On Tue, Sep 11, 2012 at 9:34 AM, Paul Tanner <firstname.lastname@example.org> wrote: > On Tue, Sep 11, 2012 at 10:28 AM, Joe Niederberger > <email@example.com> 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 flat-out 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.
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.
When it comes to models, one of the big questions is do we show (1/2) of (one sphere) as the sphere shrinking to 1/2 the volume (not that big a change in radius then), or do we show a separate sphere showing up, as though the result were additional substance? I know this sounds vague and philosophical but in "computer science" as some call it (a neighborhood in STEM), objects may change state "in place" or... a method may produce an entirely new object.
Concretely, in Python the strings are immutable, meaning s.lower( ) produces a new string, does not lowercase all the letter of s "in place". The list object, on the other hand, may be re-ordered in place, such that the previous state is no longer in memory. L.sort() is the syntax, where L is a list (similar to a set, but with implicate order and no restriction on duplicates).
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 (R-Q) where (R-Q) = 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.
You do this sidebar soliloquy to the audience saying how very important the different brands of Real really are. The Transcendentals, the Irrationals... such very different species. You recite the many differences you know about. You prove you are good at science. However, I think you should admit that those who don't miss a beat going from Q to R when explaining multiplication, are not doing violence to the spirit of the growing / shrinking / scaling demo. It's *supposed* to cover "all Reals" and to decry a coverup, as if a difference were being denied, is to forget that this is precisely what models do: they suppress the differences that make no difference (like "race" in some circles).
I think we have established there is no one thing that multiplication *is*. We also have many examples of systems in which a binary multiplication operation vis-a-vis set objects has no representation in the sense of scaling. Permutations may be multiplied. Multiplication becomes a kind of composition, a sequential piping of functions, the output of one comprising the input of the next. Matrix multiplication is isomorphic to a sequence of rotations, where order matters i.e. the same numbers of left and right turns don't guarantee the same outcome. Yet undoing exists. It's like a maze. With a minotaur maybe.
Classrooms which lack imagination and should be slated for closure, in the cartoons of the better schools, offering caricatures of what to avoid (the shysters, the snake oilers (except what if I like to oil snakes, make 'em shine? -- my freedom)), are those which only stick to numbers and number sets, when yakking about multiplication. Take a traditional American school for example, probably built in the 1930s, stuck in a time warp. Any posters about Unicode? Have they ever even heard of ASCII? Is everything digital left to some "computer club" with an aging parent blowing out ancient PCs with dust busters from Office Depot? Been there, right? You're talking walking dinos, moribund to the core, might as well have tumble weeds blowing through, a teenage wasteland of the mind. So hold those up as examples to avoid, ridicule them, cut them no slack. Cuz you don't want junior to turn out like *that*.