
Re: Visual Presentation of Real Number System
Posted:
Nov 27, 2013 3:18 PM


>Isn't that why we dropped the physical metaphors, because you were having problems with them?
I'm just having tongueincheek fun playing with the notions here to illustrate the problems with mixing physical situations with mathematical. When one says "pick a real number at random from (0,1)" that's a sloppy statement. There is no mathematical process of picking, or throwing, or anything else. But those metaphors make their way into the erudite of mathematical texts. A related example is the axiom of choice  in the formal statement there is no "choosing".
In the dart example, if you were just visiting these notions for the first time, then I'm impressed. You got the distinction between "determinable" (computable) as being relevant, and seemed (to me, at least) to be figuring out that the computables are countable and therefore of measure zero.
If you look at link I posted to the "other" discussion, you'll see that this notion of "events" that are "zero probability" but still "possible" is more or less the current official explanation of what's going on here. Chaitin himself (who I enjoy reading) says the probably is "infinitesimally small". I hope he smiled to himself when he wrote that.
I think "event", and "possible" (and perhaps even "probability") are all thoroughly contaminated with "process thinking" which has teaches a kind of thinking that mathematical objects travel, undergo change, make choices, "happen", etc.
In my view, its only in the physical world that such things occur, but since we are using mathematical objects to model such processes, the language often bleeds across the lines. It also seems to me that this occurs in probability more than in any other branch. But even in a more fundamental definition, that of "function", some people just cannot accept the notion of a function as an immutable set relation, but insist that the function houses some mysterious "action". Witness Devlin's insistence that there really is a "process" behind real number multiplication.
So, what does it really man to "pick a real number at random from (0,1)?" If you insist on a real, honest to goodness process to do the picking, you are in the realm of computability (and are only dealing with computable numbers anyway,) or physical quantum effects that can be measured, (and thus finite precision numbers, and Heisenberg uncertainty to complicate matters.) There is no conceivable picking process up to the task if you really want to deal with "real" numbers.
All that probability and measure theory really tell you here is individual points and countable collections have measure zero. Saying things like "hitting a particular real number with a dart is an event of zero probability but still possible" is metaphysical nonsense at worst and completely unnecessary at best.
Cheers, Joe N

