Where do I go wrong with Presburger “multiplication”? This is a speculation about the Presburger arithmetics, the Peano axioms with only addition added, vis-à-vis the same axioms with both addition and multiplication added.
In the first case I understand that any proposition is decidable but not in the second. 
My problem is: how is it possible that I can sneak the multiplication in - without any other actions than appealing to the reader’s feel for geometrical figures – and seemingly go from one system to the other? 
Let us assume we are equipped with the natural numbers, the Peano axioms and one composition rule: addition. This will allow us to count objects, numbers for example. We first discover the one-to-one correspondence between the natural numbers such as n, and the geometrical concept of rectangles with length n and width 1, by associating the number n with a rectangle of length n consisting of n unit area element. 
We now consider the number of unit area elements of a rectangle of length n and width m, we can count the unit areas line by line and we can denote the result $n*m$ (assuming $*$ to be an unused symbol). We can now see, by applying a physical or geometrical principle of relativity, that (imagining the rectangle placed on the floor) if we move about 90 degrees in a circle around the rectangle it will look like another rectangle of $m*n$ units of area. Assuming our movement has not changed the rectangle, we have $n*m=m*n$.  
We can also make a cut in a $n*m$ rectangle perpendicular to the length dimension, after k units of length ( 0 < k < n) which will give us $n*m = k*m + (n-k)*m$ so that $a*(b+c) = a*b + b*c$.  I haven’t described all the components of multiplication here but they don’t seem too difficult to “discover”.
Observe that prime numbers – numbers that fail to appear as the number of rectangles - can be found easily. If we wonder about 7 for example we can start by observing that any rectangle with length 7 or higher, and width >1 will correspond to a number >7. We therefore have a finite number of rectangles to consider, with length < 7 and width < 7 since $n*m=m*n$ (interpreted as numbers of unit areas for a rectangle), i.e. not too different from  solving the same problem in Peano arithmetic.
 A: In Presberger arithmetic, the only objects that you have to work with are natural numbers. So the only way to work with a rectangle would be to represent it, somehow, as one or more natural numbers. For example, you could represent a rectangle by a pair of numbers $(a,b)$ representing the length and width.  But, regardless of the representation, you will not be able to define a function $f(a,b)$ which gives the area of that rectangle. In the question, the idea was to define $f$ by counting the area "line by line". However, there is no way to do that within the language of Presberger arithmetic. The formula that defines multiplication would not be able to mention $f$ at all, and cannot use the sort of inductive definition that you have in mind, which says $f(a+1,b) = b+f(a,b)$. 
If you were working in a stronger system, it would indeed be possible to define multiplication from addition. In set theories like ZFC it is possible to define both addition and multiplication from just the successor function $s(n) = n+1$. But that is because set theory is much more expressive, which provides stronger ways of defining new functions. The language of Presberger arithmetic is simply too impoverished to define multiplication from addition. 
