
Re: NonEuclidean Arithmetic
Posted:
Sep 11, 2012 2:27 PM


On Tue, Sep 11, 2012 at 1:07 PM, kirby urner <kirby.urner@gmail.com> wrote: > On Tue, Sep 11, 2012 at 9:34 AM, Paul Tanner <upprho@gmail.com> wrote: >> On Tue, Sep 11, 2012 at 10:28 AM, Joe Niederberger >> <niederberger@comcast.net> wrote: >>>>But scaling works for all of the reals easily and perfectly and exactly, no redefinition ever needed. >>> >>> Scaling as you are using it is just a synonym for real number multiplication. >>> >> >> No, it's a property of binary multiplication on any subset of the real numbers. >> >>> So give us a detailed mathematical account of the "process" (Devlin) that multiplication is that is not flatout circular. >>> >> >> Its not circular since it's not a synonym. Repeated addition and >> scaling are properties of the operations respectively on some or all >> subsets of the reals, and that these properties are not what these >> operations *are*. They are models of properties of some or all of the >> operations, not the operations themselves. > > There is nothing that these operations *are* as distinct from our many > models thereof. > > Even computation is modeling.
If you want to promote and believe the philosophy that a thing and a model of a thing are one and the same thing, then you go right ahead and promote and believe it, but don't expect many to join you in that. There's more than one school of philosophy in town, you know.
> > The notation is quite happy with adding irrationals repeatedly by > means of infinite sums. The dot dot dot notation is well established. > All you need is a rule for generating successive quotients, smaller > and smaller differences, convergent, and you've got your real > irrational. These include the Ramanujan engines, summations and > continued fractions, for such as 1/pi, one of my favorites.
This doesn't hold up, since almost all irrationals are not computable numbers, according to the standard definition of computable numbers.
This shows that repeated addition fails utterly as a model for almost all the reals.
The reader might enjoy some of what can be found on the Internet on this. Here's one nice exposition:
"Numbers that cannot be computed" http://igoro.com/archive/numbersthatcannotbecomputed/
Partial quote:
"Interestingly, there are numbers that are noncomputable. A number is noncomputable if there is no program that prints its infinite decimal expansion (adding trailing zeros if a finite expansion is possible). How do we know that there are such numbers? The key insight is that there are more real numbers than there are C# programs. That is pretty surprising, given that both the number of real numbers and the number of C# programs are infinite. Nevertheless, it is true.
C# programs are countable. That means that we can assign a different positive integer to each program. The shortest valid C# program will be 1, the next shortest will be 2, and so forth. If there are multiple valid programs of the same length, we will sort the programs lexicographically and assign integers in that order. By the way, there are many different ways in which to define a "valid program". One approach is to say that to be valid, the program must compile, run without exceptions, and print an infinite sequence of digits, separated at exactly one place with a decimal point.
Unlike C# programs, real numbers are uncountable. There is no way to assign a different integer to each real number... there are just too many real numbers! This fact was proved by Georg Cantor in a couple different ways, the most famous of which is the diagonalization argument.
Not only do noncomputable numbers exist, but in fact they are vastly more abundant than computable numbers. Many, many real numbers are simply infinite sequences of seemingly random digits, with no pattern or special property. But, even though there are so many uncountable numbers, their examples tend to be weird and a little strenuous to explain.
As one such example, consider a number whose part before the decimal point is 0. We choose ith digit after the decimal point to be different from the digit in the same position in the number printed by program i (by "program i", I mean the program associated with integer i, as described earlier in the article). So, each digit after the decimal point guarantees that the constructed number will differ from the number printed by a particular program. This demonstrates that the constructed number will be different from any number printed by a computer program! By the way, this construction is basically Cantor's diagonalization argument, only recast in a different terminology.
Theoretical foundations of computer science, which underlie everything we do as programmers, are nothing short of amazing. If you would like to read more about computability and related concepts, check out Charles Petzold's book, The Annotated Turing. The original Alan Turing's paper that introduces computable numbers is available here, but Petzold's book is a lot easier to read." > >> >> I reiterate everything I said in my last post in this thread >> >> http://mathforum.org/kb/message.jspa?messageID=7887658 > > If I show you a setting on the number line and say that's my scale > factor, and then you watch a blob get bigger or smaller (scaling, > length) by that scale factor amount, there's no visible / conceptual > distinction between said scale factor being in Q or in (RQ) where > (RQ) = the Reals with all Rationals removed. The visual demo is the > same, and one could argue the visible number line indicator, metalic > and old, is incapable of pointing finely enough to really tell us > which kind of number is being visited. Ditto with the balloon's > volume. Is it rational or irrational? There's no difference in the > model.
That's' precisely why it works as a model of multiplication on the real numbers and its subsets.
> > You do this sidebar soliloquy to the audience saying how very > important the different brands of Real really are.
Again, the above link gives a nice explanation of how different noncomputable numbers really are and the fact that almost all reals are not computable.
Again, this shows that repeated addition fails utterly as a model for almost all the reals.
> I think we have established there is no one thing that multiplication > *is*.
There is nothing that binary multiplication *is*  it's merely one of the names we give to a binary function on a set S, S x S > S. When we have another such binary function on the same set, call it by another name, and then have certain algebraic properties over that set and these functions, we can derive all the models or properties in question of either of those operations.
No, we can't tell kids all this, but this is fact, regardless.
And for R, we can use models that have to be changed for each of the subsets of R in question, or we can use a model that does not need to be changed, or both.
> We also have many examples of systems in which a binary > multiplication operation visavis set objects has no representation > in the sense of scaling. >
Exactly.
But if we want one that holds up generally and universally and unchanged for any set under an order such that we have magnitude or absolute value, then scaling is it  it holds up without having to be changed for any and all such sets.
And if we want one that promotes proportional reasoning in students, especially if we think that proportional reasoning in students is important enough to be the sooner the better, then scaling is it, the sooner the better.

