Errol Morris on photography.

This is part three of a five-part series.

3.

HIPPASUS OF METAPONTUM

All-History.org

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.”

Wikimedia Commons

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, via Wikimedia Commons

*******

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], via Wikimedia Commons

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.

Errol Morris

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), via Wikimedia Commons

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 Pythagore-ans. 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

Biblioteca Ambrosiana, Milan, Italy/The Bridgeman Art Library

Erich Lessing / Art Resource, NY

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

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]

*******

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

*******

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]

Errol Morris

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.

Errol Morris

[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] //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

[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] //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.”

Note: An earlier version of this article referred to the “giant cockroach” in “The Metamorphosis”; that has been changed to “giant insect.”