by Rebecca Newberger Goldstein
My most recent book, to be published in January 2010, is called 36 Arguments for the Existence of God: A Work of Fiction. The sub-title, meant to be a joke, also happens to be true. The book is a novel. But it also has an appendix that is not meant as fiction, though it is ostensibly written by the novel's main character, Cass Seltzer. A psychologist of religion, Cass has recently become an intellectual celebrity following the success of his own book, entitled The Varieties of Religious Illusion. Cass does not think much of any of the arguments for God's existence, but he does nevertheless have some sympathy for the religious impulse, especially when it expresses a sense of amazement at the improbability of existence, his own and the world's. This sympathy earns him the sobriquet "the atheist with a soul."
In his appendix, Cass formulates arguments for the existence of God—all the arguments he can think of, some of which are sufficiently well-known to have acquired names, such as "The Cosmological Argument," "The Ontological Argument," and "The Argument from Design." Others Cass has to christen himself; for example, "The Argument from The Improbable Self," "The Argument from The Intolerability of Insignificance," "The Argument from The Unreasonableness of Reason." Two of the arguments in the appendix attempt to deduce God's existence from mathematics, which is not to say that these are mathematical arguments.* Rather, they argue that there is a certain mystery to mathematics, and that this mystery can best be resolved by positing God's existence as an explanation.
The first of the arguments focuses on the non-empirical nature of mathematical knowledge as the mysterious element. Mathematics is derived through pure reason—what the philosophers call a priori reason—which means that it cannot be refuted by any empirical observations.** The fundamental question in the philosophy of mathematics is this: how can mathematics be true but not empirical? Is it because mathematics describes some trans-empirical reality—as mathematical realists (often called "Platonists") believe? Or is it rather that mathematics has no descriptive content at all and is a purely formal game consisting of stipulated rules and their consequences, as formalists believe? This mystery forms the basis of what Cass calls "The Argument from Mathematical Reality":
1. Mathematical truths are necessarily true. (There is no possible world in which, say, 2 plus 2 does not equal 4, or in which the square root of 2 can be expressed as the ratio of two whole numbers.)
2. The truths that describe our physical world, no matter how fundamental, are empirical, requiring observational evidence. (So, for example, we await some empirical means to test string theory, in order to find out whether we live in a world of eleven dimensions.)
3. Truths that require empirical evidence are not necessary truths. (We require empirical evidence because there are possible worlds in which these are not truths, and so we have to test that ours is not such a world.)
4. The truths of our physical world are not necessary truths (from 2 and 3).
5. The truths of our physical world cannot explain mathematical truths (from 1 and 4).
6. Mathematical truths exist on a different plane of existence from physical truths (from 5).
7. Only something which itself exists on a different plane of existence from the physical can explain mathematical truths (from 6).
8. Only God can explain mathematical truths (from 7).
9. God exists.
Since Cass Seltzer doesn't believe that any of the arguments for God's existence are sound, his aim, after formulating an argument, is to lay bare its weakest links. Here, very briefly, is what he says of "The Argument from Mathematical Reality":
Flaw 1: The inference of 5, from 1 and 4, does not take into account the formalist response to the non-empirical nature of mathematics.
Flaw 2: Even if one Platonistically accepts the derivation of 5 and then 6, there is something fishy about proceeding onward to 7, with its presumption that something outside of mathematical reality must explain the existence of mathematical reality. Lurking within 7 is the hidden premise that mathematical truths must be explained by reference to non-mathematical truths. But why? If God can be self-explanatory, as this argument presumes, why then can't mathematical reality be self-explanatory—especially since the truths of mathematics are, as this argument asserts, necessarily true?
Flaw 3: Mathematical reality—if indeed it exists—is, admittedly, mysterious. But invoking God does not dispel this puzzlement; it is an instance of "The Fallacy of Using One Mystery to Bury Another." The mystery of God's existence is often used, by those who assert it, as an explanatory sink hole.
The other argument that makes reference to mathematics focuses on the mystery of infinity. Cass calls it "The Argument from Human Knowledge of Infinity":
1. We are finite, and everything with which we come into physical contact is finite.
2. We have a knowledge of the infinite, demonstrably so in mathematics.
3. We could not have derived this knowledge of the infinite from the finite, from anything that we are and come in contact with (from 1).
4. Only something itself infinite could have implanted knowledge of the infinite in us (from 2 and 3).
5. God would want us to have a knowledge of the infinite, both for the cognitive pleasure it affords us and because it allows us to come to know him, who is himself infinite.
6. God is the only entity that is both infinite and that could have an intention of implanting the knowledge of the infinite within us (from 4 and 5).
7. God exists.
Flaw: There are certain computational procedures governed by what logicians call recursive rules. A recursive rule is one that refers to itself, and hence can be applied to its own output ad infinitum. For example, we can define a natural number recursively: 1 is a natural number, and if you add 1 to a natural number, the result is a natural number. One can, in principle, apply this rule an indefinite number of times and thereby generate an infinite series of natural numbers. Recursive rules allow a finite system (a set of rules, a computer, a brain) to draw conclusions about infinity.
The fundamental nature of mathematics is sufficiently mysterious that mathematicians, though agreeing on what has been mathematically proved, disagree on what the results of those proofs amount to. Mathematical truth, and our knowledge of it, presents genuine philosophical questions, as profoundly baffling as any good philosophical problems are. But do these mathematics-generated philosophical questions have anything to do with God? I can't help agreeing with my own fictional character that theology based on mathematics amounts, in the end, to a kind of fiction.
*In E. T. Bell's Men of Mathematics, a story is told of an encounter between the great Swiss mathematician Leonhard Euler and the French encyclopaedist, Denis Diderot, in which Euler advanced a pseudo algebraic proof of the existence of God in order to embarrass the atheist Diderot. "Sir, (a+b^n)/n = x; hence God exists, answer please!" The story, although awfully good, appears to be apocryphal. See Dirk J. Struik's A Concise History of Mathematics, Third Revised Edition, Dover, 1967, in which he asserts that the "story seems to have been made up by the English mathematician De Morgan (1806-1871)." P. 129.
**The question of which mathematics can be applied to our physical world is an empirical question. So, for example, after non-Euclidean geometry was developed in the nineteenth century by, among others, Karl Friedrich Gauss, the question arose whether our physical space was Euclidean or non-Euclidean, a question for physicists, not mathematicians.
This paper was commissioned by the Philoctetes Center as part of a grant from The John Templeton Foundation. This and three other commissioned pieces are featured in a special issue of Dialog, the newsletter of the Philoctetes Center.