The Princeton Companion to Applied Mathematics

The book reveals a subject both ancient and modern, in a constant state of flux and advancement

editor Nicholas J. Higham associate editors Mark R. Dennis, Paul Glendinning,  Paul A. Martin, Fadil Santosa and Jared Tanner published 2015 by Princeton University Press Hardcover, 1016 pages ISBN 9780691150390
editor Nicholas J. Higham
associate editors Mark R. Dennis, Paul Glendinning, Paul A. Martin, Fadil Santosa and Jared Tanner
published 2015 by Princeton University Press
Hardcover, 1016 pages
ISBN 9780691150390

“What is Applied Mathematics?” This is the heading of an early section of the magisterial Princeton Companion to Applied Mathematics (PCAM). An easy question to ask, but as revealed by the collection of quotations from mathematicians who have tried to define applied mathematics, the editor of PCAM is in august company when he writes on its very first page that “[i]t is difficult to give a precise definition of applied mathematics.” Writing in 1965, Richard Courant believed that “[a]pplied mathematics is not a definable scientific field, but a human attitude.” We learn that a little later, in 1972, William Prager attempted this “next to impossible” task, and found that the best he could do was “to describe applied mathematics as the bridge connecting pure mathematics with science and technology”. But perhaps the best definition is attributed to physicist Lord Rayleigh by Garrett Birkhoff, who in 1977 said “mathematics becomes ‘applied’ when it is used to solve real-world problems”.

This is a satisfyingly broad definition. It covers the basics: if you count how many socks you have, you are doing applied mathematics. Now, if you study the mathematics of quantum mechanics, you are again doing applied mathematics—so it also covers extremely deep and difficult mathematics. But if so much human endeavour counts as applied mathematics, how can a book like PCAM ever aim to summarise it?

The short answer is that it cannot, hence the decision to call the book a “companion” to applied mathematics, rather than a “survey”, “summary”, or “introduction”. Nevertheless, the PCAM editors still set themselves a difficult task: explain key concepts, outline important research areas, cover modelling, simulation, and applications, and finally give some insight into the philosophy and style of applied mathematics, while also covering some history. As if that was not enough, PCAM also aims at being accessible to undergraduates, interesting to graduate students, useful for researchers, and a resource for those in other fields who wish to “apply mathematics”. The book certainly succeeds in these stated aims of being an “indispensable resource” and a “user-friendly reference”. But let us return to the central issue of what it means to apply mathematics.

This question is so full of intrigue that the editors take what might be called an instrumentalist approach to the definition of applied mathematics: applied mathematics is what applied mathematicians do, and applied mathematicians are simply those who use mathematics to solve problems in the real world, as we saw above. But what is it that gives validity to the work done by applied mathematicians? Here, I am not asking about what justifies the time and money spent on doing applied mathematics, but rather on what gives us hope that the results of all that hard work are worth our while. Why is applied mathematics possible, and what makes it “true”?

Why is applied mathematics possible,
and what makes it “true”?

In 1960, physics Nobel Laureate Eugene Wigner wrote about “the unreasonable effectiveness of mathematics in the natural sciences”.1 This phrase is not about mathematics being merely useful in science and engineering: it is about it being both indispensable and spookily powerful. It is the central tool used by millions of scientists and engineers around the world to investigate, explain, and manipulate the natural world. It is indispensable: without mathematics, there would be no modern science, and none of the technological wonders of the modern world. The power of mathematics comes from its ability to conjure a concise abstraction of the world, formally manipulate the symbols which constitute that model, and interpret the results in a way which predict scientific experiments with startling accuracy. How is such a thing possible?

That question is a philosophical one. Historically, the philosophy of mathematics has largely concerned itself with the philosophy of pure mathematics. Principal among questions in the philosophy of mathematics are the nature of truth, what gives mathematical statements validity, the nature of number, the reality of arithmetic, and the properties of infinity. There is, however, another question, one which subsumes all of the principal questions in the philosophy of mathematics and which also touches on related questions in the philosophy of science, and of the mind: why is applied mathematics possible?

One response is to take the applicability of mathematics as an axiom. Axioms are statements regarded as facts on which larger theories are based and which are taken to be self-evident and therefore not requiring an explanation. Though this might seem to be dodging the issue, on the contrary it means that this axiom can be used to give validity to mathematics: the only mathematics we should believe in is the mathematics which is applicable. This is a sketch of the so-called “indispensability argument” of the 20th century analytic philosophers Willard Van Orman Quine and Hilary Putnam. The basis of their argument is that mathematics and the physical sciences both assume the existence of objects which our senses cannot directly perceive, such as integers and atoms. Since our atomic theory predicts things which match our experience, we should take atoms as being “real” in some sense. Likewise, we expose our mathematics to the “tribunal of experience”, and take to be real precisely those mathematical objects on which our successful physical theories are based. One strange consequence of the Quine/Putnam position is that a mathematical concept is considered real only for as long as it is useful. The status of mathematical objects is forever conditional on their applicability.

Mathematicians tend to find any notion of conditional truth uncomfortable, since there is a real perception of discovery when doing mathematics. Mathematicians describe a sense of exploring real things when they do research in mathematics, that the objects of scrutiny are entirely objective, and that the knowledge gained of these objects is eternal and unchanging. It is this very beauty and permanence, this independence from the material world, which draws so many to advanced mathematics. However, this perspective is not a fact, but rather a belief known as platonism, in which there is an eternal unchanging world of “forms” to which we mysteriously have mental access, and which cast physical shadows in the material world. But how do our physical brains give us mental access to an entirely Other realm, and what does it mean for that realm to be approximated in our physical world?

Thus, while platonism might feel natural, it has these big questions to answer. Other beliefs about the nature of mathematics include logicism, which essentially takes mathematics to be applied logic, or rather logic in disguise. But why then should mathematics be applicable to the physical world unless logic itself somehow has a platonic existence? Another school of thought on the nature of mathematics is formalism, in which mathematics is taken to be one fecund example of a purely formal system in which essentially meaningless symbols are manipulated according to arbitrarily chosen and meaningless rules: mathematics as a game. By why should applying the meaningless symbols and manipulations of any game teach us about the physical world? Both logicism and formalism are rotten at the core as revealed by the logician Kurt Güdel in the middle of the 20th century, when he shocked the mathematical world by proving that any system of mathematics sophisticated enough to include basic arithmetic would necessarily be incomplete: it would contain true statements which were unprovable within that system. Finally, the intuitionists believe that the only mathematical objects which exist are those which can be mathematically constructed. This is a philosophy of the human mind, since only those things constructed by and capable of being grasped within the human mind are real. But why should these purely mental objects, which do not exist until constructed, ever be useful in the real world which existed for many eons before the human mind?

These are deep waters, and perhaps unsurprisingly, PCAM barely dips a toe into them, but does point out that a few have charted those waters. On page 58 of PCAM, in the entry titled “The History of Applied Mathematics” authors June Barrow-Green and Reinhard Siegmund-Schultze lament that they have little room to discuss “the history of philosophical reflections about mathematical applications”. They go on to describe in a brief but fascinating paragraph some of the recent developments in the area. They say that the rise in popularity of “mathematical modelling” since the 1980s prompted a concomitant growth of a broad literature addressing the philosophical issues that modelling uncovers. These issues include “the specificity of mathematics as a language, as an abstract unifier and as a source of concepts and principles for various scientific and societal domains of application.” Though each one of these examples deserves an essay in its own right, the paragraph sweeps us on to the rise of a “maverick tradition” (a fascinating oxymoron if ever there was one) which sought to study the working practices of applied mathematicians: studying applied mathematics by studying what it is that applied mathematicians do. This maverick tradition has two other defining features: first, it fights to distance itself from the so-called “foundationalist tradition” which sought a solid philosophical basis for mathematics in logic, as we saw in brief above. Second, it completely avoids discussing prematurely any “big questions”, such as why mathematics is applicable.

This maverick tradition, or the “new wave” as it is sometimes called, raises important questions which lie at the heart of the life of applied mathematicians, and which can be seen on almost every page of PCAM. One such question is the very question we are focussing on here, namely the relationship between mathematics and the physical world. This question is obviously of concern to platonists, logicists, formalists, intuitionists, and a whole host of other –ists seeking firm ground on which to base our faith in mathematics. The question is, if our physical theories rest on mathematical justifications, on what does our mathematics rest? In applying mathematics there is a constant and ongoing dialogue between the mathematics and the real world. In order to put a little flesh on the bones of this idea I will focus on a case study dear to my heart and presented in PCAM: boundary layers and the asymptotic analysis to which they gave birth.

The concept of the boundary layer was formulated by the aerodynamicist Ludwig Prandtl at the start of the 20th century. Prandtl wrestled with a key paradox of fluid dynamics, the study of the motion of anything which flows. The paradox came from the mathematician Jean Le Rond D’Alembert who observed in 1752 that from a mathematical perspective a body moving through a fluid should experience zero drag. This is quite clearly wrong: try moving your hand through water and see if you feel zero drag! Prompted by a renewed interest in flight, Prandtl’s experiments of 1904 aimed at understanding how fluids behaved near a boundary such as the wing of an aircraft. He showed that far from the boundary fluid behaves as if it were nearly frictionless, and so satisfies the inviscid fluid equations first formulated by Leonhard Euler some 147 years earlier. However, the fluid very close to the boundary is dominated by large gradients in velocity. The discovery of this boundary layer led to a resolution of a great many paradoxes. Indeed, without the concept of the boundary layer we would have no good theory of flight.

One reason for the difficulty in developing a theory of flight is that while we have a superb theory of fluid dynamics represented by the famous Navier–Stokes equations, these equations are so difficult to solve that we are unable to simply apply them to a given situation and calculate the answer. The NS equations were formulated by the mathematician and physicist Sir George Gabriel Stokes and the engineer and physicist Claude-Louis Navier in the first half of the 19th century. The equations are not much more than Newton’s laws applied to a fluid considered as a continuum. But while they are believed to be the final word on a mathematical description of fluid dynamics, they are very difficult to solve.

There is a constant dialogue between the real world and applications and the development of mathematics

That is an understatement. It is important to note that no solution to the full Navier–Stokes equations in a situation which involves fluid flow past a boundary has ever been found. Not one. And given that one of the mottos of fluid dynamicists is panta rhei (Greek for “everything flows”), fluid flows are ubiquitous, and always involve a boundary. Yet we cannot find such solutions to the Navier–Stokes equations, other than in highly idealised situations when our fully three-dimensional world is reduced to a two-dimensional one, when fluid compressibility or viscosity are ignored, or when other idealised symmetries are in play. More than that, we do not even know whether feeding reasonable boundary data and initial conditions into the Navier–Stokes equations will always yield well-behaved solutions (if we knew how to find them). This question is one of the famous seven Clay Mathematics Institute Millennium Prize problems, for each of which a prize of US $1 million is offered. PCAM has a fine section on the Navier–Stokes equations, written, as with most of its over 200 sections, by a world-leading expert. It contains a subsection entitled “Difficulties with the Navier–Stokes Equation”—and how.

So when Prandtl set about trying to find a mathematical theory of the boundary layer, he knew he had no hope of solving the Navier–Stokes equations directly. He therefore had to dramatically simplify the existing Navier–Stokes equations to apply to his boundary layer. Through a combination of great physical intuition and mathematical insight, he boldly started neglecting terms in the Navier–Stokes equations, until he had formulated a smaller, easier set of equations. Though we still use Prandtl’s equations to model the flow in a boundary layer, to this day there is not a formal derivation of Prandtl’s boundary layer equations from the Navier–Stokes equations.

There is now, however, a mathematical theory behind the process Prandtl followed. This theory, asymptotic analysis, was first formulated as a generalisation of Prandtl’s process, but has since been studied in its own right and has been applied to a wide range of areas. PCAM covers both asymptotic analysis and boundary layers in multiple sections. Asymptotic analysis is sometimes called “the art of rational approximations”, since it aims at finding approximate solutions to equations while retaining control over and awareness of the level of approximation. It is still somewhat of an art, since although there are standard means of approach, these do not always work, and experience, intuition, and creativity must be called upon to successfully apply the method.

The story of the boundary layer and asymptotic analysis is typical of applied mathematics as a whole. A need from the real world drives a mathematical investigation which of necessity is somewhat loose in its formulation. In this case, the invention of fixed wing flight required a resolution of long-standing paradoxes in fluid dynamics in order for a theory of flight to develop. Prandtl loosely but brilliantly formulates his boundary layer theory in order to begin the process. Next, as the idea is pushed forward in its area of application, it is also generalised and formalised by mathematicians, which eventually leads to its wider adoption and application into new areas. As the idea is used more widely it throws up new concerns and difficulties, which drives further development of the theory. This enables it to be more widely used and applied. There is in this way a constant dialogue between the real world applications and the development of the mathematics, between the application and the theory.

This type of dialogue is clear from this example and from many others presented in PCAM. It is at the heart of the study of applications of mathematics by the maverick tradition, and it raises many intriguing questions. For example, when was belief in Prandtl’s boundary layer theory justified? When first formulated it had no mathematical justification yet it clearly worked. Is that enough? It certainly was enough for the calculus which lacked mathematical rigour when first formulated by Gottfried Leibniz and Isaac Newton in the 17th century—that didn’t come until the 19th century, by which time the industrial and scientific revolution had been built upon calculus. Now that we know that there is at least mathematical justification for Prandtl’s process, should we have more faith in it, or is it the continued success of the concept which matters? Do we wait until we have a mathematical derivation of Prandtl’s equations from the NS equations? Do we wait until we know that the NS equations are well-behaved? How do these questions relate to the Quine/Putnam indispensability argument, and what consequences do they have for other schools in the philosophy of mathematics?

The answers to these and other philosophical questions are far beyond us at present, and beyond the scope of the book itself, because PCAM has other aims in mind. It seeks to be a companion to the vast world of applications of mathematics, and in that it succeeds beautifully. It reveals a subject both ancient and modern, in a constant state of flux and advancement as a dialogue continues between the coded rationality of mathematics and the crucible of experiment in the real world. As human knowledge advances into the sea of the unknown, behind the craggy coastline grows an ever-solidifying ground of understanding, built on the bedrock of mathematics.

Phil Wilson is at the School of Mathematics & Statistics, University of Canterbury, New Zealand.

Students at the graduate level will find this volume useful in discovering the areas of study they have aptitude for.

Over the ages, various civilizations have developed notions of numbers, geometry, trigonometry and approximations, for practical needs in areas such as agriculture—computing phases of the moon to figure out seasons was important—taxation, and trade, among others. Somewhere along the way the “pure” separated from mathematics and post-World War II, the “applied” re-emerged from mathematics and now we have the “pure” and “applied” parts of mathematics. These groupings were exemplified by the formation of societies such as the Society for Industrial and Applied Mathematics. We are now at a stage where there is a compelling need to label oneself (though not explicitly) as “pure” or “applied” for funding and career needs.

As a consequence of this, the same class of mathematical questions are looked at differently, with different degrees of depth and abstraction by practitioners of pure versus applied mathematics. Currently, a broad class of problems have come to be identified as “applied mathematics” and the volume, The Princeton Companion to Applied Mathematics, edited by Nicholas J. Higham, is an effort to collate and present concisely the collection of areas of applied mathematics.

In the introduction, applied mathematics is discussed in generality: what it is, its relation to the “pure” variety and what an applied mathematician does, the subject, with its practices presented through a few well-chosen examples. Anyone who would like to get a quick idea about applied mathematics and nothing more may well stop here.

Reading further, I noticed that the topic of “shocks” is discussed in at least three different chapters. The concept of shock is introduced in section II.30 as those points in space-time at which the solutions of a differential equation of interest fail to be more regular than Lipschitz continuous. There is a description of the nature of shocks that are acceptable for a specific problem at hand. From section V.20, “A shock wave is a sudden or even violent change in pressure occurring in a thin layer-like region of a continuous medium such as air.” This quickly forces me to think that “pressure” is, as time varies, a solution of a differential equation and this function has non-smoothness at some points in space-time. A qualitative description of shocks of certain types is presented here in the context of “pressure” of some fluids, without actually writing down any differential equations.

In the later article, a specific context is presented with no attempt at a serious mathematical analysis of shocks valid in that context, while in the former a more general formulation is given as a precise mathematical condition that constitutes the definition of a shock. These two articles on shocks suggest the qualitative difference in the perception of its practitioners, the former “pure” and the latter “applied”.

“Formulation is the most important part of applied mathematics,” writes John Wilder Tukey (page 77), “yet no one started to work on the theory of formulation—if we had one, perhaps we could teach applied mathematics.” In the last couple of decades, applied and pure mathematics are again merging as suggested by a 1998 National Science Foundation report, which says: “ Nowadays all mathematics is being applied …”.2

Going through this book would be an enriching experience similar to going through a beautiful forest looking at the multitude of flora, rather than just viewing the woods from above and seeing its vastness. This volume is excellently arranged with a Part each on concepts, laws and equations, presentations of various areas of study, modelling, example problems and some areas of application.

For example, the concept of homogenization is explained in section II.17 better than I could ever learn from other presentations—that it is the process of approximating a partial differential equation with rapidly varying coefficients and that it is relatively easily done when the coefficients are periodic. Section III.16 gives an account of the Korteweg-de Vries equation, KdV equation for short, which was formulated taking into account gravity and surface tension on the propagation of water waves in a canal. The equation is described, giving a brief account of the periodic, soliton solutions to it. Its two-dimensional version, the KP equation, is also mentioned. The great deal of work in recent times on the KdV hierarchy of equations, the solutions coming out of inverse spectral curves associated to Schrüdinger operators, are not touched upon.

Working in algebraic geometry is almost always thought of as an abstract mathematical exercise with no connection to reality, but the section IV.39 completely changes this assumption. Here it is shown how “variety”, which is at the heart of algebraic geometry, is an extremely useful object in robotics. The need to compute the zeros of a set of polynomials to be able to understand the movement of linked rods, such as the one modelling an arm, is presented very well here. Another good example of the lucidity of the writing is the section VI.1 on cloaking, where the authors present how cloaking is a transformation applied to optics. They construct a region where the conductivity is transformed in such a way that the hole in the middle is “cloaked” or hidden from an observer. This idea is explained with a collection of diagrams. Among the applications discussed, the example of imaging the Earth using Green’s theorem (section VII.16) has the construction of the Green’s function from its boundary values at the heart of its analysis, and could be used at the undergraduate level.

I could not resist reading the sections on quantum mechanics and random matrix theory to see how they are presented. David Griffiths presents the basics of quantum mechanics, such as the states, the observables, uncertainty and the spectrum of the hydrogen atom in three dimensions discussing the level spacings. He also talks about bosons and fermions, spin-statistics relation and the fine structure of the spectrum. There is no discussion of technical issues like the domains of operators or how addition of two operators makes sense, but I did not feel their absence in reading the article. I liked the smooth flow of ideas and details and enjoyed the coverage of an entire subject in a few short pages. The article on random matrix theory, on the other hand, has a broad overview, but did not give me the feeling that it conveyed sufficient details. I would have liked to have seen something mentioned on what is responsible for obtaining the Wigner semicircle law or the Pastur–Marchenko law as limits for different kinds of ensembles of random variables.

The final Part of this book addresses various attendant issues such as writing, reading and teaching technical material, something that could fit in any collection, not just one on applied mathematics.

Overall, this volume is indeed a companion for a mathematician and even for a practitioner of applications of mathematics. It comes in handy to quickly get introduced to techniques from other areas of application. I found the choice of topics interesting, representing widely studied areas, some which are still at the formative stage. Students at the graduate level will find this volume useful in discovering the areas of study they have aptitude for. This volume is most certainly a must for the library of any institution that has teaching and research at the graduate level.


  1. E. Wigner. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, 1960. 13:1.
  2. Report of the Senior Assessment Panel of the International Assessment of the U.S. Mathematical Sciences, Appendix 2. March 1998. National Science Foundation, USA.
Krishna Maddaly is at Ashoka University, Rai, Haryana.