Meaning Before Language: 4 Mathematics Without Representation

Constraint amplified beyond language

If language is a specialisation that amplifies relational constraint through symbolic portability, mathematics is a different, and in some ways more austere, specialisation. Mathematics is often treated as the purest form of representation: a mirror of structure itself, or a language spoken by the universe. Both views mistake its role.

Mathematics does not reveal structure by representing it. It amplifies constraint by stripping meaning down to what can be stabilised with maximal precision.

Mathematics is not discovered

It is tempting to say that mathematics is “out there,” waiting to be uncovered. But this temptation repeats the representational mistake in a subtler form. Mathematical structures do not pre-exist as objects awaiting description. They are actualised through cuts that impose extraordinarily tight constraints.

This is what gives mathematics its peculiar authority. Once the constraints are fixed, the consequences follow inexorably. Nothing arbitrary remains — but that necessity is conditional on the cut that established the system in the first place.

As Eddington put it, the mathematics is not there until we put it there.

Constraint without reference

Unlike ordinary language, mathematics does not primarily trade in reference. Symbols in mathematics do not stand for things in the world in any straightforward sense. They stand in relations to one another under rigorously defined constraints.

A mathematical expression is intelligible even when it refers to nothing physical, nothing empirical, nothing imaginable. Its meaning lies entirely in its place within a constrained relational system.

This is not a defect. It is mathematics’ defining feature.

Why mathematics feels objective

Mathematics feels uniquely objective because its constraints are explicit and unforgiving. Once a system is defined, any instance that violates its constraints simply fails to be an instance. There is no room for interpretation in the ordinary sense.

But this objectivity is not a view from nowhere. It is the product of maximal constraint stabilisation. Mathematics achieves universality not by escaping perspective, but by making the perspective so tightly specified that it becomes shareable without remainder.

Mathematics and physics

Physics exploits this specialisation relentlessly. Mathematical formalisms allow physical systems to be explored at the level of constrained possibility rather than empirical happenstance. But the mathematics does not describe nature directly. It articulates the space of possible instances that a given physical cut makes intelligible.

This is why different mathematical formalisms can describe the same physical phenomena, and why no formalism is uniquely forced by reality alone. The cut comes first; the mathematics follows.

Mathematics after meaning

Seen relationally, mathematics does not precede meaning. It presupposes it. The intelligibility of a mathematical system depends on prior constraints that determine what counts as a valid distinction, operation, or proof.

Mathematics is meaning made rigid.

Looking ahead

With language and mathematics now situated as distinct specialisations of relational constraint, the final step is to address semiosis directly. Signs, symbols, and codes can now be approached without mystification — not as the origin of meaning, but as technologies for transporting and coordinating constraints.

The next instalment will examine signs themselves, asking how semiosis operates once representation is no longer treated as foundational. Meaning came first. Signs came later.