Multiplication Of Integers

Multiplication of whole numbers, and how we think about it ...

I want to take a – relatively minor – exception to Keith Devlin’s post, What Exactly is Multiplication? , on multiplication as repeated addition, and then largely agree with what he writes, with some amplification.

Devlin claims that multiplication is not repeated addition and that “the mathematician’s concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.).”

I believe he is wrong on both counts.

First, both addition and multiplication are defined notions on the integers, rational numbers, and real numbers. For the positive integers, multiplication is defined recursively as .

This makes multiplication exactly repeated addition.

Secondly, the commutative property then follows as a logical consequence of this definition and is not an axiom.

It is an axiom for general structures such as rings or fields, where one posits a commutative operation of multiplication, but it is a provable property of the defined notion of multiplication of natural numbers.

One can prove properties of multiplication from the definition of multiplication of natural numbers – essentially defined as repeated addition – but, as Devlin points out very well, this is not a productive or creative way of thinking about multiplication.

The inductive definition of multiplication of whole numbers is like a lawyer’s definition: it’s designed for safety reasons, to ensure that we are able, at some low level, to prove statements we make about multiplication.

However, the legalistic inductive definition of multiplication is not really helpful in thinking in other ways about multiplication as an operation on whole numbers.

What is needed for younger students is a sound psychological – not simply logical – way of thinking about multiplication.

Keith Devlin is, to my mind, essentially blurring over the distinction between concept definition and concept image.

The inductive definition of multiplication is a concept definition and, as such, is a part of advanced mathematical thinking , as defined and elucidated by David Tall .

Defined notions are, in general, not especially suitable for younger students, who do not yet see the necessity for such rigor.

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