On Mon, Sep 3, 2012 at 12:09 PM, kirby urner <firstname.lastname@example.org> wrote: > On Sun, Sep 2, 2012 at 6:27 PM, Paul Tanner <email@example.com> wrote: > > << snip >> > >> Let's generalize this all the way from a field to a ringoid, so we can >> really see what's going on. A ringoid is simply a set under two binary >> operations such that they are "connected" with a distributive >> property, one of these operations distributing over the other. *By >> convention*, we call the one being distributed over "addition" and the >> other one "multiplication". >> > > Note that when we linger around GCD, play more with primes, bigger > primes, do more with totatives, stray into Group Theory, I'm not > suggesting a full semester on those topics necessarily, as if "in for > a penny, in for a pound" is always operative. No, you can be in for a > penny, a quarter, five dollars, ten... i.e. your level of time/energy > investment in group theory / abstract algebra is controllable. You > don't have to get into "ringoids" just because you watched 'Lord of > the Rings' in your planetarium / movie theater (what every school has > by fictional 1999+). >
A field is a ringoid. If we are dealing with two binary operations on a set such that we have a distributive property, then a ringoid is the most general case.
A group is a groupoid (or a magma if you wish that term - I prefer "groupoid" since that is the more universal usage [the term "magma" is the one those of the Bourbaki school use]). If we are dealing with one binary operation, then a groupoid is the most general case.
I prefer to deal in the more general case - I like to "weed out" unnecessary algebraic properties as much as possible, to try to see what is generally going on.
Since the discussion is about one binary operation in terms of another binary operation being repeated via a distributive property, it's clear that we need two and not just one binary operation - a group or groupoid (recall again they require only one binary operation) won't help us. So we need to be a context of a field or most generally a ringoid.
> >>> >>> (a/b)(c/d) = a (bc/d) so you can always isolate an integer and then >>> say you're adding (bc/d) to itself a times. >>> >> >> But this is a serious redefinition of viewing "multiplication" as >> "repeated addition". >> > > It may seem like radical surgery to you but I assure you the avid > teachers of the "repeated addition" meme took this step long ago.
I'm talking about how it must seem to the average kid being hit with this. If you think that having to break up one of the factors before being able to apply the repeated addition model on two factors neither of which is one of the two original factors is not for the average kid a radical departure from being able to apply it directly to the two original factors, then be my guest.
> As > I was saying above, even your "scaling" which you present in cartoon > form *in contrast* to repeated addition, is just more repeated > addition (of some unit of vector / length / distance).
For purposes of calculation, that is true, and Devlin also I think would agree. But this is about whether we should try to get them to see that with multiplication, there is fundamentally something more going on than merely the calculative part of repeated addition, to see that multiplication is in fact a different animal an addition. Especially as we jump to reals, specifically the irrationals and most especially the transcendental irrationals.
> >> Jump to >>> Reals. >>> >> >> I don't think so. >> >> Take e(pi). >> >> What does it mean to have pi *instances* of e or e *instances* of pi? >> > > You have an ungodly huge Avogadro Number of atoms and you divvy them > so that 3.14159... of them (some trillions) get to be a "unit" in some > way, with respect to some multiplier. Numbers like pi and e are just > fractions that "never end" in this vague hand-wavy extension of Q to > R.
This does not appreciate at all what an irrational - especially a transcendental irrational - is.
> > >> Talk about having to bend over backwards to redefine things! >> > > I think you're just culturally isolated and to you it's big news that > some people think this way.
I'd say that if there is any isolation going on here, it would be in people being isolated from understanding just how fundamentally different an animal an irrational number is from a rational number, and this especially applies to transcendental irrationals, which are so fundamentally different they are fundamentally different even in comparison to algebraic irrationals.
Not only that, to see more of just how fundamentally different a transcendental irrational actually is, the number of elements in the set of all reals being an uncountable infinity and the set of all reals being a continuum is due to the number of elements in the set of all transcendental irrationals being an uncountable infinity. The number of elements in the set of all algebraic numbers, which is the union of the set of all rational numbers (all of which are algebraic) and the set of all algebraic irrationals, is merely a countable infinity, which means that this union cannot be a continuum. (A set is a continuum only if it contains an uncountable infinity of elements.)
> >> >> I'd say that we have to really redefine what it means to do something >> n "times". >> > > Well, maybe we should really do that then.