Sunday, February 10, 2008

Less blogging; more slogging!

Well, today's attempt to approximate quality by quantity didn't seem to work, so what I'll do next is less blogging; more slogging out of rigorous arguments, that's what I need to do (for a while anyway). E.g. why do I think that hash, # = 1/0, is a proper number? At the moment my attitude is well, why do we think that a half, 1/2, is a proper number? Because it's an answer to "How much?" (and also, halves being primitive numbers, an answer to "How many?"), e.g. if 3 apples are divided equally between 2 boys, how many apples do they each get? One and a half. Apples being smoothly extended objects (more or less), 1 apple can be divided into 2 equal parts, into halves. Similarly (and more generally) if a unit amount of any continuum is divided into N equal parts, those parts should have measures of 1/N.
......And in particular, if # is the cardinality of the continuum (the number of points in a line full of points) then the parts will be points, with measures of 1/# = 0. That 0 is not, as the cardinal 0 and (intuitively) the more primitive half were, an answer to "How many?" but rather to "How much?" (cf. how likely we are to hit any particular point if an infinitely fine dart is thrown at random at a line of points, none of which it would be impossible to hit); and note how differently the primitive 1/2 and the rather more discovered 0 (and similarly #) relate to 'many' and 'much.' Anyway, if continua are full of points, # is a proper number (just like 0 is). Not very compelling, perhaps; but then, that's why I'll be blogging less, for a bit (if I hold myself to that)...