Why Do We Abstract?

The concept of abstraction is one of the building blocks to advanced understanding of our world.

A simple mathematical example

Everyone knows that the probability of getting heads when you flip a coin is 50%.

But a mathematician might say that the probability of getting a series of heads when flipping coins is

P = r^n,

where P is the probability of getting a series of heads, r is the probability of getting heads with a single coin toss, and n is the number of repetitions. In our single coin toss, let’s say when a football referee lets the winner choose which end zone to defend, we set r=0.5 (a fair, unbiased coin) and n=1. The probability P then becomes 0.5, or 50%. We already knew this!

While this is a trivial equation for mathematicians, for many people this already looks intimidating. It is unfortunate that our education system teaches a majority of us to become anxious when faced with mathematical equations. For example, Sheila Tobias documented this phenomenom in the 70s and 80s, by showing how professionals were able to solve underlying problems verbally but froze as soon as the same problems were written in mathematical notation. But why, one might then ask, did we write out such a difficult equation, if we already knew the answer or could get there without equations? To answer that, let’s look at the case of two coin flips.

One could still argue that it might just be as easy to just count the scenarios. With two coin flips, there are four possible events: heads-heads, heads-tails, tails-heads, and tails-tails. Given a fair coin, these are all equally likely. So the probability of getting heads twice is 25%. This easy mental exercise is likely faster than conceptualizing and writing out the equation and then plugging in the values.

But what about a thousand coin flips? This wouldn’t intimidate mathematicians: everything just follows naturally from the equations. And this is the crux of the matter: abstraction allows us to generalize to circumstances that are unseen, unfamiliar, and hard to grasp using everyday mental techniques.

Once we realize that by pulling out the general structure of this event, it doesn’t matter anymore if this is about coins. Similar formulas govern any kind of probabilistic repetition. This is where we move from mere generalization to abstraction proper.

When abstraction discovers

The coin toss formula is useful, but it is essentially a shortcut. It gets you to an answer faster than counting would. The abstraction is doing the work of a very efficient clerk.

But sometimes abstraction does something more unsettling: it tells you something true about the world that you had no way of knowing from observation alone.

In the early 1860s, the Scottish physicist James Clerk Maxwell was working with a set of equations describing electricity and magnetism — two phenomena that seemed, on the surface, to be separate. As he manipulated the equations abstractly, something unexpected fell out: a prediction that electromagnetic disturbances should propagate through space as a wave, and that this wave should travel at approximately 300,000 kilometres per second.

Maxwell recognized that number. It was the speed of light.

Nobody told the equations that light existed. Nobody put light into the model. The abstraction, followed honestly to its conclusion, discovered light as an electromagnetic phenomenon — decades before anyone could measure it directly. Heinrich Hertz confirmed the existence of the predicted waves experimentally in 1887, nearly twenty years after Maxwell’s death.

This is what abstraction can do at its most powerful. It is not a filing system for things we already know. It is a vehicle for arriving at things we did not know were there — and could not have found by looking harder at the world in front of us.

The coin-flip formula tells you the answer to a question you already understood. Maxwell’s equations told him the answer to a question nobody had thought to ask.

This is why many popular equations in many fields are generalized principles from physics, such as the Black-Scholes equation used in financial markets is derived from Brownian motion.

Abstraction as a universal cognitive tool

The coin example is mathematical, but abstraction is not the exclusive property of mathematicians. Consider a map of a major city’s metro network. It tells you which stations connect to which lines, and in what order. What it does not tell you is the exact distance between stations, what the streets above look like, or whether the escalator at a station is operational. All of that has been deliberately stripped away.

Is the map wrong? No — it is usefully simplified. A map that showed everything would be as large and complicated as the city itself, and therefore useless. The abstraction is the point.

This is the second reason we abstract, beyond generalization: abstraction manages complexity. The world contains more information than any mind can hold at once. To think clearly about anything, we have to decide what to ignore, and what the truly defining general characteristics are.

Abstraction as imagination

There is a deeper reason we abstract, beyond mere convenience. Abstraction is what allows us to escape the present entirely.

Every other animal is largely trapped in “what is” — the immediate signals of hunger, danger, warmth, and threat that the senses deliver. Humans, uniquely, can inhabit “what could be.” We can reason about a million coin flips we have never performed, a bridge that has not been built, a disease that has not yet spread. This capacity — to model situations we have never encountered — is the engine behind science, engineering, law, and art. None of it is possible without abstraction.

Put differently: a dog knows it is raining. Only a human can wonder whether it will still be raining tomorrow, and build a roof in case it is.

The mind as an analogy machine

Cognitive scientists Douglas Hofstadter and Emmanuel Sander, in their book Surfaces and Essences, make a striking claim: that analogy is not a rhetorical flourish, or a tool we occasionally reach for when explaining something difficult. It is the core mechanism of all human thought. Every time we recognize something — a chair, a greeting, an unfair situation — we are silently asking: what is this like? And then mapping the new thing onto something we already understand.

This is abstraction by another name.

When George Lakoff and Mark Johnson argued in Metaphors We Live By that all human thought is structured by metaphor — that we talk about time as if it were money, about arguments as if they were battles, about ideas as if they were objects we can hold, examine, and hand to someone else — they were pointing at the same mechanism. We do not encounter the world directly. We encounter it through a dense web of analogies, mappings, and comparisons that allow us to apply old understanding to new situations.

The coin-flip formula is an analogy, in this sense. It says: this new situation (a million flips) is structurally the same as this familiar situation (one flip), just scaled. The abstraction works because the underlying pattern transfers.

What makes humans unusual, then, is not just that we think, but that we think at multiple levels of generality simultaneously — and that we can deliberately choose to move between them.

The ladder

The linguist S.I. Hayakawa had a useful image for this in his 1949 book Language in Thought and Action. He called it the ladder of abstraction.

Imagine a specific cow — let’s call her Bessie. Bessie is a particular animal, with a particular weight, a particular temperament, and a white patch above her left ear. At the bottom rung of the ladder, we are talking about Bessie specifically. One rung up, we call her a cow, which loses the patch and the temperament but keeps the essential bovine-ness. Another rung up, she becomes livestock, a category she now shares with pigs and chickens. Higher still, she is a farm asset. At the top, she has become wealth — a pure abstraction that tells you almost nothing about Bessie, but says everything about her relationship to an economy.

At each step, we throw away detail and gain generality. At each step, we can say things about a wider set of things, but with less precision about any one of them.

The coin-flip formula lives high on this ladder. It says nothing about any particular coin, or any particular referee, or whether the coin was minted in Singapore or struck in Birmingham in 1987. It discards all of that to preserve only the one relationship that matters: probability scales with repetition at a fixed rate. That is why it can tell you about a million flips, even though no specific million flips were in mind when it was written.

The metro map does the same thing spatially. It climbs the ladder away from the messy physical reality of tunnels, crowds, and broken escalators, and preserves only the topological skeleton: which stations connect, and in what order.

Knowing which rung you are on — and why — is most of what it means to think clearly.

The cost of abstraction

This is where abstraction becomes genuinely interesting, and occasionally dangerous.

Every abstraction is a bet. When we model a coin toss as having exactly two outcomes, we are betting that the coin never lands on its edge, that the tosser isn’t cheating, and that gravity works the same way it always has. These are sensible bets. But they are still bets.

In 2008, financial institutions were using abstract mathematical models to price mortgage-backed securities. The models were elegant. They generalized beautifully. What they abstracted away — what they chose to ignore — was the possibility that house prices across the entire United States could fall simultaneously. That particular scenario had not happened in the historical data the models were trained on, so it was, in effect, thrown out.

The abstraction was not wrong in the way that 2+2=5 is wrong. It was wrong in the way that a map is wrong when it leaves out the flooded road.

Why we do it anyway

None of this is an argument against abstraction. It is an argument for doing it carefully.

The alternative — trying to reason about the world without abstracting at all — is not available to us. Our brains are abstraction machines by necessity. When you recognize a friend’s face, you are not processing every photon that bounced off them; you are matching a compressed pattern against stored compressed patterns. When you understand the word “justice,” you are working with an abstraction so old and so layered that philosophers have spent millennia arguing about what it actually refers to.

We abstract because we have no choice. The question is whether we abstract well — whether we know what we have thrown away, and whether what remains is still true enough to be useful.

Further reading

1. S.I. Hayakawa — Language in Thought and Action (1949).
2. George Lakoff & Mark Johnson — Metaphors We Live By (1980).
3. Douglas Hofstadter & Emmanuel Sander — Surfaces and Essences (2013).
4. Sheila Tobias - Overcoming Math Anxiety (1978).