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In the 6th century BC Pythagoras' followers knew that the square root of two was irrational (couldn't be expressed as the ratio of two integers). The story goes

by smanek 18y ago

In the 6th century BC Pythagoras' followers knew that the square root of two was irrational (couldn't be expressed as the ratio of two integers). The story goes that this messed with a lot of their theories so it was ordered to be kept a secret, and anyone who dared reveal the fact was killed. It seems kind of funny to me that they actually had a cult/religion based on math. Instead of just accepting the existence of irrationals and attempting to update their theories - they tried to suppress the new evidence that contradicted their teachings.

tome 18y ago
But what would "accepting the existence of the irrationals" mean? As far as the Greeks knew, a "number" was a/b where a and b were integers. "Accepting" that the length of a triangle was not a "number" could have all sorts of consequences, including rejection of their belief in validity of mathematical reasoning, so I'm not surprised they were scared of it.
- smanek 18y ago
  The Greeks did know about irrational numbers. They could draw a right triangle whose legs were 1 and 1. They knew that the length of the hypotenuse (a number that we call the square root of two) couldn't be represented as the ratio between any two integers, yet it had a length.
- bdr 18y ago
  Actually, length was primary since geometry was the foundation of their math.
antiform 18y ago
In the book The Heart of Mathematics by Edward Burger and Michael Starbird, there is a quotation of a translated passage from Proclus, a 5th-century AD philosopher: "It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."
jgrahamc 18y ago
Cantor was very troubled by his discovery of the transfinite numbers because they conflicted with the infinite nature of God (hence the name transfinite). Shouldn't be any surprise to anyone that when a theory conflicts with _beliefs_ there's trouble.
- deleted 18y ago
  [deleted]
- henning 18y ago
  I love how mathematicians use the term "discover" instead of "invent" the way anyone else would.
  - mkn 18y ago
    They use the term 'discover,' I would presume, because it's the most technically accurate term for what they do. As near as I can tell, as a relative layperson, mathematicians look at the existing state of mathematics, embodied in proven theorems, and consider the implications of those theories, looking for unanswered questions or outright contradictions. They then attempt to find the new theory that will provably answer the question or resolve the contradiction. The only 'invention' that goes on seems, to me, to be on the order of brainstorming, i.e., inventing hypotheticals to test their implications or shed new light on an existing conundrum. Fundamentally, I think the use of 'discovery' terminology, rather than 'invention' terminology arises from the nature of the new thing produced. Discoveries, in the scientific or mathematical sense, were 'always there' within the corresponding realm of inquiry (e.g., the set of axioms that comprise mathematics, the physical world for scientific discoveries, & etc.) and represent a mere formulation of an existing but previously hidden truth. Inventions, on the other hand, represent a novel application of existing principles for some external purpose. The implementation cannot be said to have flowed from existing knowledge in any meaningful way. Nor do inventions have any general theoretical utility. (Mousetraps are fun little inventions, but don't contribute to a 'Theory of Pest Control' and we certainly would not refer to the 'discoverer of the mouse trap.') Additionally, other scientists do in fact use the terms 'discover' and 'invent' in precisely the same sense as mathematicians do, and for similar reasons. In fact, these are the commonly accepted notions of the terms even among nonspecialists. Using 'mathematical invention' rather than 'mathematical discovery' erroneously (or maliciously) ascribes arbitrary subjectivity to mathematical thought.
wallflower 18y ago
Reminds me of one of my favorite films "Donald Duck in Math-Magic Land" (Yes, I own the DVD) Pythagoras and his secret crew have a cameo.
- nazgulnarsil 18y ago
  you can watch this on youtube. i recommend it.
sethg 18y ago
The Stanford Encyclopedia of Philosophy has a long article on Pythagoras that doesn't mention the whole rutti-tutti-irrationality issue. http://plato.stanford.edu/entries/pythagoras/ http://plato.stanford.edu/entries/pythagoras/ They do say that we don't have anything written by Pythagoras or his contemporaries--the earliest extant works mentioning him were written more than a century after his death--and then a lot of people attributed their own views to Pythagoras to give themselves the benefit of his fame. So it's really hard to know what the historical Pythagoreans did or believed. I think it is safe to say that if the Pythagoreans kept the irrationality of √2 a closely guarded secret, it wasn't their only closely guarded secret.
olavk 18y ago
The Pythagoreans had an elaborate philosophy based on the idea that the harmony of the universe was due to relations between the natural numbers. For example, Pythagoras discovered that musical notes and harmonies was based on relations between real numbers. He concieved the notion that the relations between the movements of the planets would create a similar harmonius symphony, the music of the spheres. All that beauty and harmony fell apart when it was discovered that irrational numbers existed.

by smanek 18y ago

tome 18y ago

But what would "accepting the existence of the irrationals" mean? As far as the Greeks knew, a "number" was a/b where a and b were integers. "Accepting" that the length of a triangle was not a "number" could have all sorts of consequences, including rejection of their belief in validity of mathematical reasoning, so I'm not surprised they were scared of it.

smanek 18y ago
The Greeks did know about irrational numbers. They could draw a right triangle whose legs were 1 and 1. They knew that the length of the hypotenuse (a number that we call the square root of two) couldn't be represented as the ratio between any two integers, yet it had a length.
bdr 18y ago
Actually, length was primary since geometry was the foundation of their math.

antiform 18y ago

In the book The Heart of Mathematics by Edward Burger and Michael Starbird, there is a quotation of a translated passage from Proclus, a 5th-century AD philosopher: "It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."

jgrahamc 18y ago

Cantor was very troubled by his discovery of the transfinite numbers because they conflicted with the infinite nature of God (hence the name transfinite). Shouldn't be any surprise to anyone that when a theory conflicts with _beliefs_ there's trouble.

deleted 18y ago
[deleted]
henning 18y ago
I love how mathematicians use the term "discover" instead of "invent" the way anyone else would.
- mkn 18y ago
  They use the term 'discover,' I would presume, because it's the most technically accurate term for what they do. As near as I can tell, as a relative layperson, mathematicians look at the existing state of mathematics, embodied in proven theorems, and consider the implications of those theories, looking for unanswered questions or outright contradictions. They then attempt to find the new theory that will provably answer the question or resolve the contradiction. The only 'invention' that goes on seems, to me, to be on the order of brainstorming, i.e., inventing hypotheticals to test their implications or shed new light on an existing conundrum. Fundamentally, I think the use of 'discovery' terminology, rather than 'invention' terminology arises from the nature of the new thing produced. Discoveries, in the scientific or mathematical sense, were 'always there' within the corresponding realm of inquiry (e.g., the set of axioms that comprise mathematics, the physical world for scientific discoveries, & etc.) and represent a mere formulation of an existing but previously hidden truth. Inventions, on the other hand, represent a novel application of existing principles for some external purpose. The implementation cannot be said to have flowed from existing knowledge in any meaningful way. Nor do inventions have any general theoretical utility. (Mousetraps are fun little inventions, but don't contribute to a 'Theory of Pest Control' and we certainly would not refer to the 'discoverer of the mouse trap.') Additionally, other scientists do in fact use the terms 'discover' and 'invent' in precisely the same sense as mathematicians do, and for similar reasons. In fact, these are the commonly accepted notions of the terms even among nonspecialists. Using 'mathematical invention' rather than 'mathematical discovery' erroneously (or maliciously) ascribes arbitrary subjectivity to mathematical thought.

wallflower 18y ago

Reminds me of one of my favorite films "Donald Duck in Math-Magic Land" (Yes, I own the DVD) Pythagoras and his secret crew have a cameo.

nazgulnarsil 18y ago
you can watch this on youtube. i recommend it.

sethg 18y ago

The Stanford Encyclopedia of Philosophy has a long article on Pythagoras that doesn't mention the whole rutti-tutti-irrationality issue. http://plato.stanford.edu/entries/pythagoras/ http://plato.stanford.edu/entries/pythagoras/ They do say that we don't have anything written by Pythagoras or his contemporaries--the earliest extant works mentioning him were written more than a century after his death--and then a lot of people attributed their own views to Pythagoras to give themselves the benefit of his fame. So it's really hard to know what the historical Pythagoreans did or believed. I think it is safe to say that if the Pythagoreans kept the irrationality of √2 a closely guarded secret, it wasn't their only closely guarded secret.

olavk 18y ago

The Pythagoreans had an elaborate philosophy based on the idea that the harmony of the universe was due to relations between the natural numbers. For example, Pythagoras discovered that musical notes and harmonies was based on relations between real numbers. He concieved the notion that the relations between the movements of the planets would create a similar harmonius symphony, the music of the spheres. All that beauty and harmony fell apart when it was discovered that irrational numbers existed.