I need to give a lot of quite basic background to this question because I think (at least from conversing with fellow graduate students) that most mathematicians have not *really* thought about fractions for a long time. I think that there is an interesting germ of an idea in here somewhere, but I cannot exactly pinpoint it. Essentially there seems to be two canonical ways to solve division problems and there does not seem to be a "natural isomorphism" relating the two ways. I am interested in framing this duality formally: is there a "categorification" of the rational numbers where this duality can be precisely framed?

I TA a class for future elementary school teachers. The idea is to go back and really understand elementary school mathematics at a deep level. Hopefully this understanding gets passed on to the next generation.

We were discussing division of fractions. Rather than say "Yours is not to wonder why, just invert and multiply", we try to make sense of this question physically, and then use reasoning to solve the problem. Take (3/4) / (2/3). When doing this there seems to be two reasonable interpretations:

1) 3/4 of a cup of milk fills 2/3 of a container. How much milk (in cups) does it take to fill the entire container?

This is a "How many in each group" division problem, analogous to converting 6/3 into the question "If I have six objects divided into three equal groups, how many objects will be in each group?"

The solution that stares you in the face if you draw a picture of this situation is the following: 3/4 of a cup fills 2 thirds of a container. That means there must be 3/8 cups of milk in each third of the container. One container must have 9/8 cups of milk then, because this is 3 of these thirds. Note that the solution involved first dividing and then multiplying.

2) I have 3/4 cups of milk, and I have bottles which each hold 2/3 of a cup. How many bottles can I fill?

This is a "How many groups" division problem, analogous to converting 6/3 into the question "If I have six objects divided into groups of two, how many groups do I have?"

The solution suggested by this situation is the following: 3/4 of a cup of milk is actually 9 twelfths of a cup . Each twelfth is an eighth of a bottle. So I have 9/8 of a bottle. This solution involved first multiplying and then dividing.

Now I come to my question. This pattern persists! Every real world example of a "how many in each group" division problem suggests a solution by first dividing and then multiplying, whereas each "How many groups" division problem involves first multiplying and then dividing. It seems that solving the problem in the other order does not admit a conceptual realization in terms of the original problem. This is interesting to me! It suggests that the two solution methods are fundamentally different somehow. The standard approach to rational numbers (natural numbers get grothendieck grouped into integers, which get ring of fractioned into rational numbers) ignores this kind of distinction. Is there a "categorification" of the rational numbers which preserves the duality between these two types of question?

UPDATE 1: In the category of sets, if you wanted to express $(\frac{6}{2})(3) = \frac{(6)(3)}{2}$ you would have to do something like this:

Let $A$ be a set with 6 elements, $B$ a set with 3 elements, $\sim$ an equivalence relation on A where each equivalence class has 2 elements, and $\cong$ an equivalence relation on $A \times B$ where each equivalence class has 2 elements. Then there is no canonical morphism from $(A/ \sim) \times B \to (A \times B)/ \cong$. This seems to explain things somewhat on the level of integers, but we are talking fractions here.

UPDATE 2: Qiaochu points out in a comment to his answer that the order of operations is not the most essential thing here. You can solve the first problem by observing that 9/4 cups of milk fill 2 containers, so 9/8 must fill one. Torsors give a formal distinction between the two situations, but it still feels like UPDATE 1 should go through in a suitable category of "fractional sets".

UPDATE 3: For a very nice discussion of the ideas related to Theo's answer see http://golem.ph.utexas.edu/category/2008/12/groupoidification_made_easy.html