The God of the Old Testament is a God of power, the God of the New Testament is a God of love; but the God of the theologians, from Aristotle to Calvin, is one whose appeal is intellectual: His existence solves certain puzzles which otherwise would create argumentative difficulties in the understanding of the universe.

-- Bertrand Russell, A History of Western Philosophy

It is in a way a forlorn and perhaps even hopeless objective -- to demonstrate the existence of God by numerical coincidences to an uninterested, to say nothing of a mathematically unenlightened public.

-- Carl Sagan, Broca's Brain

Euler strode up to Diderot and proclaimed: "Monsieur, (a + bn)/n = X, donc Dieu existe!" ["Sir (a + bn)/n = X, therefore God exists!"]

-- Michael Guillen, Bridges to Infinity

The cave is bright in the morning light. A few drops of cool liquid fall from a stalactite onto your outstretched palm. You turn toward Theano and Mr. Plex who are sitting with their backs against the cave wall.

"Today, I want to tell you about the use of computers, mathematics, and logic to prove the existence of God."

Mr. Plex's eyes seem to light up like little lanterns. "Is that possible, Sir?"

"Many have tried. Several 20th-century proofs involved mathematics, but some of the most famous proofs of God's existence started with Saint Thomas Aquinas. He lived from 1225 to 1274 and gave five proofs of God's existence in his Summa Theologica. The flavor of many of his arguments is as follows. In this universe, there are things which are only moved, and other things which both move and are moved. Whatever is moved

is moved by something, and since an endless regress is impossible, we must arrive somewhere at something which moves without being moved. This unmoved mover is God."

Mr. Plex scratches his head. "Not very persuasive."

You nod. "While this argument may not seem very persuasive today, during Aquinas' time it caused quite a stir. Interestingly, Aquinas loved to make lists of things which God cannot do. He cannot be a body or change Himself. He cannot fail. He cannot forget or grow tired, or repent or be angry or sad. He cannot make a man without a soul or make the sum of the angles of a triangle less than 180 degrees. He cannot make another God, cannot kill Himself, and cannot undo the past or commit sins."

Mr. Plex shifts his position against the cave wall as a drop of clear water falls upon his forelimb. "Sir, what about Aristotle?"

"Both Aquinas and Aristotle didn't like an infinite regression of causes, and they used their dislike to demonstrate God's existence. But these great minds lived before infinite mathematical series were commonplace concepts. How would the history of religion have changed if Aristotle and Aquinas knew integral calculus and infinite sequences? Would Aquinas have been better prepared to conceive of an infinitely old universe requiring no creator?" You pause. "Carl Sagan, a great 20th-century science popularizer, often talked about Aquinas' and Aristotle's unmoved mover God. I recall from Sagan's book Broca's Brain, 'As we learn more and more about the universe, there seems less and less for God to do. Aristotle's view was of God as an unmoved prime mover, a do-nothing king who establishes the universe and then sits back and watches the intertwined chains of causality course down through the ages.' "

Theano is stirring a pool of water with her finger. The ripples make faint splashing sounds as they contact a central stalagmite. She looks at you. "Sounds like lots of people have tried to prove God's existence using mathematical proofs, but has anyone tried to disprove the existence of God?"

"Yes. The early Christian writer Lucius Lactantius quotes the Greek atheist Epicurus in The Anger of God:

God either wishes to take away evil, and is unable, or He is able, and is unwilling; or He is neither willing nor able, or He is both willing and able. If He is willing and is unable, He is feeble, which is not in accordance with the character of God. If He is able and unwilling, He is envious, which is equally at variance with God. If He is neither willing nor able, He is both envious and feeble, and therefore not God. If He is both willing and able, which alone is suitable to God, from what source then is evil? Or why does He not remove evil?

The wind grows stronger near the mouth of the cave, and the blue of the sky outside the cave was fast turning to beige. The dwindling light tinged the cave the color of salmon. Occasionally, you hear strange animal cries and the sounds of insects. You look at Theano who appears to have a slight shiver.

"I'm OK," she says.

You continue. "In one of his more skeptical moods, St. Thomas Aquinas wrote in Summa Theologica:

It seems that God does not exist; because if one of two contraries be infinite, the other would be altogether destroyed. But the name God means that He is infinite goodness. If,

Theano looks up from the pool of blind cave fish. "Seems like Gutberlet should like Cantor's ideas because they make God's universe all the more impressive."

"Right. Gutberlet subsequently made use of Cantor's ideas and claimed God ensured the existence of Cantor's transfinite numbers. God also ensured the ideal existence of: infinite decimals, the irrational numbers, and the exact value of א. Gutberlet also believed that God was capable of resolving various paradoxes which seem to arise in mathematics. Furthermore, Gutberlert argued that since the mind of God was unchanging, then the collection of divine thoughts must comprise an absolute, infinite, complete closed set. Gutberlet offered this as direct evidence for the reality of concepts like Cantor's transfinite numbers."

You turn your attention from the fish to Mr. Plex and Theano. "Cantor's own religiosity grew as a result of his contact with various Catholic theologians. In 1884, Cantor wrote to Swedish mathematician Gösta Mittag-Leffler explaining that he was not the creator of his new work, but merely a reporter. God had provided the inspiration, leaving Cantor only responsible for the way in which his papers were written, for the organization and style, but not for their content. Cantor claimed and believed in the absolute truth of his 'theories' because they had been revealed to him. Thus, Cantor saw himself as God's messenger, and he desired to use mathematics to serve the Christian Church."

You throw a tiny pebble into the pool, and even before it hits, the fish seem to swim away. "Like Pythagoras, Cantor also believed that numbers (particularly his transfinite numbers) were externally existing realities in the mind of God. They followed God-given laws, and Cantor believed it was possible to argue their existence based on God's perfection and power. In fact, Cantor said that it would have diminished God's power had God only created finite numbers. On the other hand, Cantor's love of the infinite had a distinctly antiPythagorean flavor. Pythagoras believed infinity was the destroyer in the universe, the malevolent annihilator of worlds. If mathematics were war, the struggle was between the finite and infinite. The Pythagoreans became obsessed with infinity, and they concluded that numbers closest to one (and finiteness) were the most pure. Numbers beyond the range of ten were further from one and were less important. Cantor would not have agreed."

"Sir, it would be intriguing to gather Pythagoras, Cantor, and Goedel in a small room with a single blackboard to debate their various ideas on mathematics and God." "You bet! What profound knowledge might we gain if we had the power to bring together great thinkers of various ages for a conference on God and mathematics? Would a round-table discussion with Pythagoras, Cantor, and Goedel produce less interesting ideas than one with Newton and Einstein? Could ancient mathematicians contribute any useful ideas to modern mathematicians? Would a meeting of time-traveling mathematicians offer more to humanity than other scientists, for example biologists or sociologists?" You pause. "These are all fascinating questions. I don't have the answers."

The three of you stare at the school of fish and watch them move in synchrony, despite their lack of eyes. The resulting patterns are hypnotic, like the reflections from a hundred pieces of broken glass. You imagine that the senses place a filter on how much humans can perceive of the mathematical fabric of the universe. If the universe is a mathematical carpet, then all reatures are looking at it through imperfect glasses. How might humanity perfect those glasses? Through drugs, surgery, or electrical stimulation of the brain? Probably our best chance is through the use of computers.

Were theologians to succeed in their attempt to strictly separate science and religion, they would kill religion. Theology simply must become a branch of physics if it is to survive. That even theologians are slowly becoming effective atheists has been documented . . .

-- Frank Tipler, The Physics of Immortality

God created the natural numbers, and all the rest is the work of man.

-- Leopold Kronecker ( 1823-1891)

In Chapter 20, I alluded to the famous mathematician Kurt Goedel and the fact that he spent the last few years of his life working on a mathematical proof of God's existence. In this Postscript, I quote colleagues from around the world who have responded to my questions on this subject, and I thank them for permission to reproduce excerpts from their comments. Some of my questions were sent through electronic mail or posted to electronic bulletin boards. Three common sources for such information exchange were "sci.math," "alt.atheism," and "soc. religion. christianity " -- electronic bulletin boards (or newsgroups) that are part of a large, worldwide network of interconnected computers called Usenet. (The computers exchange news articles with each other on a voluntary basis.) ¹

Without further ado, I present Kurt Goedel's proof of God's existence:

GOEDEUS MATHEMATICAL PROOF OF GOD'S EXISTENCE

Axiom 1. (Dichotomy) A property is positive if and only if its negation is negative.

Axiom 2. (Closure) A property is positive if it necessarily contains a positive property.

Theorem 1. A positive property is logically consistent (i.e., possibly it has some instance).

Definition. Something is God-like if and only if it possesses all positive properties

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is (logical, hence) necessary.

Definition. A property P is the essence of x if and only if x has P and is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. NE(x): x necessarily exists if it has an essential property.

Axiom 5. Being NE is God-like.

Theorem 3. Necessarily there is some x such that x is God-like.

Note: I obtained this proof from: Wang, Hao ( 1987) Reflections on Kurt Goedel. MIT Press: Cambridge, Mass. (page 195).

It is completely wrong for people to assume that a true scientist cannot simultaneously be a true man of God, believing in God as Creator and Savior and believing the Bible as God's revelation.

-- Henry Morris, Men of Science, Men of God, 1982

The theory of limits, on which modern mathematics, and as a corollary, all modern science and technology depend, is actually a secularized form of the Scriptural view of reality. Mathematics and Scripture both view the human grasp of reality as increasingly approximated to, but never identified with, the human symbols we use to represent reality.

-- Ford Lewis Battles, 1978

Clifford A. Pickover received his Ph.D. from Yale University's Department of Molecular Biophysics and Biochemistry. He graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years. He is author of the popular books Black Holes-A Traveler's Guide ( 1996) and Keys to Infinity ( 1995), both published by Wiley & Sons, and The Alien IQ Test ( 1997), published by Basic Books. He is also author of Chaos in Wonderland: Visual Adventures in a Fractal World ( 1994), Mazes for the Mind: Computers and the Unexpected ( 1992), Computers and the Imagination ( 1991), and Computers, Pattern, Chaos, and Beauty ( 1990), all published by St. Martin's Press, as well as the author of over 200 articles concerning topics in science, art, and mathematics. He is also coauthor with Piers Anthony of the forthcoming sciencefiction novel, Spider Legs.

Pickover is currently an associate editor for the scientific journals Computers and Graphics, Computers in Physics, and Theta Mathematics Journal. He is an editorial board member for Speculations in Science and Technology, Idealistic Studies, Leonardo, and YLEM. He has been a guest editor for several scientific journals. He is editor of The Pattern Book: Fractals, Art, and Nature ( World Scientific, 1995), Visions of the Future: Art, Technology, and Computing in the Next Century ( St. Martin's Press, 1993), Future Health ( St. Martin's Press, 1995), Fractal Horizons ( St. Martin's Press, 1996), and Visualizing Biological Information ( World Scientific, 1995), and coeditor of the books Spiral Symmetry ( World Scientific, 1992) and Frontiers in Scientific Visualization ( Wiley, 1994). Dr. Pickover's primary interest is in finding new ways to continually expand creativity by melding art, science, mathematics and other seemingly-disparate areas of human endeavor.

In 1990, he received first prize in the Institute of Physics' Beauty of Physics Photographic Competition. His computer graphics have been featured on the cover of many popular magazines, and his research has recently received considerable attention by the press -- including CNN's Science and Technology Week, The Discovery Channel, Science News, The Washington Post, Wired, and The Christian Science Monitor -- and also in international exhibitions and museums. Omni magazine recently described him as "Van

Leeuwenhoek's twentieth century equivalent." The July 1989 issue of Scientific American featured his graphic work, calling it "strange and beautiful, stunningly realistic." Pickover has received U.S. Patents 5,095,302 for a 3-13 computer mouse and 5,564,004 for strange computer icons.