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Its commonly said that the Pythagoreans were unbalanced by the discovery of the irrationals; since their philosophy was predicated on ratios; ratios of two finite numbers.

Still, it is natural to consider a ratio of two infinite numbers; and most of these will approach an irrational. After all, one easily consider 1,2,3,...; so one might be lead to 1/2, 11/22, 111/222 ...; and one then need only show that some infinite ratios cannot be reduced to finite ones by common techniques: for example (5 x 1111...)/(6 x 1111...) = 5/6. Now, this of course is using imprecise techniques, as far as modern contemporary mathematics is concerned; but different standards of rigor held in antiquity...

Archimedes, much later than the Pythagoreans, had developed a method of exhaustion; a precursor to the calculus.

One might argue this is an outcome of the 'irrational' discovery; but given the apeiron of Anaximander, the boundless; the idea of the infinite as something unbounded was already there.

How historically grounded is the 'standard' narrative of the irrationals and the Pythagoreans? That is their entire philosophy was disrupted:

Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them

and

Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning

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    I think this may be off-topic because it's about mathematical history, not philosophy. – user2953 Jan 17 '15 at 10:36
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    @Keelan: I can see why you say that; but I do feel in this instance there is an intersection between the history of mathematics andthat of philosophy; after all, the standard accountis that the discovery of the irrationals destabilised the Pythagoreans. I'm not asking a question in the pure history of mathematics. – Mozibur Ullah Jan 17 '15 at 10:51
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    @MoziburUllah I wholeheartedly agree with your comment. That said, the main question you posed in the last sentence seems overly broad/vague for a phil.SE question (at least for tastes). IMO this might be a very interesting topic for a question, but the question itself probably needs heavy editing. – David H Jan 17 '15 at 12:12
  • Could you try to spell out the question a bit more explicitly in the headline? – Joseph Weissman Jan 17 '15 at 16:30
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According to my understanding, it is noy "historically correct" to say that Pythagoreans discovered the irrational numbers.

Archaic Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two segments of lenght a and b respectively, it is always possible to find a segment of "unit lenght" u such that "it measures" both, i.e. such that [using modern algebraic formulae which are totally foreign to Greek math] :

a=n×u and b=m×u, for suitable n,m.

From the above assumption, it follows that :

a/b = n×u / m×u = n/m.

The assumption amounts to saying that the ratio between two magnitudes is always a ratio between integers (i.e. in modern terms: a rational number).

But note that for Greek math the only numbers are the natural ones and they must be distinguished from magnitudes : a segment, a square, ... which are "measured by" numbers expressing the ratio between the measured magnitude and the relevant "unit" magnitude.

For ancient Greeks there are no rational numbers; but only magnitudes measurable with multiples of a suitable unit one.

The discovery of the existence of irrational magnitudes, through the proof that the case where b is the side of the square and a its diagonal is not expressible as a ratio between (natural) numbers, leads Greek mathematics to the withdrawal of the above (implicit) assumption, that we may call : "commensurability assumption" and to the axiomatization of geometry, i.e. the systematic effort to explicitly lists all the needed assumptions.

In conclusion, in ancient Greek mathematics there were no "irrational" numbers, nor "inifinte" ones.

  • If I read you correctly, you are saying that there were rational and irrational magnitudes (but not of numbers). – Mozibur Ullah Jan 18 '15 at 15:03
  • @MoziburUllah - not exactly ... there were numbers : 1, 2, ... magnitudes : lines, squares, ... and ratios etween numbers. In order to "measure" two magnitudes (e.g. two lenghts) we have to find a common measure to be used as "unity". – Mauro ALLEGRANZA Jan 18 '15 at 15:07
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It's not true that antic mathematicians had different rational standards, and that's precisely the point. They were expecting rigorous demonstrations. Only when the concept of limit and infinitessimals was invented could we give a rigorous treatement to irrationals.

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(5 x 11111...)/(6 x 11111...) is not equal to 5/6

(5 x 11111...) equals infinity and (6 x 11111...) equals infinity.

(infinity)/(infinity) is always undefined.

The obscure nature of Pythagoras' School makes it hard to know who discovered irrational numbers (Hippasus?). They probably found the difficulty while examining a right triangle whose sides a and b are both equal to one. Perhaps, to their horror, they found the number two is not a perfect square (e.g. 4, 9, 16, ... are perfect squares and render an integer when their square root is taken).

Pythagoras reasoned 'all is numbers' and also, that for any right triangle, the squares of sides a and b is exactly equal to the square of its hypotenuse (line c), but this can be said another way, the square erected on the diagonal of a square has twice the area of the original square. The difficulty they had was trying to create a ratio of two integers that would account for the square root of two. They tried and tried and tried, and they couldn't find such a ratio. Someone (Euclid?) later proved such a ratio does not exist. (The Presocratics, Philip Wheelwright (Editor), 1997, p.206).

  • Something which may help with your intuition, Mozibur, is that there are multiple ways to construct "a series which approaches infinity," which is what you imply when you write "1111..." For some series which approach infinity, a finite answer can arise. For other series, the answer spirals off towards infinity. Michael's point about infinity/infinity is the most pure answer, but there are many impure implementations which have intuitive results. One shining example is L'Hopital's rule in calculus which shows how to do some of these divisions within the confines of a calculus limit. – Cort Ammon - Reinstate Monica Jan 17 '15 at 17:44

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