20 Jan 2015
Free structures are a popular idea in Haskell when speaking about
constructions like the “free monad” or notions like “lists are free
monoids”. This terminology can be misleading, however. Free structures
aren’t just “free”, they have to be “free” with respect to something.

18 Dec 2014
There's a folk-wisdom trick for converting a right fold on lists into a left fold. This may not appear to be terrifically useful at first but if you look a bit beneath the surface then it reveals a practical tool in any functional programmer's toolbelt.

04 Dec 2014
There is no royal road to mathematics, but instead an enormous shining underground city. Here, I hope to give a little tour to its districts and customs for the new traveler.

30 Sep 2014
I gave a talk
for
Papers We Love, Boston
on the 18th where I discussed Martin-Löf type theory from a historical
and programming context. The goal was to motivate and inspire reading
the (notably long) 1983 Martin-Löf lectures
*On the Meanings of the Logical Constants and the Justifications of the Logical Laws*. The video was just posted and the slides are available here.

24 Sep 2014
When studying sets in mathematics you learn a number of seemingly arbitrary constructions like, for instance, the so-called Cartesian Product. It's natural to ask why these constructions are so privileged... and while the answer is not exactly straightforward, it is very revealing! (An answer to a question posed on Quora.)

22 Aug 2014
This is a silly nit to pick, but here goes. JSON doesn't denote objects, it denotes finite, initial state which is exactly why it's useful. So, JSON isn't an "object notation", only JavaScript itself is.

10 Aug 2014
There's been a little flurry of activity around analyzing Clojure's upcoming

*transducers* using Haskell types. This post outlines an isomorphism between one such representation and list "binding functions"

`a -> [b]`

. This suggests that

`flatMap`

is central to the idea of transduction.

08 Aug 2014
The technical concept "type safety", and even the underlying technical concept "type", is often misunderstood due to being conflated with marketing terms or given undue technical weight. This post enumerates some useful intuition pumps for better understanding, talking about, and consuming material related to type safety.

04 Aug 2014
Modern programming has something that was often talked about but never formalized throughout much of modern mathematics,

*codata*. This post sketches a constructive proof of diagonalization using codata in Haskell notation.

30 Jul 2014
Enumeration of container types is a very important programming technique. In immutable languages, one mechanism for this is to have a generic "left view" interface. This interface corresponds exactly to

`Stream`

codata and Swift is powerful enough to (almost) implement all of the above directly.