Toward a Realist Modal Structuralism

PhilosoPhia Christi

Vol. 12, No. 1 © 2010

Toward a Realist Modal StructuralismA Christian Philosophy of Mathematics

Walter SchultzDepartment of PhilosophyNorthwestern UniversitySt. Paul, Minnesota

The aim of this paper is to propose a philosophy of mathematics that takes structures to be basic. It distinguishes between mathematical structures and real structures. Mathematical structures are the propositional content either of consistent axiom systems or (algebraic or differential) equations. Thus, mathematical structures are logically possible structures. Real struc-tures—and the mathematical structures that represent them—are related es-sentially to God’s plan in Christ and ultimately grounded in God’s awareness of his ability. However, not every mathematical structure has a correlative real structure. Thus, even if such structures are provably consistent, they can not be said on those grounds alone to be true per se. But neither can they be said to be false. Mathematical structures are either true or fictional, yet all are possible. Hence, this philosophy of mathematics is a Christian (not merely theistic) modal structuralism. These features are elucidated by defending four conjectures:

(1) The natural number sequence is a structure exemplified by the na-ture of God’s plan in Christ. This fact makes elementary number theory true per se.

(2) Some mathematical concepts, propositions and structures are fic-tions, though not false per se.

(3) Some other mathematical concepts and structures are true per se by virtue of empirically modeling real structures.

(4) Abstract objects depend on God’s awareness of his ability.The philosophical study of mathematics (and logic) typically addresses se-mantic questions of reference and truth, ontological questions of existence

AbstrAct: The aim of this paper is to propose a philosophy of mathematics that takes structures to be basic. It distinguishes between mathematical structures and real structures. Mathematical structures are the propositional content either of consistent axiom systems or (algebraic or dif-ferential) equations. Thus, mathematical structures are logically possible structures. Real struc-tures—and the mathematical structures that represent them—are related essentially to God’s plan in Christ and ultimately grounded in God’s awareness of his ability. However, not every mathematical structure has a correlative real structure. Mathematical structures are either true or fictional, yet all are possible.

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and nature and epistemological questions. A structuralist view of mathemat-ics may be understood as an alternative to platonism, logicism, formalism, and constructivism—each of which are characterized and differentiated by their answers to these questions. In general, a structuralist view of mathemat-ics holds that mathematical propositions are primarily about structures. How-ever, this much seems to be the extent of the agreement among structuralists. First proposed near the beginning of the nineteenth century by Dedekind (1888) and Poincare (1902),1 the last quarter of the twentieth century saw the development of structuralist views of mathematics along several distinct lines. The differentiated development mirrored the debate between realists and antirealists over the status of abstract objects in general. So, there are platonist, aristotelian, and nominalist views of mathematical structures. The prominent contemporary competitors are Shapiro’s ante rem structuralism, Resnik’s in re structuralism, and Hellman’s (nominalist) modal structural-ism, respectively.2 This paper proposes a realist modal structuralism (RMS) as an alternative to these views.

Recent Christian philosophy of mathematics has taken the form of either theistic activism or theistic conceptualism regarding the existence and nature of abstract objects in general.3 Theistic Conceptualism takes the referents of mathematical terms (and other abstract objects) to be concepts in the divine intellect; Theistic Activism takes them to be products of the divine intellect. Realist Modal Structuralism has both conceptualist and activist features. However, it differs from them in that it treats abstract objects as a matter of God’s awareness of his ability to create.4

1. Ernst Dedekind, Was sind und was sollen die Zallen? (“The Nature and Meaning of Num-bers”) trans. W. W. Beman, in Essays on the Theory of Numbers (1888; New York: Dover, 1963), 31–115. Henri Poincaré, Grandeur mathématique et l’experience (1902), 20: “. . . Math-ematicians do not study objects, but the relation between objects. To them it is a matter of indif-ference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention. They are interested in form alone.”

2. Stewart Shapiro, Philosophy of Mathematics (Oxford: Oxford University Press, 1997); Michael Resnik, Mathematics as A Science of Patterns (Oxford: Oxford University Press, 1997); Geoffrey Hellman, Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press, 1989). See also the Oxford Handbook of Philosophy of Math-ematics, ed. Stewart Shapiro (Oxford: Oxford University Press, 2007).

3. For theistic activism, see Christopher Menzel, “Theism, Platonism, and the Metaphys-ics of Mathematics,” Faith and Philosophy 4 (1987): 365–82; Christopher Menzel, “God and Mathematical Objects,” in Mathematics in a Postmodern Age, ed. Russell W. Howell and W. James Bradley (Grand Rapids, MI: Eerdmans, 2001), 65–97; Thomas V. Morris and Christo-pher Menzel, “Absolute Creation,” American Philosophical Quarterly 23 (1986): 353–62. For theistic conceptualism, see Paul Copan and William Lane Craig, Creation out of Nothing: A Biblical, Philosophical, and Scientific Exploration (Grand Rapids, MI: Baker Academic, 2004): 167–96.

4. This claim is developed in the section addressing “Conjecture Four.”

All these philosophies of mathematics face at least three problems re-garding the relation of God to mathematical “objects.”5 The first is the Prob-lem of Multiple Models. The Arabic numeral, 2, the von Neumann ordinal, {{∅}}, and the Zermelo numeral, {∅,{∅}} are all supposed to denote the same number. However, under the Zermelo system, the number designated by the Arabic symbol, 0, is a member of the number designated by the Ara-bic symbol, 2 (that is, 0 ∈ 2), but under the von Neumann system, since a number x is a member of a number y, if and only if y is the successor of x, 0 is not a member of 2 (that is, 0 ∉ 2). Moreover, the system of von Neumann ordinals, the system of Zermelo numerals and the system of Dedekind-Peano axioms express the same structure. There are equivalent, yet contradictory, models of the natural number structure. Therefore, any view that holds num-bers and sets to be distinct objects in the divine consciousness must commit itself to one of these to avoid the implication that God holds contradictory concepts. But since mathematics is invariant with respect to isomorphism,6 there seems to be no reason to hold one model rather than another.

A different, but equally-challenging problem, is described by Christo-pher Menzel as follows: “. . . the existence of everything other than God is—at every moment in every possible world—explained by the exercise of God’s creative power.”7 Since God creates and sustains every thing distinct from himself, every thing distinct from God must be dependent on God. However, if mathematical objects are abstract objects whose properties in-clude eternal and necessary existence, how can they be “explained by “the exercise of God’s creative power”? How can their necessary existence and ontologically dependence on God be explained? This is the Problem of De-pendence.

Finally, each of the Christian philosophies of mathematics ought to in-dicate at least in some minimal way what it is that makes it Christian, other than being promulgated by a Christian theorist. It ought provide at least some rudimentary indication of how it is supposed to be related to Christology.

While Realist Modal Structuralism has both conceptualist and activ-ist features, it treats abstract objects to be fundamentally a matter of God’s awareness of his ability to create. In so doing, it is able to transfer the on-tological question to mathematical structures, rather than to objects. This enables it to overcome the Problem of Multiple Models and the Problem of Dependence. It also relates mathematical structures to real structures, which are God’s infinite plan in Christ and the patterns of the ways God acts ac-

�. Sufficiently specifying these problems and the positions each view takes on them would require another paper. Space permits only that they be mentioned so as to more adequately fill out this proposal for RMS.

6. Daniel Isaacson, “Mathematical Intuition and Objectivity,” in Mathematics and Mind, ed. Alexander George (Oxford: Oxford University Press, 1994), 118–40.

7. Menzel, “God and Mathematical Objects,” 73.

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cording to his plan. In what follows, these aspects of RMS are elucidated by defending four conjectures.

Conjecture OneThe natural number sequence is a mathematical structure exemplified by the nature of God’s plan in Christ. This fact makes elementary number theory true per se.

Dedekind-Peano Arithmetic (PA2) is a second-order, axiomatic descrip-tion of the natural number sequence, N. It has eight axioms:

PA21 ∀x∀y (x′ = y′ → x = y)

(Distinct elements have distinct successors.)PA2

2 ∃!x ∀y ~(x = y′)(There is a unique element with no predecessor.)

PA23 ∀x∃!y x′ = y

(Every element has a unique successor.)PA2

4 ∀x (x + o = x)PA2

5 ∀x∀y (x + y′ = (x + y)′)PA2

6 ∀x (x ⋅ o = o)PA2

7 ∀x∀y (x ⋅ y′ = (x ⋅ y) + x)PA2

8 ∀X [(X o ∧ ∀x (Xx → Xx′)) → ∀xXx](Any class containing 0 and closed under successor con-tains every element.8)

In other words PA2 is a second-order theory. It characterizes a structure under the interpretation N = ⟨N, o, ′, +, ⋅⟩ of a language L of second-order modal logic SLS5, whose domain is N = {0, 1, 2, . . . , n, . . .}, the set of natural numbers.

N assigns the symbol, o, to the least natural number 0.N assigns the symbol, ′, to the successor function, that is, {(x, y) ∈ N2 | y = x + 1}.N assigns the symbol, +, to the addition function, that is, {((x, y, z) ∈ N3 | z = x + y}.N assigns the symbol, ⋅, to the multiplication function, that is, {((x, y, z) ∈ N3 | z = x × y}.

Here, then, is a formal proof that 1 + 1 = 2: Theorem: ⊢SLS5 o′ + o′ = o″

8. Note: “X” is a second-order variable ranging over predicates. Since this logic is exten-sional, the SL sentence, “Fo” for example, expresses the same proposition as does the set theo-retical sentence, “o ∈ F.”

{1} 1. ∀x (x + o = x) P PA2

{1} 2. ∀x∀y (x + y′ = (x + y)′) P PA25

{2} 3. ∀y (x + o′ = (x + o)′) 2 UI{2} 4. o′ + o′ = (o′ + o)′ 3 UI{1} �. o′ + o = o′ 1 UI{1,2} 6. o′ + o′ = o″ 4, � Isub

What does this proof indicate? First, since the propositional content of the axioms of PA2 is a mathematical structure, the terms of the proposition, o′ + o′ = o″, pick out places in that structure. Such places are entirely character-ized by their relation to other places. Therefore, numbers are not objects, but places in a structure. Second, the proof shows that PA2 entails the proposition expressed by o′ + o′ = o″, so that truth of o′ + o′ = o″ depends on the truth of the axiom set of PA2. Therefore, the proposition, o′ + o′ = o″ is true per se if and only if PA2 is. However, PA2 is true per se if and only if there is a real infinite sequence of objects that PA2 represents. The problem is that there is no infinity of physical objects much less a sequence of them obeying the axi-oms of PA2. Therefore, although we have an answer to the semantic question regarding reference, the question regarding truth is unanswered and depends on answers to the ontological questions regarding the existence and nature of infinite structures. Adequately answering these ontological questions has been a difficult, and controversial pursuit. It is worth noting that in 1831 in a letter to Schumacher, Carl Friedrich Gauss said, “I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking. . . .” However, RMS holds that the infinite is not “only a manner of speaking”: the mathematical structure expressed by PA2 is exemplified by the temporal sequence of moments con-stituting God’s plan in Christ.

Interlude. This latter claim begs for elucidation, but space in this paper permits only a brief synopsis. RMS proposes a five-category, dynamic on-tology: God, God’s awareness of his ability (which give us an infinite array of possible worlds), dispositions (which are God’s commitments to act on condition), powers (which are God’s constant actings), and structures (which are patterns of unifying his commitments and constant actings). We concep-tualize each of the latter three as metaphysically real dispositions, powers, and structures, respectively. All this has been developed in an earlier paper.9 What I want to do now is to briefly develop the notion of God’s plan and connect it to this ontology.

The biblical language of God’s making promises, and having purposes and plans indicates that—for some events, at least—prior to that event’s oc-currence, God had an idea of that event in mind and that God’s was aware of

9. See section 6 of my paper, “Dispositions, Capacities, and Powers: A Christian Account,” Philosophia Christi 11 (2009): 321–38.

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his ability to bring it about. Whether God’s plans taken together constitute an exhaustive inventory of every event is a matter of controversy. The point now is not to advocate for either the “open” or the “exhaustive” view of God’s plans. The point here is only that an ontology which is at once bibli-cally-controlled and adequate for mathematics and science should incorpo-rate God’s having a plan into its vision. If, as Scripture affirms, all things are created through and for Christ and Christ upholds all things by the word of his power, then what gives every event in the history of creation its structure and meaning is the Father’s redemptive purpose in Christ.10 The “logic” of everything must lie in the logic of God’s action. Thus, at some point, ontol-ogy must be grounded in divine action. This clue leads me to conjecture that the abstract objects of mathematics and logic and the ontological presup-positions of science should find an integrating ground in God’s purposeful actions in Christ. In short, is aware of the full range of his ability to create. God acts according to his plan. God’s making his plan the history of creation, suggests his rejection of alternative histories (or possible worlds). God acts in various ways according to his plan. Among God’s ways are his commit-ments, constant actings, and patterns of coordinated acting. Everything is reducible and explainable in terms of these while preserving the ontological priority of the Triune God. End of Interlude.

Therefore, RMS takes God’s plan in Christ to be an infinite, strict linear order of discrete, total world states. If God’s plan in Christ is just such an infinite structure of such states, then there is an ontological correlate of the natural numbers sequence. RMS treats it as a real structure and the propo-sitional content of PA2 as a mathematical structure. We will address these in a moment. For now, the thing to understand is that RMS provides a way for second-order Dedekind-Peano Arithmetic (PA2) to be true per se along with the sentence of PA2, o′ + o′ = o″ (that is, “1 + 1 = 2”). Here, then, we have the beginnings of a theory of truth and one that is grounded in Christology. In other words, RMS provides an answer to the semantic and ontological ques-tions (at least with respect to the natural numbers).

Conjecture TwoSome mathematical concepts, propositions and structures are fictions, though not false per se.

RMS treats mathematical propositions as being either true, false or fic-tional. To see why this distinction is important, consider the following co-

10. This is a general claim and its more specific details are matters of theological dispute be-tween competing views of organizing motifs: covenants, dispensations, promise-plan, kingdom of God, and so forth. Each of this seems compatible with the concerns of this paper.

nundrum. Shapiro writes, “. . . pure mathematics is the study of structures, independently of whether they are exemplified in the physical realm, or in any realm for that matter . . . .”11 In other words, it makes no difference whether the description of the structure fits physical reality. But something seems amiss. Just because we can conceive some particular set of objects and some particular relations between those objects does not mean that what was conceived exists. Moreover, it seems profligate to think that merely the fact that a particular human conceptualization is consistent is sufficient to say that it exists in the mind of God as a deliberate and necessary creation. Works of literary fiction would be prime examples. Many mathematical ob-jects seem fictional on philosophical consideration. For example, both Ernst Zermelo and Kurt Gödel held that the empty set is a fiction.

Yet, according to Shapiro, the distinction between truth and fiction is not important. If the distinction were not important, mathematics would be like literary theory in one sense. Literary theorists study all sorts of literature including fiction. However, while works of fiction may share certain “struc-tural” features (character, plot, and so forth), and while novels are metaphors of reality, no novel is an empirical model of the world; no novel is history. Mathematicians take mathematical theories to be categorically different than novels. Is there a more rigorous way to approach this conundrum—to ques-tion the practice of treating consistency as sufficient for existence?

We can make a distinction between mathematical propositions that are true, false or fictional by taking advantage of two kinds of models: empirical models and interpretational models. We not only perceive or individuate sin-gle situations in the world but we also abstract, idealize and project patterns of such perceived situations. We are further able to represent or describe such patterns linguistically, thereby encapsulating extensive and often complex amounts of information. As John Barrow puts it, “In practice, the intelligi-bility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call ‘laws of nature’. . . . This is why mathematics can work as a description of the physical world. It is the most expedient language that we have found in which to express those algorithmic compres-sions.”12 Examples are Newton’s three dynamical equations (p = mv, F = ma, F1→2 = −F2→1) and universal law of gravity (F = Gm1 m2 /r

2), which represent physical patterns and which according to RMS are ways God sustains cre-ation.

With this notion of algorithmic compressibility in mind, let us say that an empirical model is a formal depiction of a physical phenomenon. So,

11. Shapiro, Philosophy of Mathematics, 75.12. John Barrow, Theories of Everything: The Quest for Ultimate Explanation (Oxford: Ox-

ford University Press, 1991), 199.

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for any propositional content p of a set of mathematical sentences, p is an empirical model of y just in case (a) p is symbolic generalization of y, if y is a phenomenon, or (b) p is an algorithmic compression of y, if y is a pattern of phenomena. Set p is fictional just in case at least one of the subject terms of its sentence has no physical referent (or no one is yet justified by physical theory in believing that it has). A second kind of model is related to the se-mantics of formal languages. In general and informally, p is an interpretation model just in case p is an interpretation of a set of formal sentences under which those sentences are true.

An axiom system can involve models in either sense. For example, Euclidean geometry is an axiom system that was thought to be an em-pirical model of physical space. We showed earlier that the sentence, o′ + o′ = o″, is a logical consequence of Dedekind-Peano Arithmetic (PA2). Since N(+) = + : N3 → N, N(o′ + o′ = o″) = T iff ⟨⟨N(o′), N(o′)⟩, N(o″)⟩ ∈ +. N is an interpretation model of SLS5. Therefore, PA2 is true under the interpretation N.

With this distinction in mind, consider that in a letter to Frege, Hilbert writes,

You [Frege] write, “I call axioms propositions that are true but are not proved because our knowledge of them flows from a source very dif-ferent from the logical source, a source which might be called spatial intuition. From the truth of the axioms it follows that they do not con-tradict each other.” I found it very interesting to read this sentence in your letter, for as long as I have been thinking, writing, and lecturing on theses things, I have been saying just the opposite: if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist. For me this is the criterion of truth and existence.”13

From this, one might say that while Frege placed greater epistemic reliance on empirical models, Hilbert considered interpretation models to be suffi-cient. RMS breaks this impasse because it takes the actual world (that is, God’s plan in Christ) to include the structures for empirical models. It treats aspects of possible worlds as referents of interpretation models. So, interpre-tation models that are not also empirical models, are fictions.

We should refine this by saying that if we are not justified in believ-ing that there is a real structure that a mathematical structure or concept represents, then we should treat the mathematical formalism as a fiction. Pure mathematicians have often invented mathematical formalisms that at the time seemed to have no real correlates and only later have mathemati-

13. David Hilbert, letter to Frege, in Philosophical and Mathematical Correspondence, ed. G. Gabriel, H. Hermes, F. Kambartel, C. Theil, and A. Veraart Abr. B. McGuinness, and trans., H. Kaal (Chicago: University of Chicago Press, 1980): 39, 40.

cal physicists discovered such formalisms to be useful. Thus, categorizing propositions as true, false or fictional is not always determinable.

InterludeMathematical Structures and Real Structures

To continue the argument, the difference between mathematical struc-tures and real structures must be further specified. RMS takes mathemati-cal structures to be consistent systems of propositions. A proposition is the information expressed by an indicative sentence in a context of utterance or inscription. A theory is a set of propositions and, when coherent and consis-tent, a theory is a (complex) concept. Einstein’s General Theory of Relativity (GTR) is an example. Propositions (and hence, theories or complex con-cepts) are usually thought of as being the kinds of things that are either true (or false) and this is because what they represent to be the case, is the case (or is not the case). Real correlates that make propositions true are either aspects of God’s plan in Christ or are God’s ways of acting according to plan.

Mathematical structures, then, are the propositional content either of consistent axiom systems (which are sets of mathematical propositions) or algebraic or differential equations. Mathematical structures are logically pos-sible structures which may or may not correlate with real structures. God’s acting according to his plan in Christ is what constitutes present created real-ity.14 What true mathematical structures represent are regularities in the ways God acts. Such regularities are real structures.

Conjecture ThreeSome other mathematical concepts and structures are true per se by virtue of empirically modeling real structures.

There are mathematical structures other than Dedekind-Peano Arith-metic (PA2)—such as Zermelo-Frankel set theory (ZF)—and there are other

14. Here is a possible way to understand “present created reality” that is consistent with RMS. The Causal Set Hypothesis of Quantum Gravity together with the neo-Lorentzian physi-cal interpretation of the mathematical formalism of the Special Theory of Relativity permits us to hypothesize that, for each Planck moment, there is a single, total state of the universe. Thus, each total state represents a finite, irregular three-dimensional manifold at a Planck moment of simultaneity. (A Planck moment, according to quantum physics, is the smallest, physically-pos-sible duration, 10−43 seconds.) Each realized total state (objectively considered) is nothing but God’s acting—God’s acting at that moment, in a multifaceted, coordinated way, according to plan.

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kinds of mathematical propositions such as those expressed by mathematical sentences of physical theory. How can these be accounted for ontologically in RMS? As was noted earlier, several different concepts are expressed by various uses of the terms, “structure” and “structuralism” in mathematics and the philosophy of mathematics. Saunders Mac Lane held that “Math-ematics consists in the discovery of successive stages of the . . . structures underlying the world and human activities in that world, with emphasis on those structures of broad applicability and those reflecting deeper aspects of the world.”15 RMS holds that all but one of the “structures underlying the world” Mac Lane refers to are real structures—regularities in the ways God sustains creation. Furthermore, each of these real structures is a feature of one other real structure which is God’s plan in Christ. RMS treats these as real structures. (This commits RMS to a version of Ontic Structural Realism in philosophy of science.) God’s plan and God’s ways of acting according to plan exhaust the category of real structures.

Isaac Newton was looking for an understanding of the physical world in terms of God’s omnipotence. As William Lane Craig writes: (God’s) om-nipresence should be explicated in terms of His being aware of and causally active in at every point in space.16 This echoes Jonathan Edwards’s claim that “. . . to find out the reasons of things in natural philosophy is only to find out the proportion of God’s acting.”17 RMS is consistent with these aspirations. It treats the causal nexus to be constituted by God’s ways of acting accord-ing to plan. God’s actions, are apprehended or cognized by humans in two ways. The first is as objects having properties and standing in relations in space and time. However, according to RMS, these things are nothing more than structured unities of dispositions, capacities and powers. The second way God’s actions are apprehended or cognized by humans is as “laws of nature.” Ontologically considered, laws of nature as laws of succession are patterns or regularities in God’s acting according to plan; laws of coexistence are the coordination of God’s acting according to plan. As these regularities and coordinations of God’s actions are perceived and conceived, “laws of nature” are descriptions of phenomenal regularities, where such phenomena are the manifestations of dispositions.18 Thus, laws as phenomena depend on dispositions instead of dispositions being determined by laws. As Roland Omnès puts it,

15. Saunders Mac Lane, “Mathematical Models: A Sketch for the Philosophy of Mathemat-ics,” American Mathematical Monthly 88 (1981): 462–72.

16. William Lane Craig, Time and the Metaphysics of Relativity (Dordrecht: Kluwer Aca-demic, 2001), 241.

17. Jonathan Edwards, “The Mind,” in The Works of Jonathan Edwards, vol. 6, Scientific and Philosophical Writings, ed. Paul Ramsey (New Haven, CT: Yale University Press, 1989), 353.

18. This claim is developed in Schultz, “Dispositions, Capacities, and Powers.”

. . . the fundamental laws of nature are pure mathematical forms ac-counting for the phenomena though providing no cause for them and showing no action . . . . The laws expressing the regularities of reality are much more accessible to understanding than reality itself . . . . They are prior to mathematics, however, just as reality is absolutely prior to anything.19

God’s plan is one of many possible worlds or histories, which taken col-lectively are God’s direct awareness of his ability. Dispositions, powers, and structures are not thoughts, but ways of acting according to his plan—the actual world—which is the one possible world he ordains. God’s acting ac-cording to plan gives us the real structures for empirical models; aspects of other possible worlds (merely possible structures) are referents of interpreta-tion models. The ontological questions are, therefore, addressed by treating all structures fundamentally in terms of God’s direct awareness of his ability (which we will address shortly) and all real structures in terms of God’s plan in Christ.

Conjecture FourAbstract objects depend on God ultimately as aspects of his awareness of his ability.

To address the Dependence Problem, this section proposes a different account of abstract objects in addition to a distinction in the received concept of a proposition. The latter is intended to account for apparent differences in the nature of the content of divine and human cognitive mental states. While this section is admittedly conjectural and exploratory, because the paper is aimed at advancing a thorough-going Christian view of abstracta, the discus-sion must be extended into these areas.

(1) God’s ability to create is not identical to his awareness of his ability.Most, if not all, Christian philosophers would seem to hold that God is

exhaustively aware of himself. God’s complete self-awareness involves his being aware of his own ability to create. God’s ability to create is not identi-cal to God’s direct awareness of his ability to create. The former is the object of the latter. Thus, God’s direct awareness of his ability to create logically depends on his ability to create. It is an awareness of an infinite range of things of various kinds, which, were he to create them, would be his work ad extra (“outside” of himself). It is his awareness of what is possible for him to achieve. That is, God’s awareness is about something, but it cannot only be that he is omnicompetent and it cannot be only an awareness of sheer abil-

19. Roland Omnès, Converging Realities: Toward a Common Philosophy of Physics and Mathematics (Princeton, NJ: Princeton University Press, 2005), 18.

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ity (although it includes these). The possibilities must have content. Since God’s ability to create is logically prior to his awareness of his ability, and God’s awareness cannot only be that he is omnicompetent, God’s awareness of his ability must take the form of representing to himself the possibilities of his omnicompetence. Such representations are dependent, therefore, on his nature.

(2) God’s representations are world states, not propositions.God’s direct awareness of his ability to create must be an infinite col-

lection of representations. Let a world state be any aspect of God’s direct awareness of his ability. There seem to be at least three conceptually pos-sible types. Let us suppose that an atomic world state represents the content of a Planck cell (a three-dimensional irregular hodon) at a Planck moment (a chronon). A representation of the content of the entire universe at a Planck moment be a total world state.20 A composite world state (or world state per se) is any combination of atomic world states without a regional or temporal gap.

Therefore, composite world states are similar to propositions or states of affairs as the terms are standardly used, but are not conceptually identical to them. The standard view of propositions (and possible worlds as maximal propositions or states of affairs) is that they are the content of both created agents’ thoughts and God’s. However, while a world state is any aspect of God’s direct awareness of his ability (a complete and exact representation), a proposition is the content of a cognitive mental state of a created con-sciousness. It is a created, abbreviated and synoptic representation. The ap-parent informational content of a person’s belief cannot be as detailed as God’s self-awareness. Secondly, both physical objects and human mental states are processes. As such, both are sequential manifestations of disposi-tional properties, which in turn are structured sequences of God’s acting and rooted ultimately in God’s self-awareness.21 Therefore, in one sense, some of the sequences of God’s actings yield items that have truth values. In other words, God’s acting sequentially brings about occurrent states of conscious-ness which we experience as having timeless content and take such things to have truth value.

To put this another way from the “bottom up” so to speak, RMS takes Truth (ontologically considered) to be God’s knowledge of himself and of creation. Derivatively, God’s acting according to his plan in Christ is what

20. The universe itself could be an irregular cube of simultaneity at that moment. Compare to W. Piechocki, “The Structure of Spacetime at the Planck Scale,” Acta Physica Polonica VB21 (1990): 711–15; and Erwin Biser, “Discrete Real Space,” The Journal of Philosophy 39 (1940): 518–25.

21. Accordingly, e.g., even though a created agent may entertain an idea of a world in which God does not exist, such an idea is not one of God’s representations. The representation or world state that God has in mind, which correlates to the proposition in the mind of the fool, is represented by this: (that) person’s thinking “God does not exist.”

constitutes present created reality. Correlatively, a proposition is true when what it synoptically represents as being the case has a correlative world state. A belief (understood as an occurrent state of consciousness) constitutes oc-current knowing when its content is true. Therefore, items of our mathemati-cal knowledge—that is, the content of our occurrent states of consciousness, what we call mathematical concepts and propositions—are matters of the manifestations of the network of dispositions and powers that constitute hu-man cognitive capacities and physical reality over some duration. Our math-ematical concepts are, therefore, correlated to God’s creative/sustaining act-ing according to plan and need not be treated as direct apprehensions of God’s thoughts. Thus, what seem to us to be abstract objects are ultimately the ways we apprehend what is “created in us.”

(3) God’s representations are eternal and necessary.Since God is from everlasting to everlasting and God is unchanging as

God, so is his awareness of his ability. This means that each aspect of God’s awareness of his ability to create is eternal. Since these aspects constitute an infinite collection of representations, it (that is, the collection) is also eter-nal. Since a world state is any aspect of God’s being aware of his ability, the existence of such representations are, therefore, not voluntary for God. Aspects of God’s awareness of his ability are not produced by God as acts of imagination. They are constant features of his being God. World states are, therefore, necessary in the sense that there are no alternatives to the col-lection that constitutes God’s direct awareness of his ability. They are also necessary because God’s not being aware of his ability to create is not an alternative. Whatever is possible, therefore, is necessarily possible.22

It follows that God need not be thought to create abstract objects in the standard sense of the term. Yet, abstract objects are dependent on God as logically generated aspects of his direct awareness of his ability. This is similar to St. Thomas Aquinas’s claim that abstract objects are a matter of God’s “knowledge of his power.”23 It differs, however, in crucial respects. Ability differs from power as omnicompetence differs from omnipotence. Di-rect awareness is essentially nonpropositional. If it were not, the problem of accounting for the existence of propositions would present itself again in the very process of giving an account of propositions. Secondly, God’s direct awareness accounts for the distinction between God’s sheer ability to create and the infinite range of things that can be created. Aquinas’s view seems not to account for these.

22. God’s being aware of himself and of his ways are also invariant background features to all world states and to all sequences of such states. This grounds second-order, S5 modal logic.

23. Thomas Aquinas Summa Theologica I, q.14, a.10.

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(4) God’s representations are not aspects of God per se.The representations comprising God’s awareness of his ability to create

are aspects or contents of God’s consciousness (so to speak), but not aspects of God per se. They are not aspects of God per se because God’s awareness of his ability to create is not identical to God’s ability to create and God’s ability is an aspect of God per se. Another way to see this is to compare this to a person’s imagined scenario involving that person. John imagines himself lounging on a Tahitian beach, for example. Now, even though John imagines himself on the beach, no aspect of John’s scenario is identical to John him-self. Imagined scenarios are called, intentional objects. To generalize, no aspect of an intentional object is identical to that consciousness for whom it is intentional. By this principle, it follows that God’s representations are not aspects of God per se. They are eternal and necessary by virtue of being what God is aware of in his being aware of his ability.

(5) Some of God’s representations are God’s plans and, therefore, contin-gent.

World states are combined in various relations to constitute alternative possible histories. In other words, world states—these objects of possible creation—can stand in various relations. For example, they can be sequential, simultaneous or overlapping to form complete world histories or what has come to be called, “possible worlds.” God makes some world states things he intends to create, that is, achieve and sustain. When he makes some world states things he intends to achieve and sustain, they remain representations (eternal and necessary) but take on a new character in that they become his plans. Hence, whereas world states are eternal and necessary, the feature of also being planned is contingent. God’s making some world states his plans is entirely optional. A complete sequence of plans is a possible world-history. One such sequence is God’s plan in Christ. God’s plan in Christ is one aspect of what God knows. All other alternative histories are ones God could have effected, but rejected. God’s plan in Christ is eternal, necessary, contingent, and everlasting. It is eternal, because God was always aware of its possibil-ity, necessary because God could not not be aware of it, contingent because God freely decided to enact it, and everlasting because it will never end.

Summary Conclusion

Realist modal structuralism is a philosophy of mathematics that in-cludes (1) a referential semantics coordinate with sentences of physical theory, (2) an externalist epistemology, and (3) a four-category ontology of possible worlds, dispositions, powers and structures. RMS is “structuralist” in three senses. Mathematical objects are understood in terms of structures. It is structuralist in a second sense in that the basic structure is God’s plan

in Christ understood as an infinite, strict linear order of discrete world states each composite world state is itself a structure. RMS is structuralist in a third sense, in that every physical object is a structured-unity of dispositions and powers and every law of nature and causal mechanism is a pattern of God’s acting. RMS is “Christian” (and not merely theistic) by being grounded in the actual world which is God’s plan in Christ.

RMS is “modal” because, since it treats the propositions expressed by mathematical sentences as either true, false or fictional (that is, “having no physical referent”), the realm of mathematical actuality is a region within the space of logical possibility. In other words, the space of mathematical struc-tures represent a region within the space of logical possibility. The space of logical possibility is God’s awareness of his ability. Each divine representa-tion constituting the realm of the possible is an eternal, necessary and invari-ant feature of God’s self-awareness. Therefore, the nature of possibility and necessity (metaphysical and physical) entirely depends on God’s self-aware-ness. God’s actings are what constitutes present physical reality and God’s making the actual world his plan is what, in general, makes propositions about it true. However, such divine knowledge is not constitutive of God per se because it is of himself and things he can do, has done, is doing or will do. Rather, even though God’s knowledge of creation is ultimately a matter of self-awareness, what he is aware of is distinct from what constitutes God per se. Hence, God’s knowledge of what is possible (which includes creation as contingent) is nonconstitutive self-awareness.

The Problem of Multiple Models is overcome by treating individual numbers as places in a structure. It makes no difference whether we name them using Zermelo numerals, von Neumann numerals or the familiar Ara-bic numerals. The Problem of Dependence is addressed by recognizing that what we take to be abstract objects are matters of God’s awareness of his ability, which is logically dependent on his ability. Since God’s ability is not ontologically dependent on his awareness of his ability, abstract objects have necessary, yet ontologically dependent existence.

Objection and Rejoinder

An objection to the forgoing solution may be raised: “A central thesis of theistic modal structuralism is that N is a structure that is exemplified by the temporal sequence of situations constituting God’s plan in Christ. Set-ting aside the worries about taking possible worlds as divine plans in such a ‘neo-Leibnizian’ way, such worlds are nevertheless real (mind-independent) and, in some sense, ‘thoughts in God’s mind.’ But ‘thoughts in God’s mind,’ is a sine qua non of theistic conceptualism, which seems to be little more than platonism transplanted. RMS gains little, if anything at all, over theistic conceptualism.”

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In one sense RMS is a species of theistic conceptualism. The patterns of God’s actions detected in creation suggest the existence of thoughts in God’s mind rendering RMS to be theistic conceptualist after all. However, there are important differences. First some structures are concrete and others are contingently nonconcrete.24 Theistic conceptualism does not now account for the difference. Secondly, some mathematical objects may be nothing more than useful fictions or abstraction-idealization-projections. We need a view that accounts for this practice of mathematics, logic, and science and for our conceptual capacities. Thirdly, mathematical objects need not be objects-in-their-own-right, which God thinks necessarily and constantly. Rather, they seem to be features we abstract from God’s acting according to plan. Fourth, RMS is does not posit an “abstract object” into the divine consciousness sim-ply by virtue of lack of inconsistency or provability. It gives a purposeful and coherent meaning to such objects. It is centered in Christ through whom and for whom all things exist, which points to the final and most important dif-ference. RMS is specifically Christian, whereas theistic conceptualism per se seems not to have yet made that ontological connection.25

24. See Bernard Linsky and Edward N. Zalta, “In Defense of the Simplest Quantified Modal Logic,” Philosophical Perspectives 8 (1994): 431–58.

25. I want to thank William Eppright and Jonathan Zderad of the Mathematics department at Northwestern College for many rewarding and pointed conversations regarding this paper. I also want to thank colleagues in the Association of Christians in the Mathematical Sciences, who at the 2009 biennial meeting at Wheaton College pressed many probing issues. Finally, I want to thank the anonymous referee for bringing to my attention issues that needed more development and one significant misleading sentence.

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Toward a Realist Modal Structuralism