Nonstandard Mathematics and a Doctrine of God by Granville C. Henry, Jr. Granville C. Henry, Jr. is Associate Professor of Mathematics and Philosophy at Claremont Men’s College, Claremont, California. The following article appeared in In Jerusalem, 1964, at the International Congress for Logic, Methodology and Philosophy of Science, Abraham Robinson said: "As far as I know, only a small minority of mathematicians, even of those with Platonist views, accept the idea that there may be mathematical facts which are True but unknowable? How can we talk about the content of mathematics, of all things, as true but unknowable? Robinson and Steen are not questioning whether there are mathematical relationships, theorems, or facts that are presently true but presently unknown. This would mean something that neither accepts, namely, that we presently know all true mathematics. What they are asking is whether there is some mathematical content that is true but which in principle can In this paper I want to examine how this new question of the possibility of true but unknowable mathematics may engage contemporary discussions of a doctrine of God. But first, let us look at how old mathematics, with familiar procedures and assumptions, has affected traditional doctrines of God. God as Unchanging Concepts of God change even within religious communities that maintain close continuity with their past. Changes in a Christian doctrine of God have often paralleled changes in an understanding of the nature of the soul. The understanding of both underwent significant change between the end of the New Testament period and the culmination of theology of the early church in Augustine. The change was basically towards emphasizing an understanding of the soul as eternal and of God as unchanging, as contrasted with an understanding of the soul (or spirit) that decays or dissolves at death to be resurrected by the power of God and of a God who is active and involved in the affairs of men and, hence, who changes. Both of these latter positions are nearer Biblical emphases than the former. The God of the Bible is never presented as absolutely immutable and static ontologically. He loves, wills, acts in history, becomes incarnate, changes his mind, and knows particular changing and quite mutable men. If one accepts a real knowledge by God of a changing world, then such understanding would indicate that there is some change, perhaps minor, in God himself. This movement within Christian theology is generally recognized to have resulted from a contribution of Greek philosophy and religion. The Greek contribution, however, was not seen by the church to be incompatible with scripture or orthodox doctrine. For centuries, theologians have seen the traditional scriptural accounts of creation, of covenant, of historical deliverance, of incarnation and atonement as confirming a doctrine of God who is best understood as absolute and unchanging, while not realizing the anomaly of this doctrine with God’s activity witnessed in each scriptural account. To see a clear example of the use of mathematics for an argument for an eternal soul, and thereby for an immutable God, we need look no further than Augustine’s treatise Tertullian put his finger on the source of the doctrine of an incorporeal soul when he pointed out that it I think Tertullian’s historical analysis is sound. It was Plato’s doctrine of form which, if not the clear source, was the primary philosophical justification, of an affirmation of an eternal soul and an unchanging God -- at least in the theological period dominated by Augustine. We see a clear argument in the There is excellent evidence to believe that Plato’s doctrine of forms was precipitated by his mathematical involvement and that he modeled his understanding of forms after an understanding of mathematical figures that was then held by the burgeoning mathematical community. Aristotle, for example, claimed that Plato’s ideas had ontologically the same status as Pythagorean numbers and were used by Plato the way the mathematical society used their numbers. It was Aristotle who in revising a Platonic understanding of form championed an exclusively immutable God. He understood Platonic forms to be immanent within physical things and not in a realm transcendent over them. In ordinary substances each sensible thing was seen to be a composite of matter and form possessing a combination of fixity, the form, and potentiality for change, the matter. All such substances, according to Aristotle, are in motion, that is, changing from one form to another. The reason for change from potential to actual in the individual substances is ultimately pure form itself, i.e., God (as much as Aristotle has a God), who attracts, as it were, by virtue of being a final cause or goal, all things unto himself. The unmoved mover or pure form, is absolutely immutable. He possesses only actuality and has no trace of potentiality. Although Aristotle differs markedly from Plato in the use of forms, the forms themselves, which constitute that which is most real in any particular substance and is reality itself in God, are essentially Platonic in nature, possessing Platonic characteristics of eternality, fixity, abstractness and logical relatedness. In. addition to Platonic form, there is another presupposition attendant to arguments for both an eternal soul and an unchanging God, namely the idea that unity is not divided. Augustine’s argument above rests upon an assumed unicity of the soul, i.e., that it Greek mathematicians consistently insisted that the true mathematical It seems strange to us that the number one should not be divided -- that there should be no fractions. Although Greek mathematicians knew of the embryonic development of fractional numbers in both Egyptian and Babylonian mathematics, they did not include within the body of pure mathematics these numbers but relegated them to practical matters where they languished without benefit of theoretical consideration. I think that it was a strange turn of mathematical competence rather than naiveté that prevented Greek mathematicians from objectifying fractions. They solved the enigma of the discovery of incommensurable magnitudes by the brilliant extension of the concept of ratio which itself compares magnitudes by whole numbers, that in turn depend on an indivisible unit. They could have objectified fractions and thereby declared their existence. Indeed, they had all the theoretical development in terms of ratio to do so -- provided these fractions did not include incommensurables. The Greeks knew that if one had rational numbers, he must also have irrational ones, and they chose the theory that allowed them to have both in a consistent setting. Remember, it In the two basic philosophical positions the church had available to it, the Platonic and Aristotelian, the choice of an Aristotelian emphasis led quickly and easily to an immutable God. God as pure form did not change. In a Platonic emphasis adapted to Jewish and Christian monotheism, if one considers the realm of Platonic ideas to be part of the thought of God, as did Philo, who more than any other is the founder of classical theism, we can understand how God who possesses these ideas is necessarily, at least in part, immutable. We can understand Augustine’s statement: "For He Parmenides was the first philosopher to associate a metaphysical one with the strict immutability of Being itself. He may have been indebted to the Milesians who affirmed a generalized divine substance, the Pythagorean dualism is expressed in terms of ten contrarieties: limit and unlimited, odd and even, one and plurality, right and left, male and female, resting and moving, straight and curved, light and darkness, good and bad, square and oblong. Each of these represents manifestations of the two primary opposites leading the list, the limited and unlimited. The The limit is characterized best by precise mathematical structure. This grasping of the mathematical bounded and known, allowed an ecstatic experience that transcended the round of birth and rebirth, and, hence, effected salvation. It was the number one, identified by the Pythagoreans with the unit point, that was the epitome of exact objectification. From the unit point came all the numbers, and from numbers came the whole universe. The unit point for them, as for us, was indivisible. Their identification of the number one, however, with the unit point
Parmenides, though trained as a Pythagorean, rebelled against Pythagorean thought by intensifying the importance of the objectified
I have tried to show how mathematics’ influence on the philosophical notions of Platonic forms and metaphysical One had an effect on the traditional Christian doctrine of the immutability of God. I have limited myself to one aspect of the influence of mathematics on a doctrine of God. We could have further considered the shift that occurred when God was referred to as the Infinite as well as the One. Or we could have examined in some detail the rationalization of Christian Logos that occurred from mathematical sources. Mathematics As Changing One of the primary characteristics discovered by the Greeks about mathematical structures is that once proved they do not change. A theorem accurately proved in the Mathematicians and philosophers have not always seen, nor always maintained, that mathematics is best understood in a Platonic way -- although this has been by far the dominant position. By Platonic here, I do not mean that mathematicians accept as a matter of course the whole body of Plato’s philosophy, but I do mean that they view in a minimal way that mathematical structures and relationships exist independently of man’s construction of them and are The primary sources of influence of mathematics on Christian theology have been the result of changes in the understanding of the nature of platonic mathematical structures as a result of the changing discipline of mathematics itself. I have indicated how the platonic understanding of mathematics influenced and confirmed the doctrine of the immutability of God. This was because of the strict immutability of an assumed existing realm of rigidly connected mathematical structures. Not all, however, who believe in such a rigid realm of mathematics have affirmed a strictly immutable God. Whitehead, for example, and process theologians following him maintain a doctrine of a changing God, especially in his response to the world. This God does possess, however, an unchanging essential nature, called by Whitehead God’s Primordial Nature (the realm of eternal objects), that itself contains the rigidly connected realm of mathematical relationships. The nature of eternal objects, and hence, God’s primordial nature, was modeled by Whitehead after his understanding of the nature of mathematical existence. What if we could understand the realm of mathematical structures to be itself evolving? Would this not modify both an Augustinian and a contemporary process view of God? The chief authority for the stability of a platonic realm would be challenged, and hence one of the primary means for arguing God’s immutable essential nature questioned. There are developments in mathematics that might lead us to come to that opinion. The primary mathematical developments that appear to me to be relevant for contemporary theology are the creations of multiple models for the real numbers. There are apparently a number of The discovery of nonstandard models for arithmetic and real numbers differ in degree and kind from the discovery of non-Euclidean geometries. Non-Euclidean geometries were formulated by
The possibility of the existence of non-standard models has been evident since the announcement by Gödel to the Vienna Academy of Sciences in 1930 of his now famous Incompleteness Theorem. This theorem is an effective proof by metamathematical considerations that arithmetic is essentially incomplete: that not only are there true theorems in arithmetic that cannot be proved from the axioms of arithmetic but also that no matter how many axioms are added there always remain true theorems that cannot be proved. If one finds some true theorem that cannot be proved from the axioms, then neither can its negation be proved. What if one adds to the set of axioms not the unproved true theorem but its negation? We know that in any consistent first order theory, if some theorem A is not provable from the axioms, then the theory with the negation of A affixed to the axioms is itself consistent. Obviously this new system, or more accurately an interpretation or model of this system, is different from the previously accepted one. It differs exactly in that the accepted unprovable but true theorem in the original system is false in the newly constructed one. Yet both theories conform to the previously accepted axiom system. We can see in terms of these developments why the question arose that I mentioned in the first part of the paper, namely, whether there are true but unknowable mathematical structures. In terms of the adequately, or perhaps "vividly," known, we have shown that there are structures that conform to an axiomatic system that cannot be proved from an axiomatic system. We know that there are models of the real number axioms which may never be explicitly formulated. Traditionally we have encompassed and understood an infinitude of structures by an axiomatic system. It has been the authority for our declaring that we know all of a certain type of structure. We cannot now make any comprehensive claim to know all structures for any complicated mathematical axiomatic system. Could there be compatible interpretations of a system that are somehow in principle impossible to know? Could there be true but unknowable mathematical facts? I share Steen’s and Robinson’s skepticism about the existence of platonic mathematical structures that are true but unknowable. I find there is a certain presumption about affirming the existence of a platonic mathematical form that cannot be known -- either within a Platonic perspective or outside of it. In principle, one could never have any evidence of the form’s positive existence. Also, I find the affirmation that there is a platonic realm of mathematical structures that are eternally fixed in their relationship to each other but never growing or diminishing in totality, to be also somewhat presumptuous. Our evidence historically, certainly in terms of what we If one believes in any platonically understood realm of mathematical structures, it seems to me best to understand it as a loosely known multiplicity which is incapable of unification axiomatically and to which new relationships may be added. The addition of any new relationship would, of course, be compatible with some structures and logically incompatible with others. Instead of "true but unknowable" we might say "unknowable because not yet true." In assuming that mathematical relationships have a kind of platonic reality at least in terms of being potentials for matters of fact as known by God, we recognize that these relationships may be structures of that which is known -- or part of the structures of knowing itself. The structures of knowing, at least the means by which one can know mathematics, have traditionally been known as logic. It is a well-known fact that these structures have been objectified and made epistemological objects whose nature can be examined mathematically as structures of the known. Gödel’s theorem points out that the structures of knowing cannot all be formalized mathematically. The new developments in mathematics seem to me to allow a better understanding of what it might mean for God to have the freedom to change the totality of potentials -- both in terms of the structures of knowing among human consciousness and in terms of the objects known. This would mean that not only could man’s consciousness, as well as other structures of the world, evolve in ways hitherto unknown, and in ways impossible to know, but in ways that Almost all traditional and contemporary theologies that maintain a platonic reality for mathematical potentials insist both that the mathematical structures do not change and that they are complete in their totality as understood or envisioned by God. This doctrine is found in Augustine as well as in contemporary process theology. God, though changing in his actual consequent nature in process theology, does not change in his essential nature, that aspect of him called the primordial nature. The eternal objects that comprise the primordial nature are fixed, they are pure potentials and as such have a rigid logical structure. God may establish possibilities for actual entities by selective envisionment of, or ordering of, the realm of eternal objects, and in this role he acts as destiny or providence for actual entities. From the perspective of the actual entity, there are multiple routes to the future in terms of different potentials for actualization, but each of these routes as in the completion of Beethoven’s symphony is a choice of I would like to maintain the emphasis that platonic mathematical structures do not change, as affirmed by Whitehead and Augustine, but relax the requirement that no new potentials or structures be added to the realm of eternal objects. This relaxation is based on the simple observation that it has been primarily the axiomatic method that has given mathematicians and philosophers the authority for stabilizing the mathematical realm, for claiming it to be complete as related logically to a few unquestionable assumptions. What we have learned about mathematics since the advent of Whitehead’s philosophy is that the axiomatic method cannot adequately characterize the nature of mathematical structures that are presently known. It is true that we may know some aspects of these structures apart from the axiomatic method. This is essential. But we still know the unity of mathematics, or the unity of mathematical systems, primarily through axiomatic investigation. It may be that what unity we know we know through axiomatic systems, but that this unity is not complete. The claim that individual mathematical structures are unchanging but that new ones may be formed, new potentials added to the realm of eternal objects, entails some kind of evolution in the realm of eternal objects. Under the principle that actuality determines (at least) potentiality, we would maintain that all actual relationships in the past are now potential. The realm of eternal objects is comprised at least of those relationships that were (or are) actual -- of course understood now as potential. In addition, the realm of eternal objects is comprised of all known potential relationships and especially that vast welter of mathematical relationships created by the imagination and consciousness of man. For as known by man, these relationships do have a tie to the actual world, even though in their objective status they do not characterize or have never characterized any particular complex of events. I am sure that the realm of potentials, i.e., eternal objects, is greatly enlarged by God’s knowledge of potentials. He knows the mathematical structures that we could know but now in fact do not know. In addition, I think that his activity is the primary source of new structured relationships in the realm of eternal objects and that those relationships coming from the world comprise probably only a small portion of the total. My proposal concerning the nature and evolution of eternal objects tips the balance towards a Hartshornian rather than a strictly Whiteheadian process theology. Whitehead did model his understanding of eternal objects after his understanding of mathematical existence. Eternal objects, therefore, according to him, are exact, discrete, individual, objective and existing in themselves apart from any relationship to particular actual entities. Whitehead’s God, though not fully developed in God’s freedom in this revision of process theology not only extends to his influence on actualities but also on the limitations of that which is possible, not just in the sense of choosing those possibilities that may be most relevant in a particular set, but in creating the possibilities themselves. The realm of eternal objects grows as history progresses. Things in their true There is another aspect of the theory under consideration that I find attractive, because it conforms to my ideas of the
In the first part of this paper I tried to show how an understanding of standard mathematics conditioned the doctrine of God’s immutability. Obviously, our interpretation of contemporary nonstandard mathematics relaxes any restrictions, at least from mathematics itself, of requiring God to be strictly immutable. In the second part I have tried to show how contemporary developments in mathematics might affect a contemporary doctrine of God. In particular, I chose process theology to work with. I challenge those who may have an allegiance to another set of theological doctrines of God to work out what might be the consequence if the traditional understanding that mathematical structures are complete, unified and eternal in nature were relaxed.
Notes ^{1 Abraham Robinson, "Formalism 64," Logic, Methodology and Philosophy of Science; Proceedings of the 1964 International Congress, ed. Yehoshua Bar-Hillel (Amsterdam: North-Holland Publishing Co., 1965). p. 232.
2 Lynn Arthur Steen, "New Models 0f the Real Number Line," Scientific American 225/2 (August. 1971), 99.
3 Treatise on the Saul, Ch. III, from The Ante-Nicene Fathers, eds. Roberts and Donaldson (Grand Rapids: Wm. B. Eerdmans) III, 183.
41n the earlier works of Phaedo, Republic and Parmenides the deity and at times the soul are supreme examples of fixity and immutability, whereas in the Phaedrus and the Laws the deity is freely mobile. In the Timaeus the eternal God is immutable and the world soul is self moving.
5 Metaphysics, 987b 9-13.
6 A. E. Taylor, Varia Socratica (Oxford: James Parker & Co., 1911), pp. 187 ff.
7 See, for example, Sir David Ross, Plato’s Theory of Ideas (Oxford: Clarendon Press, 1951), p. 13.
8 Metaphysics, 1052b16--1053a5.
9 Physics, 267b25,
10 The City of God. Ch. XXI, Basic Writings of Saint Augustine. tr. M. Dods (New York: Random House. 1948), II, 162.
11 The Summa Theologica Question XI, First Article.
12 "Mathematics and Theology." Bucknell Review, 20/2 (Fall, 1972), 113-26.
13 My "Whitehead’s Philosophical Response to the New Mathematics," The Southern Journal of Philosophy, 7/4 (Winter, 1969-70). 341-49.
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