On May 10, 4:53 pm, James Burns <burns...@osu.edu> wrote: > Transfer Principle wrote: > > So by context, we conclude that byron intends the decimal reals -- > > the set of reals with only finitely many nonzero decimal digits (also > > known as the ring Z[.1]). > I understood originally the distinction that byron was making, and I > strongly suspect -- /very/ strongly suspect -- that YBM already > understood that distinction before your earlier response to him. > It is an unclearly motivated distinction no matter who makes it.
I suspect that the main motivation between finite and infinite decimals is so that elementary and middle school students can learn the algorithm for computing with decimals.
From the Common Core Standards for Mathematics, Grade 5:
"Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense."
Thus, to the young learner of decimal arithmetic, the distinction between finite and infinite decimal is crucial. Arithmetic of finite decimals becomes reduced to arithmetic of closely related whole numbers -- indeed, one can perform the operation on the whole numbers and insert the decimal point at the end. The relationship between arithmetic of infinite decimals and that of whole numbers is not nearly as direct.
Emphasizing the computable-uncomputable real distinction over the finite-infinite decimal distinction is inappropriate for students at this age.
It's not until one is learning analysis, not arithmetic -- when one learns to view the reals as D-cuts or classes of C-sequences rather than infinite decimals -- when the distinction between finite and infinite decimal becomes less important.
> > So this leaves me wondering -- can there be a "meaning_ful_" math, > > one that avoids the undesirable classical result? If such a theory > > exists, then it's not bound by classical results -- indeed its aim > > would be to avoid the result .99[bar]=1 that leads to math ending > > in "meaninglessness." > Suppose you produce a non-classical system without the > "meaningless" .99[bar]=1. Will byron thank you for taking > it away from him? I greatly doubt that. Much more likely is > that he will either ignore you (the front-runner) or he > will look around until he finds something else to call > "meaningless".
If byron considers math to be "meaningless," then what would he have us do in lieu of math?
> I tend to write them down as truth and Truth, by which I mean > (1) retail truth, or true things, such as "Snow is white", > "I am in front of my computer". We would not be able to live > without retail truth, without knowing, for example, whether > we were in front of our computers. Fortunately, /retail/ truth > seems to be almost wholely lacking in problems, at least > such problems as philosophers and erotic poets pay attention to. > We are still required to open our eyes and look around us, > though, and, yes, this can be turned into a problem > by some people. > and (2) wholesale truth, or philosophical statements of vast > consequence which, nonetheless, many people insist upon ignoring > and carrying on their lives quite well without, > even if they do so with a shocking lack of gratitude to > philosophers and erotic poets for their efforts on the behalf > of the larger community. > When I say my goal is to make everybody truthful, I am referring > to retail truth. I claim that you know very well what this is, > as do I, even if you may not be able to define it precisely. > (I know I can't.) You are also able to digest meals and I know > you can, although you may not know how you do this, and I certainly > do not know. > When byron refers to truth, it has all of the earmarks of wholesale > truth. Do you not agree? Especially telling is his application of > the word "meaninglessness" to, well, pretty much everything.
Once again, if byron considers all math to be "meaningless," then what would he have us do in lieu of math?
> It looks to me as though a search for retail truth and a search for > wholesale truth can carry on quite well without interfering with > each other.
OK. And I suppose you consider the use of certain words to which I regularly object to be an instance of "retail truth"?
> You do not believe it, apparently, not even with the evidence of your > own eyes in the threads you have been involved in, but [posters are > not labeled] for what they argue for, but for the way > they argue. Wierd conclusions and outlandish reasoning do seem to > travel together, but I suspect that they are both caused by the > [label-worthy] personality.
Whether it's what they argue for or how they argue, all I want to be able to do is discourage the use of the overused labels. I want to support or modify the label-worthy argument in a way so that the labels go away.