Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural numbers. In modern language, this means that all real numbers are rational numbers. This allowed them to find out whether any two ratios of magnitude are equal: you just find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.

But Hippassus of Metapontum discovered that the ratio of a diagonal of a square to its side is not equal to a ratio of natural numbers. So a new definition was needed for when two ratios of magnitudes are equal. This was done by Eudoxus of Cnidus. In modern language, his definition says that two real numbers are equal if the same rational numbers are less than, equal to, and greater than them (basically the idea behind Dedekind cuts).

My question is, what results of geometry did the Pythagoreans prove using the incorrect assumption that all real numbers were rational? Long ago, I read a book, whose name I don't recall, which listed some results, and described how they had to be reproved with Eudoxus' new definition.


2 Answers 2


There are errors in the assumptions of your question:

  • That history about Hippasus of Metapontum is highly doubtful. There is the version that you mentioned. There is also the version according to which what he discovered (or revealed) was how to construct a dodecahedron.
  • You claim that the Pythagoreans were able to find out whether any two ratios of magnitude are equal: they just had to “find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.” Concerning this idea, David Fowler wrote: “I know of no explicit evidence for this, early or late.” (The Mathematics of Plato's Academy: A new reconstruction, 2nd edition, Clarendon Press, 1999, § 10.1)
  • Finally, I doubt that the Pythagoreans proved any geometrical statement using the idea that all magnitudes are commensurable. And, if they did, no such proof survived.

As far as I've been able to find out, the implication for Pythagoreans usually went more the other way around. The point was that all magnitudes being commensurable was a consequence of their more fundamental belief in the notion of numbers as a fundamental element of the universe.

It can be hard to know exactly what Pythagoreans believed in that respect, since there are very few Pythagorean texts on that exact topic (as far as I know), especially during the early period of Pythagoreanism before any idea of incommensurable magnitude existed, and we mostly have their ideas from other authors, but they seemed to have believed that all things being numbers, this includes space itself, and every magnitude is "discrete", if we can apply that modern notion to whatever they believed. For a few notions on this see for instance book XIII of Aristotle's Metaphysics :

And the Pythagoreans, also, believe in one kind of number-the mathematical; only they say it is not separate but sensible substances are formed out of it. For they construct the whole universe out of numbers-only not numbers consisting of abstract units; they suppose the units to have spatial magnitude.

The Pythagorean version in one way affords fewer difficulties than those before named, but in another way has others peculiar to itself. For not thinking of number as capable of existing separately removes many of the impossible consequences; but that bodies should be composed of numbers, and that this should be mathematical number, is impossible. For it is not true to speak of indivisible spatial magnitudes; and however much there might be magnitudes of this sort, units at least have not magnitude; and how can a magnitude be composed of indivisibles?

There are many other statements which make this unclear however. They also seemed to think that points did exist, associated to the number 1, and that lines were associated with the number 2, so that points were the extremities of lines, but as this was quite a contentious point in antiquity (cf. for instance Sextus Empiricus "Against the geometers"), they may not have believed that lines were composed of points. What they exactly imagined the structure of space was like from these ideas isn't obvious (it probably did not work amazingly), but one area where they were interested in was the notion of figurate numbers. Of course, the thoughts of the Pythagoreans weren't necessarily united, unchanging (it is quite possible that incommensurable lengths were discovered by Pythagoreans, although the evidence for this aren't that clear) nor resembling modern notions of a formal system, so it is hard to figure out the exact ideas that they were thinking.

The notion that there is a smallest length of a line does appear in other geometers, like Xenocrates, (maybe) Democritus or the Epicureans (which is where a lot of the specific arguments against the notions are found later on), which are maybe based on Pythagorean ideas. Sometimes this smallest length is infinitesimal, sometimes not.

This idea that all magnitudes are discrete in this sense leads to the notion of commensurability, as for instance Proclus shows in his commentary on Euclid (quoting Geminus of Rhodes) that incommensurable magnitudes leads to infinite divisibility.

For when geometers demonstrate that there is incommensurability among magnitudes and that not all magnitudes are commensurable with one another, what else could we say they are demonstrating than that every magnitude is divisible indefinitely and that we can never reach an indivisible part which is the least common measure of magnitudes? This, then, is demonstrable, but it is an axiom that every continuum is divisible; hence a finite line, being continuous, is divisible.

What wrong theorems exactly those ideas led to for the Pythagoreans is unclear as far as I know, except for the obvious cases : the assumption that any two lengths are commensurable, such as the hypothenuse to the other sides of a triangle. In particular, a lot of Pythagorean mathematics was the theory of proportions, a theory of proportion that was only valid for figures of rational proportions, which I would guess they imagined valid for any figure. If they indeed believed in indivisible lines, this would make some basic theorems wrong, as well, such as the ability to divide any segment into two equal segments, depending on if the "length" was even or odd.

It is quite possible that the discovery of incommensurable length led to the general abandonment (possibly gradual) of the notion of those indivisible lines (or, in the case of Epicurians, the abandonment of geometry altogether), so that later Pythagoreans may have had different ideas.


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