Three o’clock in the afternoon: teatime in the Institute’s Commons Room, a warm, wood-paneled room conjuring an image of cigars and brandy in the library of an English country manor instead of an American academic research center. Teatime was an honored tradition at the IAS that everyone respected, no matter how busy or engaged in their work. The Commons Room was the waterhole of intellectual life, where faculty, visitors, and hangers-on all met on an equal footing to discuss and argue the issues of the day.

On this particular afternoon, the legendary poet, T.S. Eliot, sat in the overstuffed leather chair in the corner reading last week’s London Times. As he hid behind the paper, peering intently through rounder-than-round librarian’s spectacles, the pale, graying, emaciated-looking Eliot eagerly digested the literary news from across the Atlantic. A poet adrift in a sea

of mathematicians and physicists, Eliot wondered why he had ever given in to the blandishments of IAS Director J. Robert Oppenheimer to “spend a term in Princeton.” Where is the poetry in the concept of an electron buzzing around the atomic nucleus or in the equations of an operator algebra? Where is the rhyme or the rhythm or the meter in the symbols making up the gauge theories of particle physics? He felt as out of place there as a dustman at the King’s garden party.

Near the tea table against the wall, deeply engaged in what was really an intense debate masquerading as polite tea-time conversation, stood the German mathematician, physicist, and general polymath, Hermann Weyl (pronounced “vile”) and the energetic Viennese theoretical physicist, Wolfgang Pauli. The two were a study in contrasts. Weyl, with hair as gray as his bland, three-piece suit, gold-rimmed pedant’s glasses, and dark, tightly knotted tie spotted conservatively with white polka dots, reminded Eliot of a distant uncle or, perhaps, a type of banker Eliot had dealt with in his previous life as a banker himself, the type who can’t quite bring himself to ignore an overdraft. Pauli, outspoken and ebullient, was a short, stout, dark-haired, and often dark-tempered, antagonist. The long-term visitor from the Eidgenössische Technische Hochschule (ETH) in Zurich was certainly the sharpest critic in the entire theoretical physics community. Even Oppenheimer, no mean critic himself, had to take a back seat in the seminar room when the acerbic Austrian rose to render a judgment on some gap in an argument or a poorly presented idea. In 1945 Pauli had been awarded the Nobel Prize in physics for his work on the so-called “Pauli Exclusion Principle,” which states that no two quantum objects—two electrons, for instance—can occupy the same physical state at the same time. This result, which Pauli discovered in 1928, was central to the development of both particle and quantum physics, since it provided a physical constraint that helped

researchers sort through and eliminate many mathematically appealing, but physically unrealizable, theories of matter and energy.

Unlike the nearly somnolent Eliot, the two emigrés from Nazi tyranny rattled their teacups and waved their biscuits about with abandon, as they threw off waves of intellectual energy in a staccato-like German, arguing the difference, if any, between the type of knowledge that mathematicians recognize and that which is acknowledged by natural scientists, in particular, particle physicists like Pauli.

“In mathematics,” argued Weyl vehemently, his faint German accent still noticeable, “we have the notion of proof. This gives us a clear-cut, unambiguous way to create new knowledge from old. We start with the old knowledge—the axioms of a logical system—and employ the tools of deductive inference to generate new true statements—theorems—from the old ones.”

Pauli glowered at the simplistic nature of this mathematician’s view of the world, sputtering, “The problem with this approach is that there is no criterion by which you can claim the initial knowledge, the axioms, are really knowledge. They may or may not accord with our sense of the world and the way things are. You state the axiom that two parallel lines intersect at infinity, even though in the real world there are no such infinities and parallel lines never meet. But this obvious fact is irrelevant for mathematicians. They care only about logical consistency. No physicist would ever accept a hypothesis that runs counter to observation or laboratory measurement.”

The patrician Weyl was not at all put off by this outburst. He’d heard it all many times before. “Of course, of course,” he replied, in a good-natured attempt to calm Pauli down, speaking as if to an excited child. “The situation is really a bit worse than this, even in mathematics. We have no clear consensus about what kinds of logical operations can be used

in creating new knowledge.”

Eliot’s poetic soul cringed at this interchange, which he was following from a distance with increasing interest. The views of both Pauli and Weyl on what constituted “knowledge” certainly did not accord with the intuitive idea of knowledge that any poet, humanist, or artist would almost certainly endorse. Weyl appeared to be saying that mathematical knowledge could come only from following a set of rules by which one logically deduces a conclusion from a set of more or less arbitrary assumptions. As for Pauli, the physicist, his view of knowledge as something that can be measured was hardly much better. What a severely stunted notion of knowledge these men and their fields have! thought Eliot. Perhaps Oppenheimer brought me here to try to add a bit of scope and breadth of humanistic vision to these deliberations. It’s a pity the chasm between the scientist and the poet is as wide as the Grand Canyon—and even harder to bridge. The very possibility of a meaningful dialogue between poetry and science seems hopelessly remote, he thought sadly.

Eliot’s musings echoed perfectly the thoughts on the poetry/science divide expressed by another regular visitor to the Institute, British physicist Paul Dirac, also a Nobel laureate, honored in 1933 for his mathematical work predicting the existence of anti-particles like the positron, which have the same mass as their ordinary matter counterparts but with an opposite electrical charge and magnetic moment (spin). Such objects were later observed in actual experiments. Dirac once expressed puzzlement over Oppenheimer’s predilection for writing poetry and studying Latin, asking him seriously, “How can you do both—poetry and physics? In physics we try to tell people things in such a way that they understand something that nobody knew before. In the case of poetry it’s the exact opposite. There one takes something that everyone knows and tries to express it in ways that nobody ever saw before.”

Suddenly a cacophony of voices sounded from the hall outside the Commons Room doorway as a new group of faculty, postdoctoral students, and visitors bustled into the room. As they clustered around the tea table, pouring as much tea on the tablecloth as into their cups and quickly grabbing for the few biscuits remaining, everyone turned to the commanding figure at the center of the crowd, former head of the Manhattan Project to build the atomic bomb and now Director of the IAS, J. Robert Oppenheimer, known to one and all as “Oppie.” Charismatic in the way of a religious ascetic or mystic, yet dressed in an impeccable three-piece, gray business suit more commonly seen on diplomats or politicians than on academics, Oppenheimer had the gaunt, cadaverous look of someone who slept very little and smoked far too much. Yet a look from his razor-sharp, intensely blue eyes could cause even the most egomaniacal of the Institute’s supercharged intellects to stop in their tracks and pay attention when Oppenheimer spoke. In short, he had the type of “star quality” that one can only be born with but can never acquire.

Picking up on the conversation between Weyl and Pauli, Oppenheimer turned to Eliot and asked in a resonant directorial voice, “Well, Tom, I see that Pauli and Weyl haven’t yet managed to reconcile themselves in the realm of physics. What do you think about the aesthetic differences between the poet and the physicist?” But before the shy, retiring Eliot could overcome his surprise at being asked directly about this question and frame a reply, Oppenheimer went on, saying to the group, “There’s a very big difference between the knowledge a poet expresses in a sonnet about love and the knowledge a neurochemist would acquire if he measured the concentration of chemicals in the brain when his patient is told to think about his loved one. What Pauli is talking about is scientific knowledge; what a genius like Eliot here means by knowledge is an entirely different thing.”

Such was the majesty of Oppenheimer’s tone and bearing as he made this pronouncement that everyone in the room was struck silent. After all, who could argue with him on a matter of philosophy? Or poetry? Or physics? Finally, Oppenheimer’s former teacher, Pauli, broke the spell by questioning the distinction he had just drawn.

“How can you say that? How are you able to distinguish between the knowledge we have of the mass of an electron and what Eliot would call knowledge of a loved one or the shape and color of a rose?”

“There is something I would call ‘soul,’ ” said Oppenheimer in a strong voice that carried conviction. “Or what some mystics and practitioners of Eastern religions term The One. Here is where poets and artists draw their knowledge from, not from looking at a photograph showing the trace of an elementary particle in a cyclotron or measuring the charge of a proton in a cloud chamber. Platonists, like our colleague Gödel, would call The One by the everyday label ‘intuition’ or ‘feel.’ Whatever you call it, though, it’s every bit as real and means every bit as much as the knowledge we physicists take to be the ‘true facts’ of Nature.” Finishing his pronouncement, Oppenheimer picked up his teacup, symbolically opening the floor for discussion.

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The question was a very old and venerable one. Philosophers call it epistemology, which addresses the issues, What is knowledge? How do we obtain it? How can we verify it? What are its limits? What is the relationship between what is known and the person who knows it?

The simplest and most traditional answer to the question, “What constitutes knowledge?” is that it is true, justified belief. Regrettably, this commonsense, almost flip, reply creates more questions than it answers, and is certainly open

to debate about what is true. What is a belief? What do we mean by justified belief? The linguistic philosopher, Ludwig Wittgenstein, for instance, would say that knowledge is simply the by-product of a particular worldview rather than being some objective thing that just sits “out there” waiting for us to discover it.

The physicists engaged in this teatime Princetonian debate had also drawn attention to whether science or mathematics has a privileged position insofar as creation of knowledge goes. Is there something intrinsically different—perhaps superior, even—in the type of knowledge we create in either of these areas? In fact, what do we even mean by “mathematical knowledge” or “scientific knowledge”?

Taking advantage of Oppenheimer’s momentary withdrawal to the sidelines, one of the younger mathematicians visiting the Institute for a term entered the fray. The frail-looking young man had the stooped appearance of someone who spent too much time with books and not enough in the fresh air. Speaking in a high-pitched, strangely schoolboyish voice heavy with the accent of his native France, he pushed his lanky blond hair off his forehead before remarking that the idea of truth used in mathematics and the very same concept as understood in everyday terms are completely different. He noted that Gödel’s stunning results showed that there is an eternally unbridegable gap between the two. As he stated to the teatime group: “Everyday ‘truth’ is always a bigger concept than the mathematician’s truth.”

Wishing he could recall the name of this bright French visitor (was it Weil or Cartan?), Oppenheimer glanced over at Weyl and raised his eyebrows as if to say, “This young fellow has thrown the ball back into your court, my friend,” since he knew perfectly well that Weyl was a strong critic of the implications of Gödel’s results for mathematics. This was especially true for the type of interpretation that accepted the Law of the

Excluded Middle, by which a mathematical proposition could be only true or false, provable or unprovable. An intuitionist, like Weyl, could never accept the idea of proving something true by proving that it is not false. And his reply to the young mathematician did not disappoint.

“Gödel’s discovery places a constant drain on the enthusiasm with which I pursue my scientific work,” Weyl stated rather sadly, glancing up at the ceiling as if hoping for heavenly deliverance from the plague that Gödel had visited upon his view of mathematical truth.

“In what way?” asked Pauli, trying to draw out Weyl on a matter that Weyl was clearly uncomfortable in discussing.

“Scientists and mathematicians are not indifferent to what their work means in the context of human caring, suffering, and creative existence in the world. But Gödel’s results prevent us from gaining a full understanding of the cosmos as a necessary truth,” Weyl said by way of explanation.

“Are you saying that the only real way to prove something is true is to actually construct it from the natural numbers 1, 2, 3,…?” asked one of the physicists in the room.

“Yes,” Weyl stated with some intensity. “Nonconstructive existence proofs, which show that something exists by proving its nonexistence is false, is like informing the world that a treasure exists without disclosing its location. Mathematics should be much more definite than this about the objects it studies.”

Oppenheimer was not ready to settle for this kind of defeatist attitude toward Gödel’s incompleteness results, and asked Weyl if he thought Gödel’s work robbed him of his reason for being a scientist.

“It seems that on the strength of Gödel’s Theorem, the ultimate foundations of the constructions of mathematical physics will remain trapped forever in a level of thinking involving analogies and intuitions. This implies that there are limits

to the precision of certainty, that even in theoretical physics there is a boundary.”

“And where is this boundary residing?” Pauli asked forcefully.

“The boundary is the scientist himself, as a thinker,” shot back Weyl with equal force.

“This seems a rather self-focused, almost solipsist view of what we can know, even in science,” interjected Oppenheimer, “although it’s not very far away from what quantum theorists seem to believe when they speak about the process of observation creating properties such as the spin and position of objects like electrons.”

Weyl set his teacup down on a sidetable and gazed out to the woods beyond the window, pondering what could be known in mathematics and how it contrasts with what these physicists were claiming about physical reality. Does the world of natural science contain the very same kind of limits on what can be known that Gödel showed must necessarily exist in mathematics? And what about Gödel’s results themselves?

Eliot had finally heard enough and simply had to jump into this debate. In a soft, almost deferential manner, he stated, “I always had the idea that scientists did calculations as a way of getting at the scheme of things. Is it not possible that the theories of matter, the universe, energy, and so forth could be thought of as prescriptions for calculating and predicting something? Couldn’t these predictions then be tested in laboratory experiments to validate or refute the predictions?”

Before any of the others could open their mouths to even begin to reply, Oppenheimer’s lightning-fast mind had already formulated a fully developed position on Eliot’s query. In fact, the reply came so fast that it seemed almost as if he and Eliot had been in telepathic communion on the matter.

“What Eliot asks is whether all that we can hope to know about the world ‘as it is’ is what we can read on a measuring

instrument. That instrument might be a meter stick, the dial of a voltmeter, the track on a photographic plate, or even the sensory response of the human nervous system. ‘Instrumentalism’ is what some philosophers of science call this view of the world. But I think there is more to knowledge than just reading the position of a needle on a dial. I’m sure you agree, Pauli?”

“Yes, most definitely,” concurred the Viennese quickly. “There is more poetry in physics than in a meter stick or a particle accelerator. Measurements tell us about reality; they are not reality itself. Physics and other sciences, in general, try to understand deep reality. They are not just theories of measurement and numerology.”

At this pronouncement from the highest of high priests of theoretical science, one of the visitors to the physics group muttered soto voce that such a view was fine for a philosopher, but that Eliot was right: scientific knowledge was about the consequences of following rules, formulas, and prescriptions, and thus differed considerably from the kind of general knowledge that IAS brahmins Oppenheimer, Pauli, and Weyl were espousing. What a pity that John von Neumann was not present for this discussion. The only mind at the IAS as fast as Oppenheimer’s might have shed a very different light on the matter. Perhaps scientific knowledge, à la von Neumann, will be on tomorrow’s teatime agenda, hoped the visitor.

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The plate of biscuits having been reduced to a pile of crumbs and the teapot nearly drained, teatime was clearly over. As the group disbanded and began drifting out of the room and back to their offices, Weyl whispered to Oppenheimer that he’d like a quick word. As Eliot showed no signs of either vacating his armchair or abandoning his newspaper, the pair moved to the corner of the Commons Room for a bit of privacy.

In a low voice, Weyl began, “You know, Robert, I am very concerned about Johnny’s [von Neumann] proposal to build a computing machine here at the Institute. I respect his belief that such a device will open up whole new vistas of understanding of phenomena like the weather, fluid flow, and perhaps even economic processes. Johnny’s scientific judgment is seldom wrong and I would never bet against it. But the Institute is not the right place for such a project. I have been here since the earliest days, and I know that the founders, especially Mr. Flexner, would not have entertained for a moment the idea of an engineering project being done here.”

Oppenheimer listened impassively to this plea, nodding his head from time to time as Weyl made the argument for the Platonic nature of the IAS as a home for the most rarefied of abstract, speculative thought, and outlined his reasons for opposing von Neumann’s proposal, which had been made to the faculty of mathematics just a few weeks earlier.

After a moment’s pause, Oppenheimer said softly, “Hermann, I fully understand the principle underlying your argument. Everyone here has the highest regard and greatest respect for Johnny’s belief that building a computing machine is something worth doing. But many of the mathematicians like yourself, Marston [Morse], and Deane [Montgomery] think that such a project is out of place and out of the spirit of what this Institute is all about. Others, such as Einstein, don’t think such a device will help their work in any way and remain indifferent to the whole notion.”

“But you are the Director, Robert. How do you feel about it?” pleaded Weyl. “Your position is certain to have a great influence on how the trustees think about this proposal. And make no mistake, I’m sure this adventure of Johnny’s will ultimately have to be decided by the trustees. Faculty emotions are running so high that I don’t see how you can decide it one way or the other without alienating many of them.”

Oppenheimer had to agree. “I am really torn between two diametrically opposed principles. On the one hand, the Institute was founded on a deeply held belief in the value of purely theoretical research. In Mr. Flexner’s manifesto selling the idea of the Institute to the Bamberger family, he explicitly stated that applied work of any sort would not be welcome here.”

“Precisely,” said Weyl, moving in for the kill. “Applied work can be done in many places, and the small staff and resources of the IAS certainly cannot compete with large government labs or even university or industrial research departments in doing applied experiments and developing commercial products.”

“Yes,” agreed Oppie, “but there is another side to this coin, the principle that an IAS professor like Johnny should be free to pursue his research Muse wherever she may lead him. If the IAS stands for anything, it stands for total and complete freedom of choice on what problems to think about and on what methods to employ for their solution—even to the construction of a computing machine. I’m sure you understand.”

“Of course I understand. But we all assume that someone appointed to a professorship here at least tacitly accepts the principles under which the Institute was chartered and operates. One of the most sacrosanct is that applied work is simply not done here. Johnny is now calling this principle into question with this computing project.”

Oppenheimer thought back to his discussions with von Neumann about the computing machine and questioned whether Johnny would have agreed with the label “applied” or “engineering” for this activity. While the machine he wanted to build could certainly be thought of as a tool for doing calculations, hence applied work, the actual design and construction would be very far from a routine, well-worked-out engineering exercise. Only a handful of computing machines

had ever been constructed, each one very different in design and physical structure from the others.

The crux of the matter lay in the fact that such a machine would be a tool foreign to the ways the mathematicians and theoretical physicists traditionally plied their trade. Oppenheimer saw clearly that what bothered these folks, great as their intellect and accomplishments were, was the threat the computing machine posed to their way of practicing their profession.

Rather than confront Weyl with this directly, he said, “This is a very difficult and delicate issue, Hermann, and I am going to have to give the entire matter considerably more thought. I’m sure you would be the first to agree that Johnny is one of our preeminent faculty members and I do not want to get into a position of telling him how to carry on his research. But I am very sensitive, as well, to the concerns you and others have expressed about this project being done here at the IAS. We will just have to think a bit longer and a bit harder to see if we can come to an accommodation that everyone can live with.”

Glancing at the clock on the wall over the doorway of the Commons Room, Oppenheimer quickly closed the discussion, telling Weyl, “I’m afraid you’ll have to excuse me now. I have a visitor from the Atomic Energy Commission due in my office in just two minutes. I appreciate your expressing your views on von Neumann’s project to me directly. We will take up the matter again very soon.”