Intuitive proof of commutativity of addition and multiplication?

Question

I have seen some "proofs" of the commutativity of addition and multiplication. But they don't really express how our mind thinks of this.

I "human" proof would go like this:

"If we have a number, $x$, of things on the left and a number, $y$, of things on the right the total number of things is $x+y$. We can move the things on the left to the right and the things on the right to the left. There is the same number of things. But now we write this as $y+x$. Thus $x+y=y+x$."

This relies on the axiom that objects can be moved without disapearing. And the axiom that things can be moved from one place to another.

Similarly with multiplication. We could arrange things in a rectangular grid, and the "proof" would rely on the axiom that the number of things doesn't change under a rotation of the grid.

This seems like arithmetic might be derived from a set of physical axioms of geometry and movement rather than, say, abstract set theory.

Is there a way to make put these axioms into mathematical notation? Or is this just set theory in disguise?

Answer 1

Here are $3$ sets of $5$ objects: $$ \begin{array}{|c|} \hline \begin{array}{c} \bullet \bullet\bullet \\ \bullet\bullet \end{array} \\[10pt] \hline \begin{array}{cc} \bullet \\ \bullet & \bullet \\ \bullet & \bullet \end{array} \\[10pt] \hline \begin{array}{c} \bullet \bullet \\ \bullet\bullet\bullet \end{array} \\[10pt] \hline \end{array} $$ Grab one object from each set and move it to the other side of the room: $$ \begin{array}{lcccr} \begin{array}{|c|} \hline \begin{array}{c} \bullet \bullet\phantom{\bullet} \\ \bullet\bullet \end{array} \\[10pt] \hline \begin{array}{cc} \phantom{\bullet} \\ \bullet & \bullet \\ \bullet & \bullet \end{array} \\[10pt] \hline \begin{array}{c} \phantom{\bullet} \bullet \\ \bullet\bullet\bullet \end{array} \\[10pt] \hline \end{array} & \qquad & \qquad & \qquad & \begin{array}{c} \bullet\bullet\bullet \\ \end{array} \end{array} $$ Now do it again: $$ \begin{array}{lcccr} \begin{array}{|c|} \hline \begin{array}{c} \bullet \bullet\phantom{\bullet} \\ \bullet\phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{cc} \phantom{\bullet} \\ \bullet & \bullet \\ \bullet & \phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{c} \phantom{\bullet} \bullet \\ \bullet\bullet\phantom{\bullet} \end{array} \\[10pt] \hline \end{array} & \qquad & \qquad & \qquad & \begin{array}{c} \bullet\bullet\bullet \\[12pt] \bullet\bullet\bullet \end{array} \end{array} $$ And again: $$ \begin{array}{lcccr} \begin{array}{|c|} \hline \begin{array}{c} \phantom{\bullet} \bullet\phantom{\bullet} \\ \bullet\phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{cc} \phantom{\bullet} \\ \phantom{\bullet} & \bullet \\ \bullet & \phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{c} \phantom{\bullet} \bullet \\ \phantom{\bullet}\bullet\phantom{\bullet} \end{array} \\[10pt] \hline \end{array} & \qquad & \qquad & \qquad & \begin{array}{c} \bullet\bullet\bullet \\[12pt] \bullet\bullet\bullet \\[18pt] \bullet\bullet\bullet \end{array} \end{array} $$ And again: $$ \begin{array}{lcccr} \begin{array}{|c|} \hline \begin{array}{c} \phantom{\bullet} \phantom{\bullet}\phantom{\bullet} \\ \bullet\phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{cc} \phantom{\bullet} \\ \phantom{\bullet} & \bullet \\ \phantom{\bullet} & \phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{c} \phantom{\bullet} \bullet \\ \phantom{\bullet\bullet\bullet} \end{array} \\[10pt] \hline \end{array} & \qquad & \qquad & \qquad & \begin{array}{c} \bullet\bullet\bullet \\[12pt] \bullet\bullet\bullet \\[18pt] \bullet\bullet\bullet \\[10pt] \bullet\bullet\bullet \end{array} \end{array} $$ And again $$ \begin{array}{lcccr} \begin{array}{|c|} \hline \begin{array}{c} \phantom{\bullet\bullet\bullet} \\ \phantom{\bullet\bullet} \end{array} \\[10pt] \hline \begin{array}{cc} \phantom{\bullet} \\ \phantom{\bullet} & \phantom{\bullet} \\ \phantom{\bullet} & \phantom{\bullet} \end{array} \\[10pt] \hline \begin{array}{c} \phantom{\bullet\bullet} \\ \phantom{\bullet\bullet\bullet} \end{array} \\[10pt] \hline \end{array} & \qquad & \qquad & \qquad & \begin{array}{c} \bullet\bullet\bullet \\[12pt] \bullet\bullet\bullet \\[18pt] \bullet\bullet\bullet \\[10pt] \bullet\bullet\bullet \\[30pt] \bullet\bullet\bullet \end{array} \end{array} $$ Three fives have become five threes.