by Herbert Harris
Is mathematics created or discovered? For over two thousand years, that question has puzzled philosophers and mathematicians alike. In Plato’s Meno, Socrates encourages an uneducated boy to “discover” a geometrical truth simply by answering a series of guided questions. To Plato, this demonstrated that mathematical knowledge is innate, that the soul recalls truths it has always known. The intuitionists of the early twentieth century, however, rejected this idea of eternal forms. For thinkers like Poincaré and Brouwer, mathematics was not revelation but construction: an activity of the human mind unfolding in time.
The debate continues today in an unexpected new arena. As artificial intelligences start to generate proofs, conjectures, and even entire branches of formal reasoning, we are prompted to ask again: what does it mean to do mathematics? Current systems excel at symbol manipulation and pattern matching, but are they truly thinking in any meaningful way, or just rearranging signs? The deeper question is how humans do mathematics. What happens in the brain when a mathematician recognizes a pattern, intuitively sees a relation, or invents a new kind of number?
In what follows, I’ll trace that question from ancient philosophy to modern neuroscience and then to the newest foundations of mathematics. We’ll see that mathematical invention may be the natural expression of the brain’s recursive, embodied intelligence, and that this perspective could transform how we think about both mathematics and AI.
To understand how humans “do” mathematics, we need to examine what the brain itself is doing. Recent neuroscience indicates that a significant part of the brain’s activity involves constructing predictive models of its sensory states and actions. Through active inference, the brain constantly updates these predictions when reality surprises it. The inferential looping process is called active inference. If the loop were to turn back on itself to predict its own predictions, it would be called recursion. This would indeed be Douglas Hofstadter’s strange loop, and it might be a key event in the development of human self-consciousness.
This recursive ability transforms everything. Unlike traditional computation, which executes fixed operations on static data, human thought can step back and create second-order models of itself. We can imagine how our thoughts seem to others, simulate counterfactuals, and assess our own reasoning. Self-consciousness thus gives us a form of self-directed agency, as well as the power to choose what we think.
That reflexive flexibility gives us a remarkably fluid understanding of equality, identity, and equivalence. While a machine sees identical tokens, we perceive relationships of similarity and difference that can stretch or bend without losing coherence. We consider sameness as a transformation that preserves meaning, like a melody transposed or a theorem reframed. This same recursive ability allows us to create and manipulate mathematical objects. We don’t just record patterns; we generate them, take them apart, and recombine them.
Each new mathematical construct is a crystallized act of second-order cognition: the self modeling its own operations and turning that model into a shareable object. The mathematician’s mind becomes part of mathematics. From this vantage, the creation of number systems and the invention of space are not abstract feats but natural extensions of self-modeling and free agency, the mind rehearsing its own transformations until they take on independent form. This recursive capacity to model our own modeling sets the stage for how we build number systems and, ultimately, for how we imagine the spaces they inhabit.
The story of numbers leads naturally to the story of space. Once we can represent relations among quantities, we begin to imagine where those relations live. Geometry arises when the mind projects its own motions, reaching, rotating, containing, into a stable external order. Our sense of number and shape is literally rooted in the body: children learn to count on their fingers, and adults still map magnitude onto spatial gesture. Every geometric act is a quiet drama of embodiment: the thinker becomes the coordinate system, then steps outside it to imagine another frame of reference.
This transition exposes an ancient tension between discreteness and continuity. Zeno’s paradoxes asked how motion is possible if a path must be traversed through an infinite number of points; modern analysis resolves the puzzle symbolically, but the deeper question remains psychological. How does the mind experience continuity when it can represent only discrete moments of change? The answer may lie in what we could call free generation, our ability to interpolate, to “fill in the gaps” between percepts through imagination and prediction. Active inference lets the brain smooth over discontinuities, producing a lived sense of seamless flow.
At this level, mathematics and self-consciousness intertwine again. In picturing continuous space, the mind is not describing something outside itself but enacting a process of internal coherence. Each new dimension, a line unfolding into a plane, a plane folding into a cube, is another recursive self-model: a representation of representation. As Hofstadter observed, such “strange loops” define both cognition and mathematics. The mathematician alternates between being inside and outside a system, just as the self alternates between experiencing and reflecting. Continuity, whether spatial or mental, emerges when those perspectives stabilize one another.
If mathematics arises from these self-models, then its logic must mirror how the mind perceives sameness. Classical logic starts with Leibniz’s Law: two things are identical if everything true of one is also true of the other. In traditional computation, this principle is absolute; tokens are either identical or not, and a variable either matches a pattern or does not.
Human cognition is more supple. Self-consciousness allows us to step back from our immediate representations and notice that things can be alike in some ways and different in others. A melody transposed into another key, a friend glimpsed in a different light, a self remembered from childhood, these are not identical in the Leibnizian sense, yet we experience them as the same. Such partial identities are how the mind grasps abstractions. They let us see through the shifting surface of appearances to the continuity that matters.
This fluid logic starts with the self. Personal identity is not fixed but is both biologically and socially rooted, maintained within a shifting frame of reference created by others. In daily life, the identity of the world we see is tied to our sense of continuity. My belief that I am writing at the same desk as yesterday relies on my simultaneous belief that I am the same person who wrote there. The self is a narrative that is constantly evolving, shaped by how others see and recognize us. The coherence of objects and the coherence of the self are two parts of the same inferential process, and this is just as true for the mathematician and the mathematical object.
What is remarkable is that modern mathematics is beginning to rediscover this logic of living identity. Homotopy Type Theory (HoTT), a new foundation for mathematics emerging from topology and computer science, redefines equality not as a static equivalence but as a path of transformation. Two things are “the same” when you can continuously deform one into the other without losing structure. Sets become networks of relations rather than static collections; proofs become trajectories linking equivalent perspectives. HoTT was never designed to model human cognition, yet it unintentionally reflects it: when mathematicians redefine equality as transformation, they formalize what our minds already do naturally. In HoTT, identity is lived, a record of the transformations that maintain coherence through change.
This vision resonates deeply with theories of consciousness and agency. Just as HoTT treats sameness as a relation maintained through transformation, the conscious mind retains its sense of self through ongoing acts of prediction, correction, and mutual recognition. Mathematics, seen in this light, is not an alien abstraction but an embodied discipline, mirroring the very dynamics that make experience possible. It offers an exciting new way to understand how mathematical reasoning can be both embodied by and embedded in the human mind.
This viewpoint suggests a kind of Goldilocks ontology for mathematics. Platonism places mathematical truth in a distant, timeless realm; intuitionism confines it to private acts of construction. A naturalized, embodied mathematics steers between them, taking a path neither too intuitionistic nor too Platonistic. Mathematical structures are not “out there,” waiting to be discovered, nor are they mere psychological projections. They are patterns that emerge when embodied minds model their own modeling, stable enough to count as real, yet inseparable from the living systems that generate and recognize them.
When mathematicians talk about infinity, are they pointing to an empty placeholder that stands for an endless collection of things, or describing the mind’s own capacity for recursive self-modeling? The formalist can treat infinity as a symbol: a silent token standing for “and so on forever.” But the living act of conceiving infinity is not static. It is the mind enacting unbounded recursion, imagining itself extending beyond any limit it can name.
This distinction, between manipulating tokens for infinity and living the recursive process, may mark the boundary between computation and understanding. To grasp infinity as mathematicians do is to participate in a process of self-transcendence: to experience the self stretching into a structure it can never fully contain. The infinite is not a thing known but an act performed.
Today, a new kind of mathematician is emerging: artificial intelligence. Systems trained on vast corpora of proofs and patterns are now discovering new conjectures, solving century-old problems, and even generating results that human mathematicians struggle to interpret. Google DeepMind’s AI for Math initiative aims to accelerate this revolution, an astonishing prospect, but also a revealing test.
Can AI understand the mathematics it produces? Can it explain why a proof is true, not just that it is? The question turns on embodiment. If mathematics is the mind’s recursive self-modeling, then understanding arises only when a system models not just the world, but its own modeling. Computation alone is not enough; there must be a loop through which symbolic operations acquire meaning.
Human mathematicians do this effortlessly. They form intuitions, build metaphors, and sense the connection between a structure and the act of understanding it. Their proofs are clear because they are experienced as transformations within a living cognitive entity. An AI that produces valid proofs is a language model fluent in theorems but functions like a savant without proper understanding.
To develop AI capable of genuinely explainable mathematics, we need a form of humanistic AI: systems that can simulate themselves as knowers. Is this possible for a machine? Humans develop self-awareness through a social developmental process. We learn to see ourselves through others’ eyes. From parents and caregivers to friends and colleagues, we constantly gather information about ourselves. Others model us, and we model their models, gradually cultivating the nuanced fluidity of identity needed for mathematical reasoning.
Mathematics grew from the practical applications of counting and measuring, but it has always been a product of self-conscious reflection. Each expansion of its horizon, from number to space, from space to symmetry, reveals a new act of cognition, a new way of modeling self and world together. In doing mathematics, the mind studies its own form of understanding; it becomes the universe contemplating itself through pattern and structure.
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