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This is allotment three of a five-part series.

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

Here’s one clue. At the absolute end of “The Road Aback Structure,” a abstruse of essays on Kuhn’s work, there is an annual with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a abrupt annual of the absolute origins of his idea. Actuality is the accordant articulation of the interview.

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

V. KINDI: You beggarly in mathematics?

T. KUHN: …When I was a ablaze aerial academy mathematician and alpha to apprentice Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was array of a big two-volume Calculus book by, I can’t bethink whom. And afresh I never absolutely apprehend it. I apprehend the aboriginal genitalia of it. And aboriginal on it gives the affidavit of the applesauce of the aboveboard basis of 2. And I anticipation it was beautiful. That was awfully exciting, and I abstruse what incommensurability was afresh and there. So, it was all accessible for me, I mean, it was a allegory but it got at accurately what I was after. So, that’s breadth I got it. [28]

“It was all accessible for me.” I thought, “Wow.” The accent was suggestive. I absurd √2 provocatively dressed, its aperture rouged. But there was an abrupt surprise. The abstraction didn’t arise from the concrete sciences or aesthetics or linguistics, but from mathematics. Namely, the affidavit that √2 can not be bidding as the arrangement of two integers. “…it was a allegory but it got at accurately what I was after.”

Incommensurability in mathematics expresses the actuality that not every ambit can be abstinent with accomplished numbers or fractions of accomplished numbers. Booty a unit-square – 1 by 1. How continued is the diagonal? By the Pythagorean Theorem, if anniversary ancillary has a breadth of 1 afresh the hypotenuse has a breadth of √2. The sum of the squares of the abandon = the aboveboard of the hypotenuse. Can that breadth be bidding as a atom or as a arrangement of two integers, e.g., 99/70 or 577/408? The acknowledgment is — no. [29] [30] [31]

The affidavit accustomed that there are quantities that cannot be bidding as fractions. [32] The Pythagoreans were amiss – not aggregate was composed of accomplished numbers or ratios of accomplished numbers. As Hamlet says, “There are added things in heaven and earth, Horatio, than are dreamt of in your philosophy.”


But how does algebraic incommensurability annotate Kuhn’s abstraction of abstruse incommensurability? [33] As mathematics developed, concepts accept been conflicting and proofs accustomed and clarified. The array – you could alike anticipate of it as a allegory – of numbers (and added algebraic concepts) has developed and grown. Today, we accept aberrant numbers, abstruse numbers, circuitous numbers, ideal numbers, transfinite numbers, etc. To name a few. But we are not blow our accommodation to accept concepts from the past. A ability of circuitous numbers does not anticipate us from compassionate absolute numbers, no added than a ability of aberrant numbers prevents us from compassionate rational numbers, etc., etc. We are adorning our compassionate of mathematics. We are accretion our angle of what is possible. Of what we can imagine.

Gabriel Garcia Marquez, on annual Kafka’s “Metamorphosis,” said, “I didn’t apperceive you were accustomed to abode like that.” [34] Note the use of the chat “allowed,” as in permitted. Marquez accepted altogether able-bodied what was blow to Gregor Samsa. He was metamorphosing into a gigantic insect.

Gregor Samsa ability accept accepted it as well. [35]

This adventure illustrates allotment of the abashing about incommensurability and about archetype shifts. Is it a catechism of what can be understood? Or what can be allowed?


But let’s acknowledgment to Kuhn’s interview. He said incommensurability in mathematics was a “metaphor?” But a allegory for what? [36] I thought, aback algebraic incommensurability doesn’t abduction what Kuhn was attractive for, namely, altered meanings, conceivably I should attending for an acknowledgment in the history of the proof.I was accustomed with the accepted outline – the Pythagoreans and their adapter to accomplished numbers, the betrayal of a band abstruse and the annihilation that followed. Actuality are a brace excerpts from two accepted accounts. The aboriginal extract apropos the Pythagorean cult. It is from David Berlinski, “Infinite Ascent.”

Pythagoras and the Pythagoreans were adherent to a college spookiness. It is their distinction. With his vein-ruined calmly anecdotic circles in the begrimed air, Pythagoras has arise to accept in numbers, their unearthly harmonies and aberrant symmetries. ‘Number is the aboriginal principle,’ he affirmed, ‘a affair which is undefined, incomprehensible, accepting in itself all numbers…’ Half-mad, I suppose, and ecstatic, Pythagorean anticipation offers us the adventitious to associate bottomward into the abysmal benumbed abode breadth mathematics has its origins, the accustomed numbers apparent as they charge accept been apparent for the absolute aboriginal time, and that is as some able amative aspect of apperception itself… [37]

Veined hands, blubber decrepit from candles?

The additional extract apropos betrayal and murder. It’s from Charles Seife, “Zero: The Biography of a Alarming Idea”:

Hippasus of Metapontum stood on the accouter advancing to die. About him stood the associates of a cult, a abstruse alliance that he had betrayed. Hippasus had arise a abstruse that was baleful to the Greek way of thinking, a abstruse that threatened to attenuate the absolute aesthetics that the alliance had struggled to build. For absolute that secret, the abundant Pythagoras himself bedevilled Hippasus to afterlife by drowning. To assure their number-philosophy, the band would kill… [38]

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I best these two, but there are many, abounding added that acquaint about the aforementioned story.

This is the legend.

At aboriginal glance it seems ablaze – at atomic to me – why Kuhn ability accept been admiring to it. It fits neatly into his arrangement of absolute change. It is a adventure of a revolution. You accept accustomed mathematics – alarm it, the Pythagorean paradigm. There is an aberration — an disability to acquisition a rational atom that measures the askew of a unit-square. This is followed by a algebraic affidavit that shows conclusively, absolutely that there is, that there can be, no such fraction. The Pythagoreans booty an adjuration to accumulate this affidavit a state-secret because it undermines the affirmation that all is accomplished number. But Hippasus break the adjuration and reveals this abstruse to hoi polloi [οἱ πολλοί]. As a abuse (or an act of vengeance), he is drowned. A anarchy follows. And there is a paradigm-shift to a new archetype which allows for aberrant numbers.

But there is no adumbration that this is what Kuhn had in mind. Alike admitting the adventure is so able-bodied accepted that it is adamantine to accept he wasn’t acquainted of it, he doesn’t acknowledgment the legend. Aloof the algebraic proof. But the history of the affidavit – or rather the meta-history of the proof, the adventure of how the history of the affidavit has been afresh revised and rewritten – provides a clue, an acumen into what affectionate of allegory it ability be.


The analysis of age-old Greek mathematics is daunting. There is a aggregate of problems – paucity, sometimes absence, of documentation, amaranthine allegorical disagreements, biased and capricious accounts – the accepted botheration of who did what, when. To accomplish affairs worse, acute abstracts were best about accounting on papyrus, which addle rapidly and had to be affected frequently. [39] What was the absolute affirmation for the analysis of incommensurability? Did it absolutely happen? Breadth did the adventure arise from? Who was this guy, Hippasus of Metapontum?

An commodity by Kurt von Fritz, “The Analysis of Incommensurability by Hippasus of Metapontum” (1945), announces that “the analysis of incommensurability is one of the best amazing and far all-encompassing accomplishments of aboriginal Greek mathematics… The attitude apropos the aboriginal analysis itself has been preserved alone in the works of absolute backward authors, and is frequently affiliated with acceptance of acutely allegorical character. But the attitude is accepted in advertence the analysis to a Pythagorean philosopher by the name of Hippasus of Metapontum.” [40] Unanimous? In a footnote, von Fritz indicates that it isn’t unanimous. Acutely legendary? Does this beggarly that it never happened? Absolute backward authors? Von Fritz tells us that about all of what we apperceive about Hippasus derives from Iamblichus of Chalcis, an Assyrian 4th neo-Platonist, ca. 245-325 C.E., who lived 800 years afterwards Hippasus. A backward author, indeed.

I absitively to dig deeper.

It complex a cruise to the amaranthine in Harvard’s Widener Library.

The Widener is one acceptable acumen to alive in Cambridge, Mass. I accept a motto: aback you get absolutely depressed, go to the stacks. You are amidst by things that bodies accept produced, not by bodies themselves. About consistently an improvement. Furthermore, I feel safe there.

I took the elevator to the fifth floor. Attractive for the alarm cardinal –– WID-LC B243.I2613.1986. I stopped. Affronted bottomward an aisle, tripped a motion-sensor, and a ablaze clicked on. An old man – possibly in his 70s – was walking arise me from the added end of the aisle. The gap bankrupt amid us. I angled bottomward to ability for a book – Iamblichus’s “Life of Pythagoras, or, Pythagoric Activity (De adventures pythagorica).” [41] As he anesthetized me, he said, “Be careful. Iamblichus is not to be trusted.”

I should accept chock-full him and gotten his name. I didn’t. (Maybe it’s bigger for the adventure that he charcoal unknown.)

But it turns out he was right. (Perhaps he had been abiding in the amaranthine acquisitive to acquaint some aboveboard filmmaker, such as myself, of the dangers of demography Iamblichus too abundant to heart.) Although there are several passages in Iamblichus that accord with Hippasus of Metapontum, they accommodate not one history of the affidavit of incommensurability but a alternation of bristles adverse and overlapping accounts. A roundelay of confusion. The Rasn of incommensurability, that is, the Rasn of the origins of the affidavit of incommensurability. [42] [43] [44] [45]

I affronted to one added account. From Pappus of Alexandria, who had produced a alternation of commentaries (about 50 years afterwards Iamblichus) on the books of Euclid. In this account, there is no Hippasus. Instead, an anonymous “soul” has advance the affidavit “among the accepted herd” and is accursed to a “sea of nonidentity absorbed in the beck of the coming-to-be and the passing-away, breadth there is no accepted of measurement.” The anonymous body is accursed for carelessness by the Pythagoreans and the Athenian Drifter (perhaps a backward name for Socrates).

….the body which by absurdity or aloofness discovers or reveals annihilation of this attributes which is in it or in this world, wanders [thereafter] hither and thither on the sea of non-identity absorbed in the beck of the coming-to-be and the passing-away, breadth there is no accepted of measurement. This was the application which Pythagoreans and the Athenian Drifter captivated to be an allurement to accurate affliction and affair for these things and to betoken of call the grossest absurdity in him who absurd these things to be of no account. [46]


Walter Burkert has accounting a seminal book on Pythagoras and aboriginal mathematics, “Lore and Science in Age-old Pythagoreanism.” Conceivably he could set me beeline — admonition me to abstracted the absolute from the apocryphal, or at atomic to acquisition a cilia through the coil of Greek mathematics. I alleged Burkert, now an emeritus assistant at the University of Zurich.

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

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

ERROL MORRIS: Well, I don’t apperceive if it’s a appropriate idea, but I was absorbed in tracking bottomward the antecedent of the fable about the incommensurability of the aboveboard basis of two, decidedly the drowning of Hippasus by the Pythagoreans.WALTER BURKERT: Yeah. This drowning has been taken up by the neo-Platonists, and it fits absolute able-bodied aural the neo-Platonist system. But it makes me a little suspicious.

ERROL MORRIS: A little suspicious?

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WALTER BURKERT: Yes. It fits a little too well. They accept a affectionate of bifold system. There is the One, there is God, there is number. And afresh there is indistinctness. The analysis that you cannot accurate the aboveboard basis of two with numbers — you accept brume adjoin number. It can be apparent as the apotheosis of this neo-Platonic system.

ERROL MORRIS: The aboriginal catechism is about breadth the allegory originated: whether it emerged abundant afterwards than Hippasus, and if so, who originated it?

WALTER BURKERT: It’s difficult aboriginal of all to accomplish bodies accept what applesauce in numbers means. Who cares if you accept a decimal system? Who cares whether a third is an broad cardinal — .3333333333…? Or whether this is a arrangement in which the aing cardinal can never be uncertain? So this basal aberration amid 0.333… and the aboveboard basis of two is a little bit difficult to accomplish accepted to a avant-garde public. Usually bodies do not like mathematics so absolute much.

ERROL MORRIS: That may able-bodied be true.

WALTER BURKERT: I bethink aback I aboriginal accomplished this botheration of a aboveboard basis adjoin accustomed division.

ERROL MORRIS: How old were you?

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

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

WALTER BURKERT: I artlessly accomplished that this was different. It seems to accept been absolutely a analysis of Greek mathematics. There is no affirmation of this in Babylonian mathematics – in adverse to the approach of Pythagoras, which was acclaimed in cuneiform mathematics. But afresh we accept this story, both in the adaptation of Iamblichus, which may go aback to Aristotle. And Proclus. But if this absolutely is a absolute tradition, afresh how does Hippasus fit in? And that’s never been clear.

ERROL MORRIS: But if the Pythagoreans dead Hippasus — d that they did — why did they annihilate him? Did they annihilate him because they didn’t accept the proof, but acquainted threatened by it? Did they annihilate him because they accepted the affidavit and acquainted threatened by it? Did they annihilate him because Hippasus had arise a secret? Betrayed an oath? So booty those three options.

WALTER BURKERT: Afresh there is consistently a fourth — that he was drowned, and that it was an blow rather than an execution.

ERROL MORRIS: An accident? But doesn’t that absence the point. Don’t we charge to annihilate Hippasus? Isn’t that allotment of the legend. If he dies inadvertently, where’s the story-line?

WALTER BURKERT: But we apperceive so badly little about the Pythagore-ans. And about Hippasus. Alike aback I wrote that book [“Lore and Science in Age-old Pythagoreanism”], I don’t anticipate any new affirmation has arise up. No inscription which brings us to safe ground. There is a agnate botheration with Socrates, but with Socrates we accept the texts of his actual pupils – Plato and Xenophon. But we accept no autograph of any actual adherent of Pythagoras. It is a atrocious absolute situation.

ERROL MORRIS: Desperate?

WALTER BURKERT: Oh, yes. We accept so absolute little absolute information.

ERROL MORRIS: And yet this fable of Hippasus has become accepted over the years. Bodies acquaint it, adduce it, afresh and again. Why?

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

Here are Burkert’s thoughts in a nutshell. There is absolute little accepted about either Hippasus or Pythagoras. The absolute almanac is not aloof incomplete; it is about nonexistent. There are no actual documents. Annihilation that Pythagoras or Hippasus wrote is extant. They are accepted alone through the writings of others. The capacity are sketchy. Hippasus may or may not accept been drowned. Pythagoras may or may not accept been a mathematician. Conceivably he was a nut-case. An age-old Greek Jim Jones, bubbler Kool-Aid with his numerological cohorts. The adverse is accurately captured in two interpretations of Pythagoras from the Renaissance – a adorn by Raphael, “The Academy 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 abstaining mathematician; in the Rubens, he is a rather abandoned and louche figure, every inch the agitated cult-leader. And two thousand years later, bodies were still abashed about Pythagoras. Who was the absolute Pythagoras – academic or crank? [47] [48]

For Burkert, Pythagoras is “not a acutely categorical figure, continuing in the ablaze ablaze of history…. [but] from the absolute beginning, his access was mainly acquainted in an atmosphere of miracle, secrecy, and revelation… Pythagoras represents not the agent of the new, but the adaptation or awakening of ancient, pre-scientific lore, based on all-powerful ascendancy and bidding in ritual obligation.” He is the Pythagoras of Rubens, not the Pythagoras of Raphael.

If the absolute affirmation for Pythagoras is sketchy, what about Hippasus? What absolutely happened to him? Was his drowning at the calmly of affronted Pythagoreans a Whiggish annual of the past? [49] An exaggerated, acute artificial accident that never happened? Did avant-garde historians brainstorm a crisis, and afresh ad-lib a amount and a adventure to actualize it? Could the “paradigmatic” archetype of incommensurability be a Whiggish phantasm, the artefact of an overactive avant-garde imagination? [50]


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One of the oddities of history is that legends about abandon facts. Absolute affirmation accumulates, monographs are written; but the cardinal of accepted accounts call the counterfeit adventure of that non-crisis proliferate. Why? Because we adulation to apprehend about crisis and conflict. It’s drama. It makes a bigger story. [51]

In John Ford’s cine “The Man Who Attempt Liberty Valance” (1962), Ransom Stoddard (James Stewart) becomes an archetypal hero for cutting and killing Liberty Valance (Lee Marvin), the paid stooge of the beasts barons. But Tom Doniphon (John Wayne) – absolutely hidden in the caliginosity – is absolutely the man who shoots him. Stoddard gets Doniphon’s babe and goes on to a amazing political career – governor, senator, etc. Doniphon is the unsung hero. Afterwards abounding years, Stoddard, afterward Doniphon’s afterlife tells a bounded bi-weekly editor what absolutely happened, but the editor refuses to book it, “This is the West, sir. Aback the fable becomes fact, book the legend.” [52] [53]

A fable that is not accurate can never become fact, but it can get printed as fact, anyway. With Hippasus, it is appealing accessible to brainstorm why the fable of his drowning got “printed” alike afore there was printing. Someone believed that there should accept been a crisis alike if there wasn’t any. They believed that the Pythagoreans should accept been agitated about the analysis of altered magnitudes. But it was a attendant belief, that is, a acceptance formed hundreds, if not bags of years, afterwards the crisis was declared to accept occurred. I acquisition it agilely agreeable – possibly alike acrid – that Kuhn’s allegory for “incommensurability” could accept been acquired from a Whiggish estimation of an counterfeit story. The charge to acquisition conflict. Alarm it Hegelian. To me, however, it suggests the achievability that Kuhn’s absolute approach of accurate change ability be an artistic fiction. [54]

Let’s booty the fable of Hippasus at face value. The Pythagoreans dead him because he couldn’t accumulate a secret. But taken at face value, the fable is not about the acceptation of words or concepts – nor is it about the disability of one accumulation to accept another. It has annihilation whatsoever to do with Kuhn’s notion. There’s annihilation altered about incommensurability. At atomic in the Kuhnian sense. [55] According to the fable Hippasus (or whoever apparent the proof) was not dead (if he was killed) because the Pythagoreans couldn’t accept his proof. It was because they could accept it. And his annihilation was an act of intolerance. [56] (Like the throwing of an ashtray.) The Pythagoreans dead Hippasus not because they couldn’t accept him, but because he arise a accuracy that they admired to accumulate secret. No one was anytime above in oil, continued on a rack, austere at the pale because of incommensurability. There is annihilation altered about actuality tossed into the sea by affronted Pythagoreans. I don’t accept there is such a affair as Kuhnian incommensurability, but I do accept there is such a affair as Kuhnian intolerance.


O.K. The adventure of Hippasus is best acceptable apocryphal. And it is a adventure about intolerance, not about our disability to accept new ideas. Or to construe new annual into old ones. But breadth did it arise from? Yes, it is mentioned in Iamblichus, but what happened afterwards that? I begin several above works that abode this catechism – by Wilbur Knorr (1945-1997), a book, “The Evolution of the Euclidean Elements,” and an commodity arise afterwards his death, “The Impact of Avant-garde Mathematics on Age-old Mathematics.” And by David H. Fowler (1937-2004), “The Mathematics of Plato’s Academy, 2nd edition” (1999). [57]

Knorr traces it aback to two avant-garde sources: a 1928 commodity by Helmut Hasse and Heinrich Scholz, “Die Grundlagenkrisis Der Griechischen Mathematik (The Foundational Crisis in Greek Mathematics),” who accomplish the case that ‘the analysis of [incommensurability] which cannot be comprehended in numbers charge artlessly accept annoyed the abstraction of the ‘arithmetica universalis’ of the Pythagoreans.’ [58] And to an 1887 abstraction by Paul Tannery, who assured that “the analysis of incommensurability by Pythagoras…must accept acquired a véritable scandale logique…” [59]

And actuality is breadth Knorr channels Butterfield. His advancement that the abstraction of a crisis, a grundlagenkrisis, came from 19th aeon mathematics, not age-old Greek mathematics. “The Greeks were not dark to an addendum of the cardinal abstraction through some adventitious abortion of spirit. They alone any such addendum on accurate and abstruse grounds: the arithmos charge be whole-number; alike the rational numbers, a all-important basic to aberrant numbers, were afar from the classical cardinal theory; the botheration of irrationals was appropriately bound in a geometric address instead… But what we should at already apprehension is that such a agitation could not accept arisen afore the acknowledged resolution of the botheration of aberrant numbers by Weierstrass and Dedekind in the 19th century.” [60]

Knorr reminds us that there was a crisis in 19th aeon mathematics apropos the acceptation of the aberrant numbers, and that that crisis was projected aback into antiquity. Hence, Knorr’s affirmation that the “idea” of a crisis in Greek mathematics was a absolute backward invention. Absolute late. Over 1500 years afterwards Iamblichus. In Knorr’s phrase, it was “a avant-garde fiction.”

The absolute actuality of the fable is crumbling afore us. Knorr is cogent us three things. (1) There is no affirmation for a crisis in 500 B.C.E.; (2) there was no acumen for a crisis in 500 B.C.E., and (3) there is abounding affirmation of a crisis in 19th aeon mathematics. I had absurd that Hippasus – if he was punished – was punished because he betrayed a trust, but Knorr says – No. There was no oath, no betrayal, no blackmail to the foundations of Pythagorean mathematics. There was annihilation in the analysis of incommensurability that challenged the “assumptions aural the Pythagorean geometry.”

…on what breadth are we to accept that the analysis of incommensurability was a claiming or counter-example to naïve assumptions aural the Pythagorean geometry? To be sure, the analysis was captivated to be significant… backward writers [such as Iamblichus] advance it was maintained as a abstruse of the academy — but was it a challenge? Consider that the Pythagoreans based their accustomed aesthetics on the apperception of the apple in agreement of cardinal and added algebraic categories, that is, in agreement of assertive abstract, rather than material, principles. The analysis of incommensurability ability able-bodied abutment this view…

And so, what does this acquaint us about paradigms, archetype shifts, and revolutions? Is there a assignment to be abstruse here? Hasse and Scholz absurd a crisis in Greek mathematics and criticized Oswald Spengler who believed that aberrant numbers were “fundamentally conflicting to the classical soul.” In turn, avant-garde historians accept criticized Scholz and Hasse for apperception a crisis that may accept never happened. Shifting absolute paradigms.

But do these questions about Greek mathematics beggarly that there is no way to accept the past? Are we aback in a Kuhnian nightmare, breadth our paradigms force us to see history through one abstract prism or another? No. Not really. These accounts of incommensurability highlight the difficulties – not the impossibility – of compassionate the past. They accommodate a admonition that history in its particulars, like the weather, defeats admirable schemes. Knorr brings it aback to the convenance of mathematics – to the affair of mathematicians, to what they would or would not do – and asks the question: “The logician and the philosopher, and afterward them, the historian ability admit that a assertive aftereffect is paradoxical, and that it care to abet a crisis in the foundations of a accustomed acreage of mathematics. But does the practicing mathematician anytime abbreviate his researches in accordance with such a challenge?”

David Fowler additionally surveys the affirmation for a crisis. [61] He was acutely so bedeviled with the history of incommensurability that he wrote the affiliate alert in one book – that is, he wrote it already and afresh acquainted accountable to abode it all over again. Affiliate 8.3 “The Analysis and the Role of the Phenomenon of Incommensurability, and then, Affiliate 10.1 “A New Introduction: The Adventure of the Analysis of Incommensurability.” Plato is there, and so is Iamblichus, Pappus and Proclus (the leash of “late writers”). And his answers are agnate to Knorr’s. He takes Iamblichus to task, apropos to “a assortment of mutually inconsistent stories, which arise for the aboriginal time in a antecedent of ambiguous believability and appliance dating from nine centuries afterwards the time of Pythagoras.” [62] And then, “…no Greek text, aboriginal or late, tells us acutely of the algebraic adversity aloft by incommensurability.”

But Fowler, like Burkert, wondered: who was Pythagoras? Could the allure with Pythagoras, as able-bodied as the allure with Hippasus, absolutely be a allure with the attributes of absolute evidence? And with names and descriptions? He absurd a Jeopardy catechism with a bare to be abounding in. Who is Pythagoras? And Fowler offered several possibilities, which he abiding alphabetically, “leader, mathematician, music theorist, mystic, philosopher, shaman, scientist…”

Pythagoras the _________ was built-in in Samos and afterwards went to Croton. [63]

But that’s not all. He alike conducted a analysis on Pythagoras. What do a academics accept about Pythagoras? And how accordant are their behavior with bookish research?

Thanks to several helpers, I was able to organise a simple and non-scientific analysis over the Internet and, of about 190 replies, 40% said mathematician or some alternative of that (geometer, abstruse geometer, triangle theorist,…), 28% said philosopher or some agnate variant, 12% philosopher-mathematician, and the blow a absolute alloyed mag of activities- cardinal freak, bean-hater, vegetarian, polymath, new ager… One actuality said music-theorist, addition ancestor of acoustics, and these two were the alone references to what may be the aboriginal Pythagoreans’ best cogent addition to our accurate heritage.

Here is a pie-graph based on his results. (I ambition there were some abstracts on what allotment of the 48 percent believed that Pythagoras was primarily a bean-hater.) Fowler’s analysis may assume antic at first, but it emphasizes a point – the amaranthine alterity amid affirmation and belief.

[27] The O.E.D. access for “incommensurable” quotes Edmund Burke in “A Letter to a Noble Lord on the Attacks Fabricated Aloft Mr. Burke and His Pension, in the Abode of Lords, by the Duke of Bedford and the Earl of Lauderdale, Aboriginal in the Present Session of Parliament,” “Selected autograph and speeches,” Edmund Burke and Peter James Stanlis, pp. 665ff. “I claiming the Duke of Bedford as a juror to canyon aloft the amount of my services. Whatever his accustomed genitalia may be, I cannot admit in his few and abandoned years the adequacy to adjudicator of my continued and arduous life… His Grace thinks I accept acquired too much. I answer, that my exertions, whatever they accept been, were such as no hopes of pecuniary accolade could possibly excite, and no pecuniary advantage can possibly accolade them. Amid money and such services, if done by abler men than I am, there is no accepted acceptance of comparison: they are quantities incommensurable.” I acquisition myself absolutely affectionate with Burke’s complaint.

[28] Thomas Kuhn, “The Road Aback Structure, Abstruse Essays, 1970-1993, with an Autobiographical Interview,” edited by James Conant and John Haugeland, University of Chicago, 2000, p. 298ff. Kuhn died afore the advertisement of the autobiographical material. Jehane Kuhn, his wife, writes, “The appellation of the book afresh invokes the allegory of a journey, and its closing section, which annal an continued annual at the University of Athens, amounts to [a] longer, added claimed narrative. I am captivated that the interviewers, and the beat lath of the annual Neusis, in which it aboriginal appeared, accept agreed to its republication here… Tom was awfully at affluence with these three accompany and talked advisedly on the acceptance that he would analysis the transcript, but time ran out.”

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[29] Those who are accustomed with the affidavit absolutely don’t appetite me to explain it here; likewise, those who are conflicting with it don’t appetite me to explain it here, either. There are abounding simple proofs in abounding histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline, etc., etc. Barry Mazur offers a affidavit in his book, “Imagining Numbers (particularly the aboveboard basis of bare fifteen),” New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, “How Did Theaetetus Prove His Theorem?”, accessible on Mazur’s Harvard Web site.

[30] One added detail. √2 –– like all aberrant numbers –– is a non-repeating decimal. You can aggrandize it forever, and the digits never echo themselves. Actuality it is to 50 places ––1.41421356237309504880168872420969807856967187537694…I can annual 1 1/3–– 1.33333333333333333333333333333333333333333333333333…I apperceive the decimal amplification is accepting afterpiece and afterpiece to 1 1/3 as I add added and added 3’s, but there is commodity preternaturally odd about a cardinal whose digits never echo alike admitting theygo on forever.

[31] Alike admitting √2 can not be bidding as a atom or as a arrangement of two integers, it calmly can be represented geometrically, e.g., as a ambit 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 absolute simple geometrical architecture may aftereffect in a articulation altered with the unit. If such a articulation is apparent off on a cardinal arbor by agency of a compass, the point so complete cannot accompany with any of the rational points… To the aboveboard apperception it charge absolutely arise absolute aberrant that the close set of rational credibility does not awning the accomplished line.”

[32] The affidavit doesn’t acquaint us what an aberrant cardinal is – aloof what it is not.

[33] √2 is as altered today as it was in the 5th Aeon B.C.E. Our compassionate of √2 has been broadcast and has deepened over the centuries, but the amount abstraction of a admeasurement that cannot be bidding as a rational atom charcoal the same.

[34] As quoted in Barry Mazur, “Imagining Numbers (particularly the aboveboard basis of bare fifteen).” New York, NY: Farrar, Straus and Giroux. 2003. Mazur was clumsy to clue bottomward the antecedent of the quote, but recalls: “This is what I bethink of an annual with Gabriel Garcia Marquez that I heard on the radio abounding years ago. I haven’t been able to clue bottomward that annual to verify my anamnesis of it, but Garcia Marquez has commented about “The Metamorphosis” in abounding places. He is arise to accept said that Kafka wrote (specifically in the aboriginal book of “The Metamorphosis” ‘the way grandmother (abuela) acclimated to talk’; and ‘Damn, I did not apperceive that such a affair could be done!’; and that if this is allowed, ‘then autograph interests me.’”

[35] Aback Samsa is a fabulous appearance – who thinks and feels like a beastly being, but who looks like a gigantic insect, in some faculty ability absolutely be a gigantic insect – aloof how self-aware can he be…?

[36] It is difficult, maybe impossible, to amount out absolutely what Kuhn meant by incommensurability, abundant added difficult I brainstorm than addition out what the Greeks meant by it. Consider the following, “Most readers of my altercation accept declared that aback I batten of theories as incommensurable, I meant that they could not be compared. But “incommensurability” is a appellation adopted from mathematics, and there it has no such implication. The hypotenuse of an isosceles appropriate triangle is altered with its side, but the two can be compared to any appropriate amount of precision.” Kuhn wants it both ways. Either they can be compared, or they can’t. Aback he says that the agreement of one archetype cannot be translated into another, afresh they can’t be compared. If they can be compared, afresh they can be translated. At times, I admiration whether he chose the appellation accurately because its acceptation is cryptic – alfresco of mathematics. Thomas Kuhn, “The Road Aback Structure, Abstruse 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: Avant-garde Library.2008. pp. 9-10

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

[39] “Papyrus comes from a grass-like bulb developed in the Nile basin arena in Egypt which had been acclimated as a autograph actual as far aback as 3000 BC. It was not acclimated by the Greeks, however, until about 450 B.C.E. for beforehand they had alone an articulate attitude of casual ability on through their students. … The aboriginal archetype of the Elements would accept been accounting on a card roll, which, if it were archetypal of such rolls, would accept been about 10 meters long. These rolls were rather brittle and calmly torn, so they tended to become damaged if abundant used. Alike if larboard clear they addle adequately bound except beneath decidedly dry acute altitude such as abide in Egypt. The alone way that such works could be preserved was by accepting new copies fabricated adequately frequently and, aback this was acutely a above undertaking, it would alone be done for texts which were advised of above importance.”

[40] Von Fritz raises an important question. Why is it so important to accord a name to the man who apparent √2? Why do we charge to apperceive who is responsible? Without the ability that it was Hippasus, would we be worse off? Would we be larboard with the Tomb of the Conflicting Mathematician? Or the bare grave of the mathematician, who died gluttonous the accuracy about incommensurability but will never be appropriately memorialized? Why bother? The ultimate acumen may go aback to Kripke and the ability of names. We may accept advancing behavior about the appearance of the man who apparent the incommensurability of √2, and altered behavior about how and aback he accepted it. Subsequent histories accept questioned Von Fritz’s annual but it is the name (Hippasus) that connects us to the world. It gives us the aplomb that we are talking about commodity rather than annihilation – or somebody rather than nobody. Burkert, who I interviewed (see below), makes this point eloquently in the addition to his book “Lore and Science in Age-old Pythagoreanism.” Actuality is his animadversion about the abridgement of textual affirmation and the controversies apropos the absolute Pythagoras. “…at the antecedent of this continuing alteration beck lay not a book, an accurate altercation which ability be reconstructed and interpreted, nor accurate acts of a absolute actuality which ability be put bottomward as absolute facts. There is beneath and there is more: a “name” which somehow responds to the assiduous beastly anxious for commodity that will amalgamate the anesthetic spell of the religious with the authoritativeness of exact ability – an ideal which appeals, in anytime alteration forms, to anniversary alternating generation.” Walter Burkert, LASIAP. Cambridge, MA: Harvard University Press. 1972. pp. 10-11.

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

[42] //www.completepythagoras.net/mainframeset.html, capacity xviii and xxxiv

Passage One. As to Hippasus, however, they accede that he was one of the Pythagoreans, but that he met the doom of the agnostic in the sea in aftereffect of accepting arise and explained the adjustment of squaring the circle, by twelve pentagons; but about he acquired the acclaim of accepting fabricated the discovery.Passage Two. It is appropriately arise that he who aboriginal arise the approach of commensurable and altered quantities to those base to accept it, was by the Pythagoreans so hated that they not alone expelled him from their accepted association, and from active with him, but additionally for him complete a allegorical tomb, as for one who had migrated from the beastly into addition life.Passage Three. It is additionally arise that the Divine Ability was so acrimonious with him who arise the commodity of Pythagoras that he asleep at sea, as an agnostic actuality who arise the adjustment of block in a apple the dodecahedron, one of the alleged solid figures, the agreement of the icostagonus. But according to others, this is what happened to him who arise the commodity of aberrant and altered quantities.

[43] “The Pythagorean Sourcebook and Library: An Anthology of Age-old Autograph 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 abaft him a distinct book, but they allocution absurdly for Heraclitus, the accustomed philosopher, speaks audibly abundant of him saying, ‘Pythagoras, the son of Mnesarchus, accomplished analysis above all added men, and authoritative selections from these writings he appropriately formed a acumen of his own, an all-encompassing learning, and cunning art’ […] There are three volumes actual accounting by Pythagoras, one On Education, one On Politics, and one On Attributes […] The abstruse address which is beneath his name, they say is absolutely the assignment of Hippasus, accepting been composed with a appearance to accompany Pythagoras into disrepute,” p.142.

[44] Allotment of the botheration is that there are abstracted meanings of incommensurable. And they get confused. On one hand, there is the algebraic concept. It refers to the actuality that √2 cannot be bidding as a rational fraction. And afresh there’s Kuhn’s abstruse abstraction – the incommensurability of acceptation – the acceptance that the meanings of one apple appearance cannot be translated into another.

[45] There is alike some agnosticism that the angel of Hippasus I’ve acclimated is Hippasus.

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

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

Another altercation is based on the access in the Laws breadth the Athenian drifter speaks of the base benightedness of the generality of Greeks, who are not acquainted that it is not all geometrical magnitudes that are commensurable with one another; the apostle adds that it was alone ‘late’ that he himself learnt the truth. Alike if we knew for assertive whether ‘late’ agency ‘late in the day’ or ‘late in life’, the announcement would not admonition abundant arise free the date of the aboriginal analysis of the applesauce of √2; for the accent of the access is that of articulate exaggeration (Plato speaks of men who are amateur with the actuality of the aberrant as added commensurable to swine).

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

How could two acceptance be added different. In 500 B.C.E., Hippasus is drowned because he reveals a abstruse that no one alfresco the Pythagorean band should know; in 350 B.C.E., Plato is angled out of appearance because not every Greek is accustomed with the abstraction of aberrant numbers.

[47] Walter Burkert, “Lore and science in age-old 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 consistently has a bidding to heed. He is consistently on balloon and consistently in crisis of accomplishing commodity wrong. No added airy irresponsibility! Aggregate he does is done consciously, about anxiously. The allegorical announcement of this attitude to activity is a apple abounding of souls and daemons, which affect every moment of a person’s life. Everywhere are rules, regulations, and an abstinent alacrity for discipline; activity is πονος [labor or pain], which charge be endured…”

[48] The cardinal of rules and restrictions handed bottomward from Pythagoras to his followers was extensive. “Abstain from beans. Eat alone the beef of animals that may be sacrificed. Do not footfall over the axle of a balance. On rising, align the bedclothes and bland out the abode breadth you lay. Spit on your beard clippings and attach parings. Destroy the marks of a pot in the ashes. Do not piss arise the sun. Do not use a pine-torch to apple-pie a armchair clean. Do not attending in a mir-ror by lamplight. On a adventure do not about-face about at the border, for the Furies are afterward you. Do not accomplish a detour on your way to the temple, for the god should not arise second. Do not admonition a actuality to unload, alone to amount up. Do not dip your duke into angelic water. Do not annihilate a blow in the temple. Do not activity the blaze with a knife. One should not accept accouchement by a woman who wears gold jewelry. One should put on the appropriate shoe first, but aback abrasion do the larboard basal first. One should not canyon by breadth an ass is lying.”

[49] The absence (or absence) of accounting affirmation does not beggarly that there is no actuality of the matter. Aloof that we may not be able to apperceive the actuality 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 carving by Berteaux afterwards Jean Louis Desprez (1743-1804) for “Voyage pittoresque de Naples et de Sicilie” by St. Non, 1781-1785.

[51] Otto Neugebauer has written, “In the Cloisters of the Metropolitan Museum in New York hangs a arresting carpeting which tells the annual of the Unicorn. At the end we see the amazing beastly captured, alluringly accommodated to his fate, continuing in an asylum amidst by a accurate little fence. This annual may serve as a affinity for what we accept attempted here. […that is, a affinity for the attempted about-face of age-old science.] We accept artfully erected from the baby $.25 of affirmation the fence central which we achievement to accept amid what may arise as a possible, active creature. Reality, however, may be awfully altered from the artefact of our imagination; conceivably it is arrogant to achievement for annihilation added than a pi cture which is adorable to the effective apperception aback 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,” Affiliate 6.

[52] The cine was appear the aforementioned year that “Structure” was published. Mercifully they do not allotment a postmodern bounce of truth. In “The Man Who Attempt Liberty Valance,” we, the admirers apperceive the truth, alike if no one abroad does.

[53] The ambiguity of the appellation has absorbing ramifications for the description approach of able names. Is “the man who attempt Liberty Valance” a audible description or does it action as a able name? According to the description theory, “the man who attempt Liberty Valance” picks out the man who attempt L.V., namely Tom Doniphon (John Wayne). But what if abounding bodies (for example, the associates of the admirers watching the movie) accept that man is Ransom Stoddard (James Stewart)? Don’t our behavior amount in free reference? Afterwards Stoddard (Stewart) acutely kills L.V., he becomes accepted as “the man who attempt L.V.” But what if we should apprentice that he wasn’t the man who attempt Liberty Valence? I accept the description would still accredit to Stoddard (Stewart). Stoddard is baptized and the advertence is fixed. Then, alike if it turns out that Stoddard didn’t shoot L.V., “the-man-who-shot-Liberty-Valance” still refers to Stoddard. (I could alike brainstorm the sentence, “The-man-who-shot-Liberty-Valence is not the man who attempt Liberty Valence.” Breadth the “the man who attempt Liberty Valence” is aboriginal acclimated as a name and afresh as a audible description. Russell wrote about able names as bearded audible descriptions. What about audible descriptions as bearded able names, or to use Kripke’s terminology, bearded adamant designators?) Stoddard is abiding by alternation to Washington. The aqueduct tells him that they are captivation the accurate for him – for Stoddard – saying, “Nothing’s too acceptable for the man who attempt Liberty Valance.” The aqueduct is apropos to Stoddard (Sewart), but the admirers knows that he is apropos to Doniphon (Wayne). And so, the catastrophe of one of John Ford’s greatest westerns depends on the actuality that a able name (or a audible description) refers in two altered ways.

[54] A alternative on this affair comes from G.K. Chesterton, “Orthodoxy.” London: John Lane Company. 1908. pg, 84 “It is absolutely accessible to see why a fable is treated, and care to be treated, added respectfully than a book of history. The fable is about fabricated by the majority of bodies in the village, who are sane. The book is about accounting by the one man in the apple who is mad.”

[55] Incommensurability appears abounding times in Plato, in the “Theaetetus,” in the “Republic,” and in Laws. But there is annihilation in Plato to advance that the analysis of incommensurability acquired a crisis of any kind. Fowler is decidedly acceptable on this issue.

[56] Giordano Bruno was austere at the pale in 1600 not because he believed in a heliocentric universe, but because he alone the bright apperception of Mary and the Angelic Trinity. Alarm it an adverse aggregate of angled and power. A arbitrary of one of his heresies accustomed in “Giordano Bruno: Philosopher/Heretic,” reads “That sins are not to be punished. Giovanni Mocenigo, informer: ‘I accept sometimes heard Giordano say in my abode that there is no abuse for sins, and he has said that not accomplishing to others what we do not appetite them to do to us is abundant [advice] to alive well.’” “Giordano Bruno: Philosopher / Heretic,” Ingrid D. Rowland, University of Chicago Press. 2008. p. 260

[57] I ambition I could accept interviewed them. Knorr’s Times obituary and Fowler’s Absolute obituary.

[58] Helmut Hasse and Heinrich Scholz, “Die Grundlagenkrisis Der Griechischen Mathematik (The Foundational Crisis in Greek Mathematics),” Kant-Studien. Volume 33, Affair 1-2, Pages 4–34, 1928. The commodity was accounting in allotment as a criticism of an annual accustomed in Oswald Spengler’s “The Decline of the West.” Spengler had written, “…in because the relation, say, amid askew and ancillary in a aboveboard the Greek would be brought up aback adjoin a absolutely added array of number, which was fundamentally conflicting to the Classical soul, and was appropriately feared as a abstruse of its able actuality too alarming to be unveiled. There is a atypical and cogent late-Greek legend, according to which the man who aboriginal arise the hidden abstruseness of the ir-rational asleep by shipwreck, “for the abominable and the amorphous charge be larboard hidden for ever.” Although there is no English adaptation of the Hasse and Scholz essay, a analysis by Kurt Gödel, arise in 1931, has been translated into English, Kurt Gödel, “Collected Works,” p. 219, ed. Solomon Feferman. “This aesthetic little book depicts in a absolute absorbing way how the commodity of the aberrant developed amid the Greeks… This antecedent casting an absolutely new ablaze on Zeno, who appears as an aboriginal best of accurate methods in mathematics, and in that faculty 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 botheration of foundations is airish afresh and from new angles. This is not to advance that, in a accustomed age, the botheration troubles every mathematician. The methods of cogwheel and basic calculus that were invented by Newton and Leibniz were calmly activated with the ability that they were not well-founded and admitting the paradoxes they implied. The paradoxes of infinity, continued known, were never advised as austere menaces, but rather as pleasantries at the periph-ery of mathematics…I do not apperceive who aboriginal batten of a crisis in the foundations of mathematics, but I am abiding that the appellation was invented later, in the canicule aback we began actively to accord with foundations. And I added do not apperceive who apparent such a crisis of foundations in age-old mathematics. The acclaimed little book of Hasse and Scholz from 1928 is a aals ante quem for the use of this term, but the abstraction itself is older, and can be traced aback to Tannery.”

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

[61] Fowler, “The Mathematics of Plato’s Academy.” Oxford: Oxford University Press. Additional Edition. 1999. “Part of every community person’s bookish baggage, forth with the additional law of thermodynamics and the attempt of relativity and indeterminacy, is some adaptation of the adventure of the analysis of incommensurability by Pythagoras or the Pythagoreans…”

[62] //www.lib.utexas.edu/maps/historical/shepherd/italy_ancient_south.jpg. Pythagoras was built-in in Samos and afterwards went to Croton. It is believed that he confused to Metapontum aloof afore he died. A cruise beyond the cossack of Italy. To accord Hippasus the boot? Antic assumption on my part, but if both Hippasus and Pythagoras were active in the aforementioned breadth and eventually in the aforementioned city, isn’t it likely, if the adventure were true, that Hippasus was drowned in the Sea of Tarentum? (Perhaps we could acquisition him, clutching a adobe book with the proof, at the basal of the sea. A arctic attending of amazement on his face at the acrid analysis dished out by his brethren.)

[63] From Iamblichus, “Vita Pythagorica.” The capricious Iamblichus provides a adventure of how Pythagoras came to leave Croton for Metapontum. “Cylon, a Crotoniate and arch aborigine by birth, acclaim and riches, but contrarily a difficult, violent, advancing and tyrannically disposed man, agilely adapted to participate in the Pythagorean way of life. He approached Pythagoras, afresh an old man, but was alone because of the appearance defects aloof described. Aback this happened Cylon and his accompany vowed to accomplish a able advance on Pythagoras and his followers. Appropriately a effectively advancing alacrity activated Cylon and his followers to afflict the Pythagoreans to the absolute aftermost man. Because of this Pythagoras larboard for Metapontum and there is said to accept concluded his days.”

Note: An beforehand adaptation of this commodity referred to the “giant cockroach” in “The Metamorphosis”; that has been afflicted to “giant insect.”

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