Eight modules from Euclidean geometry through the Quadrivium to the SwiftVector kernel. Each module builds on the one before it. The formal systems sequence and the liberal arts made into interactive laboratories.
Euclid's axiomatic method applied through compass and straightedge. The four-layer architecture that SwiftVector inherits directly from the Elements.
Gradient descent as terrain. Apple's Grapher as algebraic bridge. The loss landscape rendered live. Where mathematics meets machine learning.
Matrices as transformations. Eigenvectors as invariant directions. The dot product as attention mechanism. Where the transformer architecture lives.
Distributions, hypothesis testing, Bayesian inference. The mathematical language of uncertainty that underpins every machine learning model.
Number in time. Ratio, proportion, and the detection of recurring structure in a continuous stream. Fourier, sequence models, attention as counterpoint.
Inference under radical observational constraint. You cannot manipulate the subject. Parallax, redshift, spectral classification — the epistemology of AI evaluation.
The transition from mathematics to programming. Lamport's TLA+ as the bridge. Formal specification applied to governed intelligence. Where axioms become code.
The destination. Domain-agnostic governance kernel. The same formal methods from Euclid through TLA+ applied to autonomous AI agent governance on Apple Silicon.
Separated from the core curriculum because it is context rather than prerequisite. Understanding the forge is valuable but not required for the formal systems sequence.
M-series architecture. Neural Engine. Metal Performance Shaders. The hardware substrate — understanding the forge on which the mathematics runs.
Modules are being built in sequence. The Geometry of Knowing and SwiftVector modules are active. The full programme will eventually be extracted into a book.