0. Introduction
1. The predestination/free-will discussion
2. Skolem's paradox
3. Skolem's paradox illuminates the discussion
4. Conclusions
0. Introduction
How our disciplines illuminate our faith is an important consideration in the faith-learning discussion. I believe that it is more important than the reverse question if only because I regard my faith as ultimately more important than my discipline. I also think that the question of how the disciplines illuminate faith is the harder question. Our faith is a whole world-view, which can more naturally illumine all else.
The purpose of this paper is to show that both sides of the predestination/freewill discussion are admissible in a way that is more profound than simply the wave-particle duality of light. In wave-particle duality there are two competing physical models of reality which are contradictory. I shall show below that not a contradiction but a difference in viewpoint is the fundamental issue in the discussion of predestination and free will. A discussion of Skolem's paradox is helpful in this demonstration.
1. The predestination/free-will discussion
Mr. P believes in predestination; Mr. F believes in free will. We shall break in on their discussion in the middle.
Mr. Predestination: I grant you that men appear to have free will, but it must be illusory because you have just granted me that God is omnipotent, hence in control of everything including the future and our choices. That's much more important to our discussion than His being omniscient, since it is conceivable that He knows the future without controlling it.
Mr. Free-will: It is also possible that God is so powerful that He is able to create beings with free will. Since we both have granted the Bible to be a basis for our discussion you'll grant that the God of the Bible is just. It would not be just to condemn those who have no freedom to choose their fate. Jesus says, "Whosoever will may come."
Mr. P: But He also says "You have not chosen me, but I have chosen you." And Paul says to the Ephesians that they are elect in Christ.
Mr. F: One can be chosen without responding; one can be elected without serving in office. Furthermore, if Christ is elect, one can be elect in Him by choosing to be in Him.
Mr. P: Imagine then Lady Macbeth in Shakespeare's play. As we read the play we see that she responds as though she has free will. Only as we step outside the play do we realize that her free will itself is part of the inventiveness of the author, who could have just as well chosen to cast her to be at the mercy of forces beyond her control--as in fact Greek plays did. I back off then. I agree that men's free will is not illusory--it's genuine, but only from man's perspective, not from God's perspective.
Mr. F: The next thing you'll say is that man's perspective is wrong, that God's perspective is right, and that the Christian is learning more and more to see things from God's perspective. At best you are patronizing me; at worse you are accusing me of not being Christian because I don't see things from God's perspective.
Mr. P: No! I don't say that only Christians can see things from God's perspective, and I don't for a moment doubt your sincerity . . .
Mr. F: There you see--patronizing: Your play analogy is biased in favor of your point. You underestimate God, who surely is more powerful than Shakespeare. If Shakespeare were "not willing that any should perish" then no one in his plays would perish . . .
Mr. P: For the sake of the play, he might, to prove . . .
Mr. F [beginning to raise his voice]: God "is not willing that any should perish." Experience and the Bible agree that men are perishing. God has not created robots to obey Him without love, without risk, without choice. Joy, peace, patience, faith, self-control--they all depend on choice for them to be meaningful. Shakespeare's characters are not real. Hear my analogy instead.
Mr. P: Say on.
Mr. F: Physicists know some properties of light that characterize it as a wave phenomenon--diffraction and interference for example--and other properties that characterize it as particles--mass and quantization for example. Physicists must live with two competing models of light which are contradictory, but neither of which is adequate in itself to explain all the phenomena of light. I suggest, Mr. P, that you may be right, and indeed your Scriptural support is most impressive; and that I too am right. We just happen to have two contradictory models of reality neither of which is alone adequate.
Mr. P: It is most gracious and generous of you to admit that I am right, but I cannot accept your terms. For if you are also right and that is a contradiction, then we do not even have a consistent system. Grant me a contradiction, and I can prove anything.
Mr. F: I see that you are well-schooled in logic. But aren't you putting God in a Greek box? If you understood everything about God, then He would not be as great as He is. You're just another of Job's comforters. By what standard1 are you measuring God? One of your own friends told me to think of God as the standard by which we measure justice. Shall we not also have God as our standard of truth, and not the Greek syllogism?
Mr. P: But if you deny me the law of excluded middle--that a thing is either so or it is not--then I can't reason about the matter at all.2 You must tell me what the ground rules are, if not logic. You've left me only mysticism.
Mr. F: Actually, I shall use classical logic. And if you permit me, I shall show you how your understanding of Skolem's paradox will help you to understand how we are both in fact right without contradiction.
And with this we interrupt the discussion to provide the background in logic that Mr. P and Mr. F already have.
2. Skolem's paradox
Thoralf Skolem was a mathematical logician who lived in the early part of this century, a period when Hilbert was reformulating Euclid, when Russell was reformulating Hilbert, and when Gödel was reformulating Russell. As a result of this revolution in the foundations of mathematics, marked especially by the seminal work of Gödel in 1939, we now know the following to be true of mathematical models and of proof theory:
First we assume that language allows only a finite number of symbols hence only a countably infinite number of sentences. In particular, only a countably infinite number of things can be provable. Hence, by a simple counting argument any sufficiently large set must have things about it that are true but not provable. A. Tarski formalized the notion of mathematical model using set theory (we do not need enough sets to worry about the set class distinction), and then he used this to formalize the notion of truth. Even if we were to argue that Tarski's notion of truth does not capture our intuitive notion of truth, at the moment all we need is a countability argument.
Here is the barest skeleton of those above ideas necessary to understand Skolem's theorem, and hence Skolem's paradox. Skolem's theorem says that every formal system that has a model has a countable model. What is a formal system? What is a model of it? Where do we get our function to enumerate the countable model?
By formal system we shall mean an applied first-order predicate calculus. That is, a quantification theory where quantifiers range only over number variables, not over predicates, and where in addition to the logical rules there are additional axioms. Typically we shall consider a set of first-order axioms for the real numbers as our formal system in what follows.
A model is a set D called the domain or universe of the model, and a collection of subsets of Cartesian products of that set. For every constant in the formal system there is an element in the domain. For every predicate in the formal system there is one of these subsets in the model, which corresponds to the set of lists of values which make the predicate true in that domain. For example, if is the real numbers and P(x,y) is the predicate x2+y2=1, then x together with all its subsets forms a model for a first-order theory of reals with 2-place predicate, where each constant ci of the formal system maps to a real ri and the predicate P(x,y) maps to the circle with radius 1 and center at the origin.
But a first-order theory has only countably many constants ci, so there must be a countable set which is the domain of a model for the first-order theory of reals (actually, Leon Henken's model will do, where there is an element of the domain for every existential statement3). If we take that countable domain to be the ci's themselves, then the function which enumerates the ci's enumerates the model. The enumerating function itself is not in the model (as a predicate relation; i.e., as a subset of a Cartesian product of D). One can extend the model by adding the enumerating function as a predicate (and a corresponding function symbol to the first order theory), since the predicate is consistent with the rest of the axioms.
It is this last step that gives rise to Skolem's paradox, which we are now prepared to state. Uncountable sets (the reals in our example) have countable models, hence using the enumerating function described above, they are countable. Is that not a contradiction? The customary way of dealing with this paradox is to introduce the notion of relativizable: a predicate (say, "is countable") on sets is said to be absolute if it remains true when a larger model is "cut down" to a smaller sized model, for all models. In any particular instance, the relativized form of the predicate (restricted to the smaller model) may or may not be the same as the unrelativized predicate. A predicate that is not absolute is relative. Thus it turns out that the notion of countable is not absolute. This resolves the paradox: we must always say "countable with respect to what model?" Skolem himself suggested this point of view.
From a philosophical point of view4 this solution has received varying degrees of acceptance, and the fault has been variously laid to--
3. Skolem's paradox illuminates the discussion
Mr. F: Just as Skolem's paradox allows for a model to be both countable and uncountable, relative to the context, so it is possible for man to have free will and also not to have free will, relative to the context of the particular instance.
Mr.P [somewhat disgusted]: But that is exactly what I was saying about Shakespeare and his play. From Lady Macbeth's perspective--context if you will--she has free will; from Shakespeare's, she does not. All your higher mathematics has done no more good than a layman's observation from the general culture, an observation--may I remind you--which proved my point, not yours.
Mr. F [patiently]: But I have gained an important advantage by asking you to think of a purely mathematical example. Whereas Shakespeare is real and Lady Macbeth is fiction, two mathematical models can represent different sets of axioms and be on equal status as far as which is to be preferred.
Mr. P [somewhat puzzled]: As though Shakespeare really did not exist, so that both he and Lady Macbeth are as it were on equal footing?
Mr. F: That analogy has a problem. If it proves anything, it proves too much. It opens the door to the possibility that God does not have free will rather than to the possibility that man does. Surely you do believe that God has free will.
Mr. P [relieved and with confidence]: Yes, of course. But even with your analogy of all real numbers being both countable and uncountable, there is still a preferential choice. The real numbers are really countable from a more inclusive point of view.
Mr. F: You understood the Strong Skolemite position presented by Resnik5 or you would have said "the absolute point of view" instead of being so careful to say "a more inclusive point of view." But you don't understand it well, or you would be more careful about calling real numbers "really" countable. Resnik's main point is that Skolem's paradox can be restated as simply a limitation of the power of formal predicate calculus of all orders. In one model of the real numbers, they are countable; in another model they are not. Neither model is preferred. Your preference for positive assertions and your desire to prove that predestination is a more accurate description of reality have colored your thinking.
Mr. P [now a little on the defensive]: But what have you gained over your example of the wave-particle duality of light? There also you had two models neither of which had preferential explanatory power.
Mr. F: There the two models were contradictory.6 By requiring less--mathematical models are more manageable than physical models--we avoid the contradiction.
Mr. P: But now you are in the same position that you just refuted. Don't you have the same problem as with a fictional Shakespeare? Namely, hasn't your analogy made of God less than He is?
Mr. F: So it seemed with Shakespeare because I believe that he did exist, so to assume that he didn't was to begin with a contradiction. Furthermore, even if we consider his "A Midsummer Night's Dream," in which there is a play within a play, we still have the author above both plays, as though there were something above God. In the mathematical analogy, there need be no
author of the two models, no model containing them both. I believe in both predestination and free will. I do not believe that a single model contains them both.
Mr. P [triumphantly]: First of all, thank you for conceding to me that you believe in predestination. Now it's my turn to use mathematical logic to refute your position. Surely you know by Leon Henken's results7 that every consistent first-order theory has a model. Therefore what you are saying is just as contradictory as before, for if a theory doesn't have a model it is inconsistent.
Mr. F: On the contrary, all I am saying is that reality is certainly not a first-order predicate calculus.
The discussion continues, but we must summarize.
4. Conclusions
A Christian mathematician may be able to be more tolerant of ambiguity than his Christian brothers. This point has been often made in connection with the need for axioms and the value of making axioms explicit both in science and in faith.8 I feel that the following addition contributions have been made by the above:
a. There is a value in being specific. What axioms of science? What axioms of faith? This has been done elsewhere for faith versus unbelief.9 I have attempted to do the same for a specific point of faith--two apparently competing Christian worldviews--and a specific mathematical axiomatic system.
b. But at the same time, by taking a model-theoretic rather than a proof-theoretic point of view I can emphasize the whole worldview rather than focusing on this or that particular axiom.
c. Since I am Mr. F in the above discussion (who does most of the talking), I think that it clarifies the position of those like myself who hold both to predestination and free-will without compartmentalization, without compromise, and without schizophrenia. The position is sometimes referred to as "Calminian" (Calvinistic and Arminian combined). I am convinced that Calvinism contains the same Catch-22 provisions as Freudianism. If you disagree with Freud, says the Freudian, you must be repressing something. If you disagree with Calvin, says the Calvinist, you must be predestined to do so. To remove a theory from empirical verification is to remove it from discussion, as a scientific matter. If a Calvinist claims that he knows by revelation rather than either empirically or rationally, I will concur with his method but I shall continue to seek rapproachment among these three methods of epistemology: God encourages us to use reason (Isaiah 1:18) and experience ("O taste and see that the Lord is good." Cf. John 7:17), as well as revelation.
d. A Christian mathematician, but not a theologian, I have tried to bring my discipline to bear on the statement, "A man can be responsibly exercising free will and at the same time in the same act be exercising God's predestined will." This need not be a contradiction. It need only be two different models which do not contradict each other because both predestination and free will are terms whose significance is relative to their own model. Hopefully I have clarified one issue and pointed out a direction for further such investigations.
1. Rousas John Rushdoony, By What Standard, Presbyterian and Reformed Publishing Company, Philadelphia, Penna.
2. Francis A. Schaeffer, The God Who Is There, Inter-varsity Press, Downer's Grove, Ill., 1968.
3. Joseph R. Schoenfield, Mathematical Logic, Addison-Wesley, Reading, Mass., 1967, p. 45.
4. Michael David Resnik, "On Skolem's Paradox," J. of Philosophy, LXIII, 15, August 11, 1966, pp. 425-438.
5. Ibid.
6. Quantum mechanics, as a hoped-for synthesis has its own philosophical problems. See Hilary Putnam's Mathematics, Matter and Method, Cambridge University Press, Cambridge, 1975, p.157. The situation has changed little in the 12 years since Putnam first published this essay.
7. Schoenfield, op. cit.
8. For example, Ian G. Barbour, Issues in Science and Religion, Prentice-Hall, Englewood Cliffs, N.J., 1966.
9. For example, Wayne Roberts, Assumptions and Faith, Gibbs Publishing Company, Broadview, Ill.
Author's Note:
R. E. Hobart in "Free will as involving determinism and inconceivable without it," Mind, XLIII, 169, Jan. 1934, pp. 1-27, says that determinism and free will are more than compatible: "Determinism is free will expressed in the passive voice." This reference appears in The Problems of Philosophy, second edition, edited by William P. Alston and Richard B. Brandt (Boston: Allyn & Bacon, 1974), condensed as the article "The harmony of free will and determinism." I thank Dr. Arthur Holmes for this reference.
Donald MacKay's notion of complementarity is the closest of all to my view above. See his The Clockwork Image: A Christian Perspective on Science (Inter-Varsity Press, 1974).