Seeing the Thunder: Insight and Intuition in Science, Mathematics and Religion
Insight and intuition abound in the realms of religion and the arts. They also abound in the twin realms of science and mathematics. While believers and artists may attribute them to the inspiring (in-breathing) of divinity or the wondrous workings of a daemon, scientists and mathematicians—though equally amazed and often thankful—are less wont to attribute “aha” moments to supernatural causes. Yet a close look at insight and intuition, beginning with its meaning in the highly rarified world of pure mathematics as set forth by Kurt Gödel, and progressing through the multi-sensory idea of “seeing the thunder” as found in the Book of Exodus and explored by Rabbi Abraham Joshua Heschel, reveals significant connections that may help in the search for common ground—a search that is crucial given the trajectory of present scientific research, particularly into the nature of sentience.
In Gödel (1906-1978), the brilliant mathematician whose irrational fears his friend and colleague Albert Einstein attempted to allay by taking him for daily walks on the grounds of the Institute for Advanced Study at Princeton, some of the most vexing issues of sentience take on flesh: Can the human mind think beyond the boundaries of algorithms? What is the connection between insight, intuition and intelligence? Peering suspiciously at the world throGödelugh round spectacles, Gödel showed that there can exist propositions that by insight must be true but that cannot be proven mathematically. He also wrote an ontological proof for the existence of God, all the while practically starving himself to death out of fear that his food was poisoned. (For a discussion of the ontological proof, see Hector Rosario’s article “Kurt Gödel’s Mathematical and Scientific Perspective of the Divine: A Rational Theology” on The Global Spiral)
Gödel is best known for his 1931 paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems wherein he used a variation of the liar paradox (“All Cretans are liars,” said a Cretan—attributed to Epimenides ) to show that certainty is not possible in a strong axiomatic system, thereby forcing higher mathematics to change course radically, as well as bending the direction of physical science, which is ultimately dependent on mathematics for its rigor. Less known is that Gödel was well-read in philosophy including Leibniz, Husserl and Kant, the philosophy of the latter being one of his favorite discussion topics with Einstein. Furthermore, Gödel was fascinated with the nature of cognition, particularly his own. He was intrigued not only by what he knew but by how he came to know it. He did not conceive of the human mind as merely a biologic form of an algorithmic computer; instead he saw an inherent randomness in human thought-processes that defied mathematical formulation. What he realized, was deeply troubled by, and was unable to express as concisely as he wanted (even in his famous Gibbs Lecture) was that his incompleteness theorems had profound implications for the understanding of the limitations of intelligence, making, as he put it, “continued appeals to mathematical intuition” necessary.
William Byers in How Mathematicians Think: Using Ambiguity, Contradiction and Paradox to Create Mathematics (Princeton University, 2007) writes: “Gödel’s theorem is a tour de force that operates on many levels. In the first place it is a technically brilliant theorem in mathematical logic. At a more universal level it has deep implications for the philosophy of science and mathematics and beyond that for the nature of human thought in general.”
Bestowing the term “philosopher scientist” on both Einstein and Gödel, Palle Yourgrau in his book A World Without Time: The Forgotten Legacy of Gödel and Einstein (Basic Books, 2005) considered Einstein to be a classicist who sought continuity with the past, while Gödel was “at heart an ironist, a truly subversive thinker” whose ideas are so complex and so troubling that in some cases they have been intentionally ignored.
The definitions of both insight and intuition are ambiguous, in fact, ambiguity can be taken as an essential component. Insight can mean either a deep understanding that comes slowly arising from prior knowledge, or it can mean an instantaneous breakthrough that comes ex nihilo. Intuition is frequently defined as an inner knowing, seemingly without the use of rational thought or observation. There are, however, certain emotions and feelings that pertain to both insight and intuition: 1. A sense of intensity; 2. A sense of timing and time, as in a presentiment that something momentous is happening or is about to occur; 3. The feeling of surprise, indeed joy, when it does occur (the eureka experience); 4. and finally the frustration of explaining in words the attendant thought-process.
Beginning with Kant’s perception of intuition but then going much further, Gödel considered mathematical intuition to be a kind of “knowing” that defied mathematical formulation. It was similar to a physical sense such as hearing or sight. In fact, he considered it a sixth sense:
The similarity between mathematical intuition and a physical sense is very striking. It is arbitrary to consider “This is red” an immediate datum, but not so to consider the proposition expressing modus ponens or completion induction (or perhaps some simpler propositions from which the latter follows). For the difference, as far as it is relevant here, consists solely in the fact that in the first case a relationship between a concept and a particular object is perceived, while in the second case it is a relationship between concepts.1
Furthermore, Gödel asserted that something can be known objectively even though its discernment was beyond sensory perception. Indeed, axioms could exert themselves as being true. Had that assertion been made by a religious person, it would stand as a definition of faith— “to be certain of what we cannot see” (Hebrews 11:1). However, it would be a mistake to see Gödel’s statement as a defense of Platonism, which is, as the British mathematician G. H. Hardy put it in his famous book A Mathematician’s Apology, the objective independent mathematical reality that “lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply the notes of our observations.” Gödel may have been at heart a Platonist, but his extremely subtle statements on insight and intuition grapple more with the nature of cognition than with the nature of reality.
An echo of Gödel can be heard in the writings of the Catholic theologian Bernard Lonergan. In his monumental work Insight: A Study of Human Understanding (University of Toronto Press, 1957), he stated that “the mathematician differs both from the logician and from the scientist. He differs from the logician inasmuch as he cannot grant all the terms and relations he employs to be mere objects of thought. He differs from the scientist inasmuch as he is not bound to repudiate every object of thought that lacks verification.” Lonergan maintained that just as positive integers are verifiable so also are elements in the transcendent idea. Via a philosophical argument as intellectually rigorous as Gödel’s ontological proof, Lonergan concluded that moments of insight, in and of themselves, are an indication of the existence of God. “God is the unrestricted act of understanding, the eternal rapture glimpsed in every Archimedian cry of Eureka.”
Before exploring the meaning of insight and intuition further, it is worthwhile to take a look at some classic examples of eureka moments in science and mathematics (skipping over Archimedes’ archetypal experience at the public bath in Syracuse from whence the word originates). In 1865, the German chemist Kekulé von Stradonitz was stymied in determining the atomic structure of benzene. Going for a ride on a horse-drawn omnibus, he fell into a semi-doze in which he saw a chain of atoms dancing. Suddenly the tail of the chain grabbed hold of the head, forming a ring. Kekulé had his answer. A similar experience happened to Poincaré, the French mathematician, when he was riding on an omnibus. “At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation I had used to define Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea: I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty.” So also Roger Penrose, astrophysicist and winner of the Noble Prize, tells in his book The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (Penguin Books, 1989) of a similar experience he had while walking down a street with a friend. Penrose, whose mathematical thinking is done visually and non-verbally, had been preoccupied with quasars and black-holes and whether there could be a mathematically definable criterion for a “point of no return.” However, at the time of his walk, he and his friend were carrying on an entirely different conversation. Suddenly Penrose had the idea of what he would eventually call a “trapped surface,” but he immediately forgot about it and went on with the conversation. Later in the day, after he had returned to his office, he had a strange feeling of elation for which there was no obvious explanation. Only after racking his brain did he finally recall the crucial idea that had occurred to him while on his walk.
Some cognitive scientists consider such eureka moments as incredibly complex but ultimately explicable biological processes, nothing more, nothing less. Gestalt psychologists conceptualize them as a complex form of problem-solving, in which a period of unconscious incubation leads eventually to an answer. While this describes the type of insight experienced by Kekulé von Stradonitz and Penrose, it misses the point that insight can lead to the discovery and articulation of a new problem or question, not just to the solution of an old one, which may result in a paradigm shift. Probably the best example is the fall of Isaac Newton’s apple, which led him to posit an insightful question about whether the same force that pulled the apple downward also pulled the moon. However, it would be many years (during which he invented the calculus and published ‘Of Colours” among other accomplishments) before he returned to the problem of gravitation. This type of insight that is not linked to a specific problem tends to lead to a cascade effect first within the mind of the individual and then within the entire field, as if an invisible door to an unexplored universe has suddenly materialized along with the key.
Too often psychological and biological studies of insight are based on the use of word games, visual brain-teasers, and mathematical mind-booglers. Anyone who likes to do such things knows the pleasure of an insightful breakthrough after being stymied. In fact, that pleasurable moment—a little taste of joy—is the reason people like to do brain-teasers, including the most common form—the crossword puzzle. But what is involved in the solving of such puzzles is not insight because there is always underlying methodologies and patterns that a good puzzle solver figures out (either suddenly or slowly) and which are then used as tools for coming up with the correct answers. Good puzzler solvers also tend to strategize, beginning with a specific approach which they will abandon for another it if it turns out to be fruitless, a skill which requires good memory of what has been tried. While this may look like insight, it is really just a normal form of mental processing. Take, for example, the puzzle used in 2004 by researchers at Northwestern University in Illinois (Journal of Public Library of Science Biology, Mark Jung-Beeman). While their brains were being scanned using functional magnetic resonance, eighteen people were given groups of three words and had to find a single word that linked them, e.g. for the three words pine, crab, and sauce, the correct answer was apple. The brain scan revealed that there was increased activity in the right hemisphere anterior superior temporal gyrus at the moment of insight. What the researchers didn’t take into account in designing their research was that word games are based on a well-established form of logic: the words pine, crab and sauce form a subset of an unidentified set; secondly, word games rely heavily on double meanings, e.g. pine is both a type of tree and the first syllable of the name of a fruit. Once puzzle solvers realize these “rules”—a realization that is far more indicative of true insight than coming up with the correct answers—they can begin to solve these word puzzles quickly. The other problem with this type of research is that while the right hemisphere anterior superior temporal gyrus does light up, it may do so as the result of the emotion that accompanies success and not as a result of the problem being solved. Cause can not be linked to effect easily.
As the above example makes clear, a major flaw with the insight problems used by psychologists is their intentional ambiguity: they are tricky. For example, often an important piece of data has two meanings as in the example of pine/pineapple, or the wording of the entire puzzle is slightly misleading. Consider the well-known ping pong puzzle in which a person is asked to describe how to throw a ping-pong ball so that it will travel a short distance, come to a dead stop and then reverse itself. (Ansberg and Dominowski, 2000). The answer of course is to throw it straight up so that the ball reaches a moment of suspension before gravity pulls it back down. The problem is that while the wording of the puzzle is technically correct, it is also intentionally misleading. Why a ping-pong ball, not merely a ball? Because a ping pong ball immediately calls up the mental image of horizontal movement across a table, not vertical movement as in a child throwing a baseball up and down. Or consider the wording that the ball reverses itself. The ball does not reverse itself, instead the upward momentum from the thrower’s hand is eventually overcome by the downward pull of gravitation. Nor is the term “dead stop” meant to be helpful because in common parlance it implies a longer period of time than the split second of weightlessness.
The flaws continue when researchers attempt to test for insight by using visual problems such as the often used nine-dot problem or the coin problem in which the subject is asked to arrange coins in a certain pattern. Invariably those people who do well on these type of tests have high visual skills, can move objects around in their minds’ eye, and can keep track of patterns. These are very specific intelligence skills that not all intelligent people possess. And again the solution is often outside the proverbial box, requiring the subject to realize he or she is not confined by the boundaries as presented. This is similar to the intentional ambiguity of the ping pong problem.
Another major difficulty with using puzzles of any kind to study insight is that the person trying to solve the puzzle knows there is an answer, in fact, that the puzzle has been solved numerous times before, even if he or she has trouble doing so. He or she also knows that some kind of mental restructuring of the problem from the initial assessment will probably be necessary. Essentially, they are primed to try a radically different approach and to switch strategies should their first attempt fail. When Isaac Newton asked about the nature of gravity, he did not know whether there was an answer. Often when a scientist or mathematician asks such a question, he or she can not tell whether the question itself makes any kind of sense. G. H. Hardy wrote in A Mathematician’s Apology, “In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.” He meant that more often than not, an answer would come out of an unexamined quarter, but that its arrival would be accompanied by a sense of absolute rightness and spare elegance.
Yet another serious difference between insight problems as tested by psychologists and true insight as actually experienced by scientists and mathematicians is that test problems are what is known as “knowledge-lean,” meaning the test taker does not need to possess any specific body of information in order to solve them. But scientists and mathematicians always work within a knowledge-rich mental environment. Even when an answer seems to appear out of the blue, it is like a star forming amidst a cloud of interstellar gases.
Because the psychological approach presumes a problem, in other words, a point of frustration, for which none of the normal channels of thought yield an answer, it overlooks the fact that insight need not be predicated on there being a problem at all. It can come unbidden and unburdened as Hardy maintained. Gödel was well aware of this phenomenon. In his essay The modern development of the foundations of mathematics, Gödel used the phrase “second direction” to refer to an understanding of underlying concepts separate from empirically based comprehension:
In fact, one has examples where, even without the application of systematic and conscious procedure, but entirely by itself, a considerable further development takes place in the second direction, one that transcends ‘common sense.’ Namely, it turns out that in the systematic establishment of the axioms of mathematics, new axioms, which do not follow by formal logic from those previously established, again and again become evident. It is not at all excluded by the negative results mentioned earlier that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way. For it is just this becoming evident of more and more new axioms on the basis of the meaning of the primitive notions that a machine cannot imitate.
Gödel’s assertion of a non-algorithmic basis to insight and intuition is echoed by Karl Popper who wrote in The Logic of Scientific Discovery, “There is no such thing as a logical method of having new ideas, or a logical reconstruction of this process. My view may be expressed by saying that every discovery contains ‘an irrational element,’ or ‘a creative intuition.’”
Gödel’s assertion is also supported by Penrose, who uses the term quantum mechanical. Referring to Gödel’s incompleteness theorem, Penrose states: “We must ‘see’ the truth of a mathematical argument to be convinced of its validity. This ‘seeing’ is the very essence of consciousness. It must be present whenever we directly perceive mathematical truth. When we convince ourselves of the validity of Gödel’s theorem we not only ‘see’ it, but by so doing we reveal the very non-algorithmic nature of the ‘seeing’ process itself.” What both Gödel and Penrose were asserting was that the human mind at the level of insight and intuition does not function like a computer at all. It is far beyond algorithmic calculations. “Either mathematics is too big for the human mind or the human mind is more than a machine,” Gödel stated.
Gödel’s and Penrose’s idea of insight and intuition as profoundly holistic plays out in an unusual way particularly among the super-intelligent, which is that new ideas can present themselves visually, such as von Stradonitz’s dancing chain. The uber-doodler Archimedes carried around a little box filled with sand in which he constantly draw figures. Plutarch writes: “Oftimes Archimedes’ servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney.” And it was not proofs he was drawing but perceptions. He himself wrote “Certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof.” The physicist Richard P. Feynman was famous for his diagrams that he doodled on napkins and the backs of envelopes, which helped him in an intuitive way to conceptualize interactions between particles. To Feynman, visualization was a vital part of his intuitive thinking, as he put it, “a mixture of a mathematical expression wrapped in and around, in a vague way, around the object.” The paradox was that, like Archimedes doodling in sand, for Feynman the diagram often came first and provided the insight, making the subsequent proof almost superfluous (working on the proof often led to important modifications and new ideas). So also Poincaré was in no rush to work out the math because he knew the answer with perfect certainty. Even for Gödel, the rotating and warping space-time structure (which came to be known as a Gödel universe) was intuitively visual and intuitively right.
In his book Belief in Science and Christian Life (Handsel Press, 1980), Thomas F. Torrance recalls his failed efforts at becoming a mathematician, the reason being that the conceptual world reflected in the theorems was not part of his experience. He longed to see, to understand, but he lacked the essential heuristic framework necessary to do so. He was not able to “indwell” as he termed it:
I once rejoiced under the illusion that I was a mathematician. I say ‘illusion’ because what I really was was a manipulator of mathematical formulae. After going to university to read mathematics, I soon found that whilst I could follow—I deliberately avoid saying ‘understand’—proofs of theorems in text-books, I was often unable to adapt those proofs to problems related to them….My problem—and I was not alone—was that whilst able to ‘follow’ the mechanics of the proof, in turning to related problems I found that I lacked the conceptual apparatus necessary to adapt the theorem to the new circumstances. Looking back, I now realize that whilst following the symbolic operations I had paid such close attention to the letter that I had lost the overall meaning of the letters. In Polanyi’s terms, the subsidiaries had become focal. The true mathematician reads such symbols in the same way Polanyi’s ‘accomplished linguist’ reads a letter; he assimilates its meaning by indwelling the words, but afterwards is left only with the meaning, having forgotten which language it was written in.
Torrance’s use of the term “indwelling,” which resonates theologically, is key because it implies a nonlinear, integrative process that cannot be put into words, in fact, is far beyond words. He had never been able to understand why his tutors were so unwilling to teach him “the mystical methods whereby to solve problems.” Years later, he realized that they were not unwilling, but unable. Certainly Gödel would have comprehended Torrance’s problem. He wrote that certainty could never be obtained by “the manipulation of physical symbols—but rather by cultivating (deepening) knowledge of abstract concepts themselves which lead to the setting up of these mechanical systems.”
From his observations about insight in mathematics, Torrance then turned to faith. “The question ‘need one understand God in order to write theology’ is impertinent, but the question ‘need one believe, and thus be passionately committed to the attempt to understand God in order to write theology?’ is legitimate and forms a close parallel with the mathematical case.”
There is a Biblical parallel, as suggested by the Hebrew phrase “seeing the thunder” (Exodus 20:18), when the Israelites are gathered at the base of Mt. Sinai. As God speaks to Moses on the mountain, the people hear voices and trumpets, they see the flames, clouds and lightning, but they also see thethunder. Another scriptural reference occurs in the Book of Job, in which Job demands hard evidence of his misdeeds so that he can perceive a direct causal link with his suffering. God speaks to Job through the thunder, yet in the last chapter, Job says not that he has heard God, but that he has seen God “with my own eyes.” In these examples, seeing, which can also be translated as perceiving, means that the total self, not just a single sensory apparatus, is actively involved. It is not an indication of synaesthesia.
In the Hebrew understanding of sensory perception, hearing, hence language, had primacy. It was a theological primacy that acted as a counterweight to the natural dominance of sight over all the other senses. It was because of the human tendency to place a higher value on what is visual, and in so doing, to over-value, even idolize, that God forbid the making of graven images of all “things in heaven above and on earth beneath or in the waters below.” Instead, the believer was challenged to seek God through the study of Torah, illuminated by the inner light of the mind, not the outer light of the world. Martin Buber maintained that it was the Greeks who made the sense of sight primary, but that for the Hebrews the senses were more unified. Torrance wrote that along with the conviction that trust and reliance on God is what orders the world, went another: “whereas knowledge by sight depends on the see-er who must rely on himself, obedient hearing of the Word of God gives rise to knowledge in which man does not rely on himself but on God.” However, the Hebrews were also well aware that language was woefully inadequate as was their resolve to listen. As the writer of Ecclesiasticus wrote: “Summon all your strength to declare his greatness and be untiring, for the most you can do will fall short. Has anyone ever seen him, to be able to describe him? Can anyone praise him as he truly is? We have seen but a small part of his works, and there remain many mysteries greater still.”
The other essential component in the Biblical idea of “seeing the thunder” is the necessity for watchfulness and preparedness. The verse “my soul waits for the Lord more than watchmen wait for the morning,” (Psalm 130: 6), which is repeated for emphasis, implies far more than a passive sitting around waiting for the sun to come up. It implies full sensory alertness. Furthermore, the ability to recognize the significance of the insight, once its occurs, and its subsequent integration into a body of knowledge also presumes a prepared mental state, a readiness to discern.
However, in Jewish thought, there is yet another serious consideration: the role of the action, the deed, in creating the situation in which insight can occur. In God in Search of Man: A Philosophy of Judaism (Farrar, Straus and Giroux, 1955), Rabbi Abraham Joshua Heschel describes the prophet as “a unity of inspiration and experience, invasion and response. For every object outside him, there is a feeling inside him, for every event of revelation to him there is a reaction by him; for every glimpse of truth he is granted, there is a comprehension he must achieve.” Insight and action are doubly linked, with the first leading to the second at the same time that the second is generative of the first.
In 1949, grieving the loss of many family members in the death camps while struggling to find a way to speak of faith, love and righteousness in a world that had become a bitter wasteland, Heschel wrote in his essay Pikuach Neshama: To Save a Soul:
The nations of the world have produced many thinkers who have striven to reach God by intellectual inquiry alone. They have, in fact, dived deep into the stormy waters, but have come up with naught. God cannot be grasped by the intellect. The Jews have a different way: “We will act and we shall understand.” Reaching God—the understanding—arrives together with the act, emanates from within the act (the Kotzker rebbe). When we fulfill a mitzvah and perform a desirable action, we achieve the cleaving of humanity with God. It is as if in our actions, in the depths of our existence, “we see the thunder.”
To Heschel, the deed enacts preparedness. In the doing of the act of righteousness and justice, the individual prepares herself or himself for the God-given gift of insight. Returning to the Exodus passage, the Israelites say “all that the Lord has spoken, we shall do and we shall hear,” thereby reversing the normal order of hearing first and then responding. Heschel points out that “a Jew is asked to take a leap of action rather than a leap of thought.”
From this standpoint, the person who experiences insight need not be gifted. No astronomically high IQ or graduate degree is required. Like the prophet Amos, he can be a simple shepherd who metaphorically hears “the lion roar” and can’t help but prophesy. Or like Moses, he can be “slow of speech and slow of tongue.” But before Moses knew anything of God, before receiving any insights whatsoever, he did a deed that could be deemed scientific: On seeing the burning bush while he was tending his flocks, he didn’t run away in fear; he didn’t decide that it was none of his business and go on with his work. Instead, he said to himself, “I must turn aside and look at this great sight, and see why the bush is not burned up,” (Exodus 3: 3). It was only after God saw that Moses had turned aside to determine the cause of the fire that God called out his name, and instantly everything changed. Moses’ action indicated a state of observant preparedness, and thereby opened the door to “seeing the thunder” not only for himself but for all the Hebrews.
While Gödel certainly saw the thunder in terms of his own insights, he was uneasy about publicly professing a belief in God (although he did so in a series of letters to his mother). In his highly unusual ontological proof for the existence of God, he meant the word “God” to mean “a sum of perfections.” In fact, he was deeply concerned about publishing the proof because he did not want it to be misconstrued as a statement of faith instead of a “logical investigation.” The tie to insight is found not in the proof itself but in a note Gödel wrote: “If the ontological proof is correct, then one can obtain insight a priori into the existence (actuality) of a non-conceptual object. Is this perhaps another proof: If there were nothing actual, then precisely this would in fact be something actual (which goes beyond a mere concept)?” Despite his assertions to the contrary, there are signs of an inner struggle to believe. Even his interest in Husserl and phenomenology was motivated by a longing to avoid, as he put it, “both the death-defying leaps of idealism into a new metaphysics as well as the positivistic rejection of all metaphysics.” Essentially, he wanted a middle way. It was not a way he was ever to find.
In his lecture “Some Basic Theorems on the Foundations of Mathematics and Their Philosophical Implications,” delivered at Brown University in 1951, Gödel ended with the following statement by Hermite, with which he agreed: “There exists, unless I am mistaken, an entire world consisting of the totality of mathematical truths, which is accessible to us only through our intelligence, just as there exists the world of physical realities; each one is independent of us, both of them divinely created.” Were that not so, as Gödel told his friend Hao Wang in a conversation late in his life, “mathematics would be just an ornament and the real world would be like an ugly body in beautiful clothing.”
Byers, William. How Mathematicians Think: Using Ambiguity, Contradiction and Paradox to Create Mathematics. Princeton and Oxford, Princeton University Press, 2007.
Dawson, Jr., John W. Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley, Mass.: A K Peters, 1997.
Heschel, Abraham Joshua. God in Search of Man: A Philosophy of Judaism. New York: Farrar, Straus and Giroux, 1955, Paperback 1976.
—Moral Grandeur and Spiritual Audacity. New York: Farrar, Straus and Giroux, 1996.
Penrose, Roger. The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. New York: Penguin Books, 1989.
Rosario, Héctor. Kurt Gödel’s Mathematical and Scientific Perspective of the Divine: A Rational Theology, Metanexus Institute (online).
Torrance, Thomas, ed. Belief in Science and Christian Life. Edinburgh: The Handsel Press, 1980.
Yourgrau, Palle. A World Without Time: The Forgotten Legacy of Gödel and Einstein. New York: Basic Books, 2005.