Intuitive proof of commutativity of addition and multiplication?

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?

• Did you mean commutativity? Jan 13, 2020 at 19:50
• How are you defining addition and multiplication? What is the domain over which they are being defined? In other words, what ring are you using? Jan 13, 2020 at 19:54
• Axiomatization involves abstracting away all of the unnecessary concrete information in order to describe how something "should" work. Thus the axiomatization of these two properties is: for any $x$ and $y$, (1) $x+y = y+x$ and (2) $xy = yx$. Jan 13, 2020 at 19:55
• Among the things mathematicians are willing to call "numbers", only very few can be represented by objects that are moved around. You cannot "prove" commutativity of addition or multiplication of complex or $p$-adic numbers like that. Rather, we observe those things indeed for natural numbers, then make big leaps of abstraction, write down axioms that encapsulate all the structures we find interesting for whatever reason; and then we call everything that obeys these axioms "numbers". Jan 13, 2020 at 20:00
• Now in some cases one can "prove" these things assuming less: e.g. one can prove that finite skew fields are fields i.e. their multiplication is commutative; or one can prove commutativity of $p$-adic or complex multiplication from some other things, if one is given some kind of definition of them which does not contain that commutativity already. But now we get into the territory of @DonThousand's comment: You'll have to specify what exactly you want to prove in what exact framework of reference. Jan 13, 2020 at 20:03

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.