Pythagoras believed that "all is numbers" and they maintained that all numbers can be expressed as a fraction, then Hippasus (maybe) showed some numbers cannot be expressed that way. Pythagorus thought this idea was stupid and he had him killed, but this could not stop the eventual collapse of his school.

This question could be too subjective (I'll remove it if it is), but is this story untrue or too obscure to know owing to the secretive nature of his sect and its ancient nature?

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    $\begingroup$ I really don't know but I think they were doomed the moment they began mixing mathematics (or for that matter any other science) and religious hokus pokus, which would very easily take them into barbarian fanatism and ... $\endgroup$ – DonAntonio Nov 18 '13 at 15:58
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    $\begingroup$ ... as opposed to the more successful religious movements that prudently avoid mathematics. Clearly mathematics is a dangerous thing. $\endgroup$ – Robert Israel Nov 18 '13 at 16:00
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    $\begingroup$ That’s more dramatic than the story that is sometimes told and is very unlikely to be true: Híppasos and Pythagóras probably weren’t contemporaries. The fact is that we don’t really know much of anything about him for certain. Wikipedia seems to be reasonable on the subject. German WP suggests that the conflict between him and the Pythagoreans was political, not mathematical, and notes that modern scholarship finds no evidence that the Pythagoreans had a problem with irrationals. $\endgroup$ – Brian M. Scott Nov 18 '13 at 16:02
  • $\begingroup$ Tha Pythagoreans did want to keep the irrational number hidden for a while before they could "grasp" say √2. Until then everything was though to be rational. $\endgroup$ – imranfat Nov 18 '13 at 16:05
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    $\begingroup$ Brian M. Scott, I believe you are right. Aristotle wrote, "The elements of number, according to their (the Pythagoreans) theory, are the even and the odd, the former being "unlimited", the latter "limited". The One consists of both of these, partaking of the nature of both even and odd. Number derives from unity; and numbers, as we have said, constitute for them the entire visible universe." Metaphysics 986a 15 $\endgroup$ – Michael Lee Nov 19 '13 at 1:40

See the SEP entry about Pythagoras : from ca. 570 to ca. 490 BCE, and the entry on Pythagoreanism :

Pythagoreanism is the philosophy of a group of philosophers active in the fifth and the first half of the fourth century BCE :

Many other sixth-, fifth- and fourth-century thinkers are labeled Pythagoreans in the Greek tradition after the fourth century BCE. [...] There are nonetheless a number of thinkers of the fifth and fourth century BCE, who can legitimately be called Pythagoreans, although often little is known about them except their names. The most important of these figures is Hippasus [from Metapontum fifth-century BCE].

But we have to recorc also :

Aristoxenus (ca. 375- ca. 300 BCE) is most famous as a music theorist and as a member of the Lyceum, who was disappointed not be to named Aristotle's successor. In his early years, however, he was a Pythagorean, and he is one of the most important sources for early Pythagoreanism.

This shows us that the school was "alive and well" during Aristotle's time (384–322 BCE).

Thus, I think it is not correct to speak of "collapse of his school".


The school did not collapse at all. They simply continued expressing everything in terms of natural numbers and proportions among them. As late as the 19th century, Leopold Kronecker sought to do much the same thing, incredible though it might seem to us. You are probably familiar with how one could express statements about rational numbers in terms of integers. You can similarly express statements about real numbers in terms of rationals, so from a strictly logical viewpoint you don't actually need them. For example, every time you want to say that your favorite number $b$ is less than the irrational $\sqrt{2}$, just say $(\forall x\in\mathbb{Q})(x^2>2\;\to\; b<x)$.

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    $\begingroup$ But I think it is useful to discriminate between the "original" Pythagorean school (ancient Greece), the late Neopythagoreanism (with only a moderate interest in "real math : see Proclus' Commentary to Euclid's Elements) and the Renaissance (and beyond) use of the generic attribute of pythagorean for all doctrine involvinh mathematics : form astrology to Galileo and Kepler. $\endgroup$ – Mauro ALLEGRANZA May 15 '14 at 19:39
  • $\begingroup$ @Mauro, Good point. I wonder what you would make of Edward Nelson's take on Pythagorianism: web.math.princeton.edu/~nelson/papers/rome.pdf $\endgroup$ – Mikhail Katz May 15 '14 at 19:43
  • $\begingroup$ For sure, I agree with Nelson about: "My knowledge in this field is second-hand and superficial." I think that the use of the quite-legendary Pythagoras as a "source" in the (modern) debate about the phil of math is useless. We have no source of the historical P's thinking; we only have a "slogan": "reality" is made of number (Nelson: "classical math is founded on the picture of mathematical objects eternally existing in Platonic – or better, Pythagorean – reality." For sure it sounds mystic and "religiuos", but what about The Unreasonable Effectiveness of Mathematics in the Natural Sciences ? $\endgroup$ – Mauro ALLEGRANZA May 15 '14 at 19:54
  • $\begingroup$ @Mauro, Grattan-Guinness for one thinks that it is reasonable and not that effective; see ams.org/mathscinet-getitem?mr=2437195 $\endgroup$ – Mikhail Katz Aug 12 '14 at 10:18

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