Is Mathematics the Map and the Territory?
"The truth is rarely pure and never simple." — Oscar Wilde
Introduction – The Old Debate
Is mathematics an intrinsic part of reality that we discover, or a creation of the human mind that we invent? This classic debate has engaged philosophers and mathematicians from Plato to the present day. On one side, thinkers like Galileo assert that "the great book of nature is written in mathematics," suggesting that mathematical truths exist out there waiting to be found. On the other side, many point out that math is a human language—after all, as Leopold Kronecker famously quipped, "God created the natural numbers; all the rest is the work of man". In this essay, I will argue that both perspectives hold true in a complementary way. Mathematics is both discovered and invented: we uncover real structural patterns that exist independent of us, but we also create symbolic frameworks and models to represent those patterns. In other words, reality provides the raw terrain of mathematical structure, while our minds draw the map.
Mathematics as Discovery
Even without human input, the world seems to exhibit mathematical structure. Patterns and regularities abound in nature: the spiral arrays of seeds in a sunflower or scales in a pinecone follow Fibonacci numbers, appearing long before humans recognised them. Such examples imply that certain mathematical relationships (like the Fibonacci sequence or geometric symmetries) are embedded in nature's design rather than imposed by us. Living creatures themselves hint at a pre-existing arithmetic. Studies show that infants in their first year can distinguish different quantities, and macaque monkeys can perform a rough sort of addition without any language at all. For example, if two objects are placed behind a screen one at a time and then revealed, both human babies and monkeys expect to see two objects; they are surprised if only one remains, indicating an intuitive grasp that 1 + 1 = 2. This kind of basic number sense appears to be part of our evolutionary inheritance. In fact, experiments have found that many animals – from crows to lemurs – can distinguish quantities and even understand the concept of zero in a primitive way.
Moreover, fundamental logical truths of mathematics seem to hold in the physical world. Two apples combined with two apples yield four apples regardless of one's culture or notation – a correspondence between math and reality that suggests an external truth. Early Greek philosophers like Plato took such correspondences as evidence that mathematical forms (perfect circles, triangles, numbers, etc.) have an existence of their own in an abstract realm of truth. While a drawn circle in sand is never a perfect circle, the idea of a perfect circle transcends any particular human attempt to draw it. In the same way, the fact that nature roughly conforms to mathematical ideals (planets following ellipses, crystals forming symmetric lattices) hints that we are discovering pre-existing structures. We can observe symmetry in the wings of butterflies and the hexagons of honeycombs; we find consistent ratios and shapes in physics and biology. These patterns "can sometimes be modelled mathematically" and were noted long before modern science. They seem to exist whether or not we have words for them. Thus, one aspect of mathematics is clearly a discovery of the structural order built into the universe. Some truths (like basic arithmetic or the Fibonacci spiral) predate humanity and simply await our recognition.
Mathematics as Invention
At the same time, mathematics in practice is a very human endeavour – a symbolic construction that we continuously refine. The languages and notations we use are inventions, created to capture those underlying structures we observe. History makes this clear. For instance, there was nothing inevitable about the "=" sign for equality or the "0" symbol for zero – these had to be introduced and agreed upon by people. In the 16th century, the English mathematician Robert Recorde invented the equals sign "=", which eventually beat out competing symbols (like the Latin abbreviations or other glyphs) to become standard. Likewise, different civilisations experimented with various notations for numbers and operations until a consensus emerged. The mathematical notation we take for granted today – from algebraic variables to integral signs – is the result of human creativity, debate, and even "popularity contests" among mathematicians. This symbolic apparatus did not fall from the sky; it evolved alongside our cultural and practical needs. In that sense, we invented the expressive medium of mathematics, even if the patterns it describes were discovered.
Importantly, invention in math goes beyond just symbols – it extends to the frameworks and axioms we choose. We design models and formalisms to make sense of observations, and those designs involve choices. Consider the Bellman equation, a fundamental equation in reinforcement learning and dynamic programming. This equation provides a way to break down decision-making over time by assuming an agent seeks to maximise a certain cumulative reward. But notice the assumptions baked into this formalism: time is chopped into discrete steps (or treated linearly), outcomes are evaluated by a single numerical reward, and the world is described as a set of "states" that the agent can move through. None of these modelling choices are forced by reality – we made them for convenience and analytical tractability. It's perfectly possible that an alien intelligence or an alternative AI paradigm would formalise decision-making differently (imagine one that doesn't treat time as linear, or doesn't reduce all goals to a single reward metric). The Bellman equation is extremely useful, but it is our chosen lens for viewing sequential decision problems, not a unique eternal truth. In fact, it is defined within the context of an invented framework – the Markov decision process – which itself is a human abstraction of how an "optimal" agent should behave. This equation holds true given its axioms, but those axioms are part of our invented map of decision-making. Another species or future thinkers might use a different map for the same territory.
More broadly, any advanced mathematical formalism is at least partly a product of human imagination. We assume certain axioms and definitions and then explore their consequences. When those constructs happen to correspond to reality, we celebrate a discovery; but the road to that discovery was paved with creative invention. It's telling that mathematicians can concoct entirely new systems of math that may or may not correspond to the physical world (non-Euclidean geometries, multiple infinities in set theory, etc.). Often these inventions later find a home in reality – for example, the imaginary numbers were invented as purely abstract concepts, and only later did they become essential in electrical engineering and quantum physics. This interplay of invention and discovery is nicely illustrated by imagining an alien civilisation. As some scholars have argued, extraterrestrial intelligence would likely recognise the same basic truths (such as the prime numbers or simple arithmetic), but they might package those truths in very different conceptual wrappers. Perhaps they would have entirely different notations, or their mathematics might emphasise concepts we haven't even conceived. The structures (e.g. the fact of primes or the ratio of a circle's circumference to its diameter) would be discovered by them as well, but the system they build – their mathematical language and high-level concepts – could diverge radically from ours. In short, we discover the gems of mathematical truth, but we craft the jewellery setting that displays them.
The Map and the Terrain
How can mathematics be both discovered and invented? A useful metaphor is that reality is the terrain, and mathematics is the map we draw to navigate it. A map is not the same as the land it represents, yet a good map captures the key structures of the land. In the words of philosopher Alfred Korzybski, "the map is not the territory" – an abstraction or model of something "is not the thing itself". We must not confuse our mathematical descriptions with reality, even if those descriptions are accurate. Just as multiple maps can depict the same landscape in different ways (road map, topographical map, climate map), multiple mathematical frameworks can describe the same physical phenomenon. Each is a lens highlighting certain features while hiding others.
Take quantum physics as an example. The behaviour of quantum systems can be described by various mathematical formalisms – Schrödinger's wave equation, Heisenberg's matrix mechanics, Feynman's path integrals – each a kind of "map" of the quantum terrain. Schrödinger's equation in particular gives us a powerful map: it allows us to compute a system's wavefunction and predict probabilities of different outcomes. This equation works extraordinarily well for practical predictions. But is it the reality of the quantum world, or just one useful depiction? Some thinkers have noted that just because an equation works doesn't mean it is the unique or ultimate truth of nature. In fact, historian of science John Horgan suggests we consider the possibility that the Schrödinger equation was "not a discovery but an invention, an arbitrary, contingent, historical accident," much like the particular notation we use for math. After all, humanity stumbled on quantum theory through a messy, human process of trial and error. If we re-ran the history of science (or if an alien civilisation approached quantum mechanics from a different angle), we might end up with a different but equivalent equation – another map of the same territory. The map we have (Schrödinger's formalism) is incredibly useful and reflects real structure in the world, but we shouldn't confuse it for the only reality. Indeed, even today we know Schrödinger's equation is not all-powerful – it has to be supplemented with "patches and approximations" to handle complex atoms or the interactions of many particles. This is a clue that it's a model with limits, not the ultimate description of nature.
Seeing mathematics as a map helps reconcile invention and discovery. The terrain (reality) has genuine mathematical features (those are discovered), but the map (mathematical theory) is our constructed artifact (that's invented). And like any map, mathematics can come in many forms. Some maps might be more useful for certain tasks than others, and some might be more conceptually elegant. Our current scientific mathematical map is extremely effective, yet it might not be the only way to chart the territory. This idea resonates with cognitive scientist Donald Hoffman's interface theory of perception, which argues that what we perceive is more like a user interface than an objective view of reality. "Perception is not a window on objective reality. It is an interface that hides objective reality behind a veil of helpful icons," Hoffman writes. In other words, what we take to be solid reality (space, time, objects) might be more like a convenient desktop display that evolution has given us, rather than a direct read-out of the underlying code of the universe. If our sensory perceptions are a kind of species-specific map, it stands to reason that our mathematical descriptions – which are extensions of our perception and reasoning – could also be a sort of interface. Mathematics, then, is like a refined version of the human perceptual "user interface," one that lets us predict and manipulate the world with great precision. But however precise it becomes, it's still a representation and not the territory itself. We should expect that different representations (different maps) can exist, and each might emphasise different aspects of the underlying reality.
Toward Deeper Layers of Reality
The notion that mathematics is a map or interface raises a profound question: How deep does our current map go? Are we grasping the ultimate structure of reality, or just scratching the surface? It's possible that what we call mathematics today is a kind of surface layer – a remarkably effective decoding of the patterns we've encountered so far, but not necessarily the final word on nature's mysteries. History gives us humbling examples of how a given mathematical framework can be superseded by a deeper one. Euclidean geometry reigned for millennia as the description of space, until Gauss, Bolyai and Lobachevsky showed that alternative geometries were possible – and Einstein later demonstrated that physical space itself is non-Euclidean. The lesson is that our mathematical interfaces can evolve to probe deeper truths. What seems fundamental (like Euclid's postulates) may turn out to be just an approximation or a special case of something richer.
In fact, there are indications that our current mathematical system cannot be the end-all description of reality. One indication comes from mathematics itself: Gödel's incompleteness theorem. Kurt Gödel proved that any consistent formal axiomatic system rich enough to include basic arithmetic will contain true statements that cannot be proved within the system. In plainer terms, no single fixed set of axioms can capture all mathematical truths – there will always be true facts that lie beyond what the system can derive. This theorem shook the foundation of mathematics because it showed an inherent limitation in the discovered+invented edifice of math. If we think of our mathematics as a particular map with certain axioms (our "rules" of logic and construction), Gödel guarantees that this map will never cover the entire landscape of truth. There are always further true statements (further "terrain") outside the map's scope. Some mathematicians interpret this to mean that mathematics is an ever-expanding activity: we continually invent new axioms or systems to capture truths that were previously out of reach, and then those new systems have their own unprovable truths, and so on. In other words, the process of inventing new mathematics might be endless, precisely because there is always more truth to discover.
Now, consider what this might imply for describing physical reality. If reality has mathematical underpinnings, it could be that no matter how advanced our mathematical models become, there will be phenomena or truths about the universe that demand new mathematics to be understood. Our current mathematical interface might be like the first layer of an onion. We have, for instance, wonderfully successful theories (Standard Model of particle physics, General Relativity for gravity) that are expressed in advanced mathematics, but we already know these theories are incomplete (they don't play well together at the deepest level). To go further, we might need to invent entirely new mathematics – perhaps new logical frameworks, new higher-dimensional geometries, or concepts we currently can't imagine – to uncover the next layer of physical law. It's conceivable that an artificial intelligence or a future post-human intellect could devise math that makes as much sense to us as our algebra would make to an ancient Greek: something wholly new. Such an intelligence might not be bound by the historical pathways and psychological biases that shaped human math. It could explore vast abstract territories systematically, creating formalisms that allow it to comprehend aspects of reality we find mystifying (quantum gravity, consciousness, etc.). If that happens, would we recognise their constructions as "mathematics"? Perhaps not at first. But if those constructions map to reality successfully, they would serve as new interfaces – new maps for the terrain we all inhabit.
It's also worth pondering whether some aspects of reality are currently bypassed by our mathematical sense-making because we lack the right conceptual tools. Just as our ancestors lacked the language of calculus or complex numbers and thus couldn't easily grapple with certain patterns, there may be features of the universe that await the invention of the appropriate mathematics. This isn't just fanciful speculation; it's happened repeatedly in history. Imaginary numbers were once dismissed as meaningless, yet they unlocked entire domains of physics. The concept of a "matrix" or a "tensor" was at one point an invention in pure math that later became indispensable for physics. Each time we deepened the mathematical toolkit, we discovered new layers of the cosmos. So, the dance of discovery and invention continues: to discover more, we often have to invent more.
Conclusion – Rethinking What Math Is
In light of these reflections, we can begin to rethink what mathematics truly is. It is not the external world itself, nor is it a mere whim of human culture, but rather something in-between: it is how our minds interface with the patterns of reality. Mathematics is both a discovery and an invention – we uncover the timeless relationships that exist (whether in numbers, geometry, or logic), but we do so by devising symbols, languages and frameworks of thought that are our own creative products. We find diamonds in the rough, but we cut and polish them to adorn our theoretical structures.
Understanding math as a "map" reminds us to be humble and open-minded. Our maps have proven extraordinarily effective – indeed, it's astonishing how useful math is for navigating the real world. But just as no map can capture every aspect of the territory, our mathematics is not identical with reality. It is a representation that can always be refined or re-imagined. We should be confident in mathematics as far as it goes, but also curious about what lies beyond its current frontiers. The history of math and science suggests that what we have now might be a surface layer over a deeper mathematical reality we have yet to fully glimpse. After all, if an alien or a future AI encountered our mathematical papers, would they see them as the final truth, or just one particular approximation understandable to human brains? It's a thrilling thought that our entire system of mathematics might be just one "operating system" running on the universe, and there could be other compatible operating systems that look very different.
Ultimately, embracing the dual nature of math enriches our perspective. We can appreciate the almost mystical objectivity of mathematics – how it describes galaxies and genes with equal precision – while also appreciating the ingenuity of the human mind that sculpted a language to express this order. We are both explorers and inventors in the mathematical realm. So, what would math look like to an alien mind? Would their equations and theorems be recognisable or utterly alien, even if describing the same universe? And as we push the boundaries of knowledge, is our current mathematics merely a skin over something deeper – a first interface to reality that future generations (or machines) will peel back to reveal new structures? These questions remind us that the story of mathematics is far from over. Mathematics is our link to the cosmos, a bridge between mind and world. As we stride forward, we carry with us the insight that every equation is both found and made – a truth of the world translated into the human narrative. The map is not the territory, but with each generation of mathematicians, we draw the map with finer detail, and venture further into the great unknown that is reality itself.