A philosophical point:
The linked article is at risk of being internally inconsistent. The grounds on which he rejects subtraction and division as basic operations can be generalized, making his insistence that multiplication is basic more tenuous (as well as undermining his point about exponentiation). He rejects subtraction and division as basic because they are the inverses of addition and multiplication, respectively. He also acknowledges that multiplication and repeated addition give you the same results for natural numbers.
Having said all this, he attacks the notion that multiplication is repeated addition through an argument about fractions. Unfortunately, that argument can be countered using only that which he has already admitted. Give someone addition and the natural numbers, and his views about what is basic get you subtraction (since it’s “just” the inverse of addition) and the set of integers (since we can get these from the natural numbers plus subtraction—we even get to bypass debates over whether or not 0 is a natural number).
Once we have all this, we can explain the multiplication of fractions by breaking them down into a series of additive relationships among integers. We simply need to treat the numerator and the denominator separately, acting as if we were doing two different equations. Moreover, this separation of the numerator and the denominator is precisely how the multiplication of fractions is often introduced to students. Perhaps Devlin doesn’t like this either, though I address that in the next section. The point for now is just that his argument is unstable. He would do better to focus on the differences between multiplication and summation, and to give up his rejection of subtraction and division as basic (which I notice he does do in later blog entries).
An educational point:
It is true that multiplication is not repeated addition. Multiplication is multiplication. It concerns scalar relationships, not additive ones. Education, however, is an additive relationship: you teach something new by building on what the students already know. So while Mr. Devlin may be correct that teachers should not just blithely say that multiplication is repeated addition, there is no reason why they should not point out the similarity that a great deal of multiplication problems have to repeated addition when teaching it.
Note that the linked article is actually a bit of a paper tiger: it ends by basically admitting precisely what I’ve just said. Mr. Devlin acknowledges that he is making a fairly technical point about mathematics and that he does not know exactly how it should translate into the field of education. What I will say, then, is simply this: a good teacher has several different ways of explaining any given point. Thus it would be counterproductive to not have the “multiplication as repeated addition” line in one’s bag of tricks. “Multiplication as repeated addition,” however, is different from “multiplication is repeated addition.” The former only rests on using the similarities to prompt initial understanding. It is hard to see how Mr. Devlin could reasonably object to this.
I take it that this is the point @Blackberry ended up making while I was typing this.