A kid asks you, “why is area given by multiplication?” For example, why does a rectangle of length 3 and width 2 have an area of 6? You might answer “because we can break it down into six $1 \times 1$ squares, each of area 1”.
But then we’ve got to ask: why is the area of the unit square 1? Perhaps we’ve just defined it this way. But there doesn’t seem to be any obvious reason for doing so, if there should even be one. From the point of view of physical intuition/naivete, why should length be “measured” in the same way (i.e. with numbers) as area? The idea of “units are not the same as numbers” seems unsatisfactory to me because it appears to be dodging the question and assuming the hypothesis (that the same numbers can be used to represent dimensionally different objects).
Perhaps once we’ve defined how multiplication should correspond to area, we could define an addition more naturally. Say if we’ve defined the area of a unit square to be 1, then we might “reasonably” define the sum of the areas of two unit squares to be 2.
My questions are:
Would we ever want $1 \times 1 \neq 1$?
Perhaps more meaningfully: Would we ever want to think of multiplication as an operation taking two objects in a space and producing an object in a different space? A binary operation on a set $X$ is by definition a map from $X \times X$ to $X$; but, as in the case of area above, it doesn’t seem “obvious” why the codomain should be $X$. It seems that something more general like $X \times X \to Y$ is more appropriate.
In this sense, is addition different from multiplication? I.e. could or should these two operations be meaningfully defined on objects belonging to different spaces? In particular, if we would like our notion of multiplication to take two objects in $X$ and output an object in a different space $Y$, then we may want to think of addition as taking two objects in $Y$ and outputting an object in $Y$ as well. E.g. in the area example above, we multiply two lengths to get an area, whereas we add two areas to get another area.
How does all this reconcile with the desire to have a geometric argument for things like why area is given by multiplication? In particular, if it is just a matter of definition, shouldn't we admit that instead of reducing it to seemingly circular arguments like "a unit square has area 1"?
No, because $m\times n$ means $\sum_{k=1}^mn$ for $m,\,n\in\Bbb N$, which implies $1\times1=1$.
What happens when I want to multiply together arbitrary numbers of values? One famous option with just two spaces involves $c$-numbers and Grassmann numbers, and a product of these is a $c$-number (Grassmann number) if an even (odd) number of the original factors are Grassmann numbers. Another, with infinitely many spaces, is encountered in dimensional analysis.
Going back to dimensional analysis, we pretty much need addition to "keep us where we are" in a way multiplication needn't.
I'd prefer to take my definition of multiplication on $\Bbb N$ as fundamental, then note this implies a rectangle with integer sides can be rearranged to give the obvious total area as a single width-$1$ strip. Bigger number systems aren't a problem: with non-integer sides, a rectangle motivates how we can generalize all this. Flipping edges motivates $(-a)b=a(-b)=-(ab),\,(-a)(-b)=ab$. Using smaller units naturally gives us $\times$ on $\Bbb Q$, after which we handle $\Bbb R$ by continuity.