December 2025 — extracted during a wiki-skimming session

Mathematics is what happens when humans repeatedly apply a small set of cognitive transformations to values, relations, and structures; then demand consistency. This is our attempt to describe them, so others can try them out.

I was trying to read Wikipedia, honest. Just a few paragraphs of a few math concepts, that’s all.

A bit abstract, sure, and most I don’t really know, so we hover over the tags for every last keyword. And Kat just doesn’t quit really.

Apparently they’re only doing simple things really, despite the terminology -twist, fold, split and a bunch more. So, y’know what, lets just find out what those moves are, before she self-promotes into an earworm.

Before any actual mathematicians explode when reading these…I see these as a meta-observation, not a theorem. However, it does feel like a small collection of cognitive moves actually generates an awful lot of mathematical concepts doesn’t it?

Kat’s Primitive Mathematical Moves

Iz m finkz list of primitive conceptual movements to make new math fields iz no b dat big u know

This is not a definition of mathematics. It is a sketch of how humans seem to generate mathematical ideas before formalisation.

The generators below operate, broadly, on these three kinds of things:

• values
• paths
• structures

Imagine we talk about apples, a relation between apples like where they sit in relation to each other, and finally the basket we carry them in.

These are self referencing potentially as well, making maths self refractive. You can use paths to examine themselves, or structure of structures and so on.

The Generators

The Insight

Category theory suggests there are surprisingly few universal constructions — limits, colimits, adjunctions, Kan extensions. Everything else is combinations of these applied to different objects.

This list represents the generators of mathematical practice — a finite alphabet of conceptual moves that, when composed, produce all the fields.

Iz small set of moves innit? Might find some more but iz no many missing me finks.

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Math name: composition, factorisation, decomposition 
Shows up as:
function composition 
tensor products vs direct sums
prime factorisation
categorical products / coproducts
This is the fundamental move.

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Math name: monodromy, holonomy, torsion
Shows up as:
fibre bundles
gauge theory
knot theory
complex phase rotations
Kat’s “add a dimension” intuition is spot-on: twists often live in an extra parameter.

Math name: braid groups, mapping class groups
Shows up as:
topology
quantum groups
anyons in physics
category theory (braided monoidal categories)
This is twist + composition + constraint.

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Math name: quotient, identification
Shows up as:
modding out by equivalence
orbifolds
gluing edges of polygons to make surfaces
quotient spaces
Folding is how geometry compresses information.

Math name: restriction, subobject, ideal, condition
Shows up as:
subspaces
ideals in rings
boundary conditions
sheaves (local restriction)
Zariski topology lives here.

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Math name: duality, localization
Shows up as:
group inverses
adjoints
Fourier transform
categorical duals
“Invert” is often change perspective, not just algebraic reversal.

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Math name: functor, transform
Shows up as:
Fourier / Laplace
representation theory
pushforward / pullback
coordinate changes
This is movement between worlds.

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Math name: morphism, relation
Shows up as:
functions
homomorphisms
natural transformations
correspondences
Category theory is basically “take this move seriously”.

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Math name: iteration, limit, fixed point
Shows up as:
recursion
dynamical systems
renormalization
convergence
“Spiral to a limit” is a poetically accurate phrase, by the way.

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Math name: exactness, obstruction
Shows up as:
exact sequences
homology
residue classes
failure of invertibility
When division fails, new theory appears.

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Math name: extension / restriction of structure
Shows up as:
ℝ → ℂ
special cases → universal objects
abstraction ladders
This is changing the rules of the game while keeping the moves.

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Math name: equivalence, correspondence, duality
Shows up as:
Nullstellensatz
Fourier duality
Langlands program
geometry ↔ algebra
Bridges are why maths feels unified instead of fragmented.