How historically grounded is the standard narrative of the Irrationals in Antiquity?

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

(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).