The usual story I've heard is that Numbers were held in great esteem by the Pythagoreans, that the discovery of numbers that were not ratios were held to be contra reason and thus irrational, as suggested for example by wikipedia which explains that this discovery was made by (apocraphally) Hippasus and:

[who], however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios. Another legend states that Hippasus was merely exiled for this revelation.

An alternative reading is that irrational means simply not a ratio. This is a clear, definitative and unambiguous description of what such a number is.

(It also appears a signifier of a break between geometry and number, with geometry taking the lead as it could deal with magnitudes that were not rational).

The question is, if this is true, when did irrational actually denote something that is against reason? And was that read back into the discovery of irrational magnitudes?

  • If you mean "Irrational" as in "He exhibited irrational behavior, officer!" then no, they are not irrational - rationality in that sense is only a property of Agents (And things that you can conveniently shorthand as Agents, e.g. Societies, Ant Colonies etc.) – medivh Sep 18 '13 at 11:25
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    According to wiki ( en.wikipedia.org/wiki/Irrational_number#Ancient_Greece ) ancient Greeks used alogos, which clearly means "against reason" – Dave Sep 3 '15 at 17:47
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    As @medivh addressed, irrationality, in that respect is not a property numbers can have. However, Plato certain considered them to be "nonsensical." – Daniel Goldman Sep 3 '15 at 18:40
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    Irrational numbers are "not ratios," i.e. cannot be expressed as a/b for a,b in Z. – James Kingsbery Sep 8 '15 at 15:44

re: when did irrational actually denote something that is against reason?

The word "irrational" denoted "without reason" at least as far back as the 15th century according to http://www.etymonline.com/index.php?term=irrational

re: And was that read back into the discovery of irrational magnitudes?

No - we have no better story than the usual story ( as you described in your question ), in which the irrational aspect of "irrational numbers" stems from them being contrary to the pythagorean belief that all nature could be explained in terms of the counting numbers alone.

Obviously, you can't rule out that somewhere exists some evidence that the usual story is false.


Is "that bank is not a bank" a paradoxical statement? Not if the first 'bank' means 'financial institution' and the second 'bank' means 'land sloping down to a river'. As a philosopher and especially as a mathematician you have to be extremely precise with your words.

Both the idea of 'rational' as in 'rational agent' and as in 'rational number' have specific definitions in mathematics. These definitions have nothing at all in common; most importantly, the 'rational agent' definition doesn't even always correspond to the reasonable common-sense meaning of the word. As such, you are correct in seeing that 'irrational number' just means 'a number that is not the ratio of two integers'.

People often have the same trouble with 'real' and 'imaginary' numbers, neither is more real in a modern ontological sense than the other.

Could we interpret irrational numbers as somehow contrary to reason? Well, a better name would be 'immeasurable numbers' since at the time, a measurement was seen as the number of multiples of some measuring stick. As such, the diagonal of an equilateral right triangle with side lengths one, could not be measured by any measuring stick that could also be used to measure the side lengths. This was the difficulty for early philosophers.

Are some irrational real numbers against reason today? Yes, if we define 'reason' reasonably. One of the ways that a computer scientists or algorithmic philosopher would define 'reason' is as something that can be done by a computational device (such as our brain). If we subscribe to the Church-Turing thesis, then any computational device is equivalent in power to a Turing Machine (TM). There are only countably many Turing Machines, and thus only countably many 'reasonable' numbers. However, there are uncountably many real (or even irrational) numbers, and so there are irrational numbers (every rational number is obviously computable) that cannot be computed. We can say that these non-computable numbers are against reason.

  • Actually, calling irrational numbers immeasurable would be a bit wrong-headed. They were discovered via geometry, and the lengths of the triangle would be in principle measurable. – Mozibur Ullah Sep 19 '13 at 0:06
  • @MoziburUllah I explained in my answer what immeasurable means. It means that the sqrt(2) and 1 can't be measured by the same measuring stick. – Artem Kaznatcheev Sep 19 '13 at 0:22
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    @ArtemKaznatcheev great answer, and good point on warning people about misusing words. However, weren't imaginary numbers called that precisely for the pejorative value of the word? – R. Barzell Sep 3 '15 at 18:58

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