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I'm working in the field of applied mathematics (optimization and numerics) and I meet a lot of students saying that they are allergic to applied mathematics or that they hate it or some quotes like "I'm here to learn how to think and become a thinker and that's why I only want to do pure mathematics", "Can't the engineers do all that?". I think, the students split in two groups: One group contains poor students who somehow worked out to pass basic courses, but are not able to apply the basics and want to blame something for that. The other group consists normally on very good students who like the concept of Definition -> Theorem -> Proof and miss that concept when it comes to modelling, looking at specific algorithm, compare ideas etc.

Of course, due to the curriculum, they have to attend lectures about applied math, so I could say that and be fine. But somehow I feel that one can argue at least with the second group of students.

How should one react on such quotes? Should one even react or could it be even worsening? If yes, what are good arguments?

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You could tell them about John von Neumann who I understand started as a pure mathematician then became an applied mathematician later. –  Joel Reyes Noche Apr 5 at 15:21
Perhaps use the mixture of ignorance and arrogance which frames their question against them. Ask them, do they really want to do pure math without even having the basics any applied mathematician knows? –  James S. Cook Apr 5 at 20:18
Perhaps there should be a similiar question: "How to react to students saying that they are allergic to pure mathematics?" (For the record, I usually teach mathematics to engineering students.) –  J W Apr 6 at 11:00

6 Answers 6

In the first place, it is "only" common course-naming conventions (and the AMS and NSF subject classifications) are to blame for the impression that there is some meaningful schism between something called "pure" and something called "applied" mathematics. Second, as noted, as much as anything people rationalize their own limitations or failings by blaming something external. And identifying an "other" and vilifying it has a long tradition in human history.

You could explain to the students that the definition-lemma-proof-theorem-proof picture is usually created with much hindsight, after many bouts of experiment, heuristics, revision. So our modern definition often came last in real life, while the desired/needed proto-theorem was presented to us by exterior circumstances. That is, "axiomatic mathematics" is most often done after a long period of experiment and revision.

Sure, some parts of school-math present only the end-product, but it would promote a methodological error to allow people to take a "Whiggish" view of these developments. That is, it's not that pre-axiomatic ideas were foolish or became irrelevant or obsolete. Rather, they constitute the phenomena-of-interest that were later organized by axiomatization.

Terry Tao https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ and others, such as http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html have reasonably labelled developmental stages of mathematical thinking as pre-rigorous, rigorous, and post -rigorous, in application to individuals. This has observable substance in application to individuals, but application to the larger enterprise is less clear.

And we can point out to our students that few of the historically well-known mathematicians made a distinction between "pure" and "applied", nor did they artificially confine themselves either in action or self-labelling. Most of the really interesting developments in mathematics arguably help us understand both very literal objects as well as idealized objects, and the whole range between.

And there is also the caution that some of the mid-to-late 20th century sense of "applied math" really meant numerical solution of PDEs, whether truly directly "applied" to create marketable products and local jobs or not. It may be that the WWII and study of non-linear waves to make big bombs leant special gravitas to numerical solution of PDE. But by this year we find that nearly all parts of mathematics have bluntly crass utility in business settings. Certainly it is not the case that these applications necessarily follow anyone's whimsical set-ups of axiom systems, but mathematics itself never really did random things, either.

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Very good answer! Applied mathematics is just pure mathematics which happens to have some applications. So if the students are really strict about this, they should stop learning any field as soon as it has an application. –  Mon Kee Poo Apr 16 at 17:32

When in doubt, I often decide simply to quote others! A nice choice, in this case, would be someone who started as a pure mathematician, then worked in applied mathematics, and ultimately moved into mathematics education. Luckily, precisely such a person exists in Henry Pollak: Ph.D under Lars Ahlfors, then Director of the Mathematics and Statistics Research Center at Bell Laboratories, and, along the way, wrote a fair bit about mathematics education (and has been a long-time "visiting" professor at Teachers College Columbia University).

There is a wonderful interview with Pollak that touches upon some issues related to applied mathematics. Before pasting a few relevant excerpts, one citation of the interview is:

Pollak, H., Albers, D. J., & Thibodeaux, M. J. (1984). A conversation with Henry Pollak. The College Mathematics Journal, 15(3), 194-217. Link.

Three excerpts that I hope can help shape your responses to such students:


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I do not have a true answer, but only a personal anecdote that backs up the point of teaching applied maths to mathematicians-to-be.

I work on fundamental mathematics, mostly differential geometry. My best work to date (in collaboration with Greg Kuperberg) has a differential geometric statement, but uses heavily linear programming which, as a fundamental mathematician, I have never been taught.

Of course, I managed to learn what I needed for the project (and I had already met with LP in previous research), but I now consider a bit sad not to have been more educated in what is often called applied mathematics, because it is really mathematics (and often very beautiful mathematics at that).

I should stress that the use of LP in our work is related to the so-called "Delsarte method". Delsarte was French, one of the founders of Bourbaki, but it is Greg that learned me about him. He was really amazed to see how little known he and is work is in France, among pure mathematicians.

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This quote from Jerry Marsden's 1980 AMS Bulletin review of Dieudonné's Treatise on Analysis is relevant:

History screams at pure mathematicians not to ignore applications; the origins of such important topics as calculus, Fourier series, operator theory and dynamical systems were all closely related to applications. Some of the greatest mathematicians, Gauss, Hilbert, Poincaré, and von Neumann, were all rooted in applications.

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Question: What's different about applied mathematics that you don't enjoy?

Notice the difference between that and asking why they don't like applied mathematics. One demands an accurate description of applied mathematics, the other can be answered in terms of one's aspirations or perception of value, and that distinction can be explained if it's unclear or not acknowledged.


Deep theorems often come from intuitions. Intuitions have to come from somewhere. Sometimes you get them from working elementary examples' features into your brain, but this can be near-sighted. Instead, they often come from an appeal to a natural connection between two subjects, and the overdetermining of properties induced by the multiple sets of semantics - every group is a permutation group, for example, is a formalization of the intuition about groups being transformers as well as algebras. The Sylow theorems and several properties of interest are capturing periodicity and self-similarity decompositions of group structure arising from viewing them as or expecting them to be the collections of symmetries of an object. Applied mathematics functions always in a context, and cross-pollination of semantics is an inherent part of such naturalistic environments. As such, it's an important source of intuition.

Projective geometry, for instance, arose from applied mathematics and the intuition it brought, and its importance in the structure of abstract objects (algebraic geometry, projective representations, manifolds, homology, hyperbolic geometry, continued fractions, the modular group) is worth the thinking it takes to justify without any trace of solipsism.

Applied mathematics can lead to these new fields because it poses problems that wouldn't be considered problems or be thought up by someone working only off their existing knowledge. It can also pose problems that are unmanageably difficult, which in mathematics suggests there are underlying paradigms left to be uncovered.

For a modern example, crack analysis can now be done using fractal measures and other abstract ways of reasoning, except when it comes to intersections of propagating cracks, these theories tend to break down abruptly and the finite-element method must be used to proceed with the description of the cracks approximately.

It's also one of the best ways to motivate a special case, informing the specific calculations that will allow one to become adept at a field. Most (classes of) differential equations are posed as tasks for their representation of a physically observed behavior, and in that way arbitrary calculations obtain a life of their own through application. So, for a pure mathematician - and by that I mean one which isn't strictly concerned with real-world tasks, applied contexts are rich interfaces to mathematics. Knowing any field in such detail helps gauge your depth of knowledge in other fields, and an application will evade giving a sense of security until that point is reached and the motivating problems have been solved.

All mathematics is descriptive of something, so how is non-applied mathematics to be intrinsically distinct from applied mathematics? It might be they disagree with having a non-mathematical goal, or have the impression that someone primarily does applied mathematics when mathematical goals aren't inherently valuable to them. It might be because educators often treat students as if mathematics is opaque to them whenever it isn't shrouded in a real-world scenario. These are cultural reactions. Perhaps their problem is actually with the experience they expect of applied mathematics, or the type of output they expect to be giving. The diagnostic should come before the prescription. Much of what people are trying to avoid by professing these alignments can be a problem in pure mathematics, in a generalized form, and if you have the wrong reasons to avoid something you narrow your tools for gaining experience and proficiency in a field in which you're mostly left to your own devices.

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If you want to motivate your students, you have to address that issue.

Maybe, you can motivate them via modelling. Show them, that what they thought is pure math, is already modelling.

  • Set theory is modelling the human psyche.
  • Peano is modelling linear, stepwise algorithms.
  • The axioms and basic theorems about real numbers are modelling human reception of the surrounding.

Maybe you can catch them with Gabriel's horn or Banach-Tarski-paradox and the gap between math and physics and the need to close that.

Then you can continue with "truly applied math" by motivating with the need to model something.

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Would you please explain more about what you mean by "Set theory is modelling the human psyche"? It is interesting for me. –  Saint Georg Apr 6 at 14:35
@SaintGeorg Human psyche is capable of differentiating between a set and it's individual elements. Some animals cannot, that's why fish swarms are a defensive measure against attackers. Other example: Discrimination, alliances and other social activities work because we treat other humans in sets with union, intersection, complements, … –  Toscho Apr 6 at 16:24

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