In Modernity, the capacity for effortless knowing is denied, ignored, or misunderstood. As a result, the origin of all knowledge is taken as unaided human effort and activity…
If we lack a word for an experience, we obviously cannot talk to others about it, and the experience, no matter how intense or unsettling, will fade from lack of understanding, and later we will be unable to judge whether it was significant or not.
Every scientist and mathematician, professional and student, whom I have known can give countless occurrences in their own lives of hard, fruitless labor preceding effortless knowing. In the classroom or around the seminar table, I have heard an excited “I see it” innumerable times, but never in my life as a theoretical physicist did I reflect with colleagues about the meaning of such an experience. As a result, for years, I remained in the dark about the deepest aspects of what we are. My enlightenment came only after reading Plato and Aristotle. Through straightforward reflection about the interior life, accompanied by clear thinking, Plato and Aristotle discovered two aspects of the mind, which they called nous and dianoia.
We commonly speak of knowing as defining, comparing, and analyzing wholes into component parts, and drawing conclusions from first principles. Such discursive or step-by-step thinking the ancient Greek philosophers called dianoia. Sherlock Holmes is the epitome of dianoia at work. From the analysis of a cigar ash ground into a carpet, a scuff mark on a door, and the residue in a wineglass, he concludes the criminal is a lame aristocrat who resides in Kensington.
In discursive thinking, we either apply first principles or draw out their consequences. Dianoia operates step-by-step, almost in a mechanical fashion, repeating the same procedure again and again: A = B, B = C, therefore A = C; premise 1, premise 2, therefore, conclusion 1; etc.
What we frequently call thinking is not dianoia but the association of ideas. Consider the following scenario. In a seminar, the opening question is “What does Tocqueville mean by the equality of conditions?” Three minutes later the participants are discussing the siege of Richmond. Through associations the discussion moved rapidly from the equality of conditions to political equality to slavery in the South to Ken Burn’s Civil War to the Siege of Richmond. And soon, if the seminar leader did not intervene, the discussion would have moved on to the film Gone with the Wind and Clark Gable’s last line: “Frankly, lady. I don’t give a damn.” There is, of course, no logical, physical, or psychological connection between the equality of conditions and Gable’s exit line. If we step back and examine our “thinking,” we would find, perhaps to our amusement, that much of what we take for thinking is the mere association of ideas that has nothing to do with the actual structure of the world or the interior life.
Nous is the capacity for effortless knowing — to behold the truth the way the eye sees a landscape. The most famous story of sudden insight — of the “light bulb going off” — is the one told about Archimedes. The ruler Hiero II asked Archimedes to determine if the royal crown was truly made of pure gold or alloyed with silver. Archimedes knew that if the crown were not irregularly shaped, he could easily measure its volume and then check if its density was that of gold. But Archimedes could not figure out how to determine the volume of the crown. He was stumped; until one day when he stepped into his bath, the solution suddenly appeared to him: a given weight of gold displaces less water than an equal weight of silver. He shouted, “Eureka! Eureka!” and ran naked through the streets of Syracuse.
Henri Poincaré, in his book Science and Method, describes how he discovered the relationship of Fuchsian functions to other branches of mathematics. After he had worked out the basic properties of Fuchsian functions he left Caen, France, where he was living at the time, to take part in a geological conference at Coutances. The incidents of the journey made him forget his mathematical work. As a scheduled break from the conference, attendees went on a bus excursion. Poincaré reports, “Just as I put my foot on the step [of the bus], the idea came to me, though nothing in my former thoughts seemed to have prepared me for it, that the transformations I had used to define Fuchsian functions were identical with those of non-Euclidean geometry.”
When Poincaré returned home from the conference, he turned his attention to the study of certain “arithmetical questions without any great apparent result, and without suspecting that they could have the least connection with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidean geometry.”
Mathematician Carl Friedrich Gauss tried unsuccessfully for two years to prove an arithmetical theorem. In a letter to a colleague, he revealed, “Finally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible.”
Astrophysicist Fred Hoyle relates how the solution to what seemed an intractable mathematical problem in quantum physics came to him while driving a car on the road over Bowes Moor, much as “the revelation occurred to Paul on the Road to Damascus: My awareness of the mathematics clarified, not a little, not even a lot, but as if a huge brilliant light had suddenly been switched on. How long did it take to become totally convinced that the problem was solved? Less than five seconds.”
What Hoyle describes as a “huge brilliant light” and Poincaré calls a “sudden illumination” and Gauss terms “a flash of lightning,” the ancient Greek philosophers named nous, whose characteristics are “conciseness, suddenness, and immediate certainty,” as Poincaré noted. We, clearly, cannot command nous; the direct grasp of a truth is a gift, not something we can turn on or off at will, for if we could, we would.
Sometimes the direct grasp of a truth is astonishing. Extraordinary mathematicians such as Gauss and Riemann wrote down theorems without having any idea of how to prove them. Other mathematicians struggled to demonstrate those theorems and eventually did, although the proofs required pages and pages of complex reasoning. The Indian mathematician Srinivasa Ramanujan, because of his limited education, had no clear idea of what it means to prove a theorem. Yet, Ramanujan directly grasped many remarkable mathematical relations. The English mathematician Godfrey H. Hardy was astonished by the theorems that Ramanujan sent him: “I have never seen anything in the least like them before. A single look at them is enough to show they only could be written down by a mathematician of the highest class.” Some of Ramanujan’s theorems Hardy proved with great difficulty, others eluded him, although he was convinced they were true, for “no one would have the imagination to invent them.”
The direct grasp of a truth is generally accompanied by joy, especially when preceded by intense, fruitless discursive thinking, as the following example shows. Astronomer Maarten Schmidt describes his discovery of quasars as a “once-in-a-lifetime experience… the biggest in astronomy, I suppose, since the discovery that there exist galaxies besides our own. I remember the day, the hour, it happened. It was early in the afternoon of February 5, 1963. Until then, for months and months, I had been puzzling over this inexplicable thing, and then I had this sudden perception, and within half an hour everything was clear to me.”
Overjoyed, Schmidt desired to share the insight he received. He called his colleague Mitchell Wilson into his office. Wilson remembers how Schmidt “showed me his photographic plate of the spectrum of the stellar object coincident with the radio source called 3C-273, which I had also looked at, and then he gave me this remarkable explanation of it! He made it so obvious I couldn’t bear it! I screamed! Literally, I shouted! I was that excited! It was marvelous!”
After receiving a sudden insight, every physicist and mathematician I have known experiences two impulses, one from the interior life and the other from the marketplace. The first impulse is to rush out and show what he or she has received to anyone who will listen. What motivates the recipient is an experience of a beauty or a truth that is so elegant or so profound that no person should miss out on having it. The other impulse is immediately to write a paper to ensure that he or she receives recognition for “his” or “her” discovery, not someone else.
The interior impulse to share beauty and truth is universal. Many years ago, my children and I hiked up Pat’s Peak in New Hampshire for the first time. It was one of those beautiful fall days in New England that you read about. The air was cool and crisp, the sky cloudless, and the foliage ablaze with color. On our way up the mountain, we met a stranger coming down the trail. He excitedly told us about a magnificent view on the other side of the mountain, and gave us detailed instructions, so we would not miss finding the special spot he discovered. A complete stranger doing us a good turn! He did not ask anything in return; he gave to us freely. Truth and beauty draw human beings together and impel them to share the interior life.
Our knowing, generally, entails both dianoia and nous. We know directly that “the whole is greater than the part” and that “every human being by nature desires to know.” The child knows “triangle” as well as any mathematician; yet, it will take years of study and laborious toil before he or she will understand algebraic topology. Nous immediately grasps the whole of the argument in a flash. When we comprehend how all the parts fit together and why they are there, we spontaneously shout, “I see it.” The phrase “I see it” accurately describes the experience of effortlessly grasping the whole of an argument or the connection between seemingly disparate things, for nous, as we have seen, is like the physical eye gazing upon a landscape.
To know in the deepest sense is not to possess accurate information, say the names of the anatomical parts of a rabbit, or to have a method that turns out correct answers, say a mass spectroscope that determines molecular masses. To know profoundly is to experience an insight, a revelation of a deep and significant truth. Such knowing cannot be summoned by the will, although insight is usually preceded by hard work and the clearing away of major errors and wishful expectations. Sometimes, insight occurs only after giving up. In contrast to the truths arrived at through hard labor, intuitive insights are gifts. Consequently, dianoia usually results in pride, nous almost always in gratitude.
Dianoia and nous are not confined to doing mathematics and science, but are fully present in any intellectual activity. Mozart, in a letter, describes how he composes music. He informs his correspondent that musical composition involves two elements: the musical parts are put together according to the rules of composition (dianoia); and the joy the composer receives when he sees the composition as a whole, much as the eye sees a beautiful landscape (nous): “When I feel well and in good humor, or when I am taking a drive or walking after a good meal, or in a night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Where do they come from? I do not know, and I have nothing to do with it. Those which please me I keep in my head and hum them; at least others have told me that I do so. Once I have my theme, another melody comes linking itself with the first one in accordance with the needs of the composition as a whole; the counterpoint, the parts for each instrument and all the melodic fragments at last produce the complete work. Then my soul is on fire with inspiration. The work grows; I keep expanding it, conceiving more and more clearly until I have the entire composition finished in my head although it may be long. Then my mind seizes it, as a glance of my eye would a beautiful picture or a handsome youth. It does not come to me successively, with various parts worked out in detail, as they will later on, but it is in its entirety that my imagination lets me hear it.”
Those poets in Modernity who explore the depths of the interior life acknowledge publicly that all artistic creation rests upon gifts. The Polish poet Czeslaw Milosz, in the autobiographical essay “Catholic Education,” confessed, “I felt very strongly that nothing depended on my will, that everything I might accomplish in life would not be won by my own efforts but given as a gift.”
Theodore Roethke in “On ‘Identity’,” a lecture delivered at Northwestern University, told how at the age of forty-four he was in a particular hell for a poet, a longish dry period; he thought he was finished as a poet. For weeks at the University of Washington, he had been teaching five-beat line to aspiring poets, but felt a total fraud because he could write nothing himself. “Suddenly, in the early evening, the poem ‘The Dance’ started, and finished itself in a very short time — say thirty minutes, maybe in the greater part of an hour, it was all done. I felt, I knew, I had hit it. I walked around, and I wept; and I knelt down — I always do after I’ve written what I know is a good piece.” The gift arrived and Roethke thanked the source, and “wept for joy.”
We must not be misled into thinking that nous only operates in geniuses, giving them direct grasp of profound truths. When a student of Euclid’s Elements truly grasps a demonstration, he or she shouts with joy, “I see it.” Through nous, the student, in a flash, immediately grasped the whole of the argument. Without nous, any rational argument cannot be apprehended. In other tutorials, scientific, poetic, and philosophic insights occur. A student may suddenly grasp how the Iliad forms an integral whole, or why Socrates in the Phaedo, on the day of his execution, spins philosophical “tales” about the immortality of the soul to comfort his distraught friends.
The above discussion of dianoia and nous rests on straightforward observations of the interior life and consequently should not be controversial. As predictable, the ancient and modern ways of understanding the two modes of the mind are opposed. For Plato and Aristotle, dianoia is human and nous divine.
Plato rightly contends that to directly grasp a truth is superior to arguing to it. Consequently, dianoia is always in the service of nous. In the Seventh Letter, he explains how “after practicing detailed comparisons of names and definitions and visual and other sense perceptions, after scrutinizing them in benevolent disputation by the use of question and answer without jealousy, at last in a flash of understanding . . . the mind, as it exerts all its powers to the limit of human capacity, is flooded with light.”
In Modernity, nous is denied, ignored, or misunderstood. As a result, the origin of all knowledge is taken as unaided human effort and activity. Almost without exception, scientists and mathematicians claim to be the sole source of their discoveries. When intellectual insight is acknowledged as unwilled, it is attributed to the hidden workings of an unconscious mind, not to a supernatural gift. Nothing in science or mathematics is seen as a gratuitous gift from Divine Mind, or God. Scientists and mathematicians, with their denial of nous, proclaim they are beholden to nothing beyond themselves. Instead of being thankful, they are prideful, for they believe that all knowledge and invention results from human labor alone. How different was Pythagoras! After the remarkable theorem that now bears his name was revealed to him, Pythagoras in gratitude sacrificed an ox to the gods.
In mid-life, I had left materialism behind as a pernicious myth of the Dark Ages of Science. To call nous a product of unconscious thought as Poincaré and other scientists had was to beat the same old drum — the brain alone explains all interior life, although now we know with certainty that neuronal activity by itself cannot account for even perception. It made more sense to me to understand nous as the divine element within each person than to attribute our highest intellectual life to the murky unconscious mind hidden somewhere in the brain. To me — and I was in the good company of Plato, Aristotle, Augustine, and Aquinas — dianoia was strictly human and nous was the divine element within us.
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 Poincaré, Science and Method, p. 53; the Latin words ratio and intellectus correspond to the older Greek terms dianoia and nous. The Greek is used here, because the Latin calls to mind the English words “reason” and “intellect,” which often are used as synonyms. The English words are substantially more limited and less precise than either the Latin or the Greek.
 Ibid., pp. 53-54. Italics added.
 Karl Friedrich Gauss, quoted by Jacques Hadamard, The Psychology of Invention in the Mathematical Field (Princeton: Princeton University Press, 1949), p. 15.
 Fred Hoyle, “The Universe: Past and Present Reflections,” Annual Review of Astronomy and Astrophysics 20 (1982): 24, 25. Available http://calteches.library.caltech.edu/527/2/Hoyle.pdf.
 Poincaré, Science and Method, p. 55.
 G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, (Cambridge: Cambridge University Press, 1940), p. 9.
 Maarten Schmidt, quoted by Mitchell Wilson, Passion to Know (Garden City, New York: Doubleday, 1972), p. 53.
 Ibid., p. 56
 Mozart, quoted by Hadamard, p. 16.
 Czeslaw Milosz, “Catholic Education,” in Native Realm: A Search for Self-Definition (Berkeley: University of California Press, 1981), p. 87.
 For ‘The Dance’ read by Tom Bedlam see https://www.youtube.com/watch?v=wKaoXy2KaJU.
 Theodore Roethke, On the Poet and His Craft, ed. Ralph J. Mills, Jr. (Seattle: University of Washington Press, 1966), pp. 223-24.
 Plato, Seventh Letter in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns (Princeton, NJ: Princeton University Press, 1989), 344b.