Prolegomena to Embodied Knowledge

There is a debate about whether mathematics is created or discovered. The issues uncovered by this debate are the prolegomena (προλεγόμενα)—the “pre-sayings” we need before we can speak seriously about embodied mathematics. If mathematics is mainly discovered, then it looks like a mind-independent structure we gradually uncover. If it is created, then it looks like a human practice: a way of stabilizing patterns through perception, action, representation, and agreement. Either way, the debate forces a key question for embodied mathematics: even if mathematical structure is “out there,” how does it become thinkable for creatures like us?

Popular culture makes the contrast vivid. In the movie Arrival (2016), is about first contact with aliens. For the depths of inter-gallactic space, twelve massive oblong shaped ships effortlessly hover in locations distributed around the globe. Occasionally they emit undistinct sounds. The central challenge is not a universal constant but a problem of sense-making: how would largely different biologies detect structure, form hypotheses, test them, and build shared reference across radically different bodies and perceptual worlds? The breakthrough comes from treating what seems like noise as a signal (data), then patiently constructing a workable mapping between marks, meanings, and situations. That is an embodied-math lesson: pattern is not simply “seen,” it is made knowable through disciplined interaction—through attention, iteration, error-correction, and new inter-agent representational moves.

By contrast, the movie Contact (1997) dramatizes the opposite intuition. Its extraterrestrial “hello” arrives as prime numbers—mathematics as a supposedly universal handshake—and the story later plays with the idea of a message hidden deep in the decimal expansion of π, that ratio between circumference and diameter of a circle that starts with 3.14159 etc. Here math functions as a common language that bypasses culture and embodiment. The film is built on the premise that mathematics is discovered regardless of biology, and any intelligent mind can meet there.

Taken together, the two films set up the prolegomena for embodied mathematics: yes, mathematics can be framed as universal structure, but our access to it is always mediated by bodies, tools, representations, histories, and communities of practice. “Finding the signal” is not only about what exists, it can be mainly about what we are taught to perceive, coordinate, and verify.

What I want to add is a third position—my position—according to which mathematics is best understood as what I am calling "constrained invention". The created vs discovered debate does not go far as a binary choice between two opposing realities. Mathematics is an inventive practice: languages, notations, models, and metaphors. Nevertheless, it is disciplined by constraints we cannot transgress. We invent ways of describing and manipulating structures that we do not invent. This is why some descriptions work and others fail. In this sense, the constraint is discovered and its representation is invented.

This is why the language of “signal” matters. A signal is not just a thing in the world; it is a pattern that survives scrutiny under a rule for distinguishing structure from accident. In Arrival, the key insight is not merely to notice repetition, but to build a stable representational bridge being two kinds of minds that are each constrained by structures outside their "heads". The constraint is the regularity that resists arbitrary interpretation; the invention is the representational scheme that lets agents with specific senses, histories, and tools coordinate on that regularity. Mathematics, here, looks less like a cosmic dictionary and more like a disciplined craft of making patterns transferable.

In his TedTalk, Roger Antonsen’s way of putting this is especially helpful: the world offers structure, but we make up the language to think that structure. There is a kind of magic that happens when we represent. Arbitrary marks—dots, strokes, grids, symbols—suddenly become handles for grasping what we can not hold directly. This is not a denial of objectivity; it is an account of access. It is what Immanuel Kant tried to achieve in his transcendental philosophy.

Even a most “discovered” mathematical moment, like the prime-number handshake in Contact, still requires invention at the level of interpretation. Primes do not arrive without subtitles. The receiver must decide what counts as a pulse, how to segment the stream, which regularities are salient, and what constitutes evidence of intention rather than coincidence. The shared structure may be mind-independent, but contact happens only through agent mediated practice of representation, decoding, and verification. The movie’s appeal depends on the fantasy that mathematics bypasses culture; the embodied-math lesson is that it never fully can.

Antonsen also clarifies what “thinkable” means. Understanding is the ability to shift perspectives. Every equation is a metaphor; an invitation to see “the same” structure under different viewpoints. To grasp x + x = 2x is to hold one thing in two perspectives at once: addition as accumulation, multiplication as scaling. Embodiment is not decoration; it is the mechanism by which abstraction becomes navigable.

So the prolegomena for embodied mathematics becomes sharper: the structure is discovered, but mathematics is the constrained invention.