The Princeton Companion to Applied Mathematics
The essential applied math guide
Description
In 2015, Princeton University Press published a book that weighed close to two kilograms and ran past nine hundred pages, edited by the Manchester numerical analyst Nicholas Higham with a small army of collaborators. Its title was deliberately modest: The Princeton Companion to Applied Mathematics. A companion, not a textbook, not an encyclopedia. The word matters. A companion is what you take on a journey when you already know roughly where you are going but keep discovering you do not know the terrain. That is more or less the situation anyone finds themselves in the moment they try to use mathematics on a real problem rather than a clean one from an exam.
The book had a famous older sibling. In 2008, Timothy Gowers had edited The Princeton Companion to Mathematics, a volume that became something of a cultural object among mathematicians — the rare reference you could read for pleasure. Higham's project took that model and pointed it at the other half of the discipline: not the mathematics studied for its own internal beauty, but the mathematics dragged out into engineering, biology, finance, medical imaging, weather forecasting, and the design of the very computers used to do the calculating. The two books share a spine and a philosophy. They part ways on the question of what mathematics is ultimately for.
What results is a strange and useful artifact — a single object that tries to survey a field which, by its nature, refuses to stay in one place. Applied mathematics is less a territory than a set of border crossings, and a reference book that takes it seriously has to decide how to draw a map of something that keeps moving.
The question we’re asking : How do you write a single reference book for a field defined by its refusal to stay within any single boundary?What we’ll see : How the Companion organizes an unruly discipline, what it chooses to explain first, where the mathematics meets the world, and what the whole enterprise says about applied math itself.
Table of contents
01Chapter 1 — A single volume tries to hold a whole field
The first problem any editor of such a book faces is brutal and structural: applied mathematics does not have a natural table of contents. Pure mathematics can be laid out along fairly stable lines — algebra, geometry, analysis, number theory — that have held for a century or more. Applied mathematics organizes itself by problem, and problems arrive from everywhere at once. A fluid dynamicist, an epidemiologist, and a cryptographer might all be using the same eigenvalue computation without ever sharing a room. Higham's answer was to organize the Companion thematically rather than by discipline, in eight broad parts that move from foundations toward concrete application.
The opening part introduces applied mathematics as a whole — its history, its relationship to pure mathematics, its characteristic ways of reasoning about approximation and error. Then come the building blocks: the equations, laws, and functions that recur across every field, from the wave equation to the Fourier transform. Only later does the book reach specific areas of research and, finally, the domains where the mathematics lands. The architecture is deliberate. It refuses to start with applications, because the whole argument of the book is that the same handful of ideas underlie wildly different uses.
02Chapter 2 — The grammar before the vocabulary
Before the Companion tells you what applied mathematics does, it insists on teaching how it thinks. This is where the book quietly reveals its priorities. A large early section is devoted not to spectacular applications but to concepts, equations, laws, and functions — the recurring furniture of the field. The choice signals something: in applied mathematics, the model you build is only as good as your grasp of the objects you build it from, and most real trouble comes not from exotic techniques but from misunderstanding the basic ones.
So the reader encounters the differential equations that describe how quantities change — the heat equation, the wave equation, the equations of fluid motion — alongside the special functions and transforms that keep reappearing across every domain. The Fourier transform, for instance, is not presented as a party trick of pure analysis but as a working tool that turns a hard problem in one setting into an easy one in another. That translation between representations, taking a problem written in a difficult language and rewriting it in a convenient one, is close to the central move of the whole discipline.
03Chapter 3 — Where the math actually goes to work
Having built the grammar, the Companion finally opens the doors onto the world, and the range is deliberately dizzying. The later parts survey areas of application that share almost nothing on the surface: modelling a beating heart, pricing a financial derivative, reconstructing an image from a medical scanner, forecasting tomorrow's weather, designing an aircraft wing, ranking web pages, compressing a photograph. What unifies them is not the subject but the method underneath — the same differential equations, the same linear algebra, the same optimization ideas turning up in costume after costume.
The book gives real weight to modelling and simulation, which is arguably the beating center of modern applied mathematics. A model is a deliberate simplification: you throw away most of reality on purpose, keeping only the parts that matter for the question at hand, and the whole skill is choosing what to discard. Simulation then runs that model forward on a computer to see what it predicts. The Companion is honest that this is a fallible craft. A model can be elegant and wrong; a simulation can be precise and meaningless. Knowing the difference is not a technicality — it is the job.
04Chapter 4 — A map is not the territory it describes
Step back from the individual entries and the Companion starts to say something about applied mathematics that no single chapter states outright. A book like this is, in effect, a snapshot of a discipline's self-image at a moment in time — what it thinks is central, what it thinks a newcomer needs, where it believes the interesting work lies. And the picture that emerges is of a field defined less by a fixed body of theorems than by a repertoire of moves for confronting problems that arrive from outside mathematics entirely.
That has a consequence the book embodies rather than argues. A reference work usually promises permanence: consult it, and the answer will be there next year and the year after. Applied mathematics resists that promise, because its center of gravity keeps shifting with the problems the world happens to be posing and the tools available to attack them. The Companion answers this not by pretending to be timeless but by front-loading what changes slowly — the concepts, the equations, the habits of approximation and error — and treating the fast-moving applications as illustrations of the durable ideas beneath.
05Conclusion
The Companion never quite loses the modesty of its title. It does not claim to contain applied mathematics; it claims to accompany someone through it, and the distinction turns out to be the whole design. Where its pure-mathematics sibling could survey a field that mostly holds still, Higham's volume had to find a way to be useful about a subject that changes shape with every new problem and every faster machine. Its solution — teach the durable grammar first, then show the shifting vocabulary as illustration — is what lets a single book serve the undergraduate meeting a term for the first time and the researcher crossing into unfamiliar territory.
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