I've inadvertently cultivated a reputation among my peers as a math guy. The truth is, I didn't really become interested in mathematics beyond a necessary evil until about 2009, when I finally resolved that I would sit down and and just *look* at the anatomy of the field until I understood it.

When I identify something of value out in the wilderness, I, like many other people, tend to evangelize about it a little bit. For math, the reactions I've gotten have ranged from gee shucks I'm so bad with numbers

to important people like me don't have to understand propellerhead crap like that

, with precious little genuine interest. I'll address the indignation elsewhere, though I suspect it's just insecurity in disguise. As for numeracy, I'm no good with numbers either. I think in shapes.

When most people hear the word *math*, they appear to conjure up experiences from high school or their undergraduate program where they were traumatized by drill after mind-numbing drill of algebra, calculus or trigonometry. Clichéd questions abound:

What's it for? When am I ever going to use this?

Answer: These branches of mathematics are for *building things*. You would use them if you were planning to build a tool-shed or nuclear submarine. As rewarding as it is to build things, however, not everybody does it for a living—or even a hobby for that matter—so it isn't especially surprising that people don't encounter in the real world the math that they were forced to choke down in school. Aside from basic arithmetic, about the only school-compulsory math a non-engineer would find useful in their day-to-day life is statistics, which, frankly, everybody should understand in order to keep politicians and bureaucrats in check, and stay out of trouble with their bookie.

Before I started to really look at it, I thought of mathematics as a sprawling, arcane labyrinth, that existed objectively, like the hard sciences to which it is so often married. It was something of an epiphany to realize that every iota of mathematics was *invented*—rather than discovered—by a person, to solve some problem or other.

The field is also spectacularly egotistical, with its participants historically being men of leisure with plenty of time to play around with intellectual puzzles. Just about every mathematical concept beyond what the average person is exposed to is named after somebody or other, and there was considerable pressure for a mathematician to seal his notoriety early on in his career, superstitiously before his 30th birthday.

Contemplating mathematics from the point of view of a mathematician gave me a sense of empathy I didn't have before. Picture this: you're hard at work trying to get famous by beating out symbols on a blackboard. The people you're trying to impress are just like you. And there's a deadline. Of course your notation is going to be inscrutable: nobody else has to understand it.

It's quaint how the paragon of precision, order and certitude is codified as ad-hoc scribbles, indexed by last name. Knowing that makes me feel just how quirky and human the discipline of mathematics really is, and that it isn't so daunting after all.

Algebra, calculus and trigonometry are useful for engineering *stuff*, but a great deal of mathematics is designed for engineering *thought*. It is the mathematics of *form* rather than *quantity*, of structure, relationships and change. There's hardly any of the boring calculations you'd be accustomed to as *math*, and a mountain of it is directly applicable to so-called *tech* work.

- Set theory
- If you spend any time at all organizing information, you're invariably sorting things into piles. This card belongs in the pile on the left, that card belongs on the pile on the right, which is a sub-pile of the pile in the middle, and the last card is driving you nuts because it belongs in both piles. Set theory fixes card sorting by letting the cards exist in more than one pile at once.
- Mathematical logic
- As far as I'm concerned, logic is just set theory with different motivations, namely that you're trying to figure out whether something goes in the
*true*pile or the*false*pile. Unsurprisingly, getting good at this makes you really good at arguing, problem-solving and the general smelling of bullshit. It also helps with understanding the behaviour of nerds. - Group theory
- When you pair a set up with an operation that produces another member of the same set, you get a group, like the set of numbers plus multiplication, which takes numbers as inputs and makes other numbers. What makes groups interesting is that they can squash what is otherwise a huge conceptual space down to a pinpoint, which is of direct use to people designing complex data exploration environments. Of course, the set component of a group doesn't have to be numbers. It can be any set, like the set of all possible configurations of a Rubik's Cube, closed under the
*twist*operation. - Category theory
- Category theory is kind of like group theory, except it's concerned with operations that take a member of one set and turn it into a member of a
*different*set. Things get extra crazy when you realize that you can operate on the operations themselves, to produce more operations (that operate on operations). Once derided as*generalized abstract nonsense*, category theory, along with set theory and logic, has come to be recognized as a foundational pillar of mathematics. Understanding it also helps to understand what computers are doing all day long, without having to resort to a metaphor. - Graph theory
- While the aforementioned branches of mathematics are concerned with
*types*of things and the relationships between them, graph theory is about*specific*things and the (specific) relationships between them. Once you construct a graph, you can perform all sorts of valuable analyses on it. Box-arrow diagrams like entity-relationship and mind maps are graphs, so are project plans. The Web is a graph, and the study of social networks, also graphs, dates back to the 1970s. Graph theory is only going to get more important and germane to everyday knowledge work. - Topology
- I think of topology—at least one facet of it—as being kind of like geometry minus the pain of calculating all those distances and angles and stuff. Topology studies how an object can maintain its identity, even after undergoing tortuous deformations. It is
*exactly*the kind of theory that is useful for generating rules around laying informational structures out on a screen, especially screens of many different shapes and sizes. - Combinatorics
- OK, I lied. There are some numbers. Big ones. Combinatorics is for understanding the quantitative attributes of everything I already mentioned. It's understanding that 13 ranks times four suits equals 52 suit-rank-pair playing cards, or how many millions of charts a 23-dimensional business intelligence database will generate. It will enable you to understand what kinds of services you can and can't offer at your startup, for lack of available computing resources in the universe. And if you're asked for an estimate on some complex information work, a rough understanding of combinatorics will help keep you from making an ass out of yourself.

Mathematical notation is optimized over centuries for writing very hastily by hand on a surface for which space is a premium. Symbols are chosen largely arbitrarily, or by tangential analogy, such that a given notation can have different meanings in as many contexts. The operations underpinning the symbols themselves are meant to be understood in advance and carried out ultimately in a person's head.

This is actually an argument for learning how to code. Not only do computers take care of the gruntwork of computation as the name suggests, the notation is actually an *improvement* over conventional math, being that it's far easier to compel a person to use a consistent grammar than design a computer to correctly interpret chicken scratch.

Of course, the act of programming is still very much centred around imagining in your head how the symbols are going to behave before you run them, then running them to see if your supposition was correct. This I believe is why even those who work so closely with information technology—and develop extreme proficiency—still don't take the last step into programming, even when they understand its value: the user experience is just too damn onerous.

Computers turn math into real-life actions. Whether we're hacking the Gibson on a command line or faux-fingerpainting on some jelly bean touch screen, it's all math under the hood. The more we understand, the more powerful we become. I'm suggesting it's time that we start demanding interfaces that expose and help us understand the underlying mathematical principles in ways that *help* us get our work done. I make them that way myself whenever I can. This isn't just for the non-programming contingency either. I use them too.