What is the philosophy or purpose of applied mathematics? [closed]

Question

The title might seem vague but please bear with me:

I always liked pure mathematics, I believe I understood the purpose of applied mathematics as well, but just until I am finishing up now my thesis. The thesis involves PDEs and numerical schemes. I spent almost now 1.5 year with the subject. I didn't read a lot of papers, but the ones that I consulted give me the impression that they are 'everywhere', and it doesn't seem like there is a purpose (a unique or many) to what is going on, I mean what are applied mathematicians doing anyway?

Are they taking problems from real life / i.e. physics and trying to solve them using the mathematical machinery? Or are they taking some mathematical machinery and looking for fields to apply it? Is there a historical or contextual explanation to the research going on?

More importantly, how do you decide if an idea is quite original and worth publishing in the field ? I can argue that any 'application' of known results proven by other Mathematicians, is just an application and not an original solution?

I assume to believe that, for example, some biologists are doing some research on live creatures and they stumble upon a problem which will take them years to crack just because they didn't have enough background in ODEs or mathematical modeling, so when they solve it they will publish the results, but it is still known to mathematicians that this ODE problem has known solutions. Of course, their publication will contain other results not just the problem solved but the consequence of these results. This is very understandable to me and I have no issue with it.

On the other hand, what I do not understand is how does an applied mathematician 'pick' his problems? say someone who doesn't come from any of 'real life' backgrounds, like physics, chemistry, biology, geology... One who is just doing applied mathematics.

Answer 1

"Applied mathematics" is a fairly broad term. While it's frequently associated with modelling and PDE's, that's really only a small portion of what falls under the umbrella. I'm employed by an Applied Math Department and consider myself an applied algebraic topologist and geometer, for instance.

If I can somewhat ambitiously offer a definition: Applied mathematics is mathematics about methods which are intended to result in or understand algorithms usable on a computer within 20 years of their inception.

You may ask at this point what's the difference between this definition of applied math and computer science. That difference seems mostly to be some accident of history as to what subfields are associated with each.

You ask what types of results are considered novel. I think I can answer this by listing a few genres of results that I personally interact with:

1. Mathematical foundations and algorithms (preferably with an actual implementation) for computing or estimating quantities which have been investigated by theorists. For a theorist, knowing a solution exists is often enough. For an applied mathematician, one needs an actually feasible algorithm to obtain a solution. This often requires entirely new theory.

2. Data analyses with domain experts to solve a problem of interest using "cutting edge" methods. This often also requires new math and/or methodological innovation. Some computational strategies are just too new or too niche for anyone except an expert to implement. There's usually a positive feedback loop between new things done to get answers in an application and theorems one proves to understand how those methods behave.

3. Efficient algorithms and software implementations of new methods applicable to a broad category of frequently recurring problems which are usable by non-experts. This is more than code monkey work. Someone has to understand a method well enough to direct how an implementation should work.

4. Entirely new theoretical results which consult (or establish) questions in the theory underpinning applied methods. The goal here is more traditional math: generalization, better frameworks for talking about and understanding objects of interest, etc. But I think there is almost always at least some motivation that the theory will precipitate computational advances on a human time scale.

How do I pick what problems to work on? Mostly like a pure mathematician, though if I have an idea and don't think it can be put on a computer within twenty years I scuttle it. For applied problems, I'm always on the look out for colleagues with data sets that could be well-served by the types of methods I develop and study.

Answer 2

"Pure" and "applied mathematics" are maybe not quite the right terms because they are most commonly used to delineate fields within mathematics, not styles of work. For example, algebraic geometry => pure; PDEs => applied. Whether the person doing research in these areas is motivated by applications or not is an entirely separate dimension, and I (and others) like to make a distinction whether some research is "applicable" or not, where "applicable" means that the research is intended to address questions motivated outside mathematics.

In this sense, I am an "applicable mathematician", and my perspective is this:

People who do "applicable math" are using mathematics to solve problems that originate elsewhere. For example, I am a numerical analyst by training (i.e., my background is the analysis of schemes that can solve ordinary/partial differential equations -- numerical analysis being a part of what most people would call "applied mathematics"); I am using this background in collaborations with people from the geosciences/chemistry/physics/nuclear engineering/biomedical imaging to solve equations that they have come up with through the modeling process, but don't know how to solve (or solve sufficiently accurately, or sufficiently quickly). I have colleagues whose background is in algebraic geometry (what most people would call "pure mathematics") but who are solving problems that relate to identifying the location and speed of objects from the data obtained via radar (definitely a problem outside mathematics) and whose research is consequently "applicable".

Many of us in these roles see ourselves as "technology transfer agents". That is, we have the skills to talk to people outside mathematics and understand the language they use to describe what they want to but can't do. Then we use our mathematical background to solve these problems. In your question you suggest that this is simply an application of known stuff to a new area. But this is not generally true, one almost always has to adapt mathematical techniques to a new area. Moreover, the street is not one-way: We frequently learn that mathematical tools can be applied to a simplified problem, but that the real problem is more difficult, and then we take that back to our more mathematically oriented colleagues to look into, analyze, and prove something. Good technology transfer is definitely a two-way street.

Answer 3

Echoing and re-emphasizing some of @WolfgangBangerth's good points: "applied" (math) versus "pure" math is more about (self-)labelling than scientific/intellectual content.

That is, there is a common tradition in which numerical or heuristic solution of PDE's is "applied math". And, surprisingly often, conversely, "applied math" means (in many peoples' minds) numerical or heuristic solution of PDE's.

So, no, the label "applied math" cannot be relied upon to refer to math that is (genuinely) applied. In many peoples' minds, it cannot refer to cryptology, or big data, or ... lots of other things that are obviously math, and obviously fairly immediately relevant to tangible things.

And, currently, "persistent homology" seems to be relevant to Big Data analysis... but the traditional gate-keepers of "applied math" do not seem to be in any hurry to count algebraic topology as "applied math". :)

As some people will admit, there is a traditional attitude in which "pure" math means "irrelevant to nearly everything", while "applied" math means "relevant ... and fund-able".

Although it does seem necessary to know the appropriate code-switching to talk to "applied" versus "pure" mathematicians, I sincerely do not see scientific grounds on which to distinguish. At my univ, the basic grad-level "applied math and modeling" course introduces the same ideas (solving canonical PDE's on non-Euclidean spaces, Sobolev spaces, singular potentials, perturbation methods, semi-classical analysis, ...) that are essential in my own work in (what I call) number theory... related to zeros of zeta and such.

Answer 4

Personally, I think that the line between abstract/pure and applied mathematics is a very thin one, because in order to be a "good" applied mathematician, you need to dig deeper into the theoretical background of your "applied problem".

what are applied mathematicians doing anyway?

An applied mathematician is one that tackles problem arising in real life using tools from mathematics. Personally, I think that the hardest step in the work of applied mathematicians and what gives value to their work is what they call mathematical modeling. Note that at the beginning, they are faced with a real life problem, which then must be converted into mathematical equations. Now, the question asked is how to construct the adequate mathematical equations and how to pick the right mathematical theory which can serve as a tool to model the real life problem. I emphasize here that they must also have a sufficient background on the subject of the problem they are modeling (biology, physics, chemistry, computer science, etc.)

I assume to believe that, for example, some biologists are doing some research on live creatures and they stumble upon a problem which will take them years to crack just because they didn't have enough background in ODEs or mathematical modeling, so when they solve it they will publish the results, but it is still known to mathematicians that this ODE problem has known solution.

I would like to mention here that most problems in pure mathematics were initially motivated by problems from applied mathematics. To illustrate this, for example, assume that you are trying to model the spread of a disease within a population, but you want to take into account different aspects such as the environmental noise, the incubation period, etc. Then, once you construct your model, you are faced with a stochastic delay differential equation (dde). Now, assume the the theory of dde has not yet been developed. Then, this gives rise to a pure mathematics problem which consists of developing the theory of ddes.

On the other hand, what I do not understand is how does an applied mathematician 'pick' his problems? say someone who doesn't come from any of 'real life' backgrounds, like physics, chemistry, biology, geology... One who is just doing applied mathematics.

By definition applied mathematics is the application of mathematics to real life problems. So, the problems addressed by applied mathematicians are by definition problems arising from real life in different fields such as biology, physics, chemistry, computer science, etc. Note that some applied mathematicians can also address pure mathematical problems if they have the necessary background.

Finally, I would like to mention that the existence of applied mathematics has always been a subject of debate, since initially some of the founders of applied mathematics (if not all) like Henri Poincaré, Daniel Bernoulli and John Von Neumann were actually pure mathematicians.