Between the integers are the fractions; between the fractions are the irrationals, ever denser, Aleph-2 of them. Yet even the irrationals are but frozen frames of a slide show. Between the irrationals, even denser yet (Aleph-3?), are the infinitesimals, elusive but real, the “non-standard” numbers, not expressible numerically, eluding all measure-ment, all but indistinguishable from their neighbour irra-tionals which had seemed to exhaust the field, i.e. they truly “connect” the points on the real number line. Only the infinitesimals finally give continuity to the number line, i.e. go from dimension zero to one, from points to line.

Space and time are almost like the number line, indefi-nitely divisible into ever-smaller intervals without becom-ing something different, almost truly continuous – until they hit the Planck limit, at which point space-time “tears” or becomes composed of “bubbles”, something radically diffe-rent from what they were above the limit. Only the continu-ity based on infinitesimals makes motion in space-time possible, in spite of Zeno’s arguments against it. And chaos is possible – instability and extreme sensitivity, because we cannot replicate the initial conditions exactly; they are on the knife-edge between irrationals. And free will is possible, exploiting the butterfly effect of chaos.

Time and space are self-similar down to the Planck limit, numbers are similar without any limit. A magnifying glass would make the interval from 0 to 1 the same as from 1 to infinity, as well as the same as the one from 0 to 0.1, from 0 to 0.01, and so on indefinitely. At very high magni-fication, would the infinitesimals swim into view, with non-numerical tails? Probably not – we cannot reach those re-gions at any conceivable magnification; but we have to assume that self-similarity still obtains, as in repeated reflections in opposing mirrors, even though we can no longer see it. Infinity is not just way off in the distance, it permeates the number line, and is unreachable both in the distance and in the interstices. The number line is infini-tely fractal. Only in this way can zero be transformed into one, the basic operation in creation.

Atoms and molecules are not like that. Divide a droplet of water, and you eventually get (in a thought experiment) a single molecule. You cannot subdivide that any further and still have water. You can divide it into hydrogen and oxygen atoms, but they are different entities from water. The word “atom” means “indivisible”, but of course it is not. How-ever, again, when you split an atom you no longer have the same substance; you now have a nucleus and an electron. Split the nucleus and get protons and neutrons – again different entities. Split protons and neutrons to get quarks – different again.

The electrons and quarks are now considered to be un-splittable, but we may yet find that they are not. In any case, matter is not indefinitely divisible, like time and space and number. Electricity is not indefinitely divisible, only down to electrons. Light is not indefinitely divisible, only down to photons. So both matter and energy have limits of divisibility, i.e. bosons and fermions, respectively. New physical laws (i.e. quantum theory) apply at these micro levels.

Space, time, and number are like the continuous frame-work of the universe. Matter and energy are like the concre-te contents, the flesh on the skeleton. The framework is like the steel skeleton of a building, the contents are like the external and internal walls and the contents of the rooms.