My advisor in grad school used to say that applied mathematics is an attitude, not a subject classification. You can’t take an area of math and say whether it is or isn’t applied. Any area of math can be applied, or not, though some areas are applied more frequently and more directly than others.

Here’s a possible definition of applied math:

An area of math is applied, for you, if you’ve been paid to use it and not just to teach it or write about it.

By this definition, some rarefied areas of pure math are applied, but only for some people and not for others.

Here’s a less personal definition, and one that’s fuzzy rather than binary:

An area of math is applied in proportion to the amount of money people have made applying it.

Why the emphasis on money? Because that’s the common way people express their desires quantitatively. It’s a way of determining how much value the problem owner, not the mathematician, finds in the solution. Applied is in the eye of the client.

An interesting feature of both definitions is that they can change over time, especially in the personal definition. I’ve applied parts of math that I never thought I would. And there are also things I thought I’d find more use for than I have.

So maths used for a DIY project, hobby or sport is pure?

If you find something personally useful, say in a DIY project or sports, then someone is probably paid to do the same thing.

I don’t want to disparage pure math. I love pure math, math developed for the joy of doing so without any immediate application in mind. (Though such math could also be applied.)

What I do want to disparage is pseudo-applied math, math that is developed for pleasure or convenience but then marketed as if it were more useful than it is.

Although I agree with your motives, I also think that money is only an imperfect proxy for the actual measurement we are looking for.

To me, Applied Math is about using mathematics as a language to describe or predict the behavior of something that exists in the world. In this definition whether you can make money on it or not you’re doing applied math as soon as there is a correspondence between the symbols and numbers you’re using and something you can point to or directly or indirectly measure.

Daniel,

“money is only an imperfect proxy”

Really? Money isn’t a perfect measurement? Throw this post in the trash heap.

I think that the point is that money is a measurement, it is something that can be measured. In contrast: “a correspondence between the symbols and numbers youâ€™re using and something you can point to or directly or indirectly measure” cannot be measured.

In other words, I cannot look at a paper in PDE’s or probability and determine how great the correspondence is between the result and things in the world that can be measured.

But by all means, suggest a better measurement, keeping in mind that measurements produce numbers.

If getting paid is the measure, then Astrology is one of the hottest fields of applied math of all time. Some people might think that’s accurate, but it doesn’t gibe with my notion of what makes math ‘applied’.

I think math is applied to the extent that it is used to accomplish things. Predict eclipses; build bridges; optimize the output of factories; reconstruct evolutionary pathways; design experiments; reject hypotheses; draw beautiful pictures. Pure math is math that has not (yet) been used to do anything but math.

Dave: There are proximate and distant applications. When courts employed mathematicians as astrologers, these mathematicians applied their math (rather successfully) to predict the location of stars and planets. The proximate application was astronomy. The astronomy was then applied to astrology. But the same astronomy has also been put to more noble use.

I wouldn’t disparage the proximate application because of the distant application. Math might be applied to optimizing internet traffic, which is then used to sent cat photos faster. But there are other benefits too.

Good post.

The two, like democratics and republicans, need each other – but don’t like to admit it.

I don’t think there is a way to define it aside from snobbery value.

I remember an interview with my math department academic advisor at the beginning of my senior year at Berkeley.

He: “Do you want the applied math or pure math degree?”

Me: “There is more than one major? Which do you recommend?”

He: “You fulfilled the requirements for both. While no applied math graduate program will deny an applicant based solely on the name of their undergraduate program, some pure math programs will throw out all applicants with applied math degrees.”

Me: “I’ll take the pure math degree then.”

Postscript: When I was writing up my PhD thesis in computational physics, I signed up for on-campus interviews and didn’t get many bites. However, an IBM recruiter did interview me.

She: “I had to fight to interview you because we normally only look at applied math majors. But, since I was also a pure math major and I was useful enough to IBM, I asked that we interview you, too.”