Above: Stained glass window depicting God speaking to Moses from the Burning Bush declaring himself “I AM”.  [Author: Rolf Kranz.  See https://commons.wikimedia.org/wiki/File:Spay,_Pfarrkirche,_Moses_und_der_brennende_Dornbusch.jpg]

“Does God exist?” On the surface, this may appear to be a rather straightforward — not to say easy — question.  But the disagreement between those who answer “yes” and those who answer “no” is not straightforward at all.  It is, rather, “one of those fundamental disagreements that extends to how the disagreement is to be characterized,” as the philosopher Alasdair MacIntyre once observed.¹  Atheists and theists not only disagree about whether God exists; they also disagree about what it means to disagree about whether God exists.

This disagreement over disagreement is at its most obvious when it comes to popular debates about science and religion.  Against those who profess belief in God, there are those who profess that modern science renders such belief irrational.  According to these “scientific atheists,” as we might call them, the existence of God is a hypothesis of the same kind as the hypotheses of natural science, and thus may be tested empirically in the same way that we test other such hypotheses.  Unsurprisingly, the “God hypothesis” fails such tests, and so scientific atheists conclude that the hypothesis is false and that belief in God is unwarranted and irrational.

Implicit here is a misunderstanding of what is at issue in the debate over God’s existence.  For theists, God is not the kind of thing that could be tested by the methods of natural science.  The reason is that God is not a part of the observable universe, but rather the transcendent cause of the universe.  So the inability of modern science to confirm God’s existence does not show that belief in God is unwarranted or irrational; but only that modern science does not have a monopoly on warranted rational belief about what exists.

A metaphysically courageous atheist might try a more direct line of attack.  He might acknowledge that God is not understood by theists to be a part of the physical universe and therefore is not the kind of thing that could be observed.  Nevertheless, he might maintain that it is not rational to believe in what is not observable.  Therefore, if by “God” we mean something inherently unobservable, then belief in him is inherently irrational.  It’s not simply that the “God hypothesis” fails an empirical test; it fails even to be a meaningful hypothesis.  Such an argument, if successful, would have the benefit (from the atheist’s point of view) of ruling out not only belief in God but also belief in a host of other things that many atheists find suspect, from minds to objective moral values.

Unfortunately for the scientific atheist, however, this argument rules out far too many things.  For if it is sound, not only gods, minds, and morals, but all non-observable objects must go by the board.  This would leave no room for mathematical objects like numbers, sets and functions or even for theoretical objects like electrons, neutrinos, and quarks.  Yet, such objects, though not directly observable, are necessary for modern science.  This argument against theism, therefore, has the drawback of yoking atheism to a disputable metaphysical thesis, one that not even many scientific atheists would want to accept.

The indispensability of theoretical and mathematical objects for modern science is what led the eminent logician and philosopher Willard Van Orman Quine to include such non-observable objects in his otherwise empiricist philosophy.  Unlike our less sophisticated scientific atheist, Quine was not dogmatic about ontology² — about what exists.  Though he was an empiricist, he allowed for the existence of theoretical objects that are not directly observable, such as electrons, neutrinos, and quarks, as well as “abstract” objects such as numbers, sets, and functions.  Might the scientific atheist thus find in Quine’s brand of empiricism a more reasonable and promising line of argument against theism?  Let us see how far it can take him.

If we want to know what there is, Quine said, we should begin by asking what our leading scientific theories say there is.  If we want to know about physical objects, we should look to our best theories of natural science, such as quantum physics or molecular biology; if we want to know about mathematical objects, we should look to our best theories of mathematics, such as set theory and algebraic topology.  All of these theories make (or at least appear to make) ontological claims — for example, about the existence of elementary particles and quantum fields or cellular molecules and RNA strands or denumerable and non-denumerable sets or manifolds and groups.  According to Quine, we can figure out what a theory says there is by translating its statements about such items into the language of symbolic logic.

For instance, the statement “Some swans are white” may be translated into logic as follows: “∃x(Sx & Wx)” or “There is an x such that x is a swan and x is white.”  That statement is true so long as there is at least one white swan.  Similarly, the statement “All swans are white” may be translated as “∀x(Sx → Wx)” or “For all x, if x is a swan, then x is white.”  This statement is true so long as there are no non-white swans.  The symbols “∃x” and “∀x” are called “quantifiers,” and we say that they “quantify over” objects, whether swans, electrons, or numbers.  Quine shows how an entire theory may be so rendered or “regimented” into formal logic using quantifiers. And he argues that we can determine what objects a theory says there are by seeing what objects it quantifies over.

For instance, it might appear as though our theory about swans is committed not only to the existence of swans but also to the existence of “whiteness,” the property of being white.  But, according to Quine, this is only apparent.  Notice that the logical versions of our statements quantify only over swans, not over attributes of swans (like being white).  Consider the statement “∃x(Sx & Wx)” or “There is an x, such that x is a swan and x is white.”  The value of the variable “x” here is an object — a swan.  “White,” by contrast, appears only as a predicate, “W.”  A “predicate” is not an object but rather tells us something about an object, in this case that the object x is white. (In the jargon of logic, the predicate W is a “function,” and Wx means that x is included in the class of white objects.) This shows how, through logical analysis, we may “eliminate” ontological baggage, so that some of what looked like ontological claims turn out not to be.

Of course, a theory’s references to objects are not always eliminable in this way.  The statement “Some swans are white,” for instance, cannot be translated into logic without quantification over swans and still come out true.  Or consider one of Quine’s own examples.³  Classical mathematics says there are prime numbers greater than one million. This may be rendered as ∃x((x is prime) & (x > 1,000,000)). There is no way to preserve the truth of this statement — or other similar statements — without quantification over numbers. We may thus conclude that classical mathematics is committed to the existence of numbers.  And numbers, of course, are not physical objects, but abstract ones.  So classical mathematics is committed to the existence of abstract objects.

Interpreting ontological commitments in this way is valuable, Quine thought, because it allows us to reframe questions about ontology in terms of questions about language.  Rather than arguing about what there is, we can first clarify what our theories say there is.  And in this way, our “basic controversy over ontology can be translated upward into a semantical controversy about words and what to do with them.” ⁴  This is what Quine called “semantic ascent” — or “withdrawing to a semantic plane.” ⁵

Semantic ascent does not mean that ontology is merely a matter of language.  For asking what there is according to some theory isn’t the same as asking what there actually is.  To say that classical mathematics is committed to the existence of numbers is not the same thing as saying numbers truly exist.  As Quine put it, we use this logical analysis “not in order to know what there is, but in order to know what a given remark or doctrine, ours or someone else’s, says there is.” ⁶  How, then, are we to decide what actually exists?

Quine’s answer is, once again: Look to our best scientific theories. Say our best physical theory is ontologically committed (in the above sense) to electrons, neutrinos, and quarks.  Then so should we be.  If our best theory of mathematics says there are numbers, then we should believe in numbers.  But must we accept the existence of all the objects to which all our best scientific theories are committed?  And, if not, how do we decide?  Electrons, neutrinos, and quarks might seem like reasonable enough objects to count within our ontology.  So too might cellular molecules and RNA strands.  But what about non-denumerably infinite sets or algebraic manifolds?

According to Quine, there is ample room for reasonable disagreement here.  Say you have a “constructivist” ⁷ view of mathematical proof.  In that case, you may have no problem with classical mathematics, but reject the higher infinities of Cantor’s set theory.⁸ Or perhaps you’re a “nominalist,” ⁹ meaning that you believe only in the existence of “particulars.”  In that case, your ontology will not accommodate abstract objects such as numbers and sets.  The appeal of “semantic ascent,” according to Quine, is not that it resolves all ontological disputes.  Its appeal, rather, is that it “carries the discussion into a domain where both parties are better agreed on the objects … and on the main terms concerning them.” ¹⁰  How so?

The idea is that Quine’s approach at least allows all parties to agree on what it means to have an ontology.  What’s more, we are in a position to assess the merits and demerits of rival ontologies by considering their broader implications, in particular what theories they allow us to hold or preclude us from holding.  For instance, if your ontology requires you to reject higher infinities, then you must also reject whole swaths of modern mathematics.  If you forswear belief in mathematical objects, then you cannot accept most of mathematics or much of natural science.  The argument then becomes whether these ontologies, whatever their merits, are worth the price of admission.  In this way, semantic ascent allows us to find common ground from which to “apprais[e] the ontological commitments of one or another of our theories.” ¹¹

In seeking to reframe ontological questions in terms of questions of language, Quine did not hope to eliminate metaphysics through the logical analysis of language.  Such was the goal of Quine’s teacher and friend Rudolf Carnap.¹²  Quine rejected as futile all attempts to draw a firm boundary between statements that are purely empirical (and thus “meaningful”) and statements that are metaphysical (and thus “meaningless”).  Our best theories are committed to the existence of all manner of extra-empirical objects.  In this sense, as Quine provocatively put it, “the physical objects [of modern science] and the gods [of Homer] differ only in degree and not in kind.” ¹³

He did not mean that there is no difference: “Let me interject that for my part I do, qua lay physicist, believe in physical objects and not in Homer’s gods; and I consider it a scientific error to believe otherwise.” ¹⁴  Yet, Quine’s reason for rejecting Homer’s gods is pragmatic rather than dogmatic: Our leading scientific theories are not committed to their existence, and so we can get by without them.  We can’t, however, get by without elementary particles or mathematical sets.  So we ought to accept those.  But elementary particles or mathematical sets are no less metaphysical than Homer’s gods. “The quality of myth … is relative.” ¹⁵

When it comes to the question of what ontology to adopt, Quine urged “tolerance and an experimental spirit.” ¹⁶  Sometimes multiplying ontological commitments contributes to overall simplicity and is therefore perfectly defensible.¹⁷ Such is the case, Quine thought, for abstract objects like mathematical sets, which might offend against certain ontological scruples, such as those of the nominalist, but nevertheless contribute to the simplification of scientific theory.  If, for whatever reason, it proved necessary or efficacious for scientific theory to reinsert the gods of Homer alongside mathematical sets, so be it.  Until such time, however, we have no reasons to believe in Homer’s gods and good reasons not to.

Of course, Quine was offering a philosophy of science, not a philosophy of religion.  But an argument for atheism may not be far to seek.

For, indeed, the God of Abraham is no more to be found among the objects postulated by our leading scientific theories than the gods of Homer.  And if it is to our leading scientific theories that we should look when deciding what there is, then there is no reason to believe that God is.  So, by the Quinean test, God does not exist. The virtue of this argument, from the atheistic standpoint, is that it is not dogmatic about ontology.  It doesn’t try to rule out the possibility that God exists, but suggests instead that such a belief is superfluous.  What’s more, it provides common ground — the “semantic plane” — from which our ontological commitments may be evaluated and appraised.  The burden is thereby pushed onto the theist to show why belief in God should be included among our ontological commitments.

Have we thus hit on a compelling argument for atheism, or at least one that puts the theist on the defensive?  Not remotely.

First of all, one could disagree with the atheist’s characterization of ontology.  For instance, Aristotle understood metaphysics to be the study of being as such.  Thomas Aquinas followed him in calling the subject of metaphysics ens commune (literally “common being”). By contrast, Quinean ontology claims only to clarify and catalogue how many and which objects exist — what Aquinas called entia.  In so doing, it remains at the level of what philosopher Martin Heidegger called the merely “ontic,” as opposed to the ontological.  The theist could argue, then, that we have not even reached an agreement about our subject matter.  Semantic ascent may offer common ground, but only to those who already agree on what counts as metaphysics.

However, it would be a mistake to leave things at that.  For indeed, how many and which objects exist certainly matters if we want to understand being in general — i.e. what it means to be.  And Quine’s approach offers a bevy of tools and techniques that might prove useful to that purpose. Thus, it would behoove the theist to tarry a while longer on the semantic plane.

As Quine points out, semantic ascent is useful for determining what theories say there is.  But what there actually is is another question.  To answer it, our Quinean atheist urges us to look to our leading scientific theories.  That may well be a start.  But why should it be the end?  There is a long litany of things — love, works of art, countries, legal contracts, promises, mental acts, moral acts, moral judgments, my aunt Gigi — that are not posited by our leading scientific theories but which might nevertheless be a part of my ontology.  Why should the theist accept that scientific theories exhaust rather than inform our beliefs about what there is?  To accept that is tantamount to accepting atheism before the argument has even begun.

For God is not to be found among the objects posited by our best scientific theories.  About that, at least, all sides agree.  But this raises an even thornier problem for our Quinean atheist: The God of Abraham is importantly unlike the gods of Homer.  For rather than a god among many gods, God is “that than which no greater can be thought,” as St. Anselm of Canterbury put it.¹⁸  Indeed, the Old Testament is at pains to differentiate the God of Israel — Elohim — from the various deities, such as Baʽal, worshipped in ancient times, e.g., by the Canaanites.  Unlike these deities, the Biblical God is not a possible being, however powerful, but that being which exists necessarily.

According to the atheist, the theist’s error is believing in one too many things.  Yet, for the theist, the disagreement is not about the existence of one particular thing, but “about everything,” as MacIntyre puts it.¹⁹  For God, as traditionally understood, is not some thing in the universe, but is that in relation to which the universe — and everything in it — finds its meaning, intelligibility, and purpose.  God is not a being among many, which may or may not exist; God is that which exists necessarily as the uncaused cause of all that exists.

So, it is not a contingent fact that God does not appear among the objects posited by our best scientific theories; God could not so appear. The kind of existence we attribute to the objects studied by natural science — electrons, neutrinos, quarks, cellular molecules, RNA strands — is contingent, whereas God’s existence is not.  The God of monotheism is not a being that happens to exist (or not). Rather God is ipsum esse subsistens — the very act of “to be” itself — as Thomas Aquinas famously put it.²⁰  For this reason, Aquinas said that God does not even fall under ens commune, the subject of metaphysics.²¹ We are speaking only analogically when we use the same word — “exist” — to characterize both electrons and God.

To be sure, such talk of existence can hardly be rendered adequately by translation into the logic of quantifiers.  But that hardly shows that God doesn’t exist, only that such logical tools are inadequate.  Logical tools may be useful in helping us clarify our beliefs about how many and which objects exist.  They might even be useful in helping us clarify our beliefs about God.  But, from the theist’s point of view, such tools must inevitably fail to capture God’s essence.  Far from vindicating atheism, this only confirms something we theists knew all along: That God’s essence may never be grasped perfectly, this side of eternity, even if we can still know that God exists.

[M. Anthony Mills is a senior fellow and director of science policy at the R Street Institute, a public policy think tank in Washington, DC. He is also a senior fellow at the Pepperdine School of Public Policy and a scholar associate of the Society of Catholic Scientists. He researches and writes about a range of topics including science, technology, philosophy, and religion. His writings have appeared in such publications as Slate, Politico, RealClearPolitics, National Affairs, City Journal, Issues in Science & Technology, The New Atlantis, and various peer-reviewed journals. He is currently writing a book about scientific expertise in the age of populism.]

References

1.. Alasdair MacIntyre, God, Philosophy, Universities: A Selective History of the Catholic Philosophical Tradition (Rowman & Littlefield 2009), p. 76.

2.. Ontology is the branch of philosophy concerned with what there is. Accordingly, an ontology is a set of claims about what entities or kinds of entities exist; and an ontological commitment is a philosophical commitment to the existence or nonexistence of something.

3.. Taken from “Logic and the Reification of Universals,” in From a Logical Point of View (Harvard University Press 1980), p. 103.

4.. “On What There Is,” in From a Logical Point of View, p. 16.

5.. Ibid. On semantic ascent, see Word and Object (Massachusetts Institute of Technology Press 1960), especially pp. 270–276.

6.. “On What There Is,” in From a Logical Point of View, p. 15.

7.. Constructivism is the view that mathematical objects exist if and only if it is possible to “construct,” in a well-defined sense, a mathematical proof of such objects.

8.. Set theory is a fundamental theory of mathematics that was developed by German mathematician Georg Cantor (1845–1918). Among other things, Cantor proved that there are different types of infinity. For instance, although there is an infinite number of real numbers and an infinite number of natural numbers, the number of real number is nevertheless “larger” than that of the natural numbers (in the sense that there is no one-to-one correspondence between the reals and the naturals). The latter are said to be “countably” infinite or “denumberable” whereas the former are “uncountably” infinite or “non-denumerable.” In standard set theory, the reals are considered the “smallest” uncountably infinite set, with uncountably many more infinities beyond.  A number of eminent mathematicians, e.g., Thoralf Skolem (1887–1963) and Henri Poincaré (1854–1912), have expressed reservations about the higher infinities of set theory.

9.. Nominalism is the view that there are no universals. So there may be particular tables or particular chairs or particular people but there are no additional “universals” — “tableness” as such, or “chairness” as such, or humankind as such — that exist over and above these “particulars.” As applied to numbers, nominalism would be the view that there could be, say, three apples in front of me, but no such thing as the number three in general, existing independently of these particular apples.

10.. Word and Object, p. 272.

11.. “Logic and the Reification of Universals,” p. 105.

12.. See Carnap, “The Elimination of Metaphysics Through Logical Analysis of Language,” Erkenntnis, pp. 60–81 (1932).

13.. “Two Dogmas of Empiricism,” in From a Logical Point of View, p. 44.

14.. Ibid.

15.. “On What There Is,” p. 19.

16.. Ibid.

17.. See, inter alia, “On Multiplying Entities,” in Ways of Paradox, pp. 259–264.

18.. See St. Anselm: Proslogium; Monologium: An Appendix in Behalf Of The Fool By Gaunilo; and Cur Deus Homo, Translated From The Latin By Sidney Norton Deane, B. A. With An Introduction, Bibliography, And Reprints Of The Opinions Of Leading Philosophers And Writers On The Ontological Argument, (Chicago, The Open Court Publishing Company,, 1903, reprinted 1926).

19.. God, Philosophy, Universities, p. 77.

20.. See Rev. Msgr. John F. Wippel, The Metaphysical Thought of Thomas Aquinas: From Finite Being to Uncreated Being (The Catholic University of America Press: 2000).

21.. Ibid., p. 123.