January 14, 2008
Current Work on God and Abstract Objects
First and foremost, thanks for all of your work, time, and effort for the Kingdom before and now here on ReasonableFaith.org. It has been and continues to be an immense blessing to me in my walk with Christ.
I am somewhat puzzled over your views on the ontological status of abstract objects. In your essay on the ontological argument in To Everyone an Answer, you suggest conceptualism for the grounding of abstract objects (pp.134-135). You similarly appeal to conceptualism in Philosophical Foundations (pp. 505-506, etc.). In Creation Out of Nothing, again you and Paul Copan tentatively embrace conceptualism in response to the problem of reconciling abstract objects and God’s aseity. Here you do however mention some problems with conceptualism, the chief of which concerns the ontological status of the divine ideas. You write, “...f we take the divine ideas appealed to in the theory as literally thoughts that God has or mental activities he performs, then such thoughts and activities are particulars, not universals” (pp. 193-194). Is this the reason you list conceptualism as an off-shoot of considering mathematical entities as concrete objects in Fig. 1 of “J. Howard Sobel on the Kalam Cosmological Argument”?
You mentioned working heavily on the topic of abstract objects in the Q&A “Causal Premiss of the Kalam Argument”--I’m excited to see the fruition of such. In the mean time, can you please clarify and elaborate on your views on the ontological status of abstract objects in general and whether they remain abstract or are rendered concrete on a conceptualist model?
Wow, am I impressed with your familiarity with the few things I’ve written on this topic! Before I answer your question, let me bring readers up to speed who aren’t so familiar with this debate as you are.
A good place to begin is by asking ourselves, “Does the number 3 exist?” Certainly there can be three apples, for example, on the table; but in addition to the apples does 3 itself exist? We’re not asking whether the numeral “3” exists (the symbol borrowed from the Arabs to represent the quantity three). Rather we’re asking whether the number 3 itself exists. Are there such things as numbers? Do numbers really exist?
Some people might think that this question is so airy-fairy as to be utterly irrelevant. But in fact it raises a fundamental theological issue whose importance can scarcely be exaggerated. For if we say that numbers do exist, where did they come from? Christian theology requires us to say that everything that exists apart from God was created by God (John1:3). But numbers, if they exist, are almost always taken to be necessary beings. They thus would seem to exist independently of God. This is the view called Platonism, after the Greek philosopher Plato.
Someone might try to avoid this problem by espousing a modified Platonism, according to which numbers were necessarily and eternally created by God. But then a problem of vicious circularity arises: explanatorily prior to God’s creating the number 3, wasn’t it the case that the number of persons in the Trinity was 3? Of course; but then the number 3 existed prior to God’s creating the number 3, which is impossible!
I remember the sense of panic that I felt in my breast when I first heard this objection raised at a philosophy conference in Milwaukee. It seemed to be an absolutely decisive refutation of theism. I didn’t see any way out.
The way out, I discovered, is to deny the Platonist view that abstract objects like numbers exist. My first inclination was to adopt some sort of Conceptualism which construes abstract objects as ideas in God’s mind. This may still be the route I’ll take, but the more I’ve studied the problem the more attracted I’ve become to various Nominalistic or anti-realist views of abstract objects which flatly deny their existence rather than re-interpret their existence in terms of conceptual realities. As you note, Conceptualism seems to be a sort of realism which identifies numbers with thoughts in God’s mind. Such thoughts are concrete objects, not abstract objects, even though they are immaterial. Such an identification seems problematic in a number of ways, which I needn’t go into here. If, on the other hand, the Conceptualist does not take numbers to be actual thoughts God is having, then he seems to be really embracing some anti-realist view like Fictionalism.
So why should we think that abstract objects like numbers exist at all? About the only argument in favor of Platonism is the so-called Indispensability Argument inspired by the late W. V. O. Quine. Quine felt obligated to admit mathematical objects, specifically sets, into his ontology (one’s account of what exists) because he thought that the truth of our best scientific theories committed us to their reality. Quine’s argument was predicated on several distinctive theses:
1. Natural science is the sole arbiter of truth and guide to reality. (Naturalism)
2. Canonically formulated statements quantifying over mathematical entities are indispensable to our best scientific theories. (Indispensability Thesis)
3. We are ontologically committed to the value of any variable bound by the existential quantifier in a first-order symbolization of a true canonically formulated statement. (Criterion of Ontological Commitment)
4. Confirmation of the truth of our best scientific theories accrues to every indispensable statement of those theories. (Confirmational Holism)
Naturalism ensures that there are no metaphysical or otherwise extra-scientific grounds for rejecting the existence of mathematical objects. What science requires to be real is real, period. The Indispensability Thesis lies at the heart of every version of the argument. It is fundamentally the claim that quantification over mathematical entities in our best scientific theories cannot be paraphrased away. Quine recognizes that statements of ordinary language, if taken at face value, would involve quantification over pseudo-objects; hence, the need for a canonical formulation of the statements of a scientific theory, ensuring that their ontological commitments are irreducible. Quine’s Criterion of Ontological Commitment is not a criterion of existence per se but tells us rather what must exist in order for a canonical statement to be true. Given Naturalism, we shall be ontologically committed only by whatever statements in our best scientific theories are true. Finally, Confirmational Holism ensures that the indispensable mathematical statements of true scientific theories are themselves true. For whatever evidence goes to confirm the truth of the theory as a whole goes to confirm every statement it comprises. Since the mathematical statements of a true scientific theory are true and indispensable, we are ontologically committed by those theories to the mathematical objects quantified over. Hence, we are required by modern science to believe in the existence of mathematical objects.
Every one of these Quinean theses is highly controverted, and none of them, I think, much less all of them, is plausibly true. In what follows, I’ll share some of the results my recent reading on these topics. I apologize in advance for the rather technical nature of the discussion.
1. Although Quine’s naturalized epistemology has become widely influential, his Naturalism, not being itself among the deliverances of natural science, is incapable of being rationally justified. The only self-referentially coherent version of Naturalism, as Michael Rea has shown (World without Design: The Ontological Consequences of Naturalism [Oxford: Clarendon Press 2002], pp. 50-73) is that which takes Naturalism to be a disposition to accept only the deliverances of the natural sciences as true. Some persons, however, may be differently disposed, being prepared, for example, to accept also rational intuition or divine revelation as guides to truth. No one is obliged to make Quine’s personal disposition his own. Someone who is not a Naturalist might dare to challenge the commitments of even our best scientific theories. In this case, I, as a Christian philosopher, think we have good theological reasons for rejecting Platonism, regardless of what ontological commitments our scientific theories may carry. Moreover, Quine’s Naturalism, ironically, cripples mathematics, Platonistically construed, because that fragment of mathematics which natural science requires is an infinitesimal part of the mathematician’s universe of discourse.
2. The Indispensability Thesis has been criticized on several grounds. Charles Chihara’s criticisms have been especially devastating (Ontology and the Vicious Circle Principle [Ithaca, N.Y.: Cornell University Press 1973], chap. 3). He points out that Quine does not provide so much as a clue as to what a canonically formulated sentence is nor a procedure for obtaining one, much less a guarantee that the statements of scientific theories can be canonically formulated so as to eliminate all the pseudo-objects quantified over in ordinary language. Without such a procedure, Quine’s proposal cannot even get off the ground. Moreover, Quine just assumes that our best scientific theories can all be adequately formulated in first-order logic, which seems very dubious. Modal logic, tense logic, and counterfactual logic seem likely to be necessary for adequately capturing the theoretical content of natural science. Since Quine’s Criterion of Ontological Commitment fails in such contexts, the Criterion will be incapable of revealing accurately the ontological commitments of those theories.
Finally, Chihara’s Constructibilism recovers classical mathematics without quantifying over mathematical entities. This is achieved by re-writing ordinary Zermelo-Fraenkel set theory by replacing the existential quantifier “∃” (meaning “there is. . .”) with what Chihara calls a constructibility quantifier, so that existence claims are replaced by claims about what is constructible. The constructibility quantifier Cx is to be understood as asserting, “It is possible to construct an x such that. . . .” The constructibility quantifier is taken as primitive, though Chihara uses possible worlds semantics merely as a heuristic device. What is constructible on Chihara’s theory are certain open sentence tokens, that is to say, sentence tokens containing unbound variables, and assertions of set membership are re-written as assertions about some individual’s satisfying an open sentence. Chihara does not claim that his semantics represents how mathematicians actually understand their language nor that it should replace standard mathematical language but merely that it shows how mathematical statements may be regarded as true without any ontological commitment to abstract objects.
Similarly, Geoffrey Hellman’s Modal Structuralism successfully avoids quantification over mathematical objects (Mathematics without Numbers: Towards a Modal-Structural Interpretation [Oxford: Oxford University Press, 1989]). Structuralism draws its inspiration from the insight that the only mathematically relevant properties of numbers are their relational properties. The intrinsic properties of the natural numbers can therefore be ignored in favor of the abstract, ordinal structure which they instantiate. It is mathematically irrelevant what sort of objects fill the positions in such an ordinal structure. So we don’t really need numbers at all. In order to avoid ontological commitment to abstract structures, Hellman affirms merely the logical possibility of such structures. So mathematical statements do not involve quantification over objects or positions in any actual ordinal structure, since it is merely the possibility of structurally interrelated objects or positions which is considered.
3. Quine’s Criterion of Ontological Commitment is perhaps the most vulnerable plank in his argument. To begin with, everyone acknowledges that the expression “there is” (which is symbolized by the existential quantifier “∃”) is not ontologically committing in ordinary language. We say such things as “There are deep differences between Republicans and Democrats” or “There is a lack of integrity in his behavior” without thinking that we thereby commit ourselves to including such things as differences and lacks in our ontology! This point cannot be exaggerated. Existential quantification in ordinary language cannot be reasonably taken to commit us ontologically to the items quantified over. Quine, of course, recognizes this. But he insists that once the sentences of our best scientific theories have been put into a canonical form and symbolized in first-order logic, then we are committed to any items bound by the existential quantifier. I’ve already remarked that Quine gives not even a hint as to a procedure for putting the sentences of ordinary language into canonical form nor any argument at all that so doing will rid them of all unwanted commitments of ordinary language nor any guarantee that our best scientific theories can be successfully symbolized in first-order logical notation.
But even if one were to succeed in carrying out Quine’s procedure, the question of whether we are ontologically committed to the values of the variables bound by the existential quantifier all depends on our interpretation of “∃.” Why think that this quantifier has any different meaning or carries any more ontological force than “there is” in ordinary language?
Philosophers have typically discriminated between two interpretations of the existential quantifier: the objectual (or referential) and the substitutional. The objectual interpretation of the quantifier conceives it as ranging over a domain of objects and picking out some of those objects as the values of the variable bound by it. The substitutional interpretation takes the variable to be a sort of place-holder for particular linguistic expressions which can be substituted for it to form sentences. The substitutional interpretation is generally recognized not to be ontologically committing. But Jody Azzouni points out that even the objectual interpretation of the quantifier is not ontologically committing until one so stipulates (Deflating Existential Consequence: A Case for Nominalism [Oxford: Oxford University Press, 2004], p. 54). The claim that it must be ontologically committing overlooks the fact that the quantifiers of the meta-language used to establish the domain of the object language quantifiers are similarly ambiguous. Whether the items in the domain D of the object language quantifier actually exist will depend on how one construes the “there is” of the meta-language establishing D. Even referential use of the quantifier in the object language need not be ontologically committing if the quantifiers in the meta-language are not ontologically committing. If, when we say that there is an element in D, we are using ordinary language, then we are not committed to the reality of the objects in D which we quantify over. (If we were, then the paradox results that a theory already commits us ontologically by the domain itself to the objects in it, so that Quine’s criterion becomes wholly superfluous.) There is no reason that one cannot set up as one’s domain of quantification a wholly imaginary realm of objects. D is then non-empty, but objectual quantification in the object language of the domain will not be ontologically committing. The existential quantifier simply serves to facilitate logical inferences.
However that may be, why can’t the Nominalist to adopt a substitutional interpretation of the existential quantifier when quantifying over abstract objects? If I assert (∃x) Px, where “P” represents the predicate “is a prime number,” then I can say that “3” can be substituted for x to yield the true sentence “3 is a prime number” without thereby committing myself ontologically to the reality of 3. Whether 3 exists will have to be decided by extra-logical arguments and can be expressed by means of an existence predicate. It might be objected that such a selective appeal to substitutional quantification is ad hoc. But as Dale Gottlieb explains, its use may be justified in the special case of quantification over abstract objects in view of the almost magical ontological consequences that are said to ensue from the objectual interpretation (Ontological Economy: Substitutional Quantification and Mathematics [Oxford: Oxford University Press 1980], pp. 53-4). For example, from the sentence “There are three apples on the table” it follows immediately that “The number of apples on the table is 3,” which, on Quine’s Criterion, commits us ontologically to the existence of 3! Discovering what exists shouldn’t be this easy! Taking the quantifier substitutionally would prevent deriving ontology by means of mere words. Thus, for the Indispensability Argument to succeed, one would have to show that the quantifier cannot be taken substitutionally, which is, in Gottlieb’s judgement “almost impossible to establish” (Ibid., p. 50).
Still another option is afforded by Stephen Yablo, who has moved through Fictionalism, which accepts Quine’s Criterion, to what he calls Figuralism, so as to be able to preserve the truth of talk about abstract objects without ontological commitment (“Go Figure: A Path through Fictionalism,” in Figurative Language, ed. Peter A. French and Howard K. Wettstein [Oxford: Blackwell, 2001], pp. 72-102). Yablo is impressed with the similarities between abstract object talk and figurative talk such as we find in understatement, hyperbole, metonymy, and metaphor. An assertion like “It’s raining cats and dogs!” is literally false, but to stop there is to miss the whole point of such language. When a speaker uses figurative language, the literal content is not what the speaker is asserting. There is what Yablo calls a “real content” to figurative statements which may well be true. This is not to say that figurative statements can always be successfully paraphrased into expressions of their real content. Numbers may be indispensable as representational aids for the expression of the real content of mathematical language. The real content of mathematical statements is logical truths, which is why mathematics seems necessary and a priori.
Yablo extends his analysis to include other sorts of abstract object talk as well. For example
|The truth value of:|
Argument A is valid
It is possible that B
There are as many Cs as Ds
There are over five Es
He did it F-ly
There are Gs which ___
She is H
|is held to turn on:|
the existence of counter-models
the existence of worlds
the existence of 1-1 functions
the number of Es
the event of his doing it’s being F
there being a set of Gs which ___
her relation to the property H-ness
The entities on the right hand side are not what the expressions on the left are really about. We simulate belief, perhaps quite unconsciously, that the entities on the right exist, but they are mere figures of speech which are vehicles of the real content. Figurative speech may be true—herein lies the difference between Figuralism and Fictionalism—but the representational aids it employs are not ontologically committing. In light of these alternative understandings of the existential quantifier, Quine’s Criterion of Ontological Commitment appears to be not merely unwarranted but even misleading and implausible.
4. Quine’s radical Confirmational Holism is an utterly implausible doctrine. Elliott Sober, who has been a persistent critic of this Quinean thesis, agrees that scientific hypotheses are never tested in isolation but rather in conjunction with certain auxiliary assumptions (“Quine’s Two Dogmas,” Proceedings of the Aristotelian Society, Suppl. Vol. 74 : 237-80). But scientists typically test one hypothesis against another competing hypothesis which shares the same set of auxiliary assumptions. An observation O favors hypothesis H1 over H2, given the assumptions A, just in case Prob (O | H1 & A) > Prob (O | H2 & A). In such a case A is not tested and so is not confirmed by O. Mathematics and logic are part of the background assumptions common to all theories and so are not confirmed by empirical evidence for the theory. Sober charges that the Quinean Holist seems to be guilty of thinking that because O confirms H and H entails S, therefore O confirms S, an inference which is fallacious.
A further indication that mathematics is not confirmed by evidence for a scientific theory is the fact that it is never disconfirmed by evidence against a theory. Yet confirmation theory requires that if O confirms H, then not-O would disconfirm H. An even more bizarre consequence of Quine’s radical Holism is that if I believe, say, Special Relativity, then a confirmation of Relativity confirms everything I believe, no matter how unrelated it may be to Relativity theory. Relativism immediately results, for if I believe X & Y and you believe X & not-Y, then the confirmation of X confirms Y for me but confirms not-Y for you!
Quine himself later backed away from this radical holism to the position that merely “largish” sets of beliefs are subjects of confirmation, rather than one’s entire system of beliefs. But this more moderate Holism is still subject to Sober’s criticisms and is, in any case, too weak to fill the role played by Confirmational Holism in the Indispensability Argument for mathematical objects. Without the Holistic thesis, the pure mathematical statements in a confirmed theory are not thereby confirmed to be true.
That opens the door to Fictionalism, which maintains that while the nominalistic content of a scientific theory may be true, the pure mathematical content, if taken literally, is false, being a useful fiction. Fictionalists have taken two routes in responding to the Indispensability Argument. One route, taken by Hartry Field (Science without Numbers [Princeton: Princeton University Press, 1980]), is to challenge the Indispensability Thesis that mathematics is indispensable for science and to provide a nominalized version of a scientific theory in which no reference to mathematical objects is made. The second route, adopted by Mark Balaguer (Platonism and Anti-Platonism in Mathematics [New York: Oxford University Press, 1998]), is to accept the Indispensability Thesis but to maintain that however indispensable mathematics may be for scientific practice, it contributes nothing of content to our knowledge of the world and that its applicability is no better explained by Platonism. Both agree that the Platonistic content of empirical science is fictional and therefore false.
Sentences like “2 + 2 = 4” are like statements concerning fictional characters, such as “Santa Claus lives at the North Pole.” Such sentences fail to correspond to reality because they have vacuous terms in them. Because they thus fail to correspond to reality, they are literally false. Since there is no such person as Santa Claus, he cannot literally live at the North Pole. Since there are no such things as two and four, it is not literally true that four is the sum of two twos. What is true to say, however, is that Santa Claus lives at the North Pole according to the usual story of Santa Claus; he does not, according to that story, make his home in East Peoria. Similarly, it is true to say that 2 + 2 = 4 according to the standard account of mathematics. This saves the Fictionalist from the embarrassment of stating flatly that “2 + 2 = 4” is false, for he agrees that such a statement is true in the standard model of arithmetic. But he denies that that model corresponds to any independent reality. It is a mistake to think that mathematical practice commits us to the literal truth of mathematical theories, for the ontological question concerning the reality of mathematical objects is a philosophical question which mathematics does not itself address. At most our practice commits us to holding that certain statements are true according to the standard account in the relevant area.
Kendall Walton, whose fascinating work on the nature of fiction finds applications outside the literary and artistic fields, speaks of “fictional truths,” which are statements taken as true within a game of make-believe (Mimesis as Make–Believe [Cambridge, Mass.: Harvard University Press, 1990]). Fictional truths are generated by a prescription or mandate to imagine something’s being the case. The agreements which participants in the game make about what to imagine serve as rules prescribing certain imaginings. These rules are generating principles of a fictional world in which certain propositions are to be imagined as true. Walton emphasizes that such an understanding of the rules of the game may not even be conscious or explicit. “It may be so ingrained that we scarcely notice it, so natural that it is hard to envision not having it” (Ibid., p. 41). Thus, one may engage in make-believe without even being aware of it.
It’s hard to imagine a more appropriate and plausible non-literary application of Walton’s theory than axiomatized infinite set theory. The intuitive concept of a set employed by Cantor was soon found to generate the paradoxes of naïve infinite set theory and therefore to be untenable. Rather than adopt a new understanding of “set,” set theorists chose simply to avoid the paradoxes by leaving the notion of set undefined but laying out several axioms governing the behavior of sets so as to prevent the paradoxes from arising. These axioms have little claim to intuitive truth, since we do not even know what they are about. The axioms of set theory seem rather to be prescribed for us to imagine as true. These axioms then serve as generating principles for the universe of sets. Within this game of make-believe quite astonishing and wondrous theorems turn out to be fictionally true. It is, for example, fictionally true that the set of natural numbers and the set of even numbers have the same number of members, even though the set of natural numbers includes all the even numbers plus an equal and infinite number of odd numbers as well. Moreover, like worlds of fiction, the universe described in infinite set theory is radically incomplete. Just as it is neither fictionally true nor fictionally false that, for example, Hamlet wears size 9 shoes, so the Continuum Hypothesis (CH), the hypothesis that the power of the continuum is ℵ1, the next highest transfinite cardinal number after ℵ0, is neither fictionally true nor fictionally false, since it has been demonstrated to be independent of the axioms of standard set theory. The realist claim that CH must be either true or false sounds strangely like the insistence that Hamlet must wear some size shoe, either 9 or not. The set theorist is free, if he wishes, to add CH or ¬CH to his axioms, to publish, as it were, a revised edition of the story, and then a whole new set of fictional truths will be generated. In so doing, as Walton observes, one may or may not be aware that one is participating in a game of make-believe.
Quine’s Confirmational Holism is, then, quite implausible, and its rejection opens the door to a Fictionalist reading of the statements of pure mathematics employed in science.
In sum, the abundance of Nominalist defeaters of the Indispensability Argument leaves the issue of the ontological status of abstract objects like numbers at least an open question and various Nominalisms (not to speak of Conceptualism) as viable alternatives to Platonism. Hence, I’m pleased to say, no successful objection to classical theism arises from this quarter.