January, 2012

Jan 12

Musicians, Mechanics, and Mathematicians

Thank you all for your comments on my previous post, I appreciate the time you all took in sharing your perspectives very much.  Many of you have brought up great analogies to demonstrate how you feel and in reading these responses I realized I must not have been very clear.

There are some musical geniuses who have composed great works without having been taught even the basics of music theory. However, this doesn’t mean they’re not doing math. The human brain excels at building approximate mathematical models and a rare few minds are capable of building exceedingly complex ones. Still, formal knowledge of the patterns of music allow a musician to both play the same song in new and interesting ways and see the underlying connections between different pieces. As a composer it informs how rhythms and melody can be juxtaposed or fitted together to create a desired effect in a way much more profound than trial and error. It expands the musician’s mind and makes them better at what they do. 

Another great example is that of the wrench wielding mechanic. There are a great many mechanics who went to trade school and learned the high level basics of engines and a long list of procedures. They might not understand the details of combustion or material science but they can replace brake pads or swap in a new timing belt without too much difficulty. After many years of experience some may have even built a mental model so superb that they can take you for a spin around the block and tell you exactly what’s wrong.

And still, as the mechanic reaches for their bolts and wrench they might not think of the underlying mathematics of what they are about to perform. Yet, you can be sure the people who made those tools worried over it greatly. If they didn’t the wrench would not fit the head or worse, the bolt might shear under the stress applied. While they surely tested many bolts before shipping their product, they certainly didn’t before creating a formal model of how the tools were shaped or how they would perform. Even if the tools might happen to work without testing, they probably wouldn’t work very well and to sell tools made in this way would be grossly negligent.

Yet, I can’t be the only one who has suffered many near catastrophes at the hands of inept mechanics over the years. From the time a post-brake change air bubble in my brake line made my car roll out into traffic or the punctured gas tank that almost left me stranded at the side of the road. One might wonder if they even bothered testing their work.

Some might think programmers shouldn’t be beholden the same strictness as our creations aren’t usually capable of quite so much damage. Instead, the worst things most are liable to do are destroying important information, providing incorrect data to critical systems, leaking private information, sharing unsalted (or worse, unencrypted) passwords or causing people to become unable to access their bank or medical records. No big deal really, just shoveling data.

I’d love to see every programmer taking the time to learn deeply about the mathematical modeling of data and programs, but I know that’s not reasonable. However, it takes just a little bit of learning to leverage tools with very complex underlying mathematics made by others. You don’t need to be an expert in category theory to use Haskell any more than you need to be an expert in set theory to use SQL. F# and Scala are even more accessible as they have access to all of the libraries and patterns you would be familiar with as a programmer who works in .NET or Java.

So, I’m not asking that you go out and spend years studying your way to the equivalent of a PhD. Instead what I ask is that you just take a little time to understand what’s possible and then use tools made by people who do have that kind of deep understanding.

I know I wouldn’t want to drive a car with parts made by someone who didn’t use models, would you?

Huge thanks to @danfinch and @TheColonial for proof reading.

Jan 12

Why do most programmers work so hard at pretending that they’re not doing math?

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.

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.