“What is applied mathematics? It’s hard to define but I know it when I see it.”

I am a PhD student in applied mathematics at Colorado School of Mines. Recently, I attended a talk by applied mathematician Paul Martin titled, “What is applied mathematics, really? and why might you need it, really?” But, I have a confession: I still don’t know what applied mathematics is. Really.

That is **not** to speak pejoratively at all about the talk or of the degree program. I’m not even qualified to say so, but Dr. Martin and the other faculty at Mines are brilliant mathematicians. But yet (and this might be provocative, though it’s certainly not a new thought), in general, being an expert at X is not a sufficient condition for being able to define X.* *In fact, time and time again, Socrates brilliantly showed that, for example, priests could not define piety, politicians could not define virtue, and lovers could not define love. Further, and much more recently, the 2005 Kitzmiller v Dover Area School Board case concerning the legitimacy of teaching intelligent design as a scientific theory illustrated that any good attempt to define science relies on much more than asking *scientists* what it is they do (that is, a good attempt also consults *philosophers of science*). So, why is it that practitioners of a discipline, in general, have trouble defining their discipline? Well, I think that, at least in part, the answer lies in observing that *practicing *X* *and *thinking* *about *a X are often two very different skill sets. Often times, the concepts in play or the abilities necessary to *do* X well are insufficient for thinking about X. For example, *doing* applied mathematics might require skills such as learning how to choose ‘appropriate’ models for physical phenomenon, learning how to solve equations, etc. But *thinking about *applied mathematics seems to require the employment of some precise ontological and epistemological concepts (e.g., platonic objects, knowledge) that are not developed in doing applied mathematics.

To be fair, it is extremely difficult to define many fundamental concepts. In fact, some read the history of philosophy as trying to get a grasp on how we might go about defining fundamental concepts (e.g., mind, matter, substance, virtue, knowledge). So, to say that, after studying or being spoken to about applied mathematics, I don’t know what it is, is hardly surprising. But, in order to get some sort of grasp, let’s consider the two ways in which Dr. Martin attempts to define applied mathematics. **First**, he considers a plausible definition proposed by C. Wiggins; **second**, he defines applied mathematics in terms of what applied mathematicians do. I will focus on the first approach to defining applied mathematics in this post; a subsequent post will discuss the second approach.

(1) ‘Working definition’ of applied mathematics:

“Using mathematics as a tool for thinking clearly about the world.”

This definition was proposed as a nice working definition, and then used as a point of departure for talking about the practice of applied mathematics. But there is something baffling about this definition. To see the issue, we must think a bit about what *pure* mathematics is (the following is obviously a severely limited overview of a few schools of thought in the philosophy of mathematics). Of the four most prominent interpretations of mathematics–Platonism, Logicism, Formalism, and Intuitionism–at least three interpretations—Platonism, Logicism, and Formalism–have either as their starting point, or as a direct consequence, that mathematical objects are not the empirical objects of our experience (Intuitionism–which holds that there are no non-experienced mathematical truths–seems quite hard to swallow since it can’t seem to account for a huge number of statements that mathematicians generally accept as true).

Very briefly, Platonism–a position held my mathematicians such as Russell, Godel, and, at times, Frege–holds that the objects of mathematics are *independent* of empirical reality. That is, mathematical objects exist (in some sense of the word exist), but not as physical objects. As odd as that sounds, it’s not really *that* odd; after all, what physical object *is* the number ‘3’? Sure, there might be instantiations of the number ‘3’ in the world (e.g., three pebbles, three pieces of pizza), but ‘3’ itself–the thing that the 3 pebbles and the 3 pieces of pizza have in common–is nowhere to be found.

Logicism–held by mathematicians such as Frege, Dedekind, and Peano–holds that mathematics is entirely reducible to logic. There are a number of really interesting questions on this view, but the one that is particularly interesting for our purposes, is the following: if mathematics is just logic, and if mathematics speaks about objects (which it seems to), what does it mean to be an *object of logic*? I certainly won’t try to answer this question, but I will say that it is not obvious at all (nor might it even be plausible to say) that the objects of pure logic are empirical objects.

Formalism–held my mathematicians such as Hilbert, Tarski and Carnap–holds, roughly, that mathematics is a game where one manipulates meaningless (but well-formed) strings of symbols according to the rules of the game. Here, since the strings are meaningless, it doesn’t seem obvious that there are any objects to speak of at all.

Given that the main interpretations of mathematics seem to hold that, if there are mathematical objects at all, the objects of mathematics are not empirical objects, it’s easier to see why the proposed definition of applied mathematics above is baffling: if mathematics deals with non-empirical objects (or no objects at all), what does it mean to *apply* mathematics to think clearly about the empirical world? For example, physicists and applied mathematicians often use this equation:

to talk about physical waves (say, a vibrating guitar string). But, given the interpretations above, either (i) this equation refers to some set of platonic objects–whatever those are, (ii) this equation is a logical truth, deduced from axioms, whose objects are not obviously empirical at all, or (iii) this equation is a string of symbols that results from from rule-abiding manipulations of well-formed formulas, and, as with (ii), it’s not obvious that it refers to any objects at all.

Since (ii) and (iii) rely on the notion of axioms, perhaps this fact will help us understand how the above equation correctly refers to the world. But, upon a bit of inspection, it doesn’t seem like axioms will help understand how equations might reference the world. The axioms of real numbers–the axioms from which this equation is constructed–do not state what real numbers *are*. Rather, the axioms state that, *whatever the set of real numbers is*, the elements of this set will act according to certain rules (e.g., associativity). Similarly, axioms for other mathematical objects (e.g., natural numbers, groups, rings, etc.) don’t directly posit objects; rather, they posit relations between abstract somethings-I-know-not-what, represented by variables.

In this post, I hope to have given some support to a very modest claim: the above definition, that applied mathematics is the use of mathematics “as a tool for thinking clearly about the world”, raises more questions than it answers. Given that Dr. Martin made it pretty clear in his talk that it was not a perfect definition, it seems reasonable to assume that the kind of inquiry that I’ve attempted here (and that I will attempt to do in further posts) is what needs to be done if we want to understand what applied mathematics is, *really.*