Author Topic: The God of the Mathematicians  (Read 416 times)

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The God of the Mathematicians
« on: September 20, 2010, 01:54:33 PM »
DAVID P. GOLDMAN explores the religious beliefs that guided Kurt Gödel's revolutionary ideas.

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Publication: First Things

Author: Goldman, David P

Date published: August 1, 2010

Kurt Gödel was a believer - or, at least, a knower - whose engagement with God included a reworking of the ontological proof of God's existence. Born in 1906, Gödel was arguably the great mathematician of his time. Certainly no twentieth-century thinker did more to show that the human mind cannot be reduced to a machine. At twenty-five he ruined the positivist hope of making mathematics into a self-contained formal system with his incompleteness theorems, implying, as he noted, that machines never will be able to think, and computer algorithms never will replace intuition. To Godel this implies that we cannot give a credible account of reality without God. But Gödel's God is not the well-behaved deity of the old natural theology, or the happy harmonizer of the intelligent-design subculture. Gödel's God hides his countenance and can be glimpsed only in paradox and intuition. God is not an abstraction but "can act as a person," as Gödel once wrote, confronting those who seek him with paradox, uplifting man through glorious insights while guarding his infinitude from human grasp. Gödel's investigations in number theory and general relativity suggest a similar theological result: that God cannot be reduced to a mere principle of the natural world. Gödel may have seen himself as a successor to Leibniz, whose critique of Spinoza's atheism set a precedent for much of Gödel's work. When we try to ascertain the theological intent underlying Gödel's mathematical investigations, though, several difficulties arise.

The first is Gödel's reticence. "Although he did not go to church," his wife Adele told the logician Hao Wang shortly after Gödel's death in 1978, he "was religious and read the Bible in bed every Sunday morning." But fear of ridicule and professional isolation made him reluctant to talk about his faith. "Ninety percent of contemporary philosophers see their principal task to be that of beating religion out of men's heads," he wrote to his mother in 1961.

A two-page draft of an ontological proof for God's existence forms the whole of Gödel's explicitly theological output. He showed his paper only to close friends, but word got out, and the clever young things on campus giggled behind his back. His biographer Rebecca Goldstein, who was a graduate student at Princeton during Gödel's last years, snickers that she and her peers "found it hilarious" that Godei "deluded himself into believing that God's existence could be proved a priori." The ambient hostility drove some of his best students out of the profession and may have worsened the eating disorder that hastened his death.

Another difficulty is that Gödel's work extends across several demanding fields, each with a high threshold of preparation.
There is also the problem that scavengers have been at work on his legacy for decades. The postmodernists have tried to claim him as an irrationalist who proved that nothing can be proved - just the opposite of what he intended. Rebecca Goldstein rightly debunks the postmodern claim, but her biography of the great mathematician makes no mention of his religious faith except to ridicule it, ignoring key facets of his work with theological implications.

Nonetheless, the ontological proof provides a point of entry into Gödel's work, linking intuitive theology with his mathematical investigation. The proof, in one form or another, has been known at least since the eleventh century, when St. Anselm of Canterbury paused to ask: If God is greater than we possibly can conceive, then how could God not exist?

In the best-known version of the argument, Anselm noted:

1. The definition of the word God is "that than which nothing greater can be conceived."

2. God exists in the understanding, since we understand the word with that definition.

3. To exist in reality and in the understanding is greater than to exist in the understanding alone.

4. Therefore, God must exist in reality.

Versions of the argument, with key twists and turns, have been introduced in Western philosophy through the centuries by thinkers from Descartes and Leibniz down to the twentieth-century American philosopher Charles Hartshorne. And, from the monk Gaunilo in Anselm's own time down through St. Thomas Aquinas, Immanuel Kant, and Bertrand Russell, these same centuries have seen philosophers who vehemently reject any form of the argument.

In all versions there exists a tension among premises. If God is "that than which nothing greater can be conceived," how can we understand God? Whatever we might conceive, God must be greater. If this is so, then how can God exist in the understanding? In his 1930 book on Anselm, Karl Barth offers a theological rather than a philosophical answer: We understand God by calling on him by his proper name, "that which is greater than anything that can be conceived." "It does not say," writes Barth, "that God is, nor what he is, but rather, in the form of a prohibition that man can understand, who he is." In effect, Barth says that "that than which nothing greater can be conceived" is something of an alternative appellation for YHWH, God's personal name in the Bible.

In Barth's reading, Anselm's statement is not only proof, but also prayer. By asking us to attempt to conceive of that than which nothing greater can be conceived, Anselm restrains our impulse to worship, instead, a projection of ourselves. This intent, Barth adds, makes irrelevant the usual critique of Anselm. Thomas Aquinas objects that simply thinking something exists does not necessarily mean that it does exist. Kant jokes that the essence of a hundred imaginary thalers is the same as the essence of a hundred real thalers in my pocket, but their spending power is quite different.

Such refutations do not apply, Barth insists, for Anselm's exercise presumes faith. But that move, in turn, raises up a theological problem. As the Jewish philosopher Michael Wyschogrod has noted, "If the ontological proof is successful, then God would be one of the things that have being. It would then be the case that the Empire State Building is, and the Eiffel Tower is, and God is." Being would be an umbrella concept, covering God and many other things - which would make being our ruling concept. Naming God as "that which is greater than anything that can be conceived," or "TTWNGCBC" rather than YHWH, gives us not a more refined faith but only a more refined atheism.

And that is just how the secular Enlightenment proceeded. Atheism in its modern form began with a revision of the ontological argument at the outset of Spinoza's Ethics. It could not have been otherwise, for to make sense of a world without God, atheism first had to usurp the attributes of God and assign them to nature. Something "than which nothing greater can be conceived" within the natural world gives us Spinoza's "infinite substance."

Spinoza begins his Ethics with an ontological argument: "God, or substance, consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists" because "if this be denied, conceive, if possible, that God does not exist: then his essence does not involve existence. But this is absurd. Therefore God necessarily exists." In Spinoza's words, "By God, I mean a being absolutely infinite - that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality." God is reduced to the "substance" of nature.

But the ontological argument turned out to be a cuckoo's egg in the nest of atheism. Spinoza abducts Anselm's TTWNGCBC from heaven and locks him up inside the natural world. The trouble is that once inside nature, TTWNGCBC consumes everything else and becomes all that there is in nature. If God is inside nature, then there can be nothing in nature outside of God. Spinoza concludes: "As God is a being absolutely infinite . . . and he necessarily exists; if any substance besides God were granted it would have to be explained by some attribute of God, and thus two substances with the same attribute would exist, which is absurd; therefore, besides God no substance can be granted, or consequently, be conceived." If we actually can conceive of TTWNGCBC within the natural world, then we can conceive of nothing else at all.

Hegel quipped that the cause of Spinoza's death "was consumption, from which he had long been a sufferer; this was in harmony with his system of philosophy, according to which all particularity and individuality pass away in the one substance." Spinoza's younger contemporary Leibniz pounced on this vulnerability, turning Spinoza's system inside out: Instead of an "infinite substance," Leibniz postulated a "pre-established harmony" controlling an infinite number of independent monads. Leibniz added a purely theistic premise: By the law of sufficient reason, he argued, God does not do anything superfluous and therefore does not create anything twice.

In a purely formal sense, the systems of Spinoza and Leibniz seem to be mirror images: Spinoza's single substance cannot explain individuality, and Leibniz' individual monads cannot communicate with each other; "pre-established harmony" has the same function as "infinite self-generating substance." Undergraduate courses misleadingly lump the two together under the rubric of "rationalism." But there is a fundamental difference: By inverting Spinoza's metaphysics, Leibniz makes room for God to return from his Babylonian captivity in natura naturans, to lordship over being. "Sufficient reason" is a theistic premise, to be sure, but it explains the world as we perceive it, rather than the single monistic glob implied by Spinoza.

If Spinoza tries to capture the ontological proof for atheism, Leibniz sets out to restore it to theism, suitably corrected to answer the objections of Thomas Aquinas. If we suppose that God possesses all positive properties, Leibniz argues, then necessary existence is a positive property and must pertain to God. If we agree that it is logically possible for a perfect being to exist because "all perfections are compatible with each other," then we must conclude that a being possessing all positive properties must exist: "There is, or can be understood, the subject of all perfections, or a most perfect being."

Kurt Godei was a lifelong student of Leibniz; during his four decades in Princeton at the Institute for Advanced Study he checked out every book on Leibniz in the university library. In an answer to a questionnaire found in his posthumous papers, he wrote, "My belief is theistic, not pantheistic, following Leibniz rather than Spinoza." He reworked Leibniz' version of the ontological proof in 1941, using the notation of modern mathematical logic (although he showed it to no one until 1970). I doubt Godei believed he had found the ultimate and irrefutable proof of the existence of God. His deep interest in the ontological proof, rather, was one facet of his commitment to defend Leibniz' theism against the new Spinozans of mathematics and physics.

Gödel's proof is written in logical notation. Like its Leibnizian model it turns on the notion of "necessary existence," although it contains additional intermediate steps. Godei begins by asserting that "positive" properties may be distinguished from properties in general. "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)," he explains.

The property that conjoins all positive properties also must be a positive property; and because "necessary existence" is a positive property, this property-of-all-positive-properties must include necessary existence; Godei defines it as the "God-like property." God therefore possesses all positive properties, and, because non-positive properties are the negation of positive properties, God cannot have any non-positive properties. Moreover, because "necessary existence" is one of these positive properties, God must exist in all possible worlds.

The objection of circularity might be raised because "positive properties" may be called positive because they are Godlike to begin with. It is not clear either that all positive properties are logically compatible, as in the paradox of divine omniscience and omnipotence.
Other objections have been raised to Gödel's argument. It seems to imply, for example, that every positive property also must exist, for all positive properties belong to the God whom the argument proves to exist. The modern philosopher Jonathan Sobel argues that we are thus driven back to Spinoza's problem, in which God is in everything and everything is in God, because we cannot distinguish between necessary existence and contingent existence. (Dean Koons has suggested a possible if problematic repair of Gödel's proof in which only the cosmos itself is considered to have necessary existence.)

Gödel's proof is best understood as an exercise within his broader Leibnizian program. If we attempt to speak in purely natural terms of "that than which greater cannot be conceived," we cannot help but refer to the mathematical concept of infinity. That is just how Spinoza thought of his infinite substance, and to embed the infinite, an attribute of God, within natura naturans was a pantheist credo. Leibniz, the coinventor of calculus, knew that an infinite number of infinitely small quantities can have a finite sum, and his infinity of infinitesimals, therefore, does not upset the finite character of created nature. But already, with Leibniz' refutation of Spinoza, mathematics was emerging as the laboratory for ontological investigation. Whether that should be the case is a source of perpetual debate, but it is hard to imagine how it could have happened otherwise in the seventeenth century.

To treat mathematical objects as the proper subject of ontological investigation requires a grand assumption, to be sure, that, in Gödel's words, "mathematical objects exist independently of our constructions." Godei worked for years, for example, on an unpublished essay refuting Rudolf Carnap's view that mathematics was no more than a syntax to manipulate man-made symbols.
The metamathematical debate continues to this day, and how it will be resolved has not yet come clear. Nonetheless, it is perhaps telling that Leibniz, the philosopher who offered the most logical rejoinder to Spinoza, was able to do so because, coincidentally, he was the mathematician who formulated the modern concept of infinitesimals.

Leibniz' infinitesimals led to another Godei discovery with deep theological implications. In the nineteenth century, mathematicians learned that the calculus discovered by Leibniz and Newton could not integrate or differentiate some classes of functions. The calculus began with the insight that an infinite series whose terms grew infinitely small might have a positive sum. But some functions resisted the calculus. These include "spiky" functions in which changes in sign, for example, occurred in arbitrarily small intervals. From the study of such functions came the disturbing insight that some infinities are "bigger," that is, more densely packed with numbers, than other infinities. And this inspired one of the nineteenth century's greatest mathematicians to attempt to treat the different orders of infinity as if they were just another kind of number - the "transfinite numbers" - and thus to domesticate infinity.

Georg Cantor was the discoverer of these transfinite numbers in the early 1870s, when he showed that some infinite collections were "larger" than others. There is a one-to-one correspondence between the integers and the rational numbers, such that the rational numbers may be thought of as a "countable" infinity. But there is no such correspondence between the integers and the real numbers, conceived of as a continuum line.

That was an insight of historic importance. But it did not satisfy Cantor, who believed that he had solved the problem that had eluded Spinoza - namely, to preserve the differentiation of individual objects within natura naturans. By bringing God's infinitude into the natural world, Spinoza was left with nothing but his single substance. At least in the realm of numbers, Cantor believed, infinity itself could be ordered with a new series of "transfinite numbers," each representing a different order of infinity. He envisioned a new kind of cardinal numbers denoting infinite sets of numbers.

The infinitesimal monads of Leibniz thus would no longer require God and the principle of sufficient reason to differentiate themselves because infinite series of numbers would arrange themselves naturally into Cantor's transfinite ordering. Cantor's theology was confused - he also hoped to reconcile his views with those of Thomas Aquinas and the Church Fathers - but his belief that he had solved not only a mathematical problem but also an ontological mystery is well documented. A draft in his letter book states, "I have examined all objections that have ever been made against the infinite numbers, and above all because I have followed its roots, so to speak, to the first infallible cause of all created things."

Cantor asserted that the infinity of the integers and rational numbers was the first transfinite number, and he named it "Aleph-zero." What, then, was the second transfinite number, or "Aleph-one"? He had proven that the infinity of the continuum of real numbers was "denser" than that of the integers; unlike that of the rational numbers, it could not be counted. He assumed that if the first transfinite number contained the integers, the second transfinite number would contain the continuum, and that no other transfinite number could be discovered between these two.

That is Cantor's "continuum hypothesis," which attempts to identify a first and second transfinite cardinal number. From there, he believed, all the possible orders of infinity could be counted, the same way the integers count groups of one, two, three, and so forth. He not only recognized, but was driven by, the ontological implications of this assertion: If the continuum hypothesis turned out to be true, Spinoza would be vindicated because God's infinity could be packaged into a neat series of numbers. Cantor spent the last thirty-five years of his life in a vain effort to prove this. He died in 1918 in a mental hospital.

It was Godei and, later, Paul Cohen who demonstrated respectively that Cantor's continuum hypothesis could be neither proved nor disproved within existing set theory. Indeed, Cantor's hypothesis remains maddeningly undecidable. Intuition, added Godei, strongly suggests that Cantor's hypothesis is wrong: Among the infinite number of transfinite numbers, there are an infinite number of cardinalities between the integers and the points on the continuum line, and mathematical investigation of the infinite will remain infinitely fruitful. God's infinitude remains safe in heaven. Mathematicians have proven that an infinite number of transfinite numbers exist but cannot tell what they are or in what order they should be arranged.

Gödel noted drily that this represents a problem for philosophy and epistemology rather than for mathematics, which can continue its investigations without ever exhausting the subject. Gödel's result shows that not even in terms of numbers, the simplest objects we can specify, can natura naturans explain the individuality that we observe. The parallel between Gödel's attack on the continuum hypothesis and Leibniz' critique of Spinoza is very strong, and it is remarkable that both hinged on foundational insights into number theory.

Whether or not we believe, as did Godei, in Leibniz' God, we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel's critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.

There is yet a third place where Kurt Gödel's mathematical work has theological purchase: in Einstein's failure to reconcile the deterministic world of general relativity with the probabilistic world of quantum mechanics. Einstein famously declared his belief in "Spinoza's God": a god, that is, who is indistinguishable from nature and who reveals himself through natural harmonies. Einstein, we might say, was a "strong Platonist" who actually believed that if one discovers the eternal forms to which natural phenomena correspond, all the world's mystery will yield itself up to science.

The often noted problem is that the intuitively intelligible world Einstein created with the deterministic equations of general relativity jars with the probabilistic world of modern quantum mechanics. Einstein and Godei were close friends, but they disagreed profoundly on religious and philosophical matters. As Godei told Hao Wang, "Einstein's religion [was] more abstract, like Spinoza and Indian philosophy. Spinoza's god is less than a person; mine is more than a person; because God can play the role of a person."

Gödel's personal God is under no obligation to behave in a predictable orderly fashion, and Godei produced what may be the most damaging critique of general relativity. In a Festschrift for Einstein's seventieth birthday in 1949, Godei demonstrated the possibility of a special case in which, as Palle Yourgrau described the result, "the large-scale geometry of the world is so warped that there exist space-time curves that bend back on themselves so far that they close; that is, they return to their starting point." This means that "a highly accelerated spaceship journey along such a closed path, or world line, could only be described as time travel." In fact, "Godei worked out the length and time for the journey, as well as the exact speed and fuel requirements."

Gödel, of course, did not actually believe in time travel, but he understood his paper to undermine the Einsteinian worldview from within. Yourgrau the contending worldviews in contemporary science "according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)." "Skepticism, materialism, and positivism stand on one side; spiritualism, idealism, and theology on the other." He dismisses "idealism, e.g., in its pantheistic form," as "a weakened form of theology in the proper sense."

"One would, for example, say that apriorism belongs in principle on the right and empiricism on the left side." But, he adds, "On the other hand, however, there are also such mixed forms as an empiristically grounded theology." Godei was at least a weak PIatonist - he considered mathematical objects to be real and his research therefore to be empirical. He thought his theology thus to be an empirical one, founded on man's experience of the infinite fecundity of the creator's mind. That is why Gödel's religious thinking is so rich and also why it is so challenging: One must actually follow the work to make sense of the conclusions, an unusual challenge for theologians, and one they have shirked for too long.

Author affiliation:
David P. Goldman is a senior editior at First Things.

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