence/transcendence of God is perhaps the fundamental mystery (c.f. Col. 1:26, 27; 2:2, 3). This mystery finds its most dramatic expression in the incarnation of God in Jesus Christ, wherein God directly enters the physical system as a human being, concurrently observer and subject, yet not constrained by the limitations of the Incompleteness Theorem (c.f. Col. 2:9).

There is another school of thought in dealing with the problem of God's sovereignty and evil that deserves attention as an alternative approach. It sees the structure of the problem as a paradox without an antinomy, and is involved with a refinement of the problem by paying primary attention to distinctions between natural and moral evil, and the extent and nature of God's power and planning.^10,11

1. Inconsistent manifestations of a singular entity (mind-brain problem).
2. Logical indeterminacy of complete knowledge (incompleteness problem).
3. God's action in a physical universe (immanence-transcendence problem).

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