In the early days programming was considered a subdiscipline of mathematics. In fact, the very first person to write an algorithm was renowned as a mathematical genius. However, somewhere along the way we forgot. We began to think of ourselves as something different, a profession not beholden to rigor or deep understanding of the models we create.

It’s easy to see how this would happen within an industry in which so much more weight is put on knowing the API of the year over understanding base principles or expressing the desire to dig deep. People can make huge amounts of money pushing methodologies when they have little to no evidence of effectiveness. We work in an environment where hearsay and taste drive change instead of studies and models. We are stumbling in the dark.

I have come to attribute our sorry state to a combination of factors. The first is the lack of formal training for most programmers. This isn’t in itself a bad thing, but when combined with a lack of exposure to mathematics beyond arithmetic, due primarily to an inadequate school system, we are left with a huge number of people who think programming and math are unrelated. They see every day how their world is filled with repeating patterns but they miss the beautiful truth that math is really about modeling patterns and that numbers are just a small part of that.

The relationship between math and programming extends far beyond SQL’s foundation in set theory or bits being governed by information theory. Math is intertwined within the code you write every day. The relationships between the different constructs you create and the patterns you use to create them are math too. This is why typed functional programming is so important: it’s only when you formalize these relationships into a model that their inconsistencies become apparent.

Most recently it seems to have become a trend to think of testing as a reasonable substitute for models. Imagine if physics was done this way. What if the only way we knew how to predict how an event would turn out was to actually measure it each time? We wouldn’t be able to generalize our findings to even a slightly different set of circumstances. But it gets even worse: How would you even know what your measurement is of without a model to measure it within? To move back into the context of programming: how useful is a test that doesn’t capture all of the necessary input state?

This is exactly why dynamic programs with mutation become such a mess. Object oriented structure only makes things worse by hiding information and state. Implicit relationships, complex hidden structure and mutation each impair reasoning alone, but when combined they create an explosion of complexity. They each conspire with each other to create a system which defies comprehensive modeling by even the brightest minds.

This is also why programmers who use typed functional languages frequently brag about their low bug count yet eschew methodologies like test driven development: tests pale in comparison to the power of actual models.

Followup Post: Musicians, Mechanics, and Mathematicians

Many thanks to my friends on twitter for the discussion leading up to this post:  @seanschade, @craigstuntz, @sswistun, @joakinen, @copumpkin, @dibblego, @taylodl, @TheColonial, @tomasekeli, @tim_g_robinson and @CarstKoenig. Special thanks to @danfinch for taking the time to proof read.