Errol Morris – The Ashtray.

1.

THE ULTIMATUM

I don’t want to die in a language I can’t understand. — Jorge Luis Borges (as quoted in Alberto Manguel, “With Borges”)

It was April, 1972. The Institute for Advanced Study in Princeton, N. J. The home in the 1950s of Albert Einstein and Kurt Gödel. Thomas Kuhn, the author of “The Structure of Scientific Revolutions” and the father of the paradigm shift, threw an ashtray at my head.

It had all begun six months earlier.

“Under no circumstances are you to go to those lectures. Do you hear me?” Kuhn, the head of the Program in the History and Philosophy of Science at Princeton where I was a graduate student, had issued an ultimatum. It concerned the philosopher Saul Kripke’s lectures — later to be called “Naming and Necessity” — which he had originally given at Princeton in 1970 and planned to give again in the Fall, 1972.

But what was Kuhn’s problem with Kripke?

Kuhn was becoming more and more famous. He would become not just a major figure in the history and philosophy of science, but an icon – and his terms “paradigm” and “paradigm shift” became ubiquitous in the culture-at-large. An astrophysicist and rock-climbing friend from Princeton, Dick Saum, later sent me a picture of a bumper sticker that said, “Shifts happen.” [1]

Errol Morris with Dick Saum

Kripke was slight, bearded, in his early thirties. He was not well known but had a reputation as a genius. He had provided a completeness proof for modal logic (which deals with necessity and possibility) while still a teenager — and in the process reinvigorated Leibniz’s ideas about possible worlds. [2] There was also the amusing anecdote of Kripke being offered a chair at Harvard when he was 16. He supposedly wrote back, “Thank you, but my mother thinks I should finish high school first.” Nonetheless, it was hard to see how Kripke’s theories had much to do with Kuhn. Or at least, it seemed so, at first.

I ignored Kuhn’s ultimatum and went to Kripke’s lectures anyway. My relationship with Kuhn ended badly. But more about that later. Kripke addressed the 20 or so graduate students and professors assembled in a small seminar room by looking at them through an empty water glass as if it were a telescope or the lens of a camera. The water glass created all sorts of optical distortions, making Kripke’s left eye distend like the eye of a flounder. I assume that the glass had the same effect for him — rendering the seminar into an aquarium of academics.

I didn’t really understand Kripke’s lectures. It was only a year or so later at the University of California, Berkeley, that I began to understand Kripke’s ideas, due to the efforts of my friend and fellow graduate student Charles Silver.

Pall Mall Cigarettes

True Blue Cigarettes

Kuhn in those days was an incredible chain-smoker. First Pall Malls and then True Blues (a low tar and low nicotine alternative). Alternating. One cigarette lighting another. Matches were irrelevant. Maybe six, maybe seven packs of cigarettes a day. All that was essential was burning and smoke. And a massive cut-glass ashtray filled with the debris of an endless series of burnt-out butts.

The seminar was filled with an odd collection of people. Some of his graduate students from the Program in the History and Philosophy of Science, and a couple of visiting academics. He had already attracted the interest of social scientists around the world, and there were a couple who made a pilgrimage to Princeton to attend his lectures.

His often repeated, most scathing complaint concerned Whiggishness — in history of science, the tendency to evaluate and interpret past scientific theories not on their own terms, but in the

context of current knowledge. The term comes from Herbert Butterfield’s “The Whig Interpretation of History,” written when Butterfield, a future Regius professor of history at Cambridge, was only 31 years old. Butterfield had complained about Whiggishness, describing it as “…the study of the past with direct and perpetual reference to the present” – the tendency to see all history as progressive, and in an extreme form, as an inexorable march to greater liberty and enlightenment. [3] For Butterfield, on the other hand, “…real historical understanding” can be achieved only by “attempting to see life with the eyes of another century than our own.” [4][5]

Princeton was sort of a consolation prize. I had not been accepted in Harvard’s history of science program, and Erwin Hiebert, a professor at Harvard, had written a letter of recommendation to Kuhn for me. I should have known that there was going to be trouble. I had imagined graduate school as a shining city on a hill, but it turned out to be more like an extended visit with a bear in a cave.

I had written a paper on James Clerk Maxwell’s displacement current for Kuhn’s seminar on 19th century electricity and magnetism. The paper might have been 30 or so double-spaced pages. Kuhn’s reply, typed on unlined yellow paper, was 30 pages, single-spaced, with Courier marching all the way from the left to the right side of the paper. No margins. He was angry, really angry.

He had written at the very end of his comments, “You have long since passed the end of the road on which you began.” I asked, “What is that supposed to mean? I’m 24 years old.” He said that I was a “good” first-year graduate student but would become “less good” in subsequent years.

Our discussion took place at West Hall, a new building at the Institute for Advanced Study. Kuhn had taken a leave of absence from Princeton to write the book “Black-Body Theory and the Quantum Discontinuity, 1894-1912.”

We began arguing. Kuhn had attacked my Whiggish use of the term “displacement current.” [6] I had failed, in his view, to put myself in the mindset of Maxwell’s first attempts at creating a theory of electricity and magnetism. I felt that Kuhn had misinterpreted my paper, and that he — not me — had provided a Whiggish interpretation of Maxwell. I said, “You refuse to look through my telescope.” And he said, “It’s not a telescope, Errol. It’s a kaleidoscope.” (In this respect, he was probably right.) [7]

The conversation took a turn for the ugly. Were my problems with him, or were they with his philosophy?

I asked him, “If paradigms are really incommensurable, how is history of science possible? Wouldn’t we be merely interpreting the past in the light of the present? Wouldn’t the past be inaccessible to us? Wouldn’t it be ‘incommensurable?’ ” [8]

He started moaning. He put his head in his hands and was muttering, “He’s trying to kill me. He’s trying to kill me.”

And then I added, “…except for someone who imagines himself to be God.”

It was at this point that Kuhn threw the ashtray at me.

And missed.

I see the arc, the trajectory. As if the ashtray were its own separate solar system. With orbiting planets (butts), asteroids and interstellar gas (ash). I thought, “Wait a second. Einstein’s office is just around the corner. This is the Institute for Advanced Study!!” [9]

I call Kuhn’s reply “The Ashtray Argument.” If someone says something you don’t like, you throw something at him. Preferably something large, heavy, and with sharp edges. Perhaps we were engaged in a debate on the nature of language, meaning and truth. But maybe we just wanted to kill each other.

The end result was that Kuhn threw me out of Princeton. He had the power to do it, and he did it. God only knows what I might have said in my second or third year. At the time, I felt that he had destroyed my life. Now, I feel that he saved me from a career that I was probably not suited for.

2.

SHIFTING PARADIGMS

Saul Kripke is considered one of the seminal thinkers of our time. Philosophers can and will endlessly debate the content of “Naming and Necessity.” Currently, there are hundreds if not thousands of journal articles devoted to this series of three lectures. [10] His lectures realigned our ideas about meaning and reference — essentially, about how language “connects” to the world. And affirmed a decidedly un-postmodern idea of meaning, reference and truth. In Kripke’s view words are attached to things in the world through an historical (or causal theory) of reference. [11] And although Kripke’s theories examined proper names, like “Julius Caesar” or “Moses” or “Kurt Gödel,” they also apply to terms like “water” and “gold.” [12] [13]

Kripke’s theory provides an alternative to what had become known as the description theory, an amalgam of ideas proposed by Gottlob Frege, Bertrand Russell and Ludwig Wittgenstein. (And to that mix, in the ‘50s and ‘60s you can add Peter Strawson and John Searle.) Here’s one way to distinguish between Kripke’s theories and the description theory that preceded it.

You have two fish in a fishbowl. One of them is golden in color; the other one is not. The fish that is golden in color, you name “Goldie.” The other fish you name “Greenie.” Perhaps you use the description “the gold fish” and point to the one that is golden in color. You are referring to the gold fish, Goldie. Over the course of time, however, Goldie starts to change color. Six months later, Goldie is no longer golden. Goldie is now green. Greenie, the other fish — the fish in the bowl that was green in color — has turned golden. Goldie is no longer “the fish that is golden in color.” Greenie is. But Goldie is still Goldie even though Goldie has changed color. The description theory would have it that Goldie means the fish that is golden in color, but if that’s true then when we refer to Goldie, we are referring to the other fish. But clearly, Goldie hasn’t become a different fish; Goldie has merely changed his (or her) appearance. [14]

It’s Kripke’s version of “Where’s Waldo.” If the description theory (courtesy of Frege, Russell and Wittgenstein) is correct, then Goldie is on the right. If Kripke’s historical-chain of reference theory is correct, then Goldie remains Goldie no matter what color Goldie is. [15]

You could also think of Goldie and Greenie in terms of beliefs, although this is not how the description theory was originally framed. Goldie is the fish that you believe is golden in color. But Goldie starts to change color. I can believe anything I want about Goldie. I can even believe that Goldie isn’t a fish, but Goldie — that fish out there swimming around in a fishbowl — remains Goldie.

Here is Kripke’s central intuition: descriptions help us to fix a reference, that is, to attach a name to a thing, but descriptions (and beliefs) do not determine reference. There is a historical connection between words and things. Our beliefs about Goldie could be all wrong, and we can still refer to Goldie. It doesn’t matter what belief or what theory we have about Goldie. We can grab a hold of Goldie independent of that belief or that theory. And we can say true or false things about him. Is it true that Goldie is green? Or gold? Or red? (Or that Goldie has two heads.) There is a historical connection between words and things.

Here’s another way of looking at it. We can reach outside our theories and pick out things in the world. [16]

Thomas Kuhn’s “The Structure of Scientific Revolutions,” however, was far more influential than “Naming and Necessity” — possibly because it fit into the pop-culture of the moment, the idea that truth is culturally determined and depends on your “frame of reference.” It produced a cottage industry around itself. And became a kind of postmodernist Bible. [17] [18] [19]

Kuhn’s book introduced its own nomenclature — normal and revolutionary science, paradigms, paradigm shifts, anomalies, etc. Here is a brief description. According to Kuhn, science is parsed into normal and revolutionary science. In normal science a group of “practitioners” have settled on a way of defining and solving problems — a paradigm. They have a way of looking at the world and are by and large happy with it. And then there are anomalies. Anomalies shatter the tranquility of the paradigm. An anomaly, for example, could be an unexpected experimental result. Something happens that prevents things from going on as before. The anomaly leads to a revolution, and a shift to a new paradigm.

The most important and most controversial aspect of Kuhn’s theory involved his use of the terms “paradigm shift” and “incommensurability.” That the scientific terms of one paradigm are incommensurable with the scientific terms of the paradigm that replaces it. A revolution occurs. One paradigm is replaced with another. And the new paradigm is incommensurable with the old one. He made various attempts to define it — changing and modifying his definitions along the way. In the 1962 edition of “Structure” incommensurability was likened to a Gestalt-flip. Presumably, it was about how we see the world.

I found this unconvincing. In a Gestalt-flip, we never lose our ability to see the rabbit or the duck, even if we can’t see them at the same time. We see the rabbit, then the duck. Or the duck, then the rabbit. Rabbit, duck. Duck, rabbit. (I’m sure Elmer Fudd figures in here, somewhere.) But then Kuhn went on to say, “What were ducks in the scientist’s world before the revolution are rabbits afterwards.” [20]

What!? Is this about our perception of reality or about reality itself? Did the ducks become rabbits?

Here is where the dangerous, slippery slope begins.

Kuhn writes, “We may want to say that after a revolution scientists are responding to a different world.”[emphasis mine]

Attribution: Jastrow, J. (1899). The Mind’s Eye. Popular Science Monthly, 54, 299-312, via Wikimedia Commons

By 1969, in his postscript to “Structure,” incommensurability had become linguistic. Kuhn wrote, “Two men who perceive the same situation differently but nevertheless employ the same vocabulary in its discussion must be using words differently. They speak, that is, from what I have called incommensurable viewpoints.” [21] People in different paradigms speak different languages, and there is no way to translate the scientific language of one paradigm into the scientific language of another. [22] Even when they use the same words. “Consider…the men who called Copernicus mad because he proclaimed the earth moved. They were not either just wrong or quite wrong. Part of what they meant by ‘earth’ was fixed position. Their earth, at least, could not be moved. Copernicus’ innovation was not simply to move the earth. Rather, it was a whole new way of regarding the problems of physics and astronomy. The proponents of competing paradigms practice their trades in different worlds…” [23] [24] [25]

Same words, different worlds!?

I had argued with Kuhn — or attempted to argue — that the concept of incommensurability is self-defeating. If paradigms are really incommensurable — as Kuhn claims they are — how can we even say they’re incommensurable? How can we look beyond the perimeter of our own paradigm and compare it with another? Radical incommensurability should be just that. It should command silence. We can’t know enough even to assert the claim. [26]

If the descriptivists linked the meaning of a name to a description or to a cluster of descriptions and its reference to objects that satisfy those descriptions, in Kuhn’s theories descriptions and clusters of descriptions multiply without end. “Earth” in one paradigm means something different than “earth” in another. And there is no way to compare them, because after all, they are incommensurable. Ultimately, there is no way to determine reference. Or truth. For Kuhn, we are trapped inside a fog of language. And there is no way out. For Kripke, there is such a thing as reference; for Kuhn, there may be no such thing. For Kripke, there are necessary truths (and essential properties); for Kuhn, there are no truths, let alone necessary ones. And on and

on and on. “Goldie” in Kuhn’s view means “the-fish-that-I-refer-to-contingent-on-my-paradigm-or-conceptual-scheme.” Or maybe reference is not involved at all. “Goldie” means “the beliefs I have about Goldie in my paradigm.”

It really doesn’t depend on how you dress it up. Paradigms, paradigm shifts, incommensurability, etc. Kuhn’s ideas lead to the relativity or even to the denial of truth — a dangerous idea.

Readers may wonder, haven’t Kuhn’s views been discredited? No. Not at all. People see paradigms and paradigm shifts everywhere. Relativism lives on.

The New Yorker Collection from cartoonbank.com. All Rights Reserved.

3.

HIPPASUS OF METAPONTUM

Hippasus of Metapontum

Incommensurable. It is a strange word. I wondered, why did Kuhn choose it? What was the attraction? [27]

Here’s one clue. At the very end of “The Road Since Structure,” a compendium of essays on Kuhn’s work, there is an interview with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a brief account of the historical origins of his idea. Here is the relevant segment of the interview.

T. KUHN: Look, “incommensurability” is easy.

V. KINDI: You mean in mathematics?

T. KUHN: …When I was a bright high school mathematician and beginning to learn Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was sort of a big two-volume Calculus book by, I can’t remember whom. And then I never really read it. I read the early parts of it. And early on it gives the proof of the irrationality of the square root of 2. And I thought it was beautiful. That was terribly exciting, and I learned what incommensurability was then and there. So, it was all ready for me, I mean, it was a metaphor but it got at nicely what I was after. So, that’s where I got it. [28]

“It was all ready for me.” I thought, “Wow.” The language was suggestive. I imagined √2 provocatively dressed, its lips rouged. But there was an unexpected surprise. The idea didn’t come from the physical sciences or philosophy or linguistics, but from mathematics. Namely, the proof that √2 can not be expressed as the ratio of two integers. “…it was a metaphor but it got at nicely what I was after.”

Incommensurability in mathematics expresses the fact that not every distance can be measured with whole numbers or fractions of whole numbers. Take a unit-square – 1 by 1. How long is the diagonal? By the Pythagorean Theorem, if each side has a length of 1 then the hypotenuse has a length of √2. The sum of the squares of the sides = the square of the hypotenuse. Can that length be expressed as a fraction or as a ratio of two integers, e.g., 99/70 or 577/408? The answer is — no. [29] [30] [31]

The proof established that there are quantities that cannot be expressed as fractions. [32] The Pythagoreans were wrong – not everything was composed of whole numbers or ratios of whole numbers. As Hamlet says, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”

*******

But how does mathematical incommensurability elucidate Kuhn’s concept of philosophical incommensurability? [33] As mathematics developed, concepts have been introduced and proofs given and clarified. The assortment – you could even think of it as a bestiary – of numbers (and other mathematical concepts) has grown and grown. Today, we have irrational numbers, imaginary numbers, complex numbers, ideal numbers, transfinite numbers, etc. To name a few. But we are not losing our capacity to understand concepts from the past. A knowledge of complex numbers does not prevent us from understanding real numbers, no more than a knowledge of irrational numbers prevents us from understanding rational numbers, etc., etc. We are enriching our understanding of mathematics. We are expanding our notion of what is possible. Of what we can imagine.

Gabriel Garcia Marquez, on reading Kafka’s “Metamorphosis,” said, “I didn’t know you were allowed to write like that.” [34] Note the use of the word “allowed,” as in permitted. Marquez understood perfectly well what was happening to Gregor Samsa. He was metamorphosing into a gigantic insect.

Gregor Samsa might have understood it as well. [35]

This story illustrates part of the confusion about incommensurability and about paradigm shifts. Is it a question of what can be understood? Or what can be allowed?

*******

But let’s return to Kuhn’s interview. He said incommensurability in mathematics was a “metaphor?” But a metaphor for what? [36] I thought, since mathematical incommensurability doesn’t capture what Kuhn was looking for, namely, incommensurable meanings, perhaps I should look for an answer in the history of the proof. I was familiar with the general outline – the Pythagoreans and their attachment to whole numbers, the betrayal of a cult secret and the murder that followed. Here are a couple excerpts from two popular accounts. The first excerpt concerns the Pythagorean cult. It is from David Berlinski, “Infinite Ascent.”

Pythagoras and the Pythagoreans were devoted to a higher spookiness. It is their distinction. With his vein-ruined hands describing circles in the smoky air, Pythagoras has come to believe in numbers, their unearthly harmonies and strange symmetries. ‘Number is the first principle,’ he affirmed, ‘a thing which is undefined, incomprehensible, having in itself all numbers…’ Half-mad, I suppose, and ecstatic, Pythagorean thought offers us the chance to peer downward into the deep unconscious place where mathematics has its origins, the natural numbers seen as they must have been seen for the very first time, and that is as some powerful erotic aspect of creation itself… [37]

Veined hands, tallow dripping from candles?

The second excerpt concerns betrayal and murder. It’s from Charles Seife, “Zero: The Biography of a Dangerous Idea”:

Hippasus of Metapontum stood on the deck preparing to die. Around him stood the members of a cult, a secret brotherhood that he had betrayed. Hippasus had revealed a secret that was deadly to the Greek way of thinking, a secret that threatened to undermine the entire philosophy that the brotherhood had struggled to build. For revealing that secret, the great Pythagoras himself sentenced Hippasus to death by drowning. To protect their number-philosophy, the cult would kill… [38]

I picked these two, but there are many, many more that tell essentially the same story.

This is the legend.

At first glance it seems clear – at least to me – why Kuhn might have been attracted to it. It fits neatly into his scheme of historical change. It is a story of a revolution. You have normal mathematics – call it, the Pythagorean paradigm. There is an anomaly — an inability to find a rational fraction that measures the diagonal of a unit-square. This is followed by a mathematical proof that shows conclusively, irrefutably that there is, that there can be, no such fraction. The Pythagoreans take an oath to keep this proof a state-secret because it undermines the claim that all is whole number. But Hippasus breaks the oath and reveals this secret to hoi polloi [οἱ πολλοί]. As a punishment (or an act of vengeance), he is drowned. A revolution follows. And there is a paradigm-shift to a new paradigm which allows for irrational numbers.

But there is no indication that this is what Kuhn had in mind. Even though the story is so well known that it is hard to believe he wasn’t aware of it, he doesn’t mention the legend. Just the mathematical proof. But the history of the proof – or rather the meta-history of the proof, the story of how the history of the proof has been repeatedly revised and rewritten – provides a clue, an insight into what kind of metaphor it might be.

Historical Atlas by William R. Shepherd

The investigation of ancient Greek mathematics is daunting. There is a combination of problems – paucity, sometimes absence, of documentation, endless exegetical disagreements, biased and unreliable accounts – the general problem of who did what, when. To make matters worse, crucial documents were most often written on papyrus, which decayed rapidly and had to be copied frequently. [39] What was the actual evidence for the discovery of incommensurability? Did it actually happen? Where did the story come from? Who was this guy, Hippasus of Metapontum?

Euclid”s Elements [papyrus]

An article by Kurt von Fritz, “The Discovery of Incommensurability by Hippasus of Metapontum” (1945), announces that “the discovery of incommensurability is one of the most amazing and far reaching accomplishments of early Greek mathematics… The tradition concerning the first discovery itself has been preserved only in the works of very late authors, and is frequently connected with stories of obviously legendary character. But the tradition is unanimous in attributing the discovery to a Pythagorean philosopher by the name of Hippasus of Metapontum.” [40] Unanimous? In a footnote, von Fritz indicates that it isn’t unanimous. Obviously legendary? Does this mean that it never happened? Very late authors? Von Fritz tells us that almost all of what we know about Hippasus derives from Iamblichus of Chalcis, an Assyrian 4th neo-Platonist, ca. 245-325 C.E., who lived 800 years after Hippasus. A late author, indeed.

I decided to dig deeper.

It involved a trip to the stacks in Harvard’s Widener Library.

The Widener is one good reason to live in Cambridge, Mass. I have a motto: when you get really depressed, go to the stacks. You are surrounded by things that people have produced, not by people themselves. Almost always an improvement. Furthermore, I feel safe there.

I took the elevator to the fifth floor. Looking for the call number –– WID-LC B243.I2613.1986. I stopped. Turned down an aisle, tripped a motion-sensor, and a light clicked on. An old man – possibly in his 70s – was walking towards me from the other end of the aisle. The gap closed between us. I bent down to reach for a book – Iamblichus’s “Life of Pythagoras, or, Pythagoric Life (De vita pythagorica).” [41] As he passed me, he said, “Be careful. Iamblichus is not to be trusted.”

I should have stopped him and gotten his name. I didn’t. (Maybe it’s better for the story that he remains unknown.)

But it turns out he was right. (Perhaps he had been lingering in the stacks hoping to warn some naive filmmaker, such as myself, of the dangers of taking Iamblichus too much to heart.) Although there are several passages in Iamblichus that deal with Hippasus of Metapontum, they provide not one history of the proof of incommensurability but a series of five contradictory and overlapping accounts. A roundelay of confusion. The Rashomon of incommensurability, that is, the Rashomon of the origins of the proof of incommensurability. [42] [43] [44] [45]

I turned to one more account. From Pappus of Alexandria, who had produced a series of commentaries (about 50 years after Iamblichus) on the books of Euclid. In this account, there is no Hippasus. Instead, an unidentified “soul” has spread the proof “among the common herd” and is condemned to a “sea of nonidentity immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement.” The unidentified soul is condemned for carelessness by the Pythagoreans and the Athenian Stranger (perhaps a late name for Socrates).

….the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of non-identity immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account. [46]

*******

Walter Burkert has written a seminal book on Pythagoras and early mathematics, “Lore and Science in Ancient Pythagoreanism.” Perhaps he could set me straight — help me to separate the real from the apocryphal, or at least to find a thread through the labyrinth of Greek mathematics. I called Burkert, now an emeritus professor at the University of Zurich.

Fyodor Bronnikov (1827 – 1902), Pythagoreans Salute the Sun

ERROL MORRIS: The people that you can talk about this with are few and far between.

WALTER BURKERT: [laughs] Maybe, yeah. Yeah. So what is your special idea about Hippasus?

ERROL MORRIS: Well, I don’t know if it’s a special idea, but I was interested in tracking down the source of the legend about the incommensurability of the square root of two, particularly the drowning of Hippasus by the Pythagoreans. WALTER BURKERT: Yeah. This drowning has been taken up by the neo-Platonists, and it fits very well within the neo-Platonist system. But it makes me a little suspicious.

ERROL MORRIS: A little suspicious?

WALTER BURKERT: Yes. It fits a little too well. They have a kind of dualistic system. There is the One, there is God, there is number. And then there is indistinctness. The discovery that you cannot express the square root of two with numbers — you have indistinctness against number. It can be seen as the epitome of this neo-Platonic system.

ERROL MORRIS: The first question is about where the myth originated: whether it emerged much later than Hippasus, and if so, who originated it?

WALTER BURKERT: It’s difficult first of all to make people understand what irrationality in numbers means. Who cares if you have a decimal system? Who cares whether a third is an indefinite number — .3333333333…? Or whether this is a sequence in which the next number can never be uncertain? So this basic difference between 0.333… and the square root of two is a little bit difficult to make understood to a modern public. Usually people do not like mathematics so very much.

ERROL MORRIS: That may well be true.

WALTER BURKERT: I remember when I first realized this problem of a square root versus normal division.

ERROL MORRIS: How old were you?

WALTER BURKERT: Well, I would say about 13 or 14.

ERROL MORRIS: And what did you make of it at the time?

WALTER BURKERT: I simply realized that this was different. It seems to have been truly a discovery of Greek mathematics. There is no evidence of this in Babylonian mathematics – in contrast to the theory of Pythagoras, which was well-known in cuneiform mathematics. But then we have this story, both in the version of Iamblichus, which may go back to Aristotle. And Proclus. But if this really is a historical tradition, then how does Hippasus fit in? And that’s never been clear.

ERROL MORRIS: But if the Pythagoreans killed Hippasus — assuming that they did — why did they kill him? Did they kill him because they didn’t understand the proof, but felt threatened by it? Did they kill him because they understood the proof and felt threatened by it? Did they kill him because Hippasus had divulged a secret? Betrayed an oath? So take those three options.

WALTER BURKERT: Then there is always a fourth — that he was drowned, and that it was an accident rather than an execution.

ERROL MORRIS: An accident? But doesn’t that miss the point. Don’t we need to kill Hippasus? Isn’t that part of the legend. If he dies inadvertently, where’s the story-line?

WALTER BURKERT: But we know so desperately little about the Pythagoreans. And about Hippasus. Even since I wrote that book ["Lore and Science in Ancient Pythagoreanism"], I don’t think any new evidence has come up. No inscription which brings us to safe ground. There is a similar problem with Socrates, but with Socrates we have the texts of his immediate pupils – Plato and Xenophon. But we have no writing of any immediate pupil of Pythagoras. It is a desperate historical situation.

ERROL MORRIS: Desperate?

WALTER BURKERT: Oh, yes. We have so very little historical information.

ERROL MORRIS: And yet this legend of Hippasus has become popular over the years. People tell it, retell it, again and again. Why?

WALTER BURKERT: Because legends are nice. Instead of thinking, what is irrationality, we can think about the legend. But we should remember legends are absolutely independent from fact.

Here are Burkert’s thoughts in a nutshell. There is very little known about either Hippasus or Pythagoras. The historical record is not just incomplete; it is virtually nonexistent. There are no surviving documents. Nothing that Pythagoras or Hippasus wrote is extant. They are known only through the writings of others. The details are sketchy. Hippasus may or may not have been drowned. Pythagoras may or may not have been a mathematician. Perhaps he was a nut-case. An ancient Greek Jim Jones, drinking Kool-Aid with his numerological cohorts. The contrast is nicely captured in two interpretations of Pythagoras from the Renaissance – a fresco by Raphael, “The School of Athens” (ca. 1510-1512), and a painting by Rubens, “Pythagoras Advocating Vegetarianism” (ca. 1618-1630). In the Raphael, Pythagoras is a scholar, a teacher, a sober mathematician; in the Rubens, he is a rather dissolute and louche figure, every inch the raving cult-leader. And two thousand years later, people were still confused about Pythagoras. Who was the real Pythagoras – scholar or crank? [47] [48]

Erich Lessing/Art Resource, NY

Detail of Pythagoras. From Raphael (Raffaello Sanzio) (1483-1520) School of Athens. Ca. 1510-1512.

Biblioteca Ambrosiana, Milan, Italy/The Bridgeman Art LibraryPythagoras, detail from the cartoon for the ‘School of Athens’, 1510-11 (pencil & chalk on paper), Raphael (Raffaello

Sanzio of Urbino) (1483-1520)

Erich Lessing / Art Resource, NY Raphael (Raffaello Sanzio) (1483-1520)

School of Athens. Ca. 1510-1512.

Peter Paul Rubens, c. 1618-1630. The Royal Collection.

Pythagoras Advocating Vegetarianism.

For Burkert, Pythagoras is “not a sharply outlined figure, standing in the bright light of history…. [but] from the very beginning, his influence was mainly felt in an atmosphere of miracle, secrecy, and revelation… Pythagoras represents not the origin of the new, but the survival or revival of ancient, pre-scientific lore, based on superhuman authority and expressed in ritual obligation.” He is the Pythagoras of Rubens, not the Pythagoras of Raphael.

If the historical evidence for Pythagoras is sketchy, what about Hippasus? What really happened to him? Was his drowning at the hands of angry Pythagoreans a Whiggish reading of the past? [49] An exaggerated, heightened melodramatic event that never happened? Did modern historians imagine a crisis, and then invent a figure and a story to embody it? Could the “paradigmatic” example of incommensurability be a Whiggish phantasm, the product of an overactive modern imagination? [50]

Copper etching by Berteaux after Jean Louis Desprez (1743-1804) for “Voyage Pittoresque de

Naples et de Sicilie” by St. Non, 1781-1785

One of the oddities of history is that legends often supersede facts. Historical evidence accumulates, monographs are written; but the number of popular accounts retelling the apocryphal story of that non-crisis proliferate. Why? Because we love to read about crisis and conflict. It’s drama. It makes a better story. [51]

The Man Who Shot Liberty Valance

In John Ford’s movie “The Man Who Shot Liberty Valance” (1962), Ransom Stoddard (James Stewart) becomes an archetypal hero for shooting and killing Liberty Valance (Lee Marvin), the paid stooge of the cattle barons. But Tom Doniphon (John Wayne) – literally hidden in the shadows – is really the man who shoots him. Stoddard gets Doniphon’s girl and goes on to a spectacular political career – governor, senator, etc. Doniphon is the unsung hero. After many years, Stoddard, following Doniphon’s death tells a local newspaper editor what really happened, but the editor refuses to print it, “This is the West, sir. When the legend becomes fact, print the legend.” [52] [53]

A legend that is not true can never become fact, but it can get printed as fact, anyway. With Hippasus, it is pretty easy to imagine why the legend of his drowning got “printed” even before there was printing. Someone believed that there should have been a crisis even if there wasn’t any. They believed that the Pythagoreans should have been upset about the discovery of incommensurable magnitudes. But it was a retrospective belief, that is, a belief formed hundreds, if not thousands of years, after the crisis was supposed to have occurred. I find it mildly amusing – possibly even ironic – that Kuhn’s metaphor for “incommensurability” could have been derived from a Whiggish interpretation of an apocryphal story. The need to find conflict. Call it Hegelian. To me, however, it suggests the possibility that Kuhn’s entire theory of scientific change might be an imaginative fiction. [54]

Let’s take the legend of Hippasus at face value. The Pythagoreans killed him because he couldn’t keep a secret. But taken at face value, the legend is not about the meaning of words or concepts – nor is it about the inability of one group to understand another. It has nothing whatsoever to do with Kuhn’s notion. There’s nothing incommensurable about incommensurability. At least in the Kuhnian sense. [55] According to the legend Hippasus (or whoever discovered the proof) was not killed (if he was killed) because the Pythagoreans couldn’t understand his proof. It was because they could understand it. And his murder was an act of intolerance. [56] (Like the throwing of an ashtray.) The Pythagoreans killed Hippasus not because they couldn’t understand him, but because he revealed a truth that they wished to keep secret. No one was ever boiled in oil, stretched on a rack, burned at the stake because of incommensurability. There is nothing incommensurable about being tossed into the sea by angry Pythagoreans. I don’t believe there is such a thing as Kuhnian incommensurability, but I do believe there is such a thing as Kuhnian intolerance.

Rare Books Collection PA3965.D6 M4

Diogenes Laertius. Diogenis Laertii De vitis, dogmatibus et apophthegmatibus clarorum philosophorum libri X … Amstelædami: Apud Henricum Wetstenium, 1692.

O.K. The story of Hippasus is most likely apocryphal. And it is a story about intolerance, not about our inability to understand new ideas. Or to translate new ideas into old ones. But where did it come from? Yes, it is mentioned in Iamblichus, but what happened after that? I found several major works that address this question – by Wilbur Knorr (1945-1997), a book, “The Evolution of the Euclidean Elements,” and an essay published after his death, “The Impact of Modern Mathematics on Ancient Mathematics.” And by David H. Fowler (1937-2004), “The Mathematics of Plato’s Academy, 2nd edition” (1999). [57]

Knorr traces it back to two modern sources: a 1928 essay by Helmut Hasse and Heinrich Scholz, “Die Grundlagenkrisis Der Griechischen Mathematik (The Foundational Crisis in Greek Mathematics),” who make the case that ‘the discovery of [incommensurability] which cannot be comprehended in numbers must naturally have shaken the idea of the ‘arithmetica universalis’ of the Pythagoreans.’ [58] And to an 1887 study by Paul Tannery, who concluded that “the discovery of incommensurability by Pythagoras…must have caused a véritable scandale logique…” [59]

And here is where Knorr channels Butterfield. His suggestion that the idea of a crisis, a grundlagenkrisis, came from 19th century mathematics, not ancient Greek mathematics. “The Greeks were not blind to an extension of the number concept through some accidental failure of spirit. They rejected any such extension on scientific and philosophical grounds: the arithmos must be whole-number; even the rational numbers, a necessary preliminary to irrational numbers, were excluded from the classical number theory; the problem of irrationals was thus resolved in a geometric manner instead… But what we should at once notice is that such a debate could not have arisen before the successful resolution of the problem of irrational numbers by Weierstrass and Dedekind in the 19th century.” [60]

Knorr reminds us that there was a crisis in 19th century mathematics concerning the meaning of the irrational numbers, and that that crisis was projected back into antiquity. Hence, Knorr’s

claim that the “idea” of a crisis in Greek mathematics was a very late invention. Very late. Over 1500 years after Iamblichus. In Knorr’s phrase, it was “a modern fiction.”

The entire substance of the legend is crumbling before us. Knorr is telling us three things. (1) There is no evidence for a crisis in 500 B.C.E.; (2) there was no reason for a crisis in 500 B.C.E., and (3) there is ample evidence of a crisis in 19th century mathematics. I had imagined that Hippasus – if he was punished – was punished because he betrayed a trust, but Knorr says – No. There was no oath, no betrayal, no threat to the foundations of Pythagorean mathematics. There was nothing in the discovery of incommensurability that challenged the “assumptions within the Pythagorean geometry.”

…on what grounds are we to believe that the discovery of incommensurability was a challenge or counter-example to naïve assumptions within the Pythagorean geometry? To be sure, the discovery was held to be significant… late writers [such as Iamblichus] suggest it was maintained as a secret of the school — but was it a challenge? Consider that the Pythagoreans based their natural philosophy on the conception of the world in terms of number and other mathematical categories, that is, in terms of certain abstract, rather than material, principles. The discovery of incommensurability might well support this view…

And so, what does this tell us about paradigms, paradigm shifts, and revolutions? Is there a lesson to be learned here? Hasse and Scholz imagined a crisis in Greek mathematics and criticized Oswald Spengler who believed that irrational numbers were “fundamentally alien to the classical soul.” In turn, modern historians have criticized Scholz and Hasse for imagining a crisis that may have never happened. Shifting historical paradigms.

But do these questions about Greek mathematics mean that there is no way to understand the past? Are we back in a Kuhnian nightmare, where our paradigms force us to see history through one subjective prism or another? No. Not really. These accounts of incommensurability highlight the difficulties – not the impossibility – of understanding the past. They provide a reminder that history in its particulars, like the weather, defeats grand schemes. Knorr brings it back to the practice of mathematics – to the issue of mathematicians, to what they would or would not do – and asks the question: “The logician and the philosopher, and following them, the historian might recognize that a certain result is paradoxical, and that it ought to provoke a crisis in the foundations of a given field of mathematics. But does the practicing mathematician ever curtail his researches in accordance with such a challenge?”

David Fowler also surveys the evidence for a crisis. [61] He was clearly so obsessed with the history of incommensurability that he wrote the chapter twice in one book – that is, he wrote it once and then felt compelled to write it all over again. Chapter 8.3 “The Discovery and the Role of the Phenomenon of Incommensurability, and then, Chapter 10.1 “A New Introduction: The Story of the Discovery of Incommensurability.” Plato is there, and so is Iamblichus, Pappus and Proclus (the trio of “late writers”). And his answers are similar to Knorr’s. He takes Iamblichus to task, referring to “a farrago of mutually inconsistent stories, which appear for the first time in a source of doubtful reliability and relevance dating from nine centuries after the time of Pythagoras.” [62] And then, “…no Greek text, early or late, tells us clearly of the mathematical difficulty raised by incommensurability.”

But Fowler, like Burkert, wondered: who was Pythagoras? Could the fascination with Pythagoras, as well as the fascination with Hippasus, really be a fascination with the nature of historical evidence? And with names and descriptions? He imagined a Jeopardy question with a blank to be filled in. Who is Pythagoras? And Fowler offered several possibilities, which he arranged alphabetically, “leader, mathematician, music theorist, mystic, philosopher, shaman, scientist…”

Pythagoras the _________ was born in Samos and later went to Croton. [63]

But that’s not all. He even conducted a survey on Pythagoras. What do contemporary academics believe about Pythagoras? And how compatible are their beliefs with scholarly research?

Thanks to several helpers, I was able to organise a simple and non-scientific survey over the Internet and, of around 190 replies, 40% said mathematician or some variant of that (geometer, mystic geometer, triangle theorist,…), 28% said philosopher or some similar variant, 12% philosopher-mathematician, and the rest a very mixed mag of activities- number freak, bean-hater, vegetarian, polymath, new ager… One person said music-theorist, another father of acoustics, and these two were the only references to what may be the early Pythagoreans’ most significant contribution to our scientific heritage.

Here is a pie-graph based on his results. (I wish there were some figures on what percentage of the 48 percent believed that Pythagoras was primarily a bean-hater.) Fowler’s survey may seem ridiculous at first, but it emphasizes a point – the endless disparity between evidence and belief.

4.

THE AUTHOR OF THE QUIXOTE

I have suggested that Kuhn had created his own reductio ad absurdum – not unlike the proof of the incommensurability of √2. If everything is incommensurable, then everything is seen through the lens of the present, the lens of now. All history is Whiggish history. There is no history. There is no truth, just truth for the moment, contingent truth, relative truth. And who is to say which version of the truth is better than any other, if we can’t look beyond the paradigm in which we find ourselves.

But there is a messier problem. Why stop at historical relativism? Why not imagine each and every person in a different island universe? And indeed, Kuhn at least in one instance seems to embrace that possibility. In one particularly bizarre passage in “The Road Since Structure,” he suggests that his critics are writing about two different Thomas Kuhns – Kuhn No. 1 and Kuhn No. 2.

I am tempted to posit the existence of two Thomas Kuhns. Kuhn No. 1 is the author of this essay and of an earlier piece in this volume. He also in 1962 published a book [The Structure of Scientific Revolutions]. Kuhn No. 2 is the author of another book by the same title…

That both books bear the same title cannot be altogether accidental, for the views they represent often overlap and are, in any case, expressed in the same words. But their central concerns are, I conclude, usually very different. As reported by his critics (the original is unfortunately unavailable to me), Kuhn No. 2 seems to make points that subvert essential aspects of the position outlined by his namesake. [64]

To me Kuhn’s claim – that there are two Thomas Kuhns plus two books by the same name and author – suggests that there may be no coherent reading of Kuhn’s philosophy. Kuhn, of course, sees it differently. For Kuhn, the multiplicity of Kuhns and Kuhn-authored-books-with-the-same-title provides further proof of his belief that people with “incommensurable”

viewpoints can’t talk to each other. That they live in different worlds. He writes, “This collection of essays…provides an extended example of what I have elsewhere called partial or incomplete communication – the talking-through-each-other that regularly characterizes discourse between participants in incommensurable points of view.” [65]

I suppose Kuhn No. 1 is the Kuhn you can criticize, and Kuhn No. 2 is the Kuhn you can’t. As the philosopher Donald Davidson, an early critic of Kuhn, has written, “Conceptual relativism is a heady and exotic doctrine, or would be if we could make good sense of it. …what sounded at first like a thrilling discovery — that truth is relative to a conceptual scheme — has not so far been shown to be anything more than the pedestrian and familiar fact that the truth of a sentence is relative to (among other things) the language to which it belongs. Instead of living in different worlds, Kuhn’s scientists may, like those who need Webster’s dictionary, be only words apart.” [66]

Kuhn’s remarks remind me of a story by Jorge Luis Borges, “Pierre Menard: Author of the Quixote.” Instead of two Cervantes, Borges imagines a second author, Pierre Menard, a fictional character, who manages to recreate “Don Quixote,” word for word with the original. And yet, creates an entirely different work of art. Borges writes, “[Menard] did not want to compose another Quixote — which is easy — but the Quixote itself. Needless to say, he never contemplated a mechanical transcription of the original; he did not propose to copy it. His admirable intention was to produce a few pages which would coincide — word for word and line for line — with those of Miguel de Cervantes.” [67] [68] Borges’s story – quite likely an elaborate literary joke – has spawned a world of commentary. What is history – a collection of real events or of subjective memories? Borges compares two identical passages from the “Quixote,” part one, chapter 9 — one written by Cervantes, the other by Menard:

Gustavo Doré

First, Cervantes.

…truth, whose mother is history, rival of time, depository of deeds, witness of the past, exemplar and adviser to the present, and the future’s counselor.

Written in the 17th century, written by the “lay genius” Cervantes, this enumeration is a mere rhetorical praise of history.

Menard, on the other hand, writes:

…truth, whose mother is history, rival of time, depository of deeds, witness of the past, exemplar and adviser to the present, and the future’s counselor. [69]

Borges elaborates on how this same passage means something completely different to Menard. According to Menard.

History, the mother of truth: the idea is astounding.

Menard, a contemporary of William James, does not define history as an inquiry into reality but as its origin. Historical truth, for him, is not what has happened; it is what we judge to have happened. The final phrases — exemplar and adviser to the present, and the future’s counselor —are brazenly pragmatic. [70]

What is being argued here? That every text is endlessly reinterpreted? That every reader re-writes the book they are reading? That our beliefs change over time? That history changes the meaning of history? And how does Borges fit into this? Does Borges see himself as Menard or Cervantes? Neither? Both? [71]

Gustavo Doré

The story appeared in May, 1939 — shortly after Madrid fell to Franco’s forces. Borges had written movingly against the fascist powers overtaking Europe. And against anti-Semitism. “…in vain I quoted the wise declaration of Mark Twain’s that a man’s race was unimportant,

for, after all he was a human being, and no one could be anything worse.” But moreover, against the denial of reality.

In Borges’s review of “Citizen Kane,” “An Overwhelming Film,” published in 1941, he called the movie “a kind of metaphysical detective story… Forms of multiplicity and incongruity abound in the film: the first scenes record the treasures amassed by Kane; in one of the last, a poor woman, luxuriant and suffering, plays with an enormous jigsaw puzzle on the floor of a palace that is also a museum. At the end we realize that the fragments are not governed by any secret unity: the detested Charles Foster Kane is a simulacrum, a chaos of appearances.” Borges concluded by quoting Chesterton, “there is nothing more frightening than a labyrinth that has no center.” [72]

Citizen Kane

I wondered about the labyrinth with no center. For me, it is a mystery without a solution. A murder without a murderer. A world without answers, without truth or falsity. It is the nightmare offered by postmodernism. It’s chaos. What is the answer to the question – What really happened? When did it happen? Who really did it? There is no answer.

Years ago, Bertrand Russell wrote “Nightmares of Eminent Persons” (1954). (Supposedly, he was trying to meet alimony payments.) Among the various nightmares – the Mathematicians’s Nightmare, Stalin’s Nightmare, the Psychoanalyst’s Nightmare, Dr. Bowdler’s Nightmare – is the Existentialist’s Nightmare. At the conclusion of the nightmare, the existentialist is screaming, “I don’t exist. I don’t exist.” Poe’s raven appears, speaking in the voice of the French poet Mallarmé: “You do exist. You do exist. It’s your philosophy that doesn’t exist.” [73]

Russell did not write a postmodernist’s nightmare – in 1954, postmodernism did not exist – but it is not difficult to imagine a possible scenario of which he might approve. The postmodernist is seated in a chair. An off-screen voice is screaming, “Truth exists. Reality exists.” The postmodernist screams, “No, they don’t.” The off-screen voice returns, “Oh, yes, they do. Reality and truth both exist. And if you don’t believe it, jump out that window. We’re on the 39th floor.”

Which brings me to Alice’s question to Humpty Dumpty in “Through the Looking-Glass.” It addresses the issue: can words mean anything we want them to? Humpty-Dumpty is suggesting an “authoritarian” theory of meaning. Words mean whatever I want them to mean. It is easy to see it as an earlier version of “The Ashtray Argument.” [74] [75]

“I don’t know what you mean by ‘glory,’ ” Alice said.

Humpty Dumpty smiled contemptuously. “Of course you don’t—till I tell you. I meant ‘there’s a nice knock-down argument for you!’ ”

“But ‘glory’ doesn’t mean ‘a nice knock-down argument’,” Alice objected.

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

“The question is,” said Alice, “whether you can make words mean different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

Alice was too much puzzled to say anything, so after a minute Humpty Dumpty began again. “They’ve a temper, some of them—particularly verbs, they’re the proudest—adjectives you

can do anything with, but not verbs—however, I can manage the whole lot! Impenetrability! That’s what I say!” [76]

John Tenniel’s illustration “Alice Meets Humpty,” from Lewis Carrol’s “Through the Looking

Glass” (1865)

The question that lingers: was Kuhn’s philosophy an assault on truth? Had he turned history into a vast prison – something like Jeremy Bentham’s panopticon – a circular building with cells along the circumferance. And then thrown away the key? “These cells are divided from one another, and the prisoners by that means secluded from all communication with each other, by partitions in the form of radii issuing from the circumference towards the centre… The apartment of the inspector occupies the centre; you may call it if you please the inspector’s lodge.” [77] No prisoner can communicate with any other prisoner, only with the inspector. Unlike Bentham’s panopticon, however, Kuhn’s prison is a prison in time, not space.

The Works of Jeremy Bentham, vol. IV, 172-3

It is a prison of historical paradigms where only the inspector at the center of the edifice can compare one paradigm with another. But who gave Kuhn this supreme job of watching prisoners in different cells (read paradigms)? The job of ultimate observer? How did he become the inspector?

I believe that the ashtray incident was motivated by Kuhn’s own doubts about his work, as well as his annoyance with me. My belief was reinforced by a recent memoir, “Little Did I Know” (2010) by Stanley Cavell, who had been at Harvard with Kuhn in the 1950s and subsequently a junior faculty member with him at the University of California-Berkeley.

Kuhn was in the process of writing “Structure.” And as it turns out, Kuhn had Wittgenstein and Hitler on his mind. Was Kuhn, like any good Jewish boy of the period (myself included), struggling with the meaning of the Third Reich? If there are no absolute value judgments to be made about one historical period (read: paradigm) or another, what about the Nazis? Following a department meeting, Kuhn had accompanied Cavell home for a drink.

…talking past midnight Tom was becoming agitated in a way I had not seen. He suddenly lurched forward in his chair with a somewhat tortured look that I had begun to be familiar with. “I know Wittgenstein uses the idea of ‘paradigm.’ But I do not see its implications in his work. How do I answer the objection that this destroys the truth of science? I deplore the idea. Yet if

instruction and agreement are the essence of the matter, then Hitler could instruct me that a theory is true and get me to agree.” My reply I cast as follows, using the words I remember using then. “No he could not; he could not educate you in, convince you of, show you, its truth. Hitler could declare a theory to be true, as an edict. He could effectively threaten to kill you if you refuse to, or fail to, believe it. But all that means is that he is going to kill you; or perhaps kill you if you do not convince him, show him, that you accept and will follow the edict. I don’t say this is clear. But it is something I cannot doubt is worth doing whatever work it will take to make clear.” Tom’s response was startling. He arose almost violently from his chair, began pacing in front of the fireplace, saying something like, “Yah. Yah.” What causes conviction? What, perhaps rather, may undo an unnoticed conviction? [78]

I asked Cavell about this passage. About Kuhn and Wittgenstein and about how Wittgenstein had opened the door to relativism.

STANLEY CAVELL: Kuhn really was terribly alarmed that Wittgenstein was denying the rationality of truth. That somehow, everything, was going to come down to agreement. It would be circling around that… Which I don’t think is a non-issue. I think it’s quite real. ERROL MORRIS: And what were your feelings about it?

STANLEY CAVELL: …that it was a genuine issue, that Wittgenstein was opening that up. Part of it was a matter of getting down, in the mud, and figuring out what “agreement” meant.

ERROL MORRIS: This would be in “Philosophical Investigations”?

STANLEY CAVELL: Yes. “Philosophical Investigations.” That’s what we talked about. The early Wittgenstein, as far as we were concerned, was frozen history. Nobody was really interested in trying to make that work, it was “Philosophical Investigations” that was really hot. The issue about what human agreement could establish, and how deep that agreement was. Wittgenstein’s quote, “We don’t agree in judgement, we agree in form of life.” Whether that meant that knowledge of the universe was relative to human forms of life. We went around the track with that a lot, and, why not?

The actual quotation is from paragraph 241 of “Philosophical Investigations”:

“So you are saying that human agreement decides what is true and what is false?” –– It is what human beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in form of life.

This passage, as well as many others in “Philosophical Investigations,” has produced extended commentary. But Wittgenstein, notoriously difficult to pin-down, at least in this one instance, seems to be saying what he’s saying. And he opens the door (or the lid of Pandora’s Box) to a relativistic notion of truth. [79] [80] In paragraph 241, it’s agreement between human beings that decides what is true or false. It suggests that we could agree that the earth is flat and that would make it so. So much for the relationship between science and the world. And yet, Kuhn made peace with this idea. He even made it the cornerstone of his philosophy of science. A

couple of years later in “Structure,” Kuhn would write, “We may, to be more precise, have to relinquish the notion, implicit or explicit, that changes of paradigm bring scientists and those that learn from them, closer and closer to the truth.” [81]

5.

THIS CONTEST OF INTERPRETATION

Thomas Kuhn

Steven Weinberg has written eloquently in The New York Review of Books about Kuhn and paradigm shifts. A Nobel Prize-winning physicist was outlining the difference between “Kuhnian science” (that is, science as Kuhn imagined it) and actual science. And argued that Kuhn’s theories did not characterize science as Weinberg knew it. Weinberg wrote, “What does bother me on rereading Structure and some of Kuhn’s later writings is his radically skeptical conclusions about what is accomplished in the work of science…conclusions that have made Kuhn a hero to the philosophers, historians, sociologists, and cultural critics who question the objective character of scientific knowledge, and who prefer to describe scientific theories as social constructions, not so different from democracy or baseball.”

Or not so different from parapsychology, astrology or witchcraft.

He was also bothered by Kuhn’s arguments against progress. “[Kuhn] went on to reason that… there can be no sense in which theories developed after a scientific revolution can be said to add cumulatively to what was known before the revolution… More recently, in his Rothschild Lecture at Harvard in 1992, Kuhn remarked that it is hard to imagine what can be meant by the phrase that a scientific theory takes us ‘close to the truth.’”

And yet, Weinberg fails to drive the stake through the heart of the vampire. If paradigms are incommensurable how can we talk about their incommensurability? If the Pythagoreans did not

understand (or could not have understood) Hippasus’s proof because it was “incommensurable” with the Pythagorean paradigm, then why bother killing him?

Weinberg was taken (as was I) with an example that appears in Kuhn’s “What Are Scientific Revolutions?” It involves Aristotle and a Gestalt-flip. It also has a breathless, “gee whiz” character. Not unlike accounts of alien abductions and the experiences that follow. You won’t be able to understand it. Just take my word for it.

I first read some of Aristotle’s physical writings in the summer of 1947, at which time I was a graduate student of physics trying to prepare a case study on the development of mechanics for a course in science for nonscientists… I was sitting at my desk with the text of Aristotle’s Physics open in front of me and with a four-colored pencil in my hand. Looking up, I gazed abstractedly out the window of my room—the visual image is one I still retain. Suddenly the fragments in my head sorted themselves out in a new way, and fell into place together. My jaw dropped, for all at once Aristotle seemed a very good physicist indeed, but of a sort I’d never dreamed possible. Now I could understand why he had said what he’d said, and what his authority had been. Statements that had previously seemed egregious mistakes now seemed at worst near misses within a powerful and generally successful tradition. That sort of experience—the pieces suddenly sorting themselves out and coming together in a new way—is the first general characteristic of revolutionary change… [82]

Alas, there is nothing here to tell us about the nature of the epiphany. Could it be like a religious conversion? Weinberg wrote to Kuhn for an explanation – excuse me, sir, just what is going on here – but got none. I guess a true epiphany cannot be explained. It is ineffable. It just happens. And yet, I had (and still have) this nagging feeling that it is possible to compare Aristotle’s science with current modes of explanation. Even to make the claim that there has been progress during the last 2,000 or so years – say in the field of dentistry. To cite Bertrand Russell, “Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives’ mouths.” Or the variant from Russell I prefer, “Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted.” [83]

*******

Norton Wise was a post-doctoral student at Princeton and eventually the director of graduate studies of the Program in the History and Philosophy of Science (from 1991 to 2000), the same program that Kuhn had directed in the ‘70s. I had spent many, many evenings talking with him and, yes, complaining about Kuhn. I hadn’t talked to him in 40 years, but at least for me, it was as if no time had passed. Except that he had young, twin daughters at the time. They are now in their forties.

Norton is much nicer than me. In our conversation, I was reminded of his openness and sincerity.

NORTON WISE: Do you remember the mode of smoking a cigarette, where Kuhn would start drawing on a newly lit cigarette? He would start a sentence, break off in the middle of the sentence, start drawing on the cigarette, and take a drag that would exhaust half the cigarette. Just an amazing drag. And then start again in the sentence. That’s such a clear image to me.

ERROL MORRIS: I remember the mountain of butts and ash.

NORTON WISE: Within one seminar.

ERROL MORRIS: Oh yes.

NORTON WISE: That’s what I remember too. I never saw anyone smoke like that before or since. Of course the emphysema nearly killed him.

ERROL MORRIS: I often think of the attraction of smoking, that it simplifies the world into three parts. There’s you, there’s the cigarette, and everything else is the ashtray.

NORTON WISE: Well, the smoking is to me expressive of the intensity of the seminars. Everyone was on edge. And that was from early on, and then continuing through successive seminars. The most apparent thing is that intensity that Kuhn somehow brought, along with the thorough preparation that he brought to it, and then expected of everyone else. And of course very few people — no one in the seminar would have read and analyzed the material in the very focused, intense way that he had done. A kind of adversarial relationship was explicit. You were challenged to be saying anything, or offering an opinion. And an opinion immediately evoked maybe agreement, but usually not. Usually some kind of challenge. And for me that meant that the relationship, while close and intense over the years, was always adversarial. I found it extremely stimulating but sometimes disturbing. Sometimes very disturbing. Right up until the end, shortly before he died, he and I were carrying on this kind of contest of interpretation.

ERROL MORRIS: What an odd expression.

NORTON WISE: I wonder if it was in that seminar that I first became aware of it. I was giving an interpretation of William Thomson’s [Lord Kelvin’s] early electromagnetic mathematical work. He sharply disagreed with my interpretation and said that it couldn’t possibly be right because it didn’t follow the steps through which Thomson had developed his argument. It was, instead, a physicist’s way of going at things as an overview, or as a re-interpretation based on later knowledge. It was the kind of thing he was constantly hammering against. Then the next week I came back with a new interpretation. And again he attempted to shoot that down. But this time I had done a great deal more homework and really had it down in a pretty focused, pretty detailed way. And that maybe was the basis of the respect we gained for each other. All of his students needed to show him that they could stand up to that adversarial mode of interaction, and challenge his own understanding — which he greatly enjoyed. For some people that worked out pretty well; for others it was a complete disaster.

ERROL MORRIS: Your use of the word “adversarial?” I experienced it differently. I would say that it went beyond adversarial. There was a threatening element.

NORTON WISE: Yes.

ERROL MORRIS: When I first came to Princeton, he had given me an ultimatum that I was not to attend Kripke’s lectures.

NORTON WISE: I’d never heard that. Very strange.

ERROL MORRIS: Yes. “You may not go to those lectures.”

NORTON WISE: So, of course, you immediately did. ERROL MORRIS: Yes.

Our discussion turned to Butterfield and Whiggishness.

ERROL MORRIS: Although you didn’t use the word, I know it was on the tip of your tongue because you studied with him. It was the supreme accusation — Whiggishness. “This is Whiggish history.” I see the argument — and I would love to know the details of the argument about Thomson. That you were Norton Wise, a recent Ph.D. in physics, thinking like a physicist and not like a historian. And that you were providing some kind of anachronistic account.

NORTON WISE: That’s precisely correct. ERROL MORRIS: There seemed to be a kind of weirdness at the heart of all of it. I don’t know how else to express it. The accusation of Whiggishness was a cudgel.

NORTON WISE: Yes, it certainly was used in that way — to change one’s rhetorical sensibilities. ERROL MORRS: As a cudgel? As a club?

NORTON WISE: Yes. I saw that I had to learn to work in a different mode. I had to learn to write sentences differently. I had to learn to interpret the way in which a text flowed from one line to the next in the course of a page.

ERROL MORRIS: But aren’t you just talking about close reading?

NORTON WISE: Yes. [84]

Close reading? Who would argue with that? Close reading is entirely commendable. Three cheers for close reading. But close reading is the opposite of incommensurability. It tells us

that by carefully studying a passage we can understand it. Incommensurability tells us that unless we inhabit the paradigm in which it was written, we can never understand it.

*******

Which brings me back to Kripke. And to “Naming and Necessity.” Why is Kripke relevant to any of this? I believe it is because his work is an attempt to create new links — between words and things and between the present and the past. Although he is rarely described in this way, he is among other things, a philosopher of history. We can believe strange things about the past, but we can still refer to things in the past. [85] There is an objective reality. There is objective truth. And there is objective history. [86] The beliefs of 6th century B.C.E. mathematicians might be inaccessible to us (or at least, difficult for us to understand), but when Hippasus or one of his contemporaries refer to √2, they are referring to the same thing we are. [87] Indeed, it may come back to close reading.

…truth, whose mother is history, rival of time, depository of deeds, witness of the past, exemplar and adviser to the present, and the future’s counselor.

Go back to Goldie. We can believe that Goldie is golden or green or some other color, but we still refer to Goldie. As time goes by, our ideas about Goldie (and history) may change, but that doesn’t mean that we no longer refer to Goldie or that Goldie refers to one fish in one paradigm and a different fish in another. Nor does it mean that reference is contingent on a paradigm – or on a conceptual scheme or disciplinary matrix. I imagine one of those very bad elementary school arguments. Some kid says that the earth is flat. I say that the earth is an oblate spheroid. We argue. It’s flat. No, it isn’t. Yes, it is. No, it isn’t. Yes, it is. No, it isn’t. Stalemate. We have reached an impasse. Kuhn steps in to adjudicate. He tells us that there is no fact of the matter. You come from different paradigms: the flat earth paradigm and the oblate spheroid paradigm. The name “earth” means something different to each of you, and you can’t compare the meanings. There is no common reference. [As Kuhn wrote in "Structure," “...proponents of competing paradigms practice their trades in different worlds."] But wait. What about the earth? There is the earth. There is that physical thing floating out in space. Is it flat, or is it an oblate spheroid? Or if it’s neither, tell me what it is. A tetrahedron? Isn’t it one thing or another? It must be something. [88]

I suppose the kid in elementary school, if he gets angry enough, could strangle me. When all else fails, there’s Humpty Dumpty’s reply to Alice. And there’s the Ashtray Argument. With Kuhn’s blinkered cosmologies there really is no way to resolve these questions. Fisticuffs may be required.

Please remember: This is not an empty intellectual exercise. It is not a matter of indifference whether it was God or natural selection that produced the complexity of life on earth. Nor whether there is such a thing as global warming. The devaluation of scientific truth cannot be laid on Kuhn’s doorstep, but he shares some responsibility for it.

One more parable. For those who truly believe that truth is subjective or relative (along with everything else), ask yourself the question – is ultimate guilt or innocence of a crime a matter of opinion? Is it relative? Is it subjective? A jury might decide you’re guilty of a crime that you haven’t committed. You’re innocent. (It’s possible. The legal system is rife with miscarriages of justice.) Nevertheless, we believe there is a fact of the matter. You either did it or you didn’t. Period.

If you were strapped into an electric chair, there would be nothing relative about it. Suppose you are innocent. Would you be satisfied with the claim there is no definitive answer to the question of whether you’re guilty or innocent? That there is no such thing as absolute truth or falsity? Or would you be screaming, “I didn’t do it. Look at the evidence. I didn’t do it.” Nor would you take much comfort in the claim, “It all depends on your point of view, doesn’t it?” Or “what paradigm are you in?” When I was investigating the murder of Robert Wood, a Dallas police officer, and the capital murder conviction of Randall Dale Adams for that murder, would it make sense to describe my viewpoint as one paradigm, and the viewpoint of the Dallas police as another? Surely, we had different ways of looking at the evidence, different interpretations of the evidence, different ways of looking at the crime. Suppose someone said, there’s no way of comparing these two paradigms. They’re incommensurable. You can’t say one is true and the other false. There is no absolute truth. Perhaps they could gussy up the claim by citing police procedures and practices. Different traditions of looking at crime scene evidence. The social construction of the ashtray. [89]

The difficulty of ascertaining the truth in history is often confused with the relativity of truth. Two very different concepts. (We may have difficulty fixing the exact date and location of the Battle of Hastings, but that doesn’t mean it didn’t happen at a specific time and place.) “The past,” as L.P. Hartley has written, “is a foreign country. They do things differently there.” [90] But when Homer speaks of the “sun,” is he speaking about a different object than T.S. Eliot? If Newton were to give Einstein a copy of the “Principia” and Einstein were to give Newton a copy of “On the Electrodynamics of Moving Bodies,” would they be unable to understand each other or their respective theories? There would be a discussion, perhaps even disagreements about ideas and principles. Clarifications would be needed. But would they look past each other in numb stupefaction? The past may be a foreign country, but I do not believe that people there speak a language that we can not understand.

*******

Acknowledgments: Charles Silver provided many of the ideas in this essay. I have benefited enormously over the years from our conversations about Kripke, Kuhn, meaning and reference. Ron Rosenbaum, who despite a crazy schedule, read a number of drafts and made many helpful editorial suggestions. I would like to thank my researchers and editors Ann Petrone, James Larkin and Julie Fischer. James Larkin also provided a translation of Hans Freudenthal’s essay on the foundational crisis in Greek mathematics. And my interview subjects, Walter Burkert, M. Norton Wise, John Burgess and Stanley Cavell. I have also benefited from a conversation with Steven Weinberg and from his essay “The Revolution that didn’t happen.” It is essential reading. And conversations with Warren Goldfarb, Louis Menand and Brian Skyrms. And I would also like to thank Tom Levin and the other Princeton faculty members who invited me to give the Spencer Trask lecture at Princeton in November, 2010. I have included an emended excerpt from the lecture as an epilogue.

*******

SPENCER TRASK LECTURE (excerpt)

ERROL MORRIS: Thank you so much. This is indeed very odd, being here. I want to thank everybody connected with this lecture series for inviting me tonight. I haven’t been back to Princeton for close to 40 years. I tried to write some kind of lecture. I don’t think I really have it in me. I can write essays and articles. I don’t think I can write lectures. Not this one. But I started to write a lecture entitled “The Ashtray,” because the ashtray is very much on my mind. It concerned an ashtray that was thrown at my head at the Institute for Advanced Study. It may not have been the only reason why I left Princeton, but it was the beginning of the end. I started digging through stuff related to my experiences here. I was very much devoted to rock

climbing. You may not be aware of this fact, but this is one of the best places to rock climb in the world. The buildings are perfect for climbing. And I was probably in the best shape that I had ever been.

I was arrested at a demonstration at the Institute for Defense Analysis [IDA]. This is in 1972. And I was taken in a paddy wagon to Trenton. [We were protesting the resumption of bombing in North Vietnam. This time, very close to the Chinese border.]

I was waiting to be booked, walked over to a window, and realized it was just a very easy matter to climb out the window, climb down the side of the building, and leave, which I did. And I always felt a little bit ashamed of myself, because I thought, “I should’ve gotten arrested and booked and fingerprinted, et cetera, et cetera, et cetera. There’s probably no record of the fact that I was arrested in this demonstration.” [It’s always been unclear to me, in social protest, is the important thing to be there, to be arrested, to be beaten, to be in the newspaper, to be booked, or to be incarcerated? Maybe all of the above.]

This last week I found an old issue of The Princetonian online, and my name was there, in the paper, and it felt kind of good. I was kind of pleased with myself, reading about the demonstration, reading about how a former president of Princeton had said about the demonstration, “This isn’t Princeton.” Later, he claimed that he wasn’t talking about the demonstrations, merely the fact that the IDA was not formally located on the Princeton campus. I guess meanings are always in flux.

A lot of these experiences informed what I became as a filmmaker. The things that I was concerned with in those days still are very much with me, still are very much part of what I do as a filmmaker.

The issue of murder, mass murder, has stayed with me over the years. It’s certainly part of the film that I made with Robert S. McNamara, “The Fog of War.” I remember sitting in the Firestone Library and reading volumes upon volumes of the transcripts of the Nuremberg War Crimes Tribunal. Ultimately I had the opportunity to go with Robert McNamara to the International Criminal Court [ICC] in the Hague, to show “The Fog of War” to the court, and to answer questions with McNamara.

And my two favorite moments from that experience – going with McNamara to visit the archivist for the ICC. McNamara told him, “I wish that they had these statutes governing war crimes back when I was secretary of defense,” and the archivist replied, “But, sir, they did.” Another completely bizarre experience, beyond Kafkaesque, seeing Milosevic on the stand.

None of the proceedings had anything whatsoever to do with the content of the charges against him. It was all procedural — procedures about procedures about procedures, epicycle upon epicycle upon epicycle. And yet, the knowledge that Milosevic’s crimes were being addressed, even if only in a vague and uncertain way, was gratifying. At least someone was doing something. [Milosevic was claiming that since he was representing himself, and since he was untrained as a lawyer, he should be given 9/5 more time than the prosecutors. The 9/5th figure was arrived at through a complicated algorithm.]

The other experiences I should mention really briefly: I was taking classes from Thomas Kuhn, who ran the program in the history and philosophy of science where I was a graduate student, and Saul Kripke, who was giving a series of lectures “Naming and Necessity.” It wasn’t the first time he gave them. This is 1972. It might have been the second, maybe even the third time they had been delivered. It was risky for me to attend the Kripke lectures, because I had been told by Kuhn, “Under no circumstances are you allowed to go to these lectures.” It was an odd kind of thing.

Years later, I’ve come to realize that there was a debate embodied here about the nature of language – of whether truth is socially constructed or whether ultimately concerns the relationship between language and reality. I feel very strongly, even though the world is unutterably insane, there is this idea that we can reach outside of that insanity and find truth, some kind of certainty –

“The Thin Blue Line” was one the most important experiences of my life, something that I remain really, really proud of – overturning the conviction of a man who had been sentenced to death for a crime he did not commit. There are endless obstacles and impediments to finding the truth – You might never find it; it’s an elusive goal. But there’s something to remember, there’s a world out there that we can apprehend, and it’s our job to go out there and apprehend it. It’s one of the deepest lessons that I’ve taken away from my experiences here.

[1] I was looking online for some information about Saum’s whereabouts and learned that he had died in 2000 of brain cancer. A terrible loss. I had spent a good part of my year at Princeton climbing — on the buildings of the university and at the Shawangunks.

[2] One way to think about possible worlds is in the context of love. For example, people often ask: would you still love me if I were poor? Would you still love me if I had no arms? And so on and so forth. They are asking a question about possible worlds. That possible world in which you have no arms. Or that possible world in which you are poor. Or perhaps even that possible world in which you are poor and have no arms.

[3] Herbert Butterfield, “The Whig Interpretation of History,” New York: W.W. Norton & Company, 1931, p.11-12. The Whigs, advocates of the power of Parliament, are distinguished from the Tories, advocates of the power of the King.

[4] Here’s another Butterfield quote, the “tendency in many historians to write on the side of Protestants and Whigs, to praise revolutions provided they have been successful, to emphasize certain principles of progress in the past and to produce a story which is the ratification if not the glorification of the present.” For an interesting discussion of the Whig theory of history, see David Hackett Fischer’s “Historians’ Fallacies.” Fischer’s book is one of my favorites.

[5] Nick Jardine in a journal article on Butterfield writes, “By the mid-1970s, it had become commonplace among historians of science to employ the terms ‘Whig’ and ‘Whiggish,’ often accompanied by one or more of ‘hagiographic,’ ‘internalist,’ ‘triumphalist,’ even ‘positivist,’ to denigrate grand narratives of scientific progress… In particular, they were suspicious of the grand celebratory and didactic narratives of scientific discovery and progress that had proliferated in the inter-war years.” Nick Jardine, “Whigs and Stories: Herbert Butterfield and the Historiography of Science,” History of Science, 41 (2003): 125-140, at pp. 127-8.

[6] The displacement current, a concept invented by Maxwell, means something different to modern-day physicists. In many ways, it is ideal for any discussion of reference and belief. Our beliefs about the displacement current have changed since Maxwell’s time, but does that mean we are referring to something different? Maxwell was initially wedded to a mechanical aether that was replaced in successive versions of his theories. Does this mean that we cannot discuss the differences between Maxwell’s early ideas about the displacement current based on various mechanical models of electricity and magnetism, and modern non-mechanical models of electromagnetism?

[7] A version of this story appears in “Predilections,” Mark Singer’s profile of me in The New Yorker, Feb. 6, 1989.

[8] “Incommensurability” is a term introduced by Kuhn in “The Structure of Scientific Revolutions.” Although it is used repeatedly in the book, Kuhn offers no clear definition. Much of this essay is an attempt, albeit an unsuccessful one, to pin it down. Here is a sample of some of Kuhn’s explanations: (1) “The normal scientific tradition that emerges from a scientific revolution is not only incompatible but often actually incommensurable with that which has gone before.” (2) “These examples point to the…most fundamental aspect of the incommensurability of competing paradigms. In a sense that I am unable to explicate further, the proponents of competing paradigms practice their trades in different worlds.” And (3) “Just because it is a transition between incommensurables, the transition between competing paradigms cannot be made a step at a time, forced by logic and neutral experience. Like the gestalt switch, it must occur all at once (thought not necessarily in an instant) or not at all.” Thomas Kuhn, “The Structure of Scientific Revolutions.” Chicago: University of Chicago Press. 1996. pgs. 103, 150

[9] The reenactment of the tossed ashtray at the beginning of the piece was shot with the Phantom Gold at 1,000 fps.

[10] The lectures were published in a volume of Synthese and circulated as xerox copies. Subsequently, they were published in a separate volume in 1980 by Harvard University Press.

[11] John Burgess, a philosopher at Princeton, has written, “Kripke’s picture has…been called the ‘historical chain’ picture. It has also sometimes been called the ‘causal chain’ picture, but this label is inappropriate. For there need not on Kripke’s view be any causal link between the initial baptist and the object baptized. This should be clear from the foregoing summary. Any object that can be described can be named, and this includes, for instance, causally inert mathematical objects, which figure in a couple of Kripke’s examples.” This is essay is concerned with a “named” mathematical object, namely, √2.

[12] Many commentators point out that Kripke never claimed to have provided a theory of reference. He only claims to have provided a “picture” of how naming works.

[13] Many concepts in philosophy are involved here: essentialism, possible worlds, etc. But there are two issues that are central to the concerns of this essay. A correspondence theory of truth. Truth is not merely linguistic, it must also involve the relationship between language and the world. That meanings (to use the philosopher Hilary Putnam’s expression) “ain’t just in the head.”

[14] This example comes from Seymour Cohen by way of my friend Charles Silver. (Cohen’s paper was written in 1966.) Some philosophers may object that this over-simplifies descriptivist accounts of naming. This is doubtlessly true.

[15] This idea is often traced back to John Stuart Mill. There is a passage in Mill’s “A System of Logic.” “A man may have been named John because that was the name of his father; a town may have been named Dartmouth, because it is situated at the mouth of the Dart. But it is no part of the signification of the word John, that the father of the person so called bore the same name; nor even of the word Dartmouth, to be situated at the mouth of the Dart. If sand should choke up the mouth of the river, or an earthquake change its course, and remove it to a distance from the town, there is no reason to think that the name of the town would be changed. … Proper names are attached to the objects themselves, and are not dependent upon the continuance of any attribute of the object.” [emphasis mine] Alas, Mill says that names are “attached to the things themselves,” but he never tells us exactly how this is done. It might be like a gummy label: “Hello, my name is John Stuart Mill.” Mill, J.S. “A System of Logic. Vol 1.” London: John W. Parker, West Strand. 1843. pg. 40.

[16] Kripke also has a “possible worlds” view of proper names, in which proper names are thought of as “rigid designators,” that is, in all possible worlds “Goldie” refers to Goldie. What keeps the historical chain intact? What keeps the links in the chain from being broken? For Kripke it is intentions. The intention to refer. So if Speaker Y intends to refer to the same thing that Speaker X refers to, there is an unbroken chain that takes us back to, for example, Goldie.

[17] Martin Seymour-Smith included “Structure” in his “The 100 Most Influential Books Ever Written” (1998). The Times Literary Supplement labeled it one of “The Hundred Most Influential Books Since the Second World War” (1995).

[18] I can’t hope to provide a definition of postmodernism here. But the essence of it, for me, is the social construction of reality and of truth. Forgive me, this definition may not capture the

many varieties of postmodernism, but it’s the best I can do. I had never really thought of Kuhn as a postmodernist, but one of my researchers returned with a syllabus from Louis Menand’s Harvard class on postmodernism and on the list of required reading, along with Lyotard, Baudrillard and Derrida, was “The Structure of Scientific Revolutions.” Structure became an “important text” of postmodernist thought.

[19] I made a couple of phone calls. “The Structure of Scientific Revolutions,” according to the University of Chicago Press, has currently sold over 1,000,000 copies and has been translated into more than 25 languages. (All of them presumably commensurable with one another.) On the other hand, “Naming and Necessity,” according to Harvard University Press, has sold only 43,350 copies. If you’re interested in buying a copy, I am sure that HUP would appreciate your business.

[20] Kuhn, “The Structure of Scientific Revolutions.” Chicago: University of Chicago Press. 1996. pg. 111 The task of interpreting Kuhn is a daunting one. One can go through his oeuvre and find instances where he affirms scientific progress and other instances where he denies it. Instances where languages in different paradigms can be partially translated; others were they are completely untranslatable. And so on and so forth. A mélange of p and not-p.

[21] Thomas Kuhn, “The Structure of Scientific Revolutions.” Chicago: University of Chicago Press. 1996. pg. 200. Here, Kuhn seems to be channeling Quine’s “Word and Object.” But Quine’s arguments about the “indeterminacy of translation” are different in kind from Kuhn’s arguments about “incommensurability,” even though historically I believe one influenced the other.

[22] Thomas Kuhn, “The Structure of Scientific Revolutions.” Clearly, on the defensive, he seemingly gave up on his “Gestalt-flip” analogy and dipped into language analysis. But the linguistic alternative is no less infirm. Hilary Putnam writes in “Reason, Truth and History,” “The incommensurability thesis is the thesis that terms used in another culture, say, the term ‘temperature’ as used by a 17th century scientist, cannot be equated in meaning or reference with any terms or expressions we possess… If this [incommensurability] thesis were really true then we could not translate other languages — or even past stages of our own language — at all… To tell us that Galileo had ‘incommensurable‘ notions and then to go on and describe them at length is totally incoherent.” [The emphasis is Putnam’s.] Hilary Putnam, “Reason, Truth and History.” Cambridge: Cambridge University Press. 1981. p. 114-115.

[23] Thomas Kuhn, The Structure of Scientific Revolutions. Chicago: University of Chicago Press. 1996 (1962). p. 150.

[24] Kuhn on many occasions tried to clarify his claims about incommensurability. In the 1969 postscript to “The Structure of Scientific Revolutions,” Kuhn suggested that the problem was with philosophers. “A number of them have purported that I believe the following: the proponents of incommensurable theories cannot communicate with each other at all, as a result, in a debate on theory-choice there can be no recourse to good reasons; instead theory must be chosen for reasons that are ultimately personal and subjective…” But in his effort to deny these complaints, he often equivocated, sometimes changing nomenclature, e.g. “paradigm” became

“disciplinary matrix,” sometimes just muddying the waters, so that it was impossible to understand exactly what he was saying. His ultimate recourse was to use his theory of incommensurability to explain why many people misinterpreted him or couldn’t understand him.

[25] Often, Kuhn uses the terms “meaning” and “reference” in disparate ways and confuses the two. He assumes that because people have different beliefs about things that they cannot be referring to the same thing. This seems clearly wrong. I can believe that the sun orbits the earth or vice versa and still be referring to the earth and to the sun. He also assumes that because people have different beliefs about things that they cannot understand each other. That they cannot communicate with each other.

[26] The philosopher Dudley Shapere writes, “…the doctrine of extreme incommensurability remains flawed; for it is fundamentally incoherent. How can any two things be completely incomparable? On the other hand, if two scientific contexts were truly incomparable (in the extreme sense that seems implied by Kuhn’s claims of the paradigm-dependence of everything), it would be impossible to call both of them ‘scientific’, or, more specifically, ‘theories’ or ‘paradigms’, or to say that they differed in their standards of explanation (that they disagreed about what it is to explain).” Dudley Shapere, “Evolution and Continuity in Scientific Change,” Philosophy of Science, Vol. 56, No. 3 (1989), pp. 419-437.

[27] The O.E.D. entry for “incommensurable” quotes Edmund Burke in “A Letter to a Noble Lord on the Attacks Made Upon Mr. Burke and His Pension, in the House of Lords, by the Duke of Bedford and the Earl of Lauderdale, Early in the Present Session of Parliament,” “Selected writing and speeches,” Edmund Burke and Peter James Stanlis, pp. 665ff. “I challenge the Duke of Bedford as a juror to pass upon the value of my services. Whatever his natural parts may be, I cannot recognize in his few and idle years the competence to judge of my long and laborious life… His Grace thinks I have obtained too much. I answer, that my exertions, whatever they have been, were such as no hopes of pecuniary reward could possibly excite, and no pecuniary compensation can possibly reward them. Between money and such services, if done by abler men than I am, there is no common principle of comparison: they are quantities incommensurable.” I find myself entirely sympathetic with Burke’s complaint.

[28] Thomas Kuhn, “The Road Since Structure, Philosophical Essays, 1970-1993, with an Autobiographical Interview,” edited by James Conant and John Haugeland, University of Chicago, 2000, p. 298ff. Kuhn died before the publication of the autobiographical material. Jehane Kuhn, his wife, writes, “The title of the book again invokes the metaphor of a journey, and its closing section, which records an extended interview at the University of Athens, amounts to [a] longer, more personal narrative. I am delighted that the interviewers, and the editorial board of the journal Neusis, in which it first appeared, have agreed to its republication here… Tom was exceptionally at ease with these three friends and talked freely on the assumption that he would review the transcript, but time ran out.”

[29] Those who are familiar with the proof certainly don’t want me to explain it here; likewise, those who are unfamiliar with it don’t want me to explain it here, either. There are many simple proofs in many histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline,

etc., etc. Barry Mazur offers a proof in his book, “Imagining Numbers (particularly the square root of minus fifteen),” New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, “How Did Theaetetus Prove His Theorem?”, available on Mazur’s Harvard Web site.

[30] One more detail. √2 –– like all irrational numbers –– is a non-repeating decimal. You can expand it forever, and the digits never repeat themselves. Here it is to 50 places –– 1.41421356237309504880168872420969807856967187537694… I can picture 1 1/3–– 1.33333333333333333333333333333333333333333333333333… I know the decimal expansion is getting closer and closer to 1 1/3 as I add more and more 3’s, but there is something preternaturally odd about a number whose digits never repeat even though they go on forever.

[31] Even though √2 can not be expressed as a fraction or as a ratio of two integers, it easily can be represented geometrically, e.g., as a distance on a line. This diagram comes from Richard Courant, Herbert Robbins and Ian Stewart, “What is Mathematics?” (revised, 1996, Oxford University Press), p. 60. “…a very simple geometrical construction may result in a segment incommensurable with the unit. If such a segment is marked off on a number axis by means of a compass, the point so constructed cannot coincide with any of the rational points… To the naive mind it must certainly appear very strange that the dense set of rational points does not cover the whole line.”

Courant, Robbins, and Stewart. What is Mathematics? Oxford University Press

[32] The proof doesn’t tell us what an irrational number is – just what it is not.

[33] √2 is as incommensurable today as it was in the 5th Century B.C.E. Our understanding of √2 has been expanded and has deepened over the centuries, but the core idea of a measure that cannot be expressed as a rational fraction remains the same.

[34] As quoted in Barry Mazur, “Imagining Numbers (particularly the square root of minus fifteen).” New York, NY: Farrar, Straus and Giroux. 2003. Mazur was unable to track down the source of the quote, but recalls: “This is what I remember of an interview with Gabriel Garcia Marquez that I heard on the radio many years ago. I haven’t been able to track down that interview to verify my memory of it, but Garcia Marquez has commented about “The

Metamorphosis” in many places. He is reported to have said that Kafka wrote (specifically in the first sentence of “The Metamorphosis” ‘the way grandmother (abuela) used to talk’; and ‘Damn, I did not know that such a thing could be done!’; and that if this is allowed, ‘then writing interests me.’”

[35] Since Samsa is a fictional character – who thinks and feels like a human being, but who looks like a gigantic insect, in some sense might actually be a gigantic insect – just how self-aware can he be…?

[36] It is difficult, maybe impossible, to figure out precisely what Kuhn meant by incommensurability, much more difficult I imagine than figuring out what the Greeks meant by it. Consider the following, “Most readers of my text have supposed that when I spoke of theories as incommensurable, I meant that they could not be compared. But “incommensurability” is a term borrowed from mathematics, and there it has no such implication. The hypotenuse of an isosceles right triangle is incommensurable with its side, but the two can be compared to any required degree of precision.” Kuhn wants it both ways. Either they can be compared, or they can’t. When he says that the terms of one paradigm cannot be translated into another, then they can’t be compared. If they can be compared, then they can be translated. At times, I wonder whether he chose the term specifically because its meaning is unclear – outside of mathematics. Thomas Kuhn, “The Road Since Structure, Philosophical Essays, 1970-1993, with an Autobiographical Interview,” edited by James Conant and John Haugeland, University of Chicago, 2000, p. 189.

[37] David Berlinski, “Infinite Ascent: A Short History of Mathematics,” New York: Modern Library. 2008. pp. 9-10

[38] Charles Seife, “Zero: The Biography of a Dangerous Idea,” New York: Penguin. 2000.

[39] “Papyrus comes from a grass-like plant grown in the Nile delta region in Egypt which had been used as a writing material as far back as 3000 BC. It was not used by the Greeks, however, until around 450 B.C.E. for earlier they had only an oral tradition of passing knowledge on through their students. … The first copy of the Elements would have been written on a papyrus roll, which, if it were typical of such rolls, would have been about 10 meters long. These rolls were rather fragile and easily torn, so they tended to become damaged if much used. Even if left untouched they rotted fairly quickly except under particularly dry climatic conditions such as exist in Egypt. The only way that such works could be preserved was by having new copies made fairly frequently and, since this was clearly a major undertaking, it would only be done for texts which were considered of major importance.”

[40] Von Fritz raises an important question. Why is it so important to give a name to the man who discovered √2? Why do we need to know who is responsible? Without the knowledge that it was Hippasus, would we be worse off? Would we be left with the Tomb of the Unknown Mathematician? Or the unmarked grave of the mathematician, who died seeking the truth about incommensurability but will never be properly memorialized? Why bother? The ultimate reason may go back to Kripke and the power of names. We may have competing beliefs about

the identity of the man who discovered the incommensurability of √2, and different beliefs about how and when he proved it. Subsequent histories have questioned Von Fritz’s account but it is the name (Hippasus) that connects us to the world. It gives us the confidence that we are talking about something rather than nothing – or somebody rather than nobody. Burkert, who I interviewed (see below), makes this point eloquently in the introduction to his book “Lore and Science in Ancient Pythagoreanism.” Here is his comment about the lack of textual evidence and the controversies concerning the historical Pythagoras. “…at the source of this continuing changing stream lay not a book, an authoritative text which might be reconstructed and interpreted, nor authenticated acts of a historical person which might be put down as historical facts. There is less and there is more: a “name” which somehow responds to the persistent human longing for something that will combine the hypnotic spell of the religious with the certainty of exact knowledge – an ideal which appeals, in ever changing forms, to each successive generation.” Walter Burkert, LASIAP. Cambridge, MA: Harvard University Press. 1972. pp. 10-11.

[41] “Iamblichus’ Life of Pythagoras,” or, “Pythagoric life: accompanied by Fragments of the ethical writings of certain Pythagoreans in the Doric dialect and a collection of Pythagoric sentences from Stobaeus and others / translated from the Greek by Thomas Taylor.” Rochester, VT: Inner Traditions, International. 1986. (Reprint. Originally published: London, J.M. Watkins. 1818)

[42] http://www.completepythagoras.net/mainframeset.html, chapters xviii and xxxiv

Passage One. As to Hippasus, however, they acknowledge that he was one of the Pythagoreans, but that he met the doom of the impious in the sea in consequence of having divulged and explained the method of squaring the circle, by twelve pentagons; but nevertheless he obtained the renown of having made the discovery. Passage Two. It is accordingly reported that he who first divulged the theory of commensurable and incommensurable quantities to those unworthy to receive it, was by the Pythagoreans so hated that they not only expelled him from their common association, and from living with him, but also for him constructed a symbolic tomb, as for one who had migrated from the human into another life. Passage Three. It is also reported that the Divine Power was so indignant with him who divulged the teachings of Pythagoras that he perished at sea, as an impious person who divulged the method of inscribing in a sphere the dodecahedron, one of the so-called solid figures, the composition of the icostagonus. But according to others, this is what happened to him who revealed the doctrine of irrational and incommensurable quantities.

[43] “The Pythagorean Sourcebook and Library: An Anthology of Ancient Writing Which Relate to Pythagoras and Pythagorean Philosophy,” ed. David Fideler, trans. Kenneth Sylvan Guthrie, p. 116. Also, Diogenes Laertius in a footnote: “Now they say that Pythagoras did not leave behind him a single book, but they talk foolishly for Heraclitus, the natural philosopher, speaks plainly enough of him saying, ‘Pythagoras, the son of Mnesarchus, practiced inquiry beyond all other men, and making selections from these writings he thus formed a wisdom of his own, an extensive learning, and cunning art’ [...] There are three volumes extant written by Pythagoras, one On Education, one On Politics, and one On Nature [...] The mystic discourse

which is under his name, they say is really the work of Hippasus, having been composed with a view to bring Pythagoras into disrepute,” p.142.

[44] Part of the problem is that there are separate meanings of incommensurable. And they get confused. On one hand, there is the mathematical concept. It refers to the fact that √2 cannot be expressed as a rational fraction. And then there’s Kuhn’s philosophical concept – the incommensurability of meaning – the belief that the meanings of one world view cannot be translated into another.

[45] There is even some doubt that the image of Hippasus I’ve used is Hippasus.

[46] There is a reference to “the Athenian stranger,” a character from Plato’s last dialogue, Laws. And is quoted in D.H. Fowler, “The Mathematics of Plato’s Academy, 2nd ed.,” Oxford: Clarendon Press. 1999. p. 296

Another account is provided by Sir T. L. Heath, who writes,

Another argument is based on the passage in the Laws where the Athenian stranger speaks of the shameful ignorance of the generality of Greeks, who are not aware that it is not all geometrical magnitudes that are commensurable with one another; the speaker adds that it was only ‘late’ that he himself learnt the truth. Even if we knew for certain whether ‘late’ means ‘late in the day’ or ‘late in life’, the expression would not help much towards determining the date of the first discovery of the irrationality of √2; for the language of the passage is that of rhetorical exaggeration (Plato speaks of men who are unacquainted with the existence of the irrational as more comparable to swine).

(T.L. Heath. “A History of Greek Mathematics, Vol. I. From Thales to Euclid.” New York: Dover. 1981)

How could two stories be more different. In 500 B.C.E., Hippasus is drowned because he reveals a secret that no one outside the Pythagorean cult should know; in 350 B.C.E., Plato is bent out of shape because not every Greek is familiar with the concept of irrational numbers.

[47] Walter Burkert, “Lore and science in ancient Pythagoreanism,” trans. E. L. Minar, Jr., Cambridge, MA, Harvard University Press. 1972. pg. 191. “Whether a Pythagorean gets up or goes to bed, puts on his shoes or cuts his nails, stirs the fire, puts on the pot, or eats, he always has a commandment to heed. He is always on trial and always in danger of doing something wrong. No more carefree irresponsibility! Everything he does is done consciously, almost anxiously. The mythical expression of this attitude to life is a world full of souls and daemons, which affect every moment of a person’s life. Everywhere are rules, regulations, and an ascetic zeal for discipline; life is πονος [labor or pain], which must be endured…”

[48] The number of rules and restrictions handed down from Pythagoras to his followers was extensive. “Abstain from beans. Eat only the flesh of animals that may be sacrificed. Do not step over the beam of a balance. On rising, straighten the bedclothes and smooth out the place where you lay. Spit on your hair clippings and nail parings. Destroy the marks of a pot in the

ashes. Do not piss towards the sun. Do not use a pine-torch to wipe a chair clean. Do not look in a mir-ror by lamplight. On a journey do not turn around at the border, for the Furies are following you. Do not make a detour on your way to the temple, for the god should not come second. Do not help a person to unload, only to load up. Do not dip your hand into holy water. Do not kill a louse in the temple. Do not stir the fire with a knife. One should not have children by a woman who wears gold jewelry. One should put on the right shoe first, but when washing do the left foot first. One should not pass by where an ass is lying.”

[49] The paucity (or absence) of written evidence does not mean that there is no fact of the matter. Just that we may not be able to know the fact of the matter.

[50] Vue des Ruines du Temple de Junon, à Metapontum, Ville Greque sittuée du Golfe de Tarente et dans la partie de l’ancienne G.de Grece que l’on nommoit autrefois Lucania, aujourd’hui la Basilicate. Copper etching by Berteaux after Jean Louis Desprez (1743-1804) for “Voyage pittoresque de Naples et de Sicilie” by St. Non, 1781-1785.

The Metropolitan Museum of Art/Art Resource, NY

The Unicorn in Captivity. Tapestry. 1495-1505. Wool warp, wool, silk, silver, and gilt wefts. 12 ft. 1 in. x 8 ft. 3 in. (368 x 251.5 cm). South Netherlandish. Gift of John D. Rockefeller Jr.,

1937 (37.80.6) | The Metropolitan Museum of Art, New York, NY, U.S.A.

[51] Otto Neugebauer has written, “In the Cloisters of the Metropolitan Museum in New York hangs a magnificent tapestry which tells the tale of the Unicorn. At the end we see the miraculous animal captured, gracefully resigned to his fate, standing in an enclosure surrounded by a neat little fence. This picture may serve as a simile for what we have attempted here. [...that is, a simile for the attempted reconstruction of ancient science.] We have artfully erected from the small bits of evidence the fence inside which we hope to have enclosed what may appear as a possible, living creature. Reality, however, may be vastly different from the product of our imagination; perhaps it is vain to hope for anything more than a pi cture which is pleasing to the constructive mind when we try to restore the past.” As quoted in D.H. Fowler, “The Mathematics of Plato’s Academy,” from Otto Neugebauer, “The Exact Sciences in Antiquity,” Chapter 6.

[52] The movie was released the same year that “Structure” was published. Mercifully they do not share a postmodern rejection of truth. In “The Man Who Shot Liberty Valance,” we, the audience know the truth, even if no one else does.

[53] The ambiguity of the title has interesting ramifications for the description theory of proper names. Is “the man who shot Liberty Valance” a definite description or does it function as a proper name? According to the description theory, “the man who shot Liberty Valance” picks out the man who shot L.V., namely Tom Doniphon (John Wayne). But what if many people (for example, the members of the audience watching the movie) believe that man is Ransom Stoddard (James Stewart)? Don’t our beliefs matter in determining reference? After Stoddard (Stewart) seemingly kills L.V., he becomes known as “the man who shot L.V.” But what if we should learn that he wasn’t the man who shot Liberty Valence? I believe the description would still refer to Stoddard (Stewart). Stoddard is baptized and the reference is fixed. Then, even if it turns out that Stoddard didn’t shoot L.V., “the-man-who-shot-Liberty-Valance” still refers to Stoddard. (I could even imagine the sentence, “The-man-who-shot-Liberty-Valence is not the man who shot Liberty Valence.” Where the “the man who shot Liberty Valence” is first used as a name and then as a definite description. Russell wrote about proper names as disguised definite descriptions. What about definite descriptions as disguised proper names, or to use Kripke’s terminology, disguised rigid designators?) Stoddard is returning by train to Washington. The conductor tells him that they are holding the express for him – for Stoddard – saying, “Nothing’s too good for the man who shot Liberty Valance.” The conductor is referring to Stoddard (Sewart), but the audience knows that he is referring to Doniphon (Wayne). And so, the ending of one of John Ford’s greatest westerns depends on the fact that a proper name (or a definite description) refers in two different ways.

[54] A variant on this theme comes from G.K. Chesterton, “Orthodoxy.” London: John Lane Company. 1908. pg, 84 “It is quite easy to see why a legend is treated, and ought to be treated, more respectfully than a book of history. The legend is generally made by the majority of people in the village, who are sane. The book is generally written by the one man in the village who is mad.”

[55] Incommensurability appears many times in Plato, in the “Theaetetus,” in the “Republic,” and in Laws. But there is nothing in Plato to suggest that the discovery of incommensurability caused a crisis of any kind. Fowler is particularly good on this issue.

[56] Giordano Bruno was burned at the stake in 1600 not because he believed in a heliocentric universe, but because he rejected the immaculate conception of Mary and the Holy Trinity. Call it an unfortunate combination of intolerance and power. A summary of one of his heresies given in “Giordano Bruno: Philosopher/Heretic,” reads “That sins are not to be punished. Giovanni Mocenigo, informer: ‘I have sometimes heard Giordano say in my house that there is no punishment for sins, and he has said that not doing to others what we do not want them to do to us is enough [advice] to live well.’” “Giordano Bruno: Philosopher / Heretic,” Ingrid D. Rowland, University of Chicago Press. 2008. p. 260

[57] I wish I could have interviewed them. Knorr’s Times obituary and Fowler’s Independent obituary.

[58] Helmut Hasse and Heinrich Scholz, “Die Grundlagenkrisis Der Griechischen Mathematik (The Foundational Crisis in Greek Mathematics),” Kant-Studien. Volume 33, Issue 1-2, Pages 4–34, 1928. The article was written in part as a criticism of an account given in Oswald Spengler’s “The Decline of the West.” Spengler had written, “…in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the ir-rational perished by shipwreck, “for the unspeakable and the formless must be left hidden for ever.” Although there is no English translation of the Hasse and Scholz essay, a review by Kurt Gödel, published in 1931, has been translated into English, Kurt Gödel, “Collected Works,” p. 219, ed. Solomon Feferman. “This stimulating little book depicts in a very interesting way how the doctrine of the irrational developed among the Greeks… This hypothesis cast an entirely new light on Zeno, who appears as an early champion of rigorous methods in mathematics, and in that sense is compared to Weierstrass.”

[59] Freudenthal, H. (1965). “Y avait-il une crise des fondements des mathématiques dans l’antiquité?” In “Classics of Greek Mathematics, ed. J. Christianidis.” Dordrecht, the Netherlands: Kluwer Academic Publishers. Freudenthal writes, “Often during the history of mathematics the problem of foundations is posed anew and from new angles. This is not to suggest that, in a given age, the problem troubles every mathematician. The methods of differential and integral calculus that were invented by Newton and Leibniz were calmly applied with the knowledge that they were not well-founded and despite the paradoxes they implied. The paradoxes of infinity, long known, were never considered as serious menaces, but rather as pleasantries at the periph-ery of mathematics…I do not know who first spoke of a crisis in the foundations of mathematics, but I am sure that the term was invented later, in the days when we began seriously to deal with foundations. And I further do not know who discovered such a crisis of foundations in ancient mathematics. The famous little book of Hasse and Scholz from 1928 is a terminus ante quem for the use of this term, but the idea itself is older, and can be traced back to Tannery.”

[60] Wilbur R. Knorr. “The Impact of Modern Mathematics on Ancient Mathematics.” Revue d’histoire des math´ematiques, 7 (2001), p. 121–135.

[61] Fowler, “The Mathematics of Plato’s Academy.” Oxford: Oxford University Press. Second Edition. 1999. “Part of every literate person’s intellectual baggage, along with the second law of thermodynamics and the principles of relativity and indeterminacy, is some version of the story of the discovery of incommensurability by Pythagoras or the Pythagoreans…”

[62] http://www.lib.utexas.edu/maps/historical/shepherd/italy_ancient_south.jpg. Pythagoras was born in Samos and later went to Croton. It is believed that he moved to Metapontum just before he died. A trip across the boot of Italy. To give Hippasus the boot? Ridiculous conjecture on my part, but if both Hippasus and Pythagoras were living in the same area and eventually in the same city, isn’t it likely, if the story were true, that Hippasus was drowned in the Sea of Tarentum? (Perhaps we could find him, clutching a clay tablet with the proof, at the

bottom of the sea. A frozen look of astonishment on his face at the harsh treatment dished out by his brethren.)

[63] From Iamblichus, “Vita Pythagorica.” The unreliable Iamblichus provides a story of how Pythagoras came to leave Croton for Metapontum. “Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontum and there is said to have ended his days.”

[64] Thomas Kuhn,”The Road Since Structure: Philosophical Essays, 1970-93.” Chicago: University of Chicago Press.2000, pp. 123-124.

[65] Thomas Kuhn, “The Road Since Structure: Philosophical Essays, 1970-93,” Chicago: University of Chicago Press. 2000. pp. 124.

[66] Donald Davidson, “On the Very Idea of a Conceptual Scheme.” Proceedings and Addresses of the American Philosophical Association, Vol. 47, pp. 5-20. 1973-1974.

[67] Jorge Luis Borges. “Pierre Menard: Author of the Quixote.” Collected Fictions. New York: Viking. 1998.

[68] The entire passage, “…historians must and ought to be exact, truthful, and absolutely free of passions, and neither interest nor fear, hatred nor love, should make them swerve from the path of …truth, whose mother is history, rival of time, depository of deeds, witness of the past, exemplar and adviser to the present, and the future’s counselor. In this account I know there will be found all that can be rightly desired in the most pleasant history, and if something of value is missing from it, in my opinion I maintain that the fault lies with the dog who was its author rather than with any defect in the subject.”

[69] This passage in Borges (and, consequently, also in Menard and Cervantes) concerns the nature of historical truth. Cervantes is a historical realist; Menard, a historical relativist. For Cervantes history is objective; for Menard, history is socially constructed.

[70] An updated version of the “conflict” between Menard and Cervantes is embodied in an quote from a Bush aide. It appeared in a New York Times Magazine article, Oct. 17, 2004, and was later attributed to Karl Rove. It pits those that are influenced by facts against those that (take your pick) create them or make them up. “The aide said that guys like me were ‘in what we call the reality-based community,’ which he defined as people who ‘believe that solutions emerge from your judicious study of discernible reality.’ I nodded and murmured something about enlightenment principles and empiricism. He cut me off. ‘That’s not the way the world really works anymore,’ he continued. ‘We’re an empire now, and when we act, we create our own reality. And while you’re studying that reality – judiciously, as you will – we’ll act again,

creating other new realities, which you can study too, and that’s how things will sort out. We’re history’s actors . . . and you, all of you, will be left to just study what we do.”

[71] Jorge Luis Borges, “Selected Non-Fictions,” New York: Viking. 1999. p. 258-9. In Alberto Manguel’s memoir, “In Pierre Menard, Author of the Quixote, he argued that a book changes according to the reader’s attributions… Pierre Menard is, of course, an invention, a superb and hilarious imagining, but the notion that a text changes according to the reader’s assumptions is old… Once, after noting that we read now Dante in ways he couldn’t have imagined…Borges recalled an observation by the 9th century mystic, Scotus Erigena. According to the author of On the Divisions of Nature, there are as many readings of a text as there are readers…” Indeed. What is novel is the claim that Menard and Cervantes cannot be compared.

[72] The actual quote is from a Father Brown mystery, “The Head of Caesar,” “‘What we all dread most,’ said the priest in a low voice, ‘is a maze with no center. That is why atheism is only a nightmare.’ ‘I will tell you everything,’ said the red-haired girl doggedly, ‘except why I am telling you; and that I don’t know.’” Borges presumably translated Chesterton into Spanish, and it was translated back into English in a slightly different form. From “a maze with no center” to “nothing is more frightening than a labyrinth with no center. Nevertheless, Borges’s interpretation of Chesterton’s English translated from Spanish back into English is perfectly intelligible. In fact, I prefer the Chesterton after the two translations. G.K. Chesterton. “The Head of Caesar,” “The Wisdom of Father Brown.” Dodd, Mead. 1924.

[73] Bertrand Russell, “Nightmares of Eminent Persons,” London: Bodley Head. 1954, pp. 36-39. The illustrations by Charles W. Stewart, an ex-ballet dancer, are extraordinary. Here is his obituary in The Independent.

[74] I have occasionally fantasized about a possible seminar with Humpty Dumpty, Wittgenstein, and Quine.

[75] A Wittgensteinian might be tempted to see this as an example of “private language,” and hence, impossible. But Humpty Dumpty’s language isn’t private, it’s publicly imposed by Humpty Dumpty.

[76] Lewis Carroll, “Through the Looking-Glass.” London: Macmillan. 1872.

[77] http://cartome.org/panopticon1.htm

[78] Stanley Cavell, “Little Did I Know: Excerpts From Memory.” Stanford, CA: Stanford University Press. 2010. Cavell, also wrote, “Once, lingering over too much coffee and many too many cigarettes, after a particularly resonant blast of disagreement from him, I replied: ‘Tom, please do not address me. I am not a convention.’ He was shocked, put his forehead to the table, and banging it gently several times, he said, in rhythm, and softly: ‘I know. I know I do that.’”

[79] This is, of course, my interpretation. With Wittgenstein, there are bound to be disagreements. There are many philosophers who would argue vociferously that Wittgenstein is not a relativist. However, here is a relevant passage from the “Stanford Encyclopedia of Philosophy” entry on Wittgenstein, “In Wittgenstein’s terms, agreement is required ‘not only in definitions but also (queer as this may sound) in judgments’ (PI 242), and this is ‘not agreement in opinions but in form of life’ (PI 241)… Used by Wittgenstein sparingly — five times in the Investigations — this intriguing concept has given rise to interpretative quandaries and subsequent contradictory readings. Forms of life can be understood as changing and contingent, dependent on culture, context, history, etc; this appeal to forms of life grounds a relativistic reading of Wittgenstein…” Definitions, judgments, agreement in opinions. I can see in Wittgenstein’s “forms of life” an early version of Kuhn’s paradigms. Groups of people who use language in the same way, disagreements and agreements about the rules, etc. Evidently, given the passage in Stanley Cavell’s memoir, Kuhn saw an early version of his paradigms in Wittgenstein’s form of life.

[80] Kuhn may have wanted to formalize Wittgenstein’s idea that we are “trapped” in language, that there is no independent way of determining “true” from “false.” But to formalize Wittgenstein correctly, Kuhn would have to abandon any distinction between “true” and “false” even within a given paradigm.

[81] Thomas Kuhn, “The Structure of Scientific Revolutions.” Chicago: The University of Chicago Press. 1996 p. 170.

[82] Thomas Kuhn, “What Are Scientific Revolutions?” Chapter 1, The Probabilistic Revolution, Vol. 1: Ideas in History. Ed. Lorenz Kruger, Lorraine J. Daston, Michael Heidelberger. 1987. “Near misses” is a curious phrase. Is Kuhn comparing – despite incommensurability – Aristotelian and contemporary physics? It’s hard to see how Aristotle’s conception of gravity is a “near miss.” Aristotle believed that a stone falls to the ground because it returns to the place from where it originated. Is this notion incommensurable with contemporary physics? I don’t think so. It is simply false. For example, I could take a stone from the earth to the moon, and presumably it would not “try” to return to the earth.

[83] Bertrand Russell, “The Impact of Science on Society.” London: Allen & Unwin. 1952. Variant from Russell, “An Outline of Intellectual Rubbish.” Girrard, Kan: Haldeman-Julius Publications. 1943. Perhaps Kuhn, given his love of incommensurability, concluded that Mrs. Aristotle had an irrational number of molars. As an example of how this kind of talk can lead to nonsense, the Nobel prize was awarded in 2005 to two researchers who established the cause of peptic ulcers. Gone were Rolaids and the claim that they could consume 47 times their weight in excess stomach acid. Stomach acid was no longer the issue. They traced the disease to the bacterium Helicobacter pylori. This has been described in many articles as “a paradigm shift.” A paradigm shift!? Is this what it amounts to? A change in belief?!

[84] Wise sent me a followup email. “You were interested in two particular features of Tom’s mode of working. The first concerns the ability to qualify his statements so as to almost make a strong claim without actually making it. An example appears in his classic article on energy conservation (p. 100 in “The Essential Tension,” where he says: ‘Unless the Naturphilosophie

indigenous to the educational environment of these seven men had a productive role in the researches of some, it is hard to see why more than half of the pioneers should have been drawn from an area barely through its first generation of significant scientific productivity. Nor is this quite all. If proved, the influence of Naturphilosophie may also help to explain why this particular group of five Germans, a Dane, and an Alsatian includes five of the six pioneers in whose approaches to energy conservation we have previously noted such marked conceptual lacunae.” In other words, “I have not done the research to find actual evidence for this role of Naturphilosophie, but it is true nonetheless, unless of course it turns out not to be.” I have never mastered this mode of writing, though I recognize its effectiveness. [Effective!? I assume that Norton is being ironic. – E.M.] Second, I cited an example of my own experience in working on the electrical theory of William Thomson, Lord Kelvin, in order to show why Kuhn’s critique of Whiggish interpretation had great power in the detailed reading of texts. I was presenting in seminar Thomson’s mathematical result on how electrical force F changes to F’ across the boundary of an insulator of inductive capacity k, so that F’/F = 1/k. This presented problems because Thomson claimed to be reasoning on the basis of a flow analogy to conduction of heat across a surface where the conductivity changed and his equation for the effect did not seem to follow from this view. I gave an account of his thinking which was based (implicitly) on a modern physicist’s understanding of electrostatic induction. Tom insisted (strenuously and adversarially) that my account could not be correct because it did not preserve the relation between Thomson’s verbal statements and his equation. It was Whiggish, which is the basic lesson here. The story continues, however, with its characteristically adversarial nature. He asserted an interpretation of his own which I in turn thought could not be right and said so. I then spent a great deal of time and effort on the problem and returned the next week to show why he was wrong and to present yet another account, which turned out to solve the difficulties and to give considerable insight into Thomson’s mode of analysis. The Whig story and the adversarial story are about equally important in my perception of the simultaneously intellectual and psychological struggle in working with Tom. [Are they two separate stories – the Whig and the adversarial – or is it all adversarial? – E.M.]

[85] Of course, it is still possible to be skeptical about Kripke’s claim, namely that a historical chain of intentions guarantees that we can refer to things in the past. But it is, at least, an attempt to provide an alternative to the descriptivist picture of how we refer to things.

[86] Kripke never tells us that we can recover beliefs from the past. I have often wondered: OK, we can refer to Pythagoras without knowing anything about him. But if we don’t know anything about him, what good does it do to be able to refer to him?

[87] “√2” refers to the same thing pre-Hippasus as it does post-Hippasus, even though our beliefs about √2 may have changed.

[88] Ralph Waldo Emerson in his essay “Self-Reliance” weighed in on this issue, “Their every truth is not quite true. Their two is not the real two, their four not the real four: so that every word they say chagrins us and we know not where to begin to set them right…”

[89] As John Burgess, a professor of philosophy at Princeton, told me in an interview: “Kuhn speaks one way when speaking to historians, and another way when speaking to philosophers.

The trouble begins when he starts talking about things being socially constructed. As far as I’m concerned, the stars can’t be socially constructed. The stars are billions of years old, and human society is not billions of years old. And so the latter cannot have constructed the former. On the other hand, if you just say astronomy is socially constructed, well, that’s a trivial truism. Who would deny that? Before there was human society, there weren’t any astronomers, and there wasn’t any astronomy. There is some philosophical issue here, but it’s these modes of expression. They are just a sloppy way of talking. Throwing around these expressions is in fact concealing issues rather than illuminating anything. It is encouraging a fallacious jump from the thought– since astronomy is constructed, therefore, the things astronomy is about are constructed, or something like that. And when speaking with philosophers, Kuhn would deny holding this outrageous belief. But when speaking to historians, he’d go back to using this language.”

[90] From L.P. Hartley, the opening lines of “The Go-Between,” London: H. Hamilton. 1953.