On Fri, Aug 31, 2012 at 10:58 PM, Jonathan Crabtree <email@example.com> wrote:
<< snip >>
> Until now, multiplication itself has never been correctly defined for the rational numbers. > > Jonathan Crabtree > 1st September 2012 > > P.S. Think of a photocopying machine. You have one letter and you need four altogether. So what button do you press on the multiplication machine? Three! ie k-1 Press the four button and you end up with five.
This may be true, and we have had threads on "what is multiplication, really?" quite a few times over the years, on this list.
My general position is it's no one thing with respect to objects in general, yet you might say there's a strange attractor that keeps it anchored to notions of inverse, associativity and multiplicative identity.
In that respect, addition and multiplication are similar, with their distinction so far coming down to choice of operator (commonly + and *) and choice of representation for an identity element (commonly 0 vs 1).
Defining multiplication in terms of addition is not a bad ingredient to throw in, but likewise thrown in should be sets wherein multiplication is defined, but not addition.
You can have one (multiply operation) and not the other (addition operation).
A good example is "permutations" as I described them earlier:
You'll have A * B -> C (closure), and every A * inverse(A) = 1, the identity permutation, (A*B)*C == A*(B*C) associativity and in this case commutativity. You'll have a group.
I think it's fine to introduce these little games of things that "multiply" in a somewhat abstract sense, to build an appreciation for the pattern.
We can write these little programs (functions) and noodle the objects interactively, programmatically.
We don't just stare at inert symbols on paper on wood pulp. The symbols have a life in an interpreted environment.
An advantage to this approach, in complement to any "multiplication as repeated addition" meme, is that when we spiral back through division and subtraction, we already have a strongly developed notion of "inverse" (relates to "identity element"). That (3/4) / (1/2) is "syntactic sugar" for (3/4) * (2/1) is obvious i.e. "to divide by" is "to multiply by the multiplicative inverse of" just as "to subtract from" is to "add the additive inverse of".
Making these connections, thanks to experiments with unary and binary operators within interactive computer game like environments is going to result in a more sophisticated appreciation for algebra in the class of 2020 etc.
As soon as + and * (addition and multiplication) make an appearance with respect to the same set (class or type of object), if we can link them via the distributive property, which we can in a field, then we can start up with the "repeated addition" meme -- before spiraling back to powering.
Speaking of "math objects" and "class or type of object" is deliberate, as we're connecting to the idea of "typed languages" of the object-oriented variety especially.
Using operator overloading, we are able to give our own meanings to the * and + operators -- part of what's driving this more abstract algebraic approach is the our new-found ability to make it concrete.