Asymmetry in interpreting integer multiplication This is a very basic / fundamental question about arithmetic.
Multiplication of natural numbers can be thought of as repeated addition, which amounts to "copying" a given number of times. For example, $2 \times 3$ can thought of as the number $3$ taken $2$ times, that is, $3 + 3$. But it can also be viewed as the number $2$ taken $3$ times, that is $2 + 2 + 2$. Visually, we can look at these two ways of multiplying as
$$\begin{array}{ccc} 
\bigcirc & \bigcirc & \bigcirc\\
\bigcirc & \bigcirc & \bigcirc
\end{array}
\quad\quad  \textrm{or} \quad \quad
\begin{array}{cc} 
\bigcirc & \bigcirc \\
\bigcirc & \bigcirc \\
\bigcirc & \bigcirc
\end{array}$$
Of course, the result is $6$ in both cases, but it seems to me that fundamentally, this is not exactly the same operation. When considering the "structure" of the repeated addition, multiplication appears to be asymmetric, although it is commutative when considering only the integer result. (Actually, I think commutativity becomes a meaningful property only if we first consider this asymmetry.)
Given this asymmetry, is there any convention on the order of arguments for integer multiplication? That is, should we write $2 \times 3 = 2 + 2 + 2$, or is $2 \times 3 = 3 + 3$ preferable?
 A: There is a well-established convention for multiplication of ordinal numbers in set theory (where it matters): $\alpha \cdot \beta$ means “count to $\alpha$, $\beta$ times”, or in other words, “$\alpha + \dots + \alpha$”.
This follows from taking the definition to be inductive in the factor on the right:
$$
\alpha \cdot (\beta+1)=\alpha\cdot \beta+\alpha
.
$$
But for integers (where it doesn't matter), opinions differ. Of course, while developing the theory, one has to choose one way or the other, until the commutativity has been proved.
See also this question at the Mathematics Educators StackExchange.
A: One can interpret this as a demonstration that multiplication is commutative: $a \times b = b \times a$, since one produces the horizontal orientation of the rectangle, the other the vertical.
(This relies on choosing a specification for which of $2+2+2$ and $3+3$ you mean by $2 \times 3$: which choice we make is unimportant, but only because the operation does turn out to be commutative (in particular, this is false for other types of object).)
