Came across this after some related Quora discussion:
http://www.maa.org/external_archive/devlin/devlin_06_08.htmlI have big issues with this rant. Mainly, it misses the main point that the definitions are, by and large, arbitrary. In some cases, some definitions simplify things, but really the definition is part of the math, and it can be personal preference on what to take as your axiom and what to take as your conclusions.
The basic claim is that multiplication is not defined through addition. But it can be. Roughly: suppose you know what binary x+y means for any real numbers x and y. Association lets you define n-ary '+'. Then you define
m*x = x*m = sum(i=1..m, x) = x + x + ... + x (m times)
for any integer m and real number x. (Note that commutivity is by definition.)
Now, given some integer m!=0, we may define a real number m^{-1} = (1/m) by the real number that satisfies
m*(1/m) = (1/m) * m = 1.
Now, suppose p and q are rational numbers, p=n/m, q=k/l, where n,m,k,l are integers (m, l nonzero). Then we define
p*q = (n*k)*(1/(m*l)).
This is perfectly defined as n*k is an integer and 1/(m*l) is a real number.
Now, what is x*y for any real numbers x and y? Well, we take a sequence of rational numbers x_n -> x and y_n -> y. (This could actually be done relative to any topology; the normal topology on the real numbers coincides with the usual Euclidean distance. You could argue that we need to know how to multiply real numbers to measure convergence, but you could do it on the level of topology. Or you could simply take the Euclidean norm as given, and it's through that norm that we define multiplication. Though maybe if you take any topology and require multiplication to be continuous wrt that topology you would get the same thing.. I'm not really sure.) Then x*y is defined to be the limit of the sequence x_n*y_n. A small amount of work shows it's well defined.
So we started with addition of reals and some notion of a topology and ended up with multiplication of reals. There is nothing to say that this definition of multiplication is not "really" multiplication---it surely is. Moreover, in statements like:
And telling them that multiplication is repeated addition definitely requires undoing later.
How much later? As soon as the child progresses from whole-number multiplication to multiplication by fractions (or arbitrary real numbers). At that point, you have to tell a different story.
"Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?" a bright kid will say, wondering how many more times you are going to switch the rules.
The author is missing a big issue. There ARE different kinds of multiplication. Just because an algebra A is a subset of some B and you have a binary operation to make B an algebra doesn't mean A with the induced operation of B is the same thing as B. In fact, A with induced operation may not even be an algebra. In other words, even if I take a "multiplication" with the real numbers as axiomatic, that doesn't necessarily mean you should have the same notion of multiplication on the integers. Sure, it turns out the natural definitions coincide, but even so (Z,*), (Q,*) and (R,*) are entirely different objects. Moreover, if you consider, for example, the set X = {0,1,2,3,4,5}, this is certainly a subset of real numbers. You can even induce R's multiplication on X... except you get into trouble. Because * does not take X into X; it takes X into some larger set. In fact, the thing we may be interested in is not R's * but a different kind of * that maps X into X, some special *_X. Maybe x(*X)y = (x*y) mod 6. In fact, there are unlimited numbers of multiplications.
Though, notably, in my example multiplication by fractions is not 'changing any rules'. It is operating on the same rule that multiplication by an integer is.
Also, this:
Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not "basic" operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.
is arbitrary. You could just as easily say that there are two things you can do to numbers: you can subtract them and you can divide them. I discount addition and multiplication , since these are just inverses. It may be more natural to define addition and multiplication as axiomatic (though as I said, you don't 'need' to define multiplication; you can define topology or metrics instead), probably because of how we count, but it doesn't *have* to be done this way; it's a choice.