Mathematics — Conditions and Consequences: 1 The Preconditions of Mathematics: The Semiotic Genesis of Abstraction

Mathematics did not begin as the study of numbers or shapes. It began as the reflexive construal of relation itself — the capacity to make difference and connection explicit, stable, and transferable. To understand how mathematics became possible, we must turn not to calculation but to semiotic relationality: the human ability to construe relations symbolically and then act upon those construals as if they were things in their own right.

1. From Pattern to Relation

Before there were numbers, there were differences that mattered — rhythms, cycles, correspondences. The hunter noticing footprints, the farmer marking seasons, the weaver balancing symmetry and tension — each enacted a kind of proto-mathematical awareness: the recognition that relations could repeat, and that such repetition could be systematised.

This was not yet number, nor even count. It was the semiotic intuition that experience could be patterned, that relational form could be abstracted from its material substrate and held stable in the mind. Counting and measuring emerged only when these relational regularities were externalised in gesture, mark, and speech — the first symbolic stabilisations of pattern.

2. Symbolic Abstraction and the Birth of the Number Concept

The shift from noticing to counting marks a decisive moment: relations are now objectified in sign form. The gesture of “one,” “two,” “three” is already an act of abstraction — a construal of quantity independent of any specific instance. What makes this possible is semiotic recursion: the ability to construe not just an object, but the relation between relations.

This recursive potential allows the symbolic system to fold back on itself: the sign no longer merely refers to an event but becomes a theory of relational potential. The concept of “number” is thus not the discovery of a property in the world but the actualisation of a new semiotic dimension — one in which relations themselves can be related.

3. Space, Form, and the Relational Eye

Parallel to number runs geometry — not as the study of things in space, but of spatial relation as construal. To trace a line, to balance a form, is to stabilise the perceived relationality between points, edges, and boundaries. Geometry transforms embodied movement into symbolic coordination — the extension of bodily relation into symbolic space.

The geometric diagram is the visual analogue of number: both are semiotic condensations of relation. Each allows pattern to become potential, relation to become form.

4. Formalisation as Relational Compression

As symbolic abstraction deepens, mathematics becomes a meta-system for stabilising relational consistency. What was once a gesture becomes notation; what was once notation becomes rule. The emergence of formal systems — from arithmetic to algebra — is a history of semiotic compression: the recursive folding of relational patterns into forms that preserve structure while extending potential.

This compression enables transmission: mathematics becomes a portable relational field, independent of the material conditions of its emergence. It can be taught, transformed, and applied — a semiotic ecology of structured potential.

5. Mathematics as the Semiotic Reflex of Relation

Thus, the preconditions of mathematics are not cognitive or empirical alone. They are semiotic and relational:

• The ability to construe relation as an object of thought.

• The capacity to stabilise and iterate that construal symbolically.

• The recursive potential to relate relations, yielding abstraction.

Mathematics arises not from the world’s order but from the symbolic capacity to construe order as such — to make potential explicit, to hold relation stable, and to extend it beyond the immediate context of experience. It is, in this sense, the reflexivity of relation itself.