The study of functions, limits and the infinite processes built on them. Begins with the foundations a calculus course assumes, what a function is, where it is defined, and how trigonometric and exponential values are computed, then follows the consequences into sequences, series and differential equations. Calculus supplies the derivative and the integral; this subject supplies what they act on and what they are used to solve.
What happens to a theory of choice when its predictions are tested against how people actually choose. Expected utility is a representation theorem, so it permits any attitude to risk and is refuted only by a pattern no utility function reproduces. Such patterns exist, the demonstration is algebraic rather than statistical, and the models built to accommodate them describe behaviour without thereby endorsing or condemning it.
The mathematics of change and accumulation. Begins with the limit, which makes both precise, then the derivative as a rate and the integral as a total, the fundamental theorem relating them, infinite series, functions of several variables, and line integrals in the plane.
Objects, arrows, and the laws that decide when a mapping between them preserves structure. A category is a composition satisfying associativity and identity; a functor is a mapping satisfying two further laws, and those laws are what the word structural means. A mapping can send objects sensibly and give every arrow the right endpoints while failing them. Naturality is a square that must commute at every arrow rather than a diagram that can be drawn.
How an analysis reaches a reader, and what the display decides on its way. A chart assigns the reader a perceptual task whose accuracy has been measured, and the scale, the baseline and the aspect ratio each change the comparison performed while leaving every number in place. The subject treats those as analytical decisions rather than presentation, because they are the ones that survive a correct calculation.
Estimating relationships from data that was observed rather than assigned. A least-squares coefficient reports how the outcome differs across units that differ in the regressor; whether that difference was caused by it depends on what else differs alongside, and the model's own diagnostics cannot see the answer. The subject is about identification: naming what would have to hold for a coefficient to be an effect, quantifying the part of the gap that can be quantified, and judging the arguments offered to close the rest.
Whether a treatment caused a change, and what evidence licenses saying so. Starts from the fact that each unit reveals only one of its two potential outcomes, builds the designs that make a comparison credible, and ends with what can and cannot be claimed when treatment was never assigned by anyone.
The arithmetic of payments that arrive at different times, and the decisions it supports. Discounting converts dated amounts to a common date; level and endless streams have closed forms that are the partial sum and the limit of a geometric series. The substance is which summary to act on, since a criterion that requires no discount rate cannot express a preference that depends on one, and the rate at which a project breaks even need not be a single number at all.
Reasoning from a sample to the population that produced it. Begins with the estimator as an object in its own right, a rule with a distribution, judged by where it is centred and how far it scatters, since the intervals and tests of classical inference are built on one, and the properties that make a procedure trustworthy are properties of the rule rather than of the number it happened to return.
Describing a domain so that a machine can draw conclusions from the description. Subsumption between concepts is a partial order, so a hierarchy is not a tree and need not have a meet or a join for every pair, which is why description logics supply constructors rather than relying on named concepts. A classifier then returns everything the axioms entail, often more than was drawn, and its silence means the ontology entails nothing either way rather than that the answer is no.
Vector spaces and the maps between them. Begins with the space axioms and the subspace test, then systems, matrices, determinants, bases and dimension, eigenvalues and diagonalization, orthogonality and projection, the singular value decomposition, and quadratic forms.
Use mathematical models to formulate, analyze and solve decision problems under constraints. Starts from what an optimization problem is, builds the geometry and linear algebra that make linear programs tractable, works through the simplex method, and goes on to integer decisions and to checking a reported solution against the model as written.
Measuring things that cannot be observed directly, and saying what the resulting numbers support. An observed score is a true score plus error, reliability is the share of variance that is not error, and validity is not a further coefficient but an argument about a particular interpretation of scores for a particular use. The subject exists because those two questions are routinely collapsed into one, and a stable instrument can measure the wrong thing consistently.
Building models that predict, and estimating how well they will. Starts from the gap between the error a model reports on the data it was fitted to and the error it will make on data it has not seen, since every later judgement, which model to prefer, how much flexibility to allow, whether a reported improvement is real, depends on estimating the second rather than the first.
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