On 17 Jul 2006 05:29:08 -0700, "Kevin Karn" <email@example.com> wrote:
>Patricia Shanahan wrote: >> >> Isn't there also a simplicity argument here? The set of all natural >> numbers, with no gaps and no end, is in many ways an easier thing to >> reason about than the set of natural numbers that someone, somewhere, >> has used or will use. > >You may be right, but your argument is incompletely formed, and I don't >trust it. It's not clear what you mean by "easier to reason". How do >you measure "ease of reasoning"? And it's not clear why ease of >reasoning (whatever that is) should be a priority. > >My main question is this: If we accept your simplicity argument, how do >we draw the line between orgies of metaphysical speculation (on the one >hand) and actual science (on the other)? It seems that the >metaphysicians would argue (according to your approach): Our aim is >ease of reasoning, i.e. *simplicity*. We're building vast castles of >useless logic and nested infinities which have no relevance whatsoever >to the real world, because it's *simpler* to proceed that way. That >seems wrong to me.
Why? As long as the mathematical tools give results that correspond to physical experiments, where is the problem? Uncountable ordinals do not predict anything about physics, so they can be safely ignored. That is not the same as throwing them away and working with something more cumbersome for no proper reason.