Chessman
Your advice to Sid, doesn’t go far enough. As Sid recognized what we have here is the old argument between Zeno and Parmenides vs. Aristotle. The concepts in question then were infinity, infinitesimals, and continuity. Zeno’s paradoxes argue that if space is continuous, there can’t be motion because even after an infinite number of divisions there would still be that pesky infinitesimal left over. Aristotle won the argument because obviously there was motion and the continuity of space was intuitively obvious. Guess what? Aristotle used the same ‘potential infinity’ argument (he claimed ‘actual infinity did not exist’) that MOM is using; it was picked up by Thomas Aquinas so in a sense the dismissal of an ‘actual infinity’ became imbedded in Catholic philosophy. I suspect that this may be MOM’s reason for arguing his particular point. And you know what, they are correct relative to Sid’s argument. The set of rational numbers (ratios of integers, fractions) is only ‘potentially infinite’.
However, mathematicians overcame the problem of the infinitesimal through the use of limits. Newton and Leibniz made use of the infinitesimal factor ‘dx’ to develop calculus. But ‘dx’ is a sticky little bugger because it is big enough to divide by and small enough to ignore. This problem was refined, but not eliminated by Cauchy and Weierstrass using the epsilon/delta trick. It wasn’t until Dedekind rigorously defined the set of irrational numbers and Cantor showed that the set of irrationals has a greater cardinality (aleph-1) than the cardinality (aleph-0) of the set of rationals that it could be argued that space is continuous because it is composed of an ACTUAL infinity of points. The mathematical argument rests on the idea of denumerability (can be counted).The rationals are denumerable; the irrationals are not.
Sid should frame his argument around Aleph-1. Time, if assumed to be continuous, can be finite with respect to extension and infinite with respect to divisibility. However, there is no doubt in my mind, from previous discussions with MOM that he will not concede. He will continue to wield his ‘ontological one’ until hell freezes over in the same manner that Kronecker (a great mathematician in his own right), who is credited with the statement, “God made the integers, man made all the rest”, fought Cantor without concession. I understand MOM’s argument, but I think he is ruling out the idea that if you proceed with divisibility eventually you will arrive at a ‘point’. It could be a ‘point’ in time or a ‘point’ in space and since science in there persistent granulation of matter will eventually arrive at ‘points’ of space —what could ‘string’ or quarks be made of— we will find that matter is substantially spatial.
My suggestion: SID, ask MOM these questions: do you consider space to be real? If so, then do you consider space to be continuous? And if you believe space to be continuously real, do you accept the mathematical assumption that space is composed of points and every point can be represented by a number and for every number there is a point? If the answer is yes to all, then space is demonstrably both real and ACTUALLY infinite If answering no to any of the questions; then the alternative is my personal belief: namely that the space that gives dimensionality to our universe, without which there can be no objective reality, is discrete.
Yppop
