Jonathan

Messages like this mean I _really_ need to finish my next post, which I’ll try to do this week.

In short (although he unfortunately never makes this clear) while Devlin acknowledges the existence of Peano Arithmetic et. al. and building multiplication off of them, that the number system is not the one we use until we reach the level of integral domain and by then our definition has to be changed.

In fact, problems start happening as soon as one adds 0. Here’s a quote from a number theory book:

The idea we followed in the last chapter was that muliplication was repeated addition, and so 2 * 3 told us when to add 3 2’s together. The idea easily covers 0 * 3. It tells us to add 3 0s together. But, using the definition above, (0 + 0) + 0 = 0 + 0 = 0, so 0 * 3 ought to be 0.

There is more of a problem with 3 * 0 however. If we apply our basic idea, we find ourselves adding together 0 3s. We have no 3s (or anything else) to work with, so we certainly can’t get a number for an answer. Our original motivation isn’t sufficient to cover this case.

…

Now, in this case, the modification is slight, but things start to get worse later.

What’s going on, essentially, is a difference of philosophy of mathematics. Either standpoints has possible arguments. I promise I will finish my full exposition soon.

Multiplication on real numbers IS repeated addition. It’s so MATHEMATICALLY. It’s the standard way to DEFINE multiplication on integers in foundations of mathematics. And it can be easily extended to all real numbers (infinite sums).

Furthermore, kids first learn multiplication on natural numbers, NOT fractions or real numbers. Teaching kids multiplication over natural numbers WHILE telling them multiplication is not repeated addition is IDIOTIC. The kids would never trust another word the teacher says.

What Devlin in effect advocates is that we should LIE to kids initially so that we can PRETEND later on that some conceptual problems do not exist.

Well he should start by outlawing the term MULTIPLY since in ordinary English it usually means to become more numerous (e.g. the Biblical go forth and multiply).

Why Devlin gets away with his crap is a mystery to me. I guess it’s because he wrote 50 books on far too many topics than he can truly understand.

Cheers,
Mike