The Decision Problem
In 1928, the mathematician David Hilbert posed a challenge he called the Entscheidungsproblem (which translates to “decision problem”).
Roughly speaking, the problem asks whether there exists an effective procedure (what we would today call an “algorithm”) that can take as input a set of axioms and a mathematical statement, and then decide whether or not the statement can be proved using those axioms and standard logic rules.
Hilbert thought such a procedure probably existed.
Eight years later, in 1936, a twenty-four year old doctoral student named Alan Turing proved Hilbert wrong with a monumental (and surprisingly readable) paper titled, On Computable Numbers, with an Application to the Entscheidungsproblem.
In this paper, Turing proved that there exists problems that cannot be solved systematically (i.e., with an algorithm). He then argued that if you could solve Hilbert’s decision problem, you could use this powerful proof machine to solve one of these unsolvable problems: a contradiction!
Though Turing was working before computers, his framework and results formed the foundation of theoretical computer science (my field), as they can be used to explore what can and cannot be solved by computers.
Over time, theoreticians enumerated many problems that cannot be solved using a fixed series of steps. These came to be known as undecidable problems, while those that can be solved mechanistically were called decidable.
The history of theoretical computer science is interesting in its own right, especially given Hollywood’s recent interest in Turing.
But in this post, I want to argue a less expected connection: Turing’s conception of decidable and undecidable problems turns out to provide a useful metaphor for understanding how to increase your value in the knowledge work economy…
Two Types of Tasks
Let’s begin by briefly turning our attention from Turing to something more mundane: tasks in a knowledge work setting.
The standard definition of a task for a knowledge worker is a clear objective that can be divided into a series of concrete next actions. The productivity guru David Allen, who introduced the “next action” terminology, emphasized in Getting Things Done that if you properly break down your tasks into concrete next actions, your day can function like a factory worker “cranking widgets” as you seamlessly shift from one action to the next.
There are parallels between this definition of a task and Turing’s notion of a decidable problem. In both cases, a clear procedure can be systematically applied to the challenge until it’s solved.
But these decidable tasks are not the only type common in knowledge work.
Another type of task are those that have a clear objective but cannot be divided into a clear series of concrete next actions.
- A theoretician trying to solve a proof.
- An creative director trying to come up with a new ad campaign.
- A novelist trying to write an award-winning book.
- A CEO trying to turn around falling revenues.
- An entrepreneur trying to come up with a new business idea.
All these examples defy systematic deconstruction into a series of concrete next actions. There’s no clear procedure for consistently accomplishing these goals. They don’t reduce, in other words, to widget cranking.
There are parallels between these types of tasks and Turing’s notion of undecidability. As with Turing’s undecidable problems, any given instance an undecidable task might be solvable, there’s just no systematic approach that’s guaranteed to always work.
On The Value of Undecidability
I argue that the ability to consistently complete undecidable tasks is increasingly valuable in our information economy.
Because these solutions cannot be systematized, this skill cannot be automated or easily outsourced.
Similarly, if you can complete undecidable tasks, you cannot be replaced by a 22-year old willing to work twice your hours at half your pay — as it’s not simply raw effort that matters.
(By contrast, if your day is composed entirely of decidable tasks you’re vulnerable to any of these above dangers.)
That’s the good news.
The bad news is that undecidable tasks are often really hard to complete. Because there’s no easy way to divide them into concrete actions you have to instead throw brain power, experience, creative intuition, and persistence at them, and then hope a solution emerges from some indescribable cognitive alchemy.
You may have guessed where I’m heading with this analysis.
What type of effort supports such difficult cognitive challenges? Deep work.
In other words, understanding the decidable/undecidable task split provides yet another argument for the value of deep work — as it’s only in the cultivation and consistent application of serious concentration that you can expect to succeed where Turing’s machines fail.