John von Neumann had the fastest, most logical mind of the twentieth century. So when he analyzed a problem there was never any doubt in anybody’s mind as to what needed to be done. This morning as he drove his shiny, new, blue Cadillac to work, that razor-sharp mind was focused on the vexing problem of Gödel’s promotion. What could he do to overcome the resistance of his colleagues in the School of Mathematics to what he saw as an open-and-shut case? Indeed, how could any of them call themselves “Professor” if Gödel could not? Passing “Von Neumann’s corner,” an intersection famed in Princeton for the number of times the great man had wrapped his car around a particularly stubborn tree there when his flights of mental fancy distracted his thoughts from the steering wheel, von Neumann suddenly had an inspira-

tion. He would organize a presentation. Not just your everyday, plain vanilla type of IAS seminar. No, nothing so dry and formal and academic. Rather, he would persuade Gödel to present a popular account of his greatest work to the faculty and Institute visitors. And he would arrange for this public relations effort to occur just before the faculty meeting at which Gödel’s case for promotion was to be considered. Von Neumann thought that by hearing directly from Gödel himself about his stunning achievements in logic, the faculty would be softened up for the pitch he would then make at the meeting in support of Gödel’s promotion.

Fired with enthusiasm, von Neumann drove through the infamous intersection, smiling just a bit when he recalled his last fiasco at this corner. When asked by the police how he had managed to run straight into a tree on a sunny, dry, autumn morning, von Neumann replied that the trees were proceeding in an orderly fashion at 60 miles per hour past his window— when one of them suddenly jumped out in front of him! No wonder he bought a new Cadillac every year. Arriving at the Institute parking area, he jumped from the car and bustled across the lot into Fuld Hall to set about implementing his plan for Gödel’s “sales presentation.”

The first hurdle I’ll have to jump, thought von Neumann, is to persuade Gödel himself to suggest giving the talk. He knew that a direct approach would be futile, because the ever more reclusive Gödel would undoubtedly cringe at the idea of making any type of public presentation, especially one that he saw as overtly self-promotional. So something far subtler would be needed to convince Gödel that his interests would be served by such a talk. Von Neumann’s lightning-quick analysis immediately presented the answer: Discreetly hint to Gödel that some of the faculty members objecting to his promotion didn’t really believe his work shed much light on the foundations of mathematics, but rather was more of a semantic parlor

trick in logic. Von Neumann was certain that would infuriate Gödel, who would then insist on presenting his work to the faculty. He would want to set it in context and outline the importance of the work for understanding the limitations of the human brain’s ability to access mathematical “deep reality.”

As fate would have it, von Neumann’s assistant happened to mention as they exchanged morning greetings how odd it was to see Gödel coming into the Institute that morning completely wrapped up in a heavy scarf and overcoat when the sun was shining and the mercury was already rising to an uncomfortable level for a late-spring day in Princeton. Knowing now that Gödel was in the Institute, von Neumann immediately set off down the corridor to Gödel’s office to put his plan into action.

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Among Gödel’s many eccentricities, the bleak sparseness of his office ranked high. Entering the room at Gödel’s not entirely friendly response, “Who’s there?” to the knock on his closed door, von Neumann saw Gödel lying on a kind of divan in the corner. The window shades were drawn, so that the room was in semi-darkness on this bright, sunny morning. Besides the divan, the only other furniture in the room was a desk with a rather worn leather desk chair, a completely bare wooden bookcase behind the desk, a blackboard wiped clean, and a simple wooden chair. All in all, a perfect imitation of a police interrogation room. All that was missing was an ashtray over-flowing with half-smoked cigarettes and a few bloodstains on the wall. Well, thought von Neumann, who needs books, a table, and chairs for small meetings, small talk, or anything else when all you do is think? This was not going to be easy, he realized, as he began his oblique pitch to the Grand Exalted Ruler of the Platonic Realm.

“I’m very pleased to see you here today, Kurt. This morning I had an idea about the philosophical basis of mathematical logic, and I wanted to get your opinion about it.”

“You know, Johnny, that I’m not as active in this field as I once was. But I will try to help. What’s on your mind?”

“Well,” von Neumann began, taking a seat in the chair in front of the desk and wondering if Gödel deliberately chose such an uncomfortable chair to discourage the few visitors he had from lingering. “I was speaking about your incompleteness results with one of our esteemed colleagues recently, who said that while he greatly admired the virtuosity of your work, he felt that what your results demonstrate is not so much the limitations of mathematical reasoning, but that your belief in the actual existence of the answer to an undecideable proposition is simply not the right way to think about the ‘reality’ of a mathematical object.”

Gödel’s face darkened, as he let loose one of his strange high-pitched laughs, something between a witch’s cackle and a giggle. Waving his hand in a dismissive gesture he said, “I suppose you were talking with some constructivist. These people have been arguing for a long time that the type of undecideable propositions that my work implies must exist have no mathematical reality at all, since they cannot be constructed in a finite way. In other words, they cannot be built up directly from the positive integers by a finite number of operations like addition, subtraction, and so forth.”

“Precisely my point, Kurt,” said von Neumann immediately, sensing an opening to put forward his idea that Gödel give a lecture straightening out the skeptics on these matters. “I think these constructivists have too narrow a view of mathematical truth. Someone needs to present a clear picture of where they have gone astray. Maybe an Institute seminar on mathematical truth would be a good way to do it. That would at least clear the air on these foundational matters here

in Princeton. Yes, this type of public airing of the dispute would be a very good thing. How do you feel about that, Kurt?”

Gödel’s face took on a thousand-yard stare, as he opened one of the shades and gazed off into the trees beyond the lawn outside his office window, pondering von Neumann’s idea. Eager to push Gödel to volunteer to deliver such a talk, von Neumann added: “You know, this lecture would be very helpful in moving forward your candidacy for a professorship here, since it would help overcome some of the obstacles raised in the faculty meetings about how important your work really is.”

This remark instantly got Gödel’s attention, and he turned to von Neumann and said sharply, “I don’t want someone else explaining my work, Johnny. If anyone is going to do that it’s going to be me.”

Smiling inwardly at how easily he had maneuvered Gödel into this declaration, von Neumann announced, “Excellent. I could not agree more. You are the only one to do it. I will set up the talk as the next colloquium lecture in the School of Mathematics. My assistant will tell you the date later today.”

Von Neumann glanced quickly at his wristwatch, eager to escape from Gödel before there could be any debate or, even worse, a change of heart. “I have a meeting with the Director now, Kurt. I’m glad we had this chat. This lecture is going to do a lot of good for the School of Mathematics in many different ways. We will speak later.”

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Von Neumann made sure everyone in the Princeton and New York mathematical communities knew about the big lecture. He personally invited the most distinguished professors of logic, including Alonzo Church from Princeton, Sammy Eilenberg from Columbia, and, of course, his IAS colleague—and

Gödel’s bête noir—Hermann Weyl. For his part, Gödel decided to focus his presentation on the famous Continuum Hypothesis, first enunciated by Georg Cantor in 1874. Following his work on mathmatical incompleteness, Gödel had devoted much of his mathematical effort to trying to settle this problem. Cantor’s hypothesis had attracted so much attention that Hilbert had put it at the top of his list of 23 problems whose solution was important for the development of mathematics, which he presented in his historic address to the International Mathematical Congress in Paris in 1900.

The Continuum Hypothesis is about the existence of different levels of infinity. Starting with the style of infinity represented by the natural numbers 1, 2, 3,…, which is called countably infinite and denoted by the first letter of the Hebrew alphabet (aleph-zero), Cantor used an ingenious argument to show that the level of infinity represented by the real numbers was strictly greater than that of the natural numbers. The set of real numbers consists of all possible subsets of natural numbers. This is now termed the power of the continuum and is represented by the symbol The Continuum Hypothesis states that there is no level of infinity that is strictly larger than and smaller than .

In 1938 Gödel showed that if you use the axioms of set theory and symbolic logic most familiar to working mathematicians, you cannot disprove the Continuum Hypothesis. In other words, the Continuum Hypothesis is consistent with these axioms. But he was never able to establish the converse„ namely, that you cannot prove it either, using this same axiomatic framework.

As news of Gödel’s upcoming lecture spread throughout the local mathematical communities, a rumor arose that perhaps Gödel had finally succeeded in settling the second half of the problem to the effect that it was also not possible to prove the Continuum Hypothesis using this standard

axiomatic framework. Because Gödel was notoriously reclusive and almost never gave lectures, this rumor acquired some currency. After all, why would the fanatically reclusive Gödel make a public presentation if he didn’t have something extraordinary to announce? The buzz associated with this possibility pumped up interest in the lecture to almost feverish proportions before the day of the lecture arrived. As he stepped to the podium to introduce the talk, von Neumann realized that he had never seen the seminar room at the IAS School of Mathematics so crowded with curious onlookers. Absolutely everyone who was anyone was here.

“Distinguished colleagues and guests,” he began. “Welcome to the School of Mathematics Colloquium. Nearly 20 years ago, our colleague, Kurt Gödel, obtained one of the most unexpected results in the history of mathematics—essentially, that deductive argumentation has inherent limitations for uncovering mathematical truth, even in the limited domain of the arithmetic of the natural numbers. In the intervening years, there has been much confusion and many differences of opinion on what these results really imply for our ability to settle every well-formulated mathematical proposition. Today, Kurt has generously offered to give us a brief account of his thoughts during the process of obtaining his results, and to present his personal opinion on these matters of mathematical philosophy. So I give the floor to him now for what I’m certain will be a fascinating and enlightening presentation.”

Moving to the front of the lecture room, Gödel peered out at the audience through his thick, perfectly circular spectacles. He looked like someone who was caught in a spotlight and wondered why. Uh-oh, thought von Neumann, this is not a good beginning. The last thing he wanted was for Gödel to project the air of a confused, slightly mad professor. That would not help him argue Gödel’s case for promotion at all.

Everyone in the audience, especially von Neumann,

began to get a bit uneasy at Gödel’s seeming reluctance to begin the talk. But suddenly his eyes snapped into focus and he seemed to realize where he was and why. Gathering his thoughts, Gödel began to speak softly—but clearly and precisely—launching into some of the history of his celebrated work on incompleteness in arithmetic and his contribution to the solution of the Continuum Hypothesis.

“In March 1928 in Vienna, I attended two stimulating lectures by L.E.J. Brouwer, the famous Dutch topologist and logician. These lectures showed me what was known and what remained to be discovered in mathematical logic at that time. About one year later, I obtained a copy of Hilbert and Ackermann’s book, Grundzüge der theoretische Logik (The Foundations of Theoretical Logic), in which they stated the problem of the completeness of predicate logic as an open problem. My doctoral dissertation settled this problem by showing that, indeed, predicate logic is complete; every valid statement one can make in predicate logic can be proved.”

Here Gödel was referring to the form of logic that underlies all of set theory. First-order predicate calculus or first-order logic is a theory in symbolic logic that formalizes quantified statements such as “There exists an object with the property that…” or “For all objects, the following is true….” First-order logic is distinguished from higher-order logic in that it does not allow statements such as “For every property, the following is true…” or “There exists a set of objects such that ….” Nevertheless, first-order logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. It is the classical logical theory underlying mathematics.

Now entering into the spirit of his talk, Gödel roamed steadily back and forth across the front of the seminar room, as relentless in pursuit of his theme as a lion stalking its prey in the African savannah.

“But predicate logic is much too weak to deal with the

questions that most interest us—those involving the relationship between whole numbers, the realm of arithmetic. Then, one year after completing my dissertation, I discovered a way to code every statement about numbers into a number itself. This coding scheme allowed me to get arithmetic to, in effect, talk about itself. Thus, I was able to use numbers in both a syntactic and a semantic fashion. On the one hand, a given number was coded for some assertion about the relationship between numbers and thus had semantic content within arithmetic; on the other hand, that number was simply a number and, thus, had no real meaning beyond that fact. This dual character of numbers allowed me to create for any consistent logical system a number-theoretical statement that was undecidable—could not be proved or disproved—using that logical system.”

By this, Gödel meant that he was able to translate the sentence, “This statement is unprovable” into a proposition about numbers. His coding scheme then provided a means to further translate this numerical proposition into a number itself. This number was then the “codeword” for an undecidable proposition, one that is provable if and only if it is not provable.

“As consistency of the logical framework was an indispensable condition for the incompleteness result to hold, I was always concerned about whether the particular system being employed could in some way actually prove its own consistency. This would have been quite a feat, akin somehow to pulling yourself up by your bootstraps. But by employing essentially the same lines of reasoning as for the incompleteness theorem, I was able to show that it is impossible for a logical system to prove its own consistency.”

At this juncture someone from the audience asked if this meant that mathematics was always tentative, relying on assumptions like consistency that could never really be demon-

strated, but only assumed. Gödel replied that he would address this kind of metamathematical question at the conclusion of his talk.

“Around 1930, I also ran across Hilbert’s outline of a proposed proof of the Continuum Hypothesis. Sensitized to the metamathematical problems of arithmetic, I immediately saw the continuum problem as a question from the multiplication table of cardinal numbers (the positive integers, 1, 2, 3,…). The problem’s intractability strongly suggested to me that the very notion of a set was in need of clarification. My own results on incompleteness and consistency had already pushed my thoughts in this direction, and the continuum problem only further convinced me that the right axioms of set theory had not yet been found.”

Gödel went on to state that not only was the Continuum Hypothesis of interest in its own right, but that it served as a catalyst and testing ground for his ideas on what was the right concept of a set. By 1937 he had stated in private correspondence that he had succeeded in proving the consistency of the Continuum Hypothesis with the usual axioms of set theory, showing that the Hypothesis could not be disproved using this axiomatic framework. At that time he showed the outline of his proof only to von Neumann and Karl Menger, a Viennese colleague who was at that time professor at the University of Notre Dame in the United States. Gödel told the colloquium audience about this discovery.

“The first step in my proof was not to prove the Continuum Hypothesis directly, but to prove only its consistency with the axioms of set theory. Next, I wanted to use only definable properties of sets and relations instead of invoking recursive definitions that create an object by an infinite process. The final step in my argument is to take all ordinal numbers (roughly, the number of elements of a set of ordered numbers, for example, the ordinal number of the set {1, 2, 3,…, n}

is n) as given rather than trying to construct them from first principles.”

Hearing this last remark, von Neumann nodded his head benevolently and looked at the audience to see how many would join him in recognition of Gödel’s resolutely Platonistic view of mathematics, in which objects like the ordinal numbers exist independent of the human mind or of any specific procedure for explicitly constructing them.

When he looked back to the front of the room, von Neumann was horrified to see Gödel facing the blackboard, scribbling some incomprehensible symbols, while muttering in a low voice that was totally inaudible to those in the room. Good God, thought von Neumann, this could turn into a first-class disaster if Gödel goes into his absent-minded professor act. If there was ever a time to be clear, direct, and communicative, it was now. Hoping to bring Gödel back to the land of the living, von Neumann interjected a question: “Kurt, could you say what the principal tool was that you used to obtain the consistency of the Continuum Hypothesis with the axioms of set theory?”

“Oh, yes. It was my creation of the notion of a constructible set.”

“And what, precisely, is that?” continued von Neumann, hoping to draw out Gödel and have him explain to this audience of non-logicians what a magical rabbit he had pulled out of his mathematical hat to prove this extremely difficult result.

To von Neumann’s dismay, Gödel turned again to the blackboard, saying, “The constructible sets are a specific example of a collection of sets that satisfy all the axioms of set theory, and with which the Continuum Hypothesis is consistent. Therefore, the Continuum Hypothesis cannot be disproved using the usual setup in logic, since it is consistent with the axioms. In the language of logicians, the constructible sets are a model for set theory.”

At this juncture, Weyl interrupted Gödel to ask his view of the reality of mathematical objects. In particular, he asked if Gödel believed that a level of infinity between the integers and the real numbers truly existed, in the same sense that the seminar room or the piece of chalk Gödel was holding had an objective existence. Continuing to peer at the blackboard as if searching in the chalk dust for the perfect answer to this almost metaphysical question, Gödel finally began to expound his Platonistic view of mathematical objects.

“I believe that we do have objectivity in mathematics. Propositions about numbers are either true or false. Facts are independent of arbitrary conventions. And theorems about numbers characterize objective facts about integers. Moreover, these facts must have a content, because the consistency of number theory is derived from higher facts, and we can’t assume any kind of set because if we did, number theory would not be consistent; we would get contradictions.”

Carl Ludwig Siegel, one of the professors at the Institute who opposed Gödel’s promotion, and a world-renowned number theorist himself, now stood up and in a booming voice overlaid with a heavy German accent, enquired, “So you do ‘objective’ mathematics? You feel that mathematicians discover objects rather than create them, much like the stars being there quite independently of the existence of astronomers to look at them?”

“Yes,” said Gödel, finally turning away from the black-board to face his interlocutor, “Mathematics is an empirical science. In my view the Continuum Hypothesis definitely has an answer—Yes or No. We have just not yet looked at the continuum hard enough to see what the answer is.”

“Would you say, then,” asked Weyl, “that the set of natural numbers has an independent existence that we can ‘see,’ in the same way, for instance, that we can look around Manhattan and see the Empire State Building?”

“Absolutely,” Gödel shot back, in perhaps his most aggressive statement in the lecture. “Anyone who takes the trouble to learn a little mathematics can ‘see’ the set of natural numbers for himself. So the natural numbers must have an independent existence as a certain abstract possibility of thought.”

One of the bright-eyed and bushy-tailed graduate students in logic who had come over from Princeton University to attend the seminar then asked naïvely, “What is the best way to perceive this pure abstract possibility?”

“First,” replied Gödel, “you must close off the other senses, for example, by lying down in a quiet, darkened room. But this passive, rather negative action is by no means sufficient. You must actively seek with the mind. Do not forget that the mind is capable of perceiving infinite sets. So don’t just imagine combinations and permutations of physical objects— finite things. Look to the infinite. The ultimate goal of such an exercise is to perceive the Absolute.”

Oh no, thought von Neumann. Now Gödel has really gone too far. Veering off into what sounds like a lot of mystical mumbo-jumbo in front of this high-powered, mathematical audience, is not going to help me make a case to the faculty for his sanity and mathematical judgment. I must derail this line of discussion immediately and move things back to the purely mathematical. But before he could even open his mouth to redirect the discussion, the unrelenting Weyl was back.

“For myself, I believe that mathematics is mind-dependent; the objects of mathematics, such as a possible ‘style’ of infinity in between the integers and the reals, is not an objective fact but rather something that we must be able to construct. If we can, it exists; otherwise, it does not.”

Gödel looked at Weyl in the same pitiless manner he might use to inspect some loathsome insect creeping from behind his refrigerator, before finally stating, “I will not argue with my esteemed colleague on this point. I simply state that

although intuition represents a relationship between the human mind and mathematical reality, the mathematical world goes beyond our perception of it—just as the physical world does. This is what it means to be mind-independent.”

The introduction of the world of tangible, physical objects into the discussion calmed von Neumann’s growing concern at the metaphysical turn the discussion had been taking. He saw that the physicists in the crowd were now ready to join in, as Gödel’s remark about our perception of the physical world set the antennae of the quantum theorists buzzing. The first to get his oar in the water was, interestingly enough, one of the youngest, the brilliant newcomer from England, Freeman Dyson, who spoke to Gödel but looked directly at von Neumann, as he said: “Your results prove that there are inherent limitations in every logical system. So regardless of the axiomatic framework we choose, there is some proposition that can be stated but neither proved nor disproved using the rules of that system. We quantum physicists also have a fundamental principle that constricts what we can know by measurement: the Heisenberg Uncertainty Principle, which limits how accurately we can simultaneously know the values of certain pairs of properties that a particle may possess, like position and momentum or time and energy. It’s impossible not to speculate about whether these two types of limiting results—yours and Heisenberg’s—have a common root. Or is it just a tempting analogy? Do you have any thoughts on this?”

Again von Neumann benevolently nodded his approval of Dyson’s query, since it was precisely the kind of question he hoped someone would ask. There was little doubt in his mind that by linking Gödel’s work to something as central to the scientific mindset in Princeton as Heisenberg uncertainty, the question would draw attention to the profundity of what Gödel had achieved. He squirmed in his seat, impatiently awaiting Gödel’s reply. Finally, the Ruler of the Platonic

Realm spoke.

“It is very important to distinguish the world of mathematical objects from that of the physical. Both Heisenberg’s result and my own are first and foremost mathematical results about mathematical objects. In Heisenberg’s case, the objects are mathematical transformations [technically: operators] representing different properties that might be measured about a quantum object like an electron. The inherent uncertainty his principle asserts comes from comparing the measurement process for two different properties, such as position and momentum. If the order in which you carry out the measurement of this pair of properties makes no difference, then the corresponding operators commute and there is no inherent problem in simultaneously measuring both properties to arbitrary accuracy. But if the order does make a difference, the operators do not commute and there is an irremovable level of uncertainty in any such pair of measurements. This commutativity is a mathematical condition that can be checked for any two pairs of operators.

“It happens that we can correlate approximately these mathematical operators with physical properties, and then transfer the noncommutativity of a pair of such objects into an inherent limitation on how accurately we can measure the corresponding properties. But then you have passed from the mathematical to the physical, introducing a host of new questions surrounding the degree to which the idealized mathematical representation of the measurement situation corresponds to the actual physical setup in the laboratory.”

Dyson, not to be put off by this basically mathematical reply, carried his question one step further.

“As a natural scientist, my interest is whether something akin to the limits you show for mathematics could ever arise in the world of real objects. This is why I ask about Heisenberg uncertainty. Every experiment ever performed confirms that

we cannot simultaneously know the position and momentum of a particle to arbitrary accuracy. This appears to me to be a limit on what can be known, just as your incompleteness results show there are limits to what we can deduce by logical argument. Can you say something about these limitations on what can be known in the very different worlds of mathematical and physical objects?”

Before Gödel could open his mouth, a buzz of loud mutterings arose as several members of the audience attempted to join the discussion. One of the visitors from the Princeton University Physics Department shouted just a bit louder than the rest, asking Gödel his view of the truth of Euclidean geometry. Distracted from Dyson’s question by the new direction this question led, Gödel seemed to go into a state of momentary paralysis, at which point von Neumann thought he had better step in and calm things down. Rising, he asked the audience to please conduct their questioning in a more orderly fashion, and allow Gödel to respond to one question before firing off another. This intercession seemed to offer just the breathing space Gödel needed to gather his thoughts and address the question about geometry.

“Geometrical intuition, strictly speaking, is not mathematical, but rather a priori physical intuition. In its purely mathematical aspect our Euclidean space intuition is perfectly correct; namely, it represents correctly a certain structure existing in the realm of mathematical objects. Even physically it is correct ‘in the small,’ that is, in the immediate neighborhood of a single point in space.

“But I want to emphasize that the congruence between the properties of mathematical objects such as points, lines, and planes and their real-world correlates is not a question of mathematics; it is more a question in the realm of mathematical epistemology or ontology, in which we investigate the relationship between the objects of the mathematical universe

and those of the world of the natural sciences.”

Von Neumann thought this was just the right moment to bring the proceedings to a close and stood up to thank the audience for their attention and lively discussion. In his closing remarks he said he thought Gödel’s exposition had brought a deeper understanding of where the incompleteness results stood in the scientific scheme of things and hoped that the audience agreed with him.

After giving the customary round of applause for Gödel’s presentation, the audience filed out of the seminar room, von Neumann and Weyl the last to go. As they left, von Neumann stopped Weyl in the corridor to ask him again about his views regarding Gödel’s promotion.

“Hermann, I think Gödel’s presentation this afternoon makes it obvious that his work plays a central role in our thinking about the relationship between mathematics and the world of matter and energy. As a mathematical physicist yourself, I’m sure you saw these connections long ago. So do you still oppose Gödel’s promotion to Professor on the grounds that his incompleteness results are somehow dangerous for mathematics?”

“You know very well, Johnny, that my objections were never about the quality of Gödel’s work. I do believe that his Platonistic view on the existence of mathematical objects is wrong-headed and sets mathematics onto an unhappy course, philosophically speaking anyway. But my real concerns over Gödel being a full professor are mostly about what I see as his otherworldly nature and, to put it bluntly, his mental instability. You know better than anyone that being a Professor at the IAS involves an enormous amount of administrative duty to keep the School of Mathematics alive and viable. Gödel’s legalistic turn of mind could paralyze this entire process. That is my principal concern. And it is the same concern expressed by others in the School, including Siegel and Montgomery.

We simply must be confident that Gödel will not be a logjam in our procedures. Do you have any thoughts on how to deal with this problem?”

“To be honest, Hermann, I do not. But I will speak with others, as well as with Gödel himself, before the faculty meeting and see if there is some middle ground that everyone can feel comfortable with. As the meeting is more than two weeks away, let us try to talk again soon about this. Maybe by next week I’ll have a solution to propose to you.”