The Greeks were very worried that some numbers irrational numbers are never ending and therefore cannot be calculated exactlywhy was that?

Answer 1

This question has depths not yet plumbed by mathematics, science or religion. In trying to answer it I will point to some of those dark holes, saying there lie answers yet to be brought to the light. The first hole is this: when an answer is brought to the light, is the person who went into the dark and emerged again into the light with a new answer a discoverer or an inventor? Much hangs on this.I will make some assumptions about this question. These are* it refers to the Ancient Greeks based in Athens(Athenians)* the 'why was that' part of the question refers to the worry, and not to the never-ending-ness* 'calculated exactly' means expressible finitely in terms of rational numbers* it will be OK to simplify the story a bit (if greater accuracy is required, please ask).The Athenians had built up an arithmetic from the counting numbers (1, 2, 3, . . . .) and made (discovered or invented?) a whole world view based on these numbers. Everything in this world was supposed to be describable in terms of these numbers. Music was, even the music of the spheres was.Ratios between the numbers were permitted, so they were called 'rational' numbers. There is no doubt that among the Athenians there were those with open minds who sought answers even from dark holes, and there were those who 'knew' what they knew, feared dark places, and had closed minds. The latter far out-numbered the former. Remember the fate of Socrates. Remember that in those days there was not the distinction between theology and science that is assumed today.So the majority of the people were certainly dismayed when Eudoxus, not one of their number but an upstart from Asia Minor, came up with the conclusion that the length of the diagonal of a square, the side of which had a rational length, could not itself be a rational number. 'Worry' is not too strong a term for how they felt. Even 'devastation' would be appropriate. How would you feel if everything you thought you knew suddenly seemed wrong? That is how I imagine they felt.The eventual recognition of 'irrational' numbers as real numbers permitted great advances in mathematics. The 'new' became ordinary, but there always was (and perhaps will always be) more 'new' numbers emerging from the dark which will be seen as 'worrying', only to become ordinary in their turn. Witness transcendendals and imaginary numbers (now both universally accepted) and transfinites (almost universally accepted).I may be one of the few still holding out against transfinites. I think that what Cantor drew out of the dark hole should be buried again. I believe Cantor was wrong. This raises a whole new question. Is the 'new' necessarily 'right'? Plus ca change, plus c'est la meme chose.