I started programming in high school as a hobby and went on to study mathematics and linguistics at the University of Chicago, where I graduated in 2006. The d

I started programming in high school as a hobby and went on to study mathematics and linguistics at the University of Chicago, where I graduated in 2006. The date is just to give a sense of the timeline: it wasn't until 2-3 years ago that these worlds felt meaningfully compatible to me. So, I'll do my best to explain why this is "practical," as someone who spent a long time trying to establish a relationship between "mathematician jfarmer" and "programmer jfarmer." For me, the #1 practical benefit is clarity. For example, you say that "the algebra of data types, as described, is simply the algebra of natural numbers." Is this really true? The mathematician's move would be to abstract out this common "algebra" and isolate that structure and that structure alone. The algebra of data types and the algebra of the natural numbers then become instances of this more universal algebra. In other words, the picture you have is that "A is an instance of B" of really it's that A and B are instances some third thing, C, that isolates their shared algebraic structure. If you're curious, the shared algebraic structure is called a semiring (http://en.wikipedia.org/wiki/Semiring http://en.wikipedia.org/wiki/Semiring). Why is this helpful? Because it makes for clearer explanations. We can understand what facts/theorems are true because of this shared structure vs. what is true because of the particularities of one of the more specific structures. We can prove things about semirings per se which will be true for any particular semiring. We then have a new strategy available to us: if we can show some structure is a semiring, a whole universe of facts become available to us without any additional work. Moreover, from this perspective, whole classes of seemingly-distinct algorithms become instances of a more abstract algorithm depending on which semiring we're operating over. For example, the probabalistic and and non-probabalistic versions of a CKY parser are instance of the same algorithm which vary only according to which semiring they're executed over. ({TRUE, FALSE}, OR, AND) and ([0,∞), +, * ), respectively. That's not just a curiosity; it's an entirely new axis of thought that reveals connections between what previously seemed like similar-but-unrelated pieces of software. It opens up the possibility of writing a function that takes a semiring specification* as input and returns a function for the "instantiated" algorithm over that semiring. * : This example requires more than just a semiring, but I'm just trying to illustrate the point.