so. saturday and sunday will count as one day, and will be the closest thing to what my proposed book will look like (as far as i can tell at this point). this will feel somewhat like an essay. and it might feel quite weird. and this is the part of the blog that i would love you (the interwebs) to comment on. am i making sense? do you care? should i stop? so, here goes:
part one - after plato
we stand in a cave you and i, facing the entrance (or should we prefer, an exit). behind us is a fire. behind that, some ropes - cut and discarded. the fire throws a dancing light on the jagged back wall where, in the corner, scrawled in chalk are the words: Plato woz 'ere.
cascading down in front of the entrance is a waterfall; we won't be leaving any time soon. at least, not without a smidgen more of an idea about what lies beyond. we stick our hands into the water and count our fingers. 1. 2. 3...
part two - some numbers are bigger than others
we count to ten and the ticker goes up: we reset the units and start the tens. when we get to one hundred, we do it again. no great revelation to you, i suspect. unless you are a minicoy islander, an ignorant robot or other entity not au fait with our collective predilection for base-10 (for that is what it is).
there isn't any better explanation for why we favour base-10, or decimal, in our counting system than simple appeal to fingers. it certainly isn't efficacy driving the system. base-12 (duo-decimal) is, mathematically, far more elegant. it is the smallest number with four non-trivial factors. it is certainly easier to use and, as anecdotal evidence, i present its prevalence in the doing parts of our life: eggs, weights + measurement, telling the time. if i've got one, two, three, four, six or twelve people (including me) at my gaffe demanding a butty, it's loads easier if i buy my eggs in twelves.
is it just our hands? well, the truth of the matter is that it is unimportant why we have ended up counting eggs and votes decimally. it is, however, vitally important to keep in mind that it needn't have been so; we gain no benefit and it bears no relation to the world of mathematics or the state of the world as expressed through mathematical statements.
part three - some numbers' mothers...
as it stands, we have a certain set of numbers that we call the counting set. one, two, three and so on. so far, so kindergarten. once we have mastered the infinity thus presented, we get another: the negative numbers. the two are, of course, held apart by mr zero - neither one nor the other.
but i've got one cake and three hungry mouths and so, after a fashion, i invent fractions: one third each: one over three. and here we have another infinity, for as well as i can have one over three, i can have one over four, over five, over six and so forth; a neat transposition of our very first set. and all of this infinity exists between a half and zero, each subsequent fraction getting incrementally smaller. of course, there are other mappings we can do with the counting numbers. where n is the counting number, we can have n minus one over n, giving us an infinity existing between a half and one, each subsequent fraction getting bigger.
because we operate in base-10, it is fairly natural for one over ten to be an easy number for us to grasp (0.1). so too is one over ten plus one over a hundred, plus one over a thousand (0.111). and here we are at the decimals. there is an infinity in between any two decimals (if you need help with that one, shout).
so we should be pretty set to deal with the world with maths, right?