The choice you've presented the Greeks is between giving up the idea that there are irrational numbers at all (presumably by denying that there can even be squares covering an area of two square units) and retaining criteria (2), or just dropping criteria (2) in favour of something restricted to the use of whole numbers in expressing the rationals only. That makes a little more sense to me, but not much. — MetaphysicsNow
I would fix the bolding part: it'd be a case of giving up the idea that there are irrational numbers by denying that such areas
can be measured ('are amenable to measurement at all'). It's actually worth quoting Heller-Roazen in full on this point:
"The Pythagoreans, however, were no strangers to the uncountable. Although they barred numberless relations from the domain of their arithmetic, they also named them in no uncertain terms. They called them “unspeakable" (αρρητoι), “irrational" (αλoγoι), and “incommensurable" (ασνμμετρoι). From such appellations, one might infer close acquaintanceship. Yet the familiarity the classical theorists of number possessed with such relations could not be knowledge, according to any classical standards of science. Infinitely eluding the rule of unity, incommensurable quantities could not be considered to number anything that was and that remained a single thing; for this reason, they could hardly be considered to number anything at all.
Of such unspeakable relations, it could only be deduced that, like the impossible root of the single tone, they could be no collections of one. They were, quite simply, immeasurable, and as long as every definition in arithmetic and music was to be numerical and every number was to be discrete, they were unrepresentable as such. They might well have been somehow manifest to the Pythagoreans, but, being uncountable, they could be no “remainders.” Their sole place was at the limits of their art of quantity".
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It's also worth noting that our conversation so far is almost like a case study in what it means for how different categorizations commit one to different parsing-out of concepts: "it's not there there can't
be squares like that; it's that they can't be
measured"; In some sense, this is a 'choice' too: perhaps one can indeed deny that there are squares covering an area of two square units; but one would have to make the corresponding move of then saying something like: 'the things you thought were squares covering two square units are not squares; they are ξquares'. Wittgenstein's 'rule-following paradox' was all about this: that every move in a game can be said to accord to a previously undiscovered rule, without breaking old ones. But these new rules are not just shuffling of goal-posts: they make one see things anew -
if done right.
So there's a kaleidoscopic or rubik's cube aspect to philosophizing: twisting a knob - a concept - in one way, ramifies throughout the whole series of 'possible' concepts and implications.