Zeno and Demokritus

Direct Media Links: Webm, Ogg Ipod ; Time: Zeno of Elea Zeno of of Elea was influenced by Parmenides who believed that the nature of the universe is immobile. This is in some sense a very modern point of view. If we look at space time, then paths of objects appear as static curves. If you draw the path of the earth in space time, then it moves on an elliptical helix. In the geometry of space-time, it is a static object.

But Zeno was interested in motion per se and wondered whether an arrow which is a fixed position at every fixed moment of time can move at all. How come that something which is at a fixed position at every moment can move at all. This is a conundrum which has been solved by calculus. We can describe parametrized curves r(t) and assign to them derivatives r'(t) which are limits (r(t+h)-r(t))/h when h goes to zero. Zeno did not know calculus and the notion (r(t+h)-r(t))/h does not make sense at h=0. The limit makes sense.

Matter: Demokritus A similar puzzle appears when thinking about matter. Is there a smallest unit of matter. Demokritus speculated in that direction. He most likely was trying to grind sand in finer and finer pieces and started to fail at some point, so concluding that there must be smallest elements of matter. His conclusion is from a modern point of view naive. He might have been able to get to a scale of maybe 1 micro meter (an optimistic estimate) given that sand has a size larger than 50 micro meters. Natural sand 60 to 200 micro meters. Dust particles are 5 microns or less. If too small they can not be seen. But even 1 micron 10^-6 meters is 4 scales of magnitudes larger than an atom 10-10 meters. Demokritus had not way to guess this. We now know that the nucleus is much smaller 10-15 meters and quarks are of the size 10 -18 meters. Space: Planck Unlike Demokritus who had been 4 orders of magnitudes off with his means to explore this experimentally, we are 15 orders of magnitudes off to explore the smallest units of space (if they exist). Now, we can extrapolate from Demokrit and today and speculate that even 10-35 is not the end. There might a new physics and a new mathematics valid at scales like 10^-100. Is space discrete? It is clear that at Planck scale 10-35 our notions of distance do not make sense any more. Now this is maybe a matter for philosophers An example where one can read things like "There is a strong intuitive argument that if causal finitism is true, then space and time are discrete". The question whether space and time are discrete is maybe not a good question because it can not be answered experimentally at least in our time. A good question to ask is "Do we really need the continuum?" I wrote a bit about this here in the context of computer science. If you look at a continuum object like a curve, then you never ever need infinity. You just draw polygons.