Metaphors We Compute By

In the book Metaphors We Live By, George Lakoff and Mark Johnson set out to show the linguistic and philosophical worlds that metaphor wasn’t just a matter of poetry and rhetorical flourish. They showed that metaphor permeates all areas of our lives, and in particular that metaphor dictates how we understand the world, how we act in it, how we live it. They showed our conceptual system is based on metaphors too, but since we are not normally aware of our own conceptual system, they had to study it via a proxy: language.

In the beginning there was the Word

By studying language they tried to understand how metaphors work by imposing meaning in our lives. The basic example they present is the conceptual metaphor of “Argument is War”. We understand the act of arguing with another person in the same way we understand wars. This means we come up with the following expressions in our daily language:

Your claims are indefensible.

He attacked every weak point in my argument.

I demolished his argument.

I never won an argument with him.

The problem with these sentences that seem innocuous is that we act and feel based on them. We end up seeing the person we are arguing with as our opponent. Someone that’s attacking our positions. This mean we structure argument in a similar way as if we were at war with the other person. This means the metaphor is not just language flourish, we live it. Lakoff and Johnson propose the exercise of imagining a culture where arguments are not viewed in terms of war, of winners and losers, but in which language is a dance, where we have to cooperate with our partner in order to achieve our desired goal, reaching conclusions as a team. The book keeps going analyzing the different aspects of language and metaphors and how they affect our concepts and view of the world. The authors give many examples to defend the thesis that our understanding of the world is based on metaphors and that those metaphors are the foundation of our behaviors.

A big takeaway from the book is that metaphors enable certain ways of thinking while it restrict others, as the example above shows. In this article we are interested in understanding how metaphors shape the way we understand computing and its related problems. We want to see what kind of problems are enabled by the metaphors in use, and for which ones we lack the right metaphors still. And no, monads are not like burritos!

In what follows we are not going to discuss the book thesis any further, on the other hand, we will accept it and bring it into Computer Science. First we will see how metaphor help us understand the world of computers, a world that didn’t exist a mere 80 years ago. Then we will try to see how metaphor works in the way we structure code, or design algorithms and data structures. We will see that we even solve problems in our discipline based on what metaphors are part of our arsenal.

“Sometimes our tools do what we tell them to. Other times, we adapt ourselves to our tools’ requirements.” – The Shallows. Metaphors are the tools of comprehension.

A metaphorical understanding

We understand new concepts by way of relating them to things we already know. Back in the late ’40s and early ’50s when today’s computers came to life, there was no word to name such invention, but we understood them as automatic brains. What’s more interesting is the fact that the word computer existed already, and it was the term used to refer to the person doing calculations for engineers. Think of an engineer having to know what would be the trajectory of a projectile, or how the wind would affect an airplane’s wing shape, so they would throw a couple of formulas and numbers to their “human computers” in order to get the answer they wanted. Now they had these new machines that worked automatically, so they called them Electronic Computers, dropping the electronic part as timed passed by. So our very own discipline was named after a metaphor.

But metaphors also obscure possibilities if we don’t understand their limitations. A common problem with new metaphors is that we use the original meaning of the word at face value. We don’t pay attention to how the word we are using to explain a new concept limits our understanding of the very concept we are trying to explain. James Gleick on his book The Information gives a fascinating account of the invention of the telegraph. The word “tele-graph” means far-writing. Lo and behold early telegraphs were strange machines that tried to actually write at a distance, using one-to-one mapping of letters of the alphabet into wires which sounds terribly impractical. Around that time, thanks to Braille we understood that language could be coded in a form different from how it sounds (or how it is written). Morse Code was the next step in improving our telegraphs, and to understand that we don’t have to write-at-a-distance to have actual long distance communications. Thanks to Claude Shannon and the many smart people before him, we managed to escape from the problems of the telegraph direct metaphor and build the whole discipline of Information Theory. We know that information must be encoded first, then sent into a channel, to be decoded at the other end of the channel so the destination receives it.

Peter Gärdenfors explains how metaphors create new knowledge in his book “The Geometry of Meaning – Semantics Based on Conceptual Spaces”:

[…] a metaphor transfer information from one conceptual domain to another; what is transferred is a pattern rather than domain specific information. A metaphor can thus be used to identify a structure in a domain that would not have been discovered otherwise. This is how metaphors create new knowledge.

Lakoff (quoted by Gärdenfors) says:

Metaphorical mappings preserve the cognitive topology of the source domain in a way consistent with the inherent structure of the target domain.

Metaphors and Code

An Unknown author in a letter to Donald Knuth said that to program is to write to another programmer about our solution to a problem. A program is our explanation of how a problem might be solved, it’s a metaphor that stands in place of our understanding, but for metaphors to be effective they need to convey meaning using concepts already known to us. We could use the explanatory power of a particular program as measure of its own elegance.

Let’s think about the following example. We could program a computer to command other computers to perform tasks respecting their arrival order. This description is already hard to understand. On the other hand we could describe the solution by talking of a Queue Server that assigns Jobs to Workers based on a First Come First Served queue discipline. Queues are a familiar concept from our daily lives, we see them at the supermarket or the bank. We wait in them at airports or train stations. We understand how they work, so for someone reading our code it might be easier to talk about queues, workers and jobs, than to go all over the verbosity required to explain this setting without using the Queue metaphor. We see that by choosing the right metaphor our program reaches a level of abstraction that requires the least effort for someone foreign to the problem to understand our solution. Also by solving our problem with queues, we get a whole mathematical theory for free. Mathematics is itself a field where often problems are only tackled once an appropriately expressive language is available to approach them. We are no longer punching blind in the dark. Now we can analyze and understand our problem with all the tools provided by Queueing Theory.

Data Structures as Metaphors

We analyze data structures to see which one would be a better fit for the performance characteristics of our particular problem, but we often forget that a data structure also contains explanatory power.

The choice of data structures helps us convey the meaning. We could store a bunch of elements in an array, but what if we actually need to ensure that the elements are unique? Isn’t that what Sets are for? The underlying representation of our set can still be backed by a plain array, but now our program express its intentions in a clearer way. Whenever another programmer reads it, they will understand that the elements in our collection must be unique. A very important realization is to see that for a computer our programs will be just another succession of bits that it needs to process. It is us that give meaning to those bits, so we have to use the right metaphor on top of them to make our programs clearer. As it is said:

no one has seen a program which the machine could not comprehend but which humans did. –– Unknown author See here for the paper with the original quote.

We must strive to make our programs as easy to understand to other programers as possible. Ease of comprehension should be the tally with which we measure our programs. We should keep in mind that we can arrange our code in many different ways to solve a computing problem, but not all those arrangements will favor human communication and understanding. We must ask ourselves: is someone reading my code going to understand how I solved this particular problem?

The same way a metaphor enables certain ways of understanding and limits others, so do data structures. We already saw the problem with using direct metaphors like the original telegraph. When it comes to data structures we can see that a Set will tell us that its elements are unique, and it would allow us to test if an element is a member of the set. But a Linked List gives us the idea of traversing its elements on after the other, without being able to skip them. An Array gives us the idea that we can address its elements by index. The same can be seen with Queues and Stacks, two of the most fundamental data structures taught in any algorithms course. Both can be implemented using an array, the difference is that one returns the elements in FIFO order, while the other does in LIFO order respectively. Even if this looks like an everyday thing for most programmers, the moment we choose to use a Stack instead of a Queue, we are deciding how we want to explain our program. The Stack is a very good metaphor for the collection of items that our program works with, because it tells a future reader of our program in which order we expect to process the items. We don’t even need to read how the stack is implemented, since we can assume we will get the items in LIFO order. This is why types are so important in computer science. Not types as in static type checking of programs, but the types as the concepts we use to describe our programs: Persons, Users, Stacks, Trees, Nodes, you name it. Types are the characters that tell the story of our programs, without types, we just have operations on streams of bytes.

Cognitive Leaps

A situation we will face once and again is finding the right metaphor that describes and explains our problem. As we saw earlier with our queueing theory example, we had to make a cognitive leap to go from tasks that we have to process in certain order to understanding this is a queueing problem. Once we managed the cognitive leap, all the mathematical tools from queueing theory were at our disposition. Graph Theory is filled with examples of mundane tasks that once converted to a graph problem, have well known solutions. Whenever you ask Google Maps to get you to your destination, Google is translating your problem to a graph representation and suggests one or more paths in the graph. Graphs are the right metaphor, understood by mathematicians and computers alike. Do we have more instances of problems that seem difficult but that we crack once we find the right metaphor? The Distributed Systems literature has a very interesting one.

In the late ’80s Alan Demers and his colleagues from Xerox tried to find a solution to database replication in unreliable networks. They classified their algorithms as “randomized”, explaining them by using the Rumor Mongering metaphor: a computer would tell two other computers about and update, then each in turn will tell two other computers about the update, and so on, until the information got replicated. This metaphor gave way to a new area of study called around so called Gossip Algorithms. The gossip metaphor makes the idea easy to explain, but we still lack the mathematical tools that would help us analyze the effectiveness of the algorithms. During their research they discovered another metaphor related to Epidemics. They understood that their algorithms replicated data like when an epidemic disseminated across a population. By using this new metaphor they got immediate access to all the knowledge in The Mathematical Theory of Epidemics, which fitted their work like a glove. Not only they named their paper “Epidemic Algorithms for Replicated Database Maintenance” but they also took the nomenclature of the discipline to explain their algorithms. It was a matter of finding the right metaphor to get access to a new world of explanatory power.

Metaphors Everywhere

We might still be skeptical and ask ourselves if we do use that many metaphors in programming. Let’s take a look at the Distributed Systems literature in general (metaphors are in italics):

Whenever nodes need to agree on a common value, we start a consensus algorithm to decide on a value. There’s usually a leader process that takes care of making the final decision based on the votes it has received from its peers. Nodes communicate sending messages over a channel, which might get congested due to too much traffic. This could create an information bottleneck, with queues at each end of the channels backing up. These bottlenecks might render one or more nodes unresponsive, causing network partitions. Is the process that’s taking long to respond dead? Why didn’t it acknowledge the heart beat and a timeout triggered?.”. I could go on, but I guess you get the point.

A Story in Code

As programmers we must be able to tell a story with our code explaining how you solved a particular problem. Like a writer we must know our metaphors. Many metaphors will be able to explain a concept, but we must have enough skill to choose the right one that’s able to convey our ideas to those future programmers that will read our code. Thus we cannot use every metaphor under our belt. We must master the art of metaphor selection, of meaning amplification. We must know when to add and when to subtract. We will learn to revise and rewrite our code like a writer does. Once there’s nothing else to add or remove we have finished our work. The problem we started with is now the solution in front of our eyes. Is that the meaning we intended to convey in the first place?

References

Lakoff, George, and Mark Johnson. “Metaphors We Live By”

Gleick, James. “The Information: A History, a Theory, a Flood”

Shannon, Claude Elwood, and Warren Weaver. “The Mathematical Theory of Communication”

Bailey, Norman T. J. “The Mathematical Theory of Epidemics”

Demers, Alan, Dan Greene, Carl Hauser, Wes Irish, and John Larson. “Epidemic Algorithms for Replicated Database Maintenance”

Gärdenfors, Peter. “The Geometry of Meaning – Semantics Based on Conceptual Spaces”

Credits

Image: https://flic.kr/p/6GRuVy License CC BY-NC 2.0

Slides

I had the honor to present this blogpost as an eighteen minute talk at this year’s Monki Gras Conference in London. Here are the slides: