I’m a theoretical mathematician, which, for those who may not know, means that I love math because it’s math (as opposed to practical mathematicians who are interested in math for its applications). So, my complaint might surprise you, because I’m annoyed that math education doesn’t provide any motivation for what I’m doing.
So here’s the thing: there is no applied math and no theoretical math. There’s only math math. It’s not a semantics problem, although that’s part of it. The very idea of such a dichotomy is logically inconsistent, and self-referentially so. But first, a brief detour through the theory of a language.
So what is a language, anyway? At this point any computer scientist worth his salt will glaze over and start talking about automata and grammars, theories of computation, etc. We can summarize this theoretical approach by saying “A language is a set of words, each word being a finite string composed from an alphabet.” This is all fine and good, but you’re not going to learn English this way.
The way people actually learn English is by using English. Suppose you are hungry. You make noises until you are fed. Some noises get you fed faster than others. Wash, rinse, repeat.
What about grammars, about i’s before e’s? You need to go no further than consulting Wikipedia to figure out that those rules are almost more hurtful than helpful. Grammar is a formalism that helps, but the language is true and real and defines the grammar, not the other way around. For a good laugh, take a look at this page. You’ve got whole universities full of incredibly bright, educated individuals arguing very hard about things that by definition have no right answer.
Another way to see this is to ask who comes up with grammars? When was the last time somebody said “New rule: X” and suddenly everybody did X? That doesn’t happen . There’s a whole field of study that investigates how changes in the way people speak eventually make their way into codified rules that some scholar puts in a book. But it’s not the rules that make the language, it’s the language that makes the rules.
It’s common to think of words in a language as disjoint sets: this is why we can tell the difference between an “English” and a “French” word. But of course, English borrows words from other languages. This is why we have bagels, fjords, chutzpah, vodka, and zebras. Sometimes we borrow these words incorrectly: see hoi polloi. It’s a hop, a skip, and a jump to constructing “languages” that have very high overlap with English, but with little shared meaning.
One such language is the one used by the “Freemen on the land“, a fringe quasi-conspiracy-theory group that assigns special meaning to certain common legal phrases. For instance, they think the word “understand” really means “stand under” as in “stand under my authority.” So the cop who says “Do you understand you are being arrested?” to them is really asking “Do you give consent to being arrested?” and so they will actually “decline” to be arrested. Similarly, if their name is IN CAPITALS it supposedly refers to their legal person, as distinct from their natural person which is the flesh-and-blood person, and under this theory they sort of ignore court orders and such. Seriously.
This is a silly example, but the point is the same word in two different languages can have two very different meanings. This is particularly a problem if the languages have a high degree of overlap and it isn’t clear in what sense the word is being used. In fact, the greater the overlap, the greater the chance of confusion.
Oh yes, math. So it turns out that math is a language–people use the vocabulary of math to express ideas. This vocabulary often takes the form of words that look like English, but they are not English words. This is why “it follows easily” in a math paper often signals the hardest, most tedious nonsense you can imagine–the idea of “easy” as we use it in English doesn’t enter into that phrase. Within the language of mathematics, axioms, functions, numbers, vectors, and pretty much everything else take on a separate, distinct meaning from their ordinary English meaning.
So this brings us to the problem: what is an application? What does “applied math” even mean? Well, obviously, we turn to the dictionary, and read that an application is “a formal request to an authority for something: an application for leave“. I’m kidding, I’m kidding. The definition we (arbitrarily, with no rule to guide us) pick out of the five in my dictionary is “the action of putting something into operation : the application of general rules to particular cases“, which an average person would tell you is what applied mathematics “means”. Except this meaning is no better than the one I trotted out as a joke, because English and math are not the same language, and all the definitions in the dictionary are equally wrong in the language of math. When you’re reading a math paper and are wondering what “onto” means, nothing written by Oxford Press that says “onto” has to do with “moving aboard a public conveyance” is going to be the least bit illuminating or authoritative. You break out a special math dictionary for the language of math. Either that or the paper you’re reading is secretly trying to prove that f(x) is a streetcar, and applied mathematicians are those guys awaiting word from the High Math Council as to whether or not their math licenses will be formally approved. Maybe you should go to one of those Freemen meetings.
OK, so what does “application” mean in the language of math? Good luck with that. An application is (according to the “pure” math guys anyway) anything that’s not math. So by definition, you can’t define it in the vocabulary of mathematics. You can’t use A to talk about something that is by definition orthogonal to A. No matter how many times you take the vector (1,0) and perform scalar operations on it, you’re never going to get (0,1). You can’t get to applications if all you have is math as a logical starting point.
So we’re left with the uncomfortable situation that “application” as it’s used within the language of mathematics has no a priori meaning. The existentialists might ask if a word that has no meaning is really a word. Practical people generally apply one of three definitions to “application”:
All of these definitions preserve the property that everything is still self-consistent, and they also preserve the property that you will be no fun at parties. But they leave open the nagging idea that something is missing. I mean, what are you going to do with this definition? It’s not a very interesting definition on its own. What, you wanted a definition with applications?
I can muddy the waters even further with just a little handwaving. Most people agree that graph theory had no applications back in the day. Most people agree it has a lot of applications now. (For some definition of “applications”, of course.) So what happened? Well, nothing to do with the math. Very literally, a mathematician can, against his will, go from being theoretical to being applied, overnight. Application rape: it can happen to you. Theoretical mathematicians are the hipsters of math: they were into graph theory back before it went mainstream, now they have to scurry for a more obscure field to work in now that it’s “popular” .
Well, yes, now that you mention it. It certainly would be nice to have some words to talk about and categorize the different philosophies and motivations of mathematicians throughout the world . But “applied” and “theoretical” have been wholly corrupted for this purpose. Only a mathematician would think to try to compress the philosophy of every mathematician in the whole world into a bistable formalism, to organize every math department of every university into this formalism, to stake claims to fields and journals and tenure based on this formalism. If my motivation for doing mathematics is to solve a physics problem, am I an applied mathematician or a theoretical one? Does it change your mind if the underlying physics problem itself is a theoretical one with no applications? Does it change your mind if it turns out that my theoretical physics system has a one-to-one mapping with ZFC and is thus really just a theoretical math problem in disguise? What if my “theoretical” physics problem leads to a real-life physics breakthrough? What if the whole universe is just a Turing simulation, the whole world is just a theory? Put me in a bin, pretty please.
Can’t help you there. But what I can say is if you’re looking only inside math–you’re doing it wrong. Science doesn’t tell you why science is important. Music doesn’t tell you why music is important. You have to look outside. It can be just a little bit outside: Math makes you smile. It can be a lot outside: Math explains brownian motion. But either way, outside is where the reasons live. Whether your reason rises to the level of an “application” or not–quite literally–is up to you.
 Like every good rule, this one has an exception.
 More accurately, it’s the other way around: hipsters are the theoretical mathematicians of ordinary life. Find me a hipster that thinks graph theory is too “mainstream.” Didn’t think so.
 However, this would require investment in personal relationships among mathematicians. For all its faults, the dichotomy does have the advantage that you can make inaccurate judgments about people at a distance, shortcutting the tedious nature of social interaction.