Imagination Games
Mathematics, science and make-believe
I want to introduce you to a new concept I’ve thinking about recently - the concept of an imagination game. It was initially inspired by the mathematics courses I’ve been taking, especially the concept of hypothetical reasoning in logic, but it feels like it could become a generally applicable mental model for analysing many different open-ended creative processes.
Thinking through the lens of imagination games helped me see the connection between how mathematical and scientific knowledge grows, and how many creative domains share similarities with childlike play. It has also helped me gain an understanding of things I couldn’t explain before, like why certain mathematical theories are the way they are, and how they could have evolved differently.
In this essay I will argue that the kind of reasoning children use to dream up fanciful worlds of princesses, trolls and heroes to play in with their friends is fundamentally the same kind mathematicians and scientists use to formulate their abstract theories and models of the universe.
Imaginative Reasoning
There’s a common pattern of reasoning that goes by different names in different disciplines: in natural deduction, the most popular proof system in mathematics, it’s called hypothetical reasoning. In science it’s called hypothesis generation or conjecture and in philosophy it’s the kind of reasoning associated with thought experiments. It exists all across the social sciences too - historians often use counterfactual reasoning when analysing past events, and in finance, speculating about what could happen in the market captures the essence of this kind of reasoning with a pejorative slant.
But it’s not just professionals or academics who use this style of reasoning - take the cognitive distortion catastrophising1 for example, or daydreaming about what you would do if you won the lottery - both of these are universal thought patterns that use the same underlying style of reasoning as the more academic examples above. In fact any time you’ve used conditional or speculative sentence constructions you’ve used this style of reasoning:
Imagine if I won the lottery, then I could travel the world.
Suppose Napoleon had not invaded Russia in 1812, then Paris would not have fallen in 1814.
Assume for the sake of contradiction that to occupy a given position in space is to be at rest, then a flying arrow is at rest.2
I group all of these terms and sentence schemas under the general category of imaginative reasoning3. In each case imaginative reasoning boils down to making one or more assumptions (without pressure to explain or justify them in detail) and exploring the consequences. In the sentences above, the words “suppose”, “assume”, and “imagine if” introduce assumptions while the second clause (after the word “then”) derives a consequence.
Imaginative reasoning by itself is quite a granular and “zoomed-in” view of reasoning because it’s defined at the level of sentences and sub-clauses. But when we zoom out from, say, how imaginative reasoning is used in the context of a single line of a single proof in propositional logic, and instead think about the role of imaginative reasoning in the creation of an entire mathematical theory, then there’s a whole new class of interesting questions to explore.
We can wonder about things like where the assumptions at the very bottom of the theory, called axioms, came from in the first place? Are they self-evident truths or something else? Can they be connected to imaginative reasoning somehow? How is it that starting from a blank page mathematical theories evolve to so perfectly capture the relationships and behaviours we wish to model while excluding those we want to forbid?
These are the questions that motivated my interest in imagination games. But before extending the idea to the highest intellectual realms of human endeavour like mathematics or science, let’s being by examining its humble roots in the minds of young children.
Make-Believe
When a 2- to 3-year-old boy plays airplane with a wooden stick, … he is at the same time spontaneously creating his own imaginary reality. Who is to say that this reality is not more "real" to the boy than the adult reality? - Desire for society: Children’s knowledge as social imagination
Do you remember when you were a young child hanging out with your friends and how you would often spontaneously conjure up elaborate fantasy worlds of princesses, trolls, and heroes to play in? I realised that I have gone my whole life taking for granted what an incredible feat of intelligence this is. The invention of a pretend world involves a great deal of imagination - to think of an interesting setting, to come up with roles, names and titles and use symbolic substitution to turn the plain, meaningless objects in the environment like sticks and stones into wands, guns and currency.4
Before they can start playing, children must negotiate a starting set of facts and rules governing how they are expected to behave during the game. Some of these will be ad-hoc rules specific to the game, while others appear to be universal, like the “finders-keepers” rule for role-playing props which is consistently reinvented by children across different cultures. Each player also brings their own set of inexplicit background rules and facts, like moral rules, knowledge of generations and gender, cultural assumptions and more.
The set of rules created at the start of make-believe games is never rigorous and fixed, but fuzzy and malleable. Once children start playing, power struggles over possessions or the relative status of their roles, and arguments about the game being boring or unfair result in re-negotiations and modifications of the rules. For example, children frequently clash over the “finders-keepers” rule which grants individuals monopolies over desirable objects. It’s common for them to reinvent the principle of “sharing through turn-taking” to resolve conflicts. In this way the game and its ruleset evolve progressively over time as problems are encountered and resolved during play.
Most make-believe games die when they are abandoned and forgotten by the small number of players who created them, but every now and then a game captures the attention of its players in a way that defies intuition - its small, finite list of facts and rules produces games with seemingly infinite depth and complexity. It feels like you could play them every day for the rest of your life and find endless novelty and enjoyment in them.
When we find games with these properties, we are eager to copy them and share them with others so they can explore them with us. Such games are replicated across different social groups and are passed down from generation to generation. Piece-by-piece, the ad-hoc conjectured rule sets existing purely in the minds of players are transcribed into explicit lists of rules and regulations, like those governing modern sports and chess. Formalisation and sharing allows others to play the game without having to go through the slow trial and error process of inventing it for themselves. Over time, these formal facts and rules become precious cultural artefacts that encapsulate a vast amount of knowledge about how to solve the problems encountered during the game’s evolution, making them harder and harder to improve upon without causing regressions.
Imagination Games
There are a few properties of make-believe play which I believe can be found across all imagination games.
Imagination
An imagination game always starts with an act of creativity to conjure up an initial set of facts, rules, or assumptions that define the game and guide the players’ behaviour. Imagination is also used throughout the game to conjecture refinements to the rules and facts in response to problems deliberately caused or stumbled across by the players that were not possible to predict in advance.
Facts and Rules
Players establish a set of facts declared to be true for the purpose of the game, and some rules governing the behaviour of the players. Don’t mistake “facts” for “true statements”, they are best thought of as “imaginary assumptions”, and could be fantastical, improbable or downright absurd. Together, they are like the chessboard and pieces in chess: they delimit the game’s boundaries and permissible actions, but unlike a rigid chess game, they can exist on a spectrum from formal and explicit to fuzzy and implicit. Whether the facts and rules of the game are constrained to what is deemed “reasonable” or “rational” depends on the game, so the subject matter of imagination games is not limited to what’s real, or even plausible but merely what’s logically possible.
Play
Playing the game constitutes exploring the make-believe world. It’s a cyclical process in which players explore and experiment with the consequences of the facts and rules. Inevitably, rules are broken and facts challenged. Through argument, debate, consensus or force, the facts and rules are adjusted to solve problems and the game is continually re-imagined in the minds of the players. While the facts and rules of the game become harder to vary and improve on over time, they never reach a point of perfection beyond which they cannot be further improved.
Understanding Mathematical Creation
I want to show you that the kind of reasoning children use to dream up their fanciful worlds of princesses, trolls and heroes is fundamentally the same kind mathematicians use to formulate their abstract theories. To that end, I’ve written a simplified piece of discovery fiction about how you might have gone about formalising the basic rules of arithmetic, to help you understand how mathematical creation works.
Your name is Giuseppe Peano, the year is 1889 and you are polishing your latest manuscript, Arithmetices principia, nova methodo exposita, “The Principles of Arithmetic, Presented by a New Method”. Influenced by the axiomatic approach of Euclid’s Elements, the goal of your work is to formalise the rules underlying the elementary mathematical operations of arithmetic like addition and multiplication. Your dream is to find a minimal set of simple facts and rules, called axioms, to derive the full theory of the counting numbers.
To an outsider it might seem silly for you to direct your genius intellect towards something that seems so inconsequential, after all, can’t we just take it for granted that 1 + 1 = 2? But to you, mathematics is the study of formality and rigour, even at the expense of frustrating your colleagues when you point out their errors in public. Even amongst mathematicians you are considered eccentric in terms of your meticulous attention to detail, going so far as to develop your own simplified dialect of Latin to communicate your ideas more precisely.5
Through a great deal of trial and error, you’ve managed to create an ingeniously simple list of axioms which you believe are sufficient to model the counting numbers.
Since it’s almost time to publish your manuscript, you scan through some of the axioms one last time, recalling how and why each became a necessary component of your theory.
Let’s start with axiom number one. It’s the simplest of the axioms which asserts that zero exists. In contrast to the other axioms, this one was pretty obvious - without it, the number line would lack a beginning, a starting point from which all other numbers could follow. Zero provided the single raw ingredient required to create all of the other numbers.
The second axiom was quite the invention, even if you say so yourself. The challenge of the theory is to model the positive side of the number line which is an infinite set of numbers. You tried listing numbers as axioms like you did for zero:
one is a number
two is a number
three is a number
…
But this wouldn’t work - even if you spent an eternity writing additional axioms for each number, you’d never reach infinity. And even if you did somehow write axioms for all numbers to infinity, you wouldn’t have specified the relationship between each number - they would be a jumbled bag of disconnected dots instead of an organised line. These problems motivated you to formalise the concept of a successor, which says that every number leads to the next number called its successor.
So with axiom one you established that zero exists, and according to axiom two, if zero exists, then so does the successor of zero, called one. And if one exists, then so does its successor, two, and so on…
zero exists
one is the successor of zero
two is the successor of one
…
In one fell swoop you took care of creating an infinite set of numbers and ordering them in a line through the successor relationship. Bravo!
Now look at the third axiom in the list. It states that given any number, that number plus one cannot be equal to zero. You realised it was necessary because without it, you ran into situations where nothing could prevent 7 + 1 being equal to 0. This produced weird effects, like causing “wrap around” issues where instead of being able to count to higher and higher numbers, you got trapped for an afternoon in an infinite loop cycling endlessly through the first 8 natural numbers with no escape.
The fourth axiom establishes a unique link between a successor and the number that comes before it, for example between the number 5 and the number 4. Only 4 + 1 can equal 5 and nothing else. Again this rule was necessary to exclude. When you spent the morning testing your theory when you didn’t have this rule, you realised that 3 + 1 and 4 + 1 could be equal to 5, and both 6 + 1 and 9 + 1 could be equal to 10! This turned the number line into something that more closely resembled a branch of a tree with twigs protruding from it.
Now back to reality. From reading textbooks I got this very misleading impression that Peano somehow sat down one day and wrote out the axioms of the natural numbers in one shot. I had this misconception that axioms were unquestionable “self-evident truths”.
But despite being a simplified caricature, don’t you think the fictional account of Peano’s formalisation of arithmetic seems like a more realistic description of mathematical creation? While Peano was uniquely suited to the challenge of formalising arithmetic because of his penchant for rigour and minimalism, it can’t have been a simple task. Since he was working at the level of axioms, there wasn’t any “raw material” to use - the axioms he came up with couldn’t be mechanically derived from other mathematicians’ work. They had to be imagined into existence.
Would it therefore be fair to say that mathematics is an imagination game? At the very least it seems reasonable to say that imaginative reasoning is heavily involved in the creation of mathematical theories, to invent axioms out of thin air and reason under the assumption they are true. But to show that it’s an imagination game we need to challenge a widely held misconception that once a mathematical theory is created, it is frozen in time as an eternal truth. Because this would mean that the growth of mathematical knowledge proceeds in a linear fashion where once it is created, it is simply added to the bedrock of established truths. We need to show that like the set of rules defining a child’s make-believe game, mathematical theories can never be considered final truths, they are always fallible and always subject to change in response to problems encountered by players.
There’s a wonderful book called Proofs and Refutations by the Hungarian mathematician Imre Lakatos which makes a strong argument in favour of this view. Lakatos’ book is structured as a socratic dialogue between students and a teacher that reconstructs the history of mathematicians’ attempts to prove Euler’s polyhedron formula, which states that the number of faces plus the number of vertices minus the number of edges is equal to 2, or concisely, that V - E + F = 2.
The students produce counterexample after counterexample to the teacher’s statements by playing with the conception of what it means to be a polyhedron. They repeatedly come up with “deviant” polyhedra that satisfy the teacher’s definitions strictly, but aren’t what you’d normally think of as polyhedra. In the picture below you can see the kinds of shapes that the students came up with to break the formula.
The teacher responds to these counterexamples, or “monsters” as he calls them, by refining his definitions to exclude these undesirable cases through a process he terms “monster-barring”. The back-and-forth dialogue shows that mathematical creation is a much messier affair than the clean, linear way it’s presented in math textbooks, instead the repeated cycles of conjectures put forward by the teacher and refutations by the students make mathematics look much more like a science.
So just like make-believe games, mathematical theories are brought into existence by an act of imagination. Just as fantasy creatures don’t need to exist for children to play and behave as if they did, nor do the axioms of mathematics need to describe physical reality for mathematicians to explore their consequences. And as Lakatos showed us, play in both scenarios is a cyclical process of trial and error: children repeatedly re-negotiate the facts and rules of their games in response to problems, while mathematicians make continual adjustments to their axioms to banish the “monstrous” inconsistencies and counterexamples from their fantasy worlds.
Reverse Engineering Science and Technology
For, as [Popper] saw it, all advances of scientific understanding begin with a speculative adventure, an imaginative preconception of what might be true... It constituted an invention of a possible world, or a fraction of that world… - Sir Peter Medawar: science, creativity and the popularisation of Karl Popper
If you accept the idea presented above that mathematical creation is an imagination game analogous to the make-believe games played by children, then the idea that science could be too should feel like less of a stretch.
Just like mathematics, the creation of a new scientific theory involves the imagination of a new possible world that is not directly derivable from past theories because if the current best theory directly implied the challenger theory, then the challenger theory wouldn’t really contain new knowledge.6 There must always be a discontinuity between the old and the new bridged by a leap of imagination, and it must always be a speculative leap into a sort of fantasy world, because by definition a new theory will not be considered “reasonable” or “rational” by the standards of the current best theory. Take Einstein’s theories of relativity published in 1905 and 1915 for example. It took at least until 1919 with Eddington’s famous experiment for his theory to become accepted as superior to Newtonian gravity, because it directly contradicted centuries of Newtonian mechanics.
Now instead of walking you through a story similar to the formalisation of arithmetic example above, I want to show you how the concept of an imagination game can be used as a tool. Whenever you recognise something that seems like an instance of an imagination game, you can often reverse engineer why it evolved into its current state. In other words, imagination games are interpretable.
Implicit in the current state of the game is a long series of decisions made by players about how the facts and rules should evolve to solve problems encountered while playing. For example, when children playing make-believe tell you that they invented the rule of sharing role-play props through turn-taking, we can come up with reasons as to why they invented this rule by imagining what problems would have arisen in its absence, like arguments about the fairness of the game. Of course, how the children came up with the rule is difficult for us to say, it could have been suggested by an adult or an original creative insight by one of the children, but understanding why the rule was retained is simpler. It should follow some sort of interpretable logic.
To perform this kind of backward reasoning, all you need to do is ask “let’s assume this did not exist, what would break?”. Recall that we already did exactly that during the fictional account of the formalisation of arithmetic.
Let’s assume the third axiom did not exist, what would happen? Ah, then I could cause ‘wrap around’ issues where I could count from 0 to 7 and wrap back around to 0.
This kind of reasoning is actually a type of imaginative reasoning known since Ancient Greece, called “reductio ad absurdum”, or “reduction to the impossible”. The Ancient Greeks relied on it heavily as a tool for their philosophical and scientific endeavours because they could use it to pinpoint misconceptions in people’s arguments. Think about the Socratic dialogues or Zeno’s paradoxes, they are both great examples of this style of reasoning.
“Assume that to occupy a given position in space is to be at rest, then a flying arrow is at rest, which is a contradiction.” - Paraphrasing Zeno. He used this as a counterargument to the Pythagorean theory of motion, showing how it leads to an absurd result.
In the context of imagination games, especially in science and technology, this technique is valuable for three reasons.
Firstly, you might discover after challenging every rule and fact of the game that each seems to play an essential role. Try as you might, it doesn’t seem possible to make an alteration and improve or simplify the game. This is exactly what happened in the formalisation of arithmetic example above. We tried to delete axioms one-by-one and found that it caused catastrophic errors. We can describe Peano’s axioms as “hard-to-vary”.7
Another possibility is that you find that the theory exists somewhere on the spectrum between hard and easy to vary. Michael Nielsen describes this in this essay where he talks about the insights he got from writing a piece of discovery fiction about Bitcoin. .
To deepen my understanding I wrote a discovery fiction essay that begins with the question: how could we design a digital currency? And then describes an extremely simple, obvious idea for such a currency – an obviously flawed back-of-the-napkin idea, the kind of thing anyone familiar with computers would come up with if they took the question seriously. Then the essay points out an obvious problem with the idea. And proposes a simple, obvious fix. Then points out another obvious problem. And another simple fix. And so on, over and over, until eventually we "discover" Bitcoin! This approach shows very concretely how and why each element of Bitcoin is introduced, and what purpose it serves; it also makes clear that some features of Bitcoin are a little arbitrary, and how they could have been different. - Discovery Fiction
By finding a mix between essential and non-essential properties you understand there was some randomness in the evolution of the technology, that if you were to re-run the imagination game from scratch you’d likely end up with a different result
But there is also a third possibility - that you uncover a hidden problem no one spotted yet.
You may, for example, find that there are difficulties in a certain presentation, something that you don’t quite understand. Others may not even notice these difficulties… There may, for example, be gaps in an explanation. Or there may be actual inconsistencies—perhaps in what a theory says, or perhaps between what a theory says and what you observe. - Lecture 3: Problems by Karl Popper
And this is the most exciting outcome. Because every new scientific theory or piece of technology starts with a problem, the seed for a new imagination game.
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Many thanks to Niko, Alex, Paul and Srdjan who gave very helpful feedback on drafts of this essay! And feel free to send me a message on Twitter if you have any thoughts.
Catastrophising is the tendency to always imagine the worst possible outcome for a scenario.
Is there an alternative satisfying name for the general category of imaginative reasoning? “Hypothetical reasoning” or “conditional reasoning” are okay, but fail to capture the idea of the creative or imaginative spark to invent novel and interesting assumptions. “Counterfactual reasoning” is a good term but deals specifically with past events.And of course I refuse to frustrate the reader with an abomination like “subjunctive conditional reasoning”.
See the book Desire for society: Children's knowledge as social imagination which contains many (hilarious) transcripts of children playing make-believe games.
See: The Math Genius Who Invented His Own Language. Peano even lectured his students in his new language, despite the fact they had never studied it.
It could have other benefits like being be more understandable though which would still make it worthwhile.
Hard-to-vary is an idea from David Deutsch.