mathematics

Mathematics is the study of structure, quantity, space and change, the language in which all formal sciences are expressed.

Mathematics is the study of structure, quantity, space and change, the language in which all formal sciences are expressed.

Machine learning draws on four of its branches: linear algebra (matrix multiplications in every layer), calculus (gradients and backpropagation), probability theory (distributions, sampling, Bayesian reasoning) and optimization (loss landscapes, gradient descent).

Compositionality (the principle that complex functions are built by composing simpler ones) is both a mathematical concept and the operational mechanism of transformers and chain-of-thought reasoning.

Statistics and machine learning ask different questions of the same data. Statistics asks what the data tells us (inference); ML asks what can be predicted from it (generalization). The boundary is blurry: regularization, cross-validation, and bias-variance tradeoffs live in both.

Applied branches with their own notes: differential equations (writing the rule: time, initial conditions, constraints, ODE and PDE), dynamical systems (state, rates, modes, stability), graph theory (Laplacians, diffusion, spectral methods), probability (Markov chains) and vector calculus.

Mathematics is unreasonably effective in ML: architectures designed from mathematical principles (attention as soft dictionary lookup, residual connections as Euler integration, normalization as variance stabilization) consistently outperform architectures designed by intuition alone.