Thursday, May 28, 2009

more greek, and: "i was right, in the sense that i was wrong."

another passage from ivor thomas's selections illustrating the history of greek mathematics:

"i think you know that those who deal with geometrics and calculations and such matters take for granted the odd and the even, figures, three kinds of angles, and other things cognate to these in each field of inquiry; assuming these things to be known, they make them hypotheses, and henceforth regard it as unnecessary to give any explanation of them either to themselves or to others, treating them as if they were manifest to all; setting out from these hypotheses, they go at once through the remainder of the argument until they arrive with perfect consistency of the goal to which their inquiry was directed.
..
"therefore i think you also know that although they use visible figures and argue about them, they are not thinking about these figures but of those things the figures represent; thus it is the square in itself and the diameter in itself which are the matter of their arguments, not that which they draw; similarly, when they model or draw objects, which may themselves have images in shadows or in water, they use them in turn as images, endeavouring to see those absolute objects which cannot be seen otherwise than by thought."

from plato, republic, vi. 510 c-e.

in other news, i was right .. in the sense that i was wrong.

my doubts were well-founded: i finally found the error i suspected. my logical quantifiers were off, and the self-similar set turned out to have positive lebesgue measure.

yes: an embarrassing mistake,
but better now than later.

thinking more carefully about it, nowhere in my most recent arguments do i use information about how a measure distributes its mass, across space.

you'd think that, in attempting to prove a result about singular measures, that i'd account for that. \-:

on the bright side, i've proven a special case of the result. the hypothesis is too technical to be worth anything, but nevertheless: a step forward.