The other day some of my fellow grad students got an e-mail forwarded from our professor. The e-mail was from a student in electrical engineering who wanted help with a system of nonlinear differential equations. In particular, the student wanted "the solution." Keep in mind that most differential equations don't work that way. When you take a class in differential equations, you always start with those basic examples that you can solve explicitly. Once you get out into the "real world," you quickly realize how futile it is to dream of finding an exact, closed-form solution. Our professor had to respond to this student by explaining something about the general theory of differential equations, to which the student replied with apparent dissatisfaction. She was only interested in "the answer." Of course, our professor is not planning on wasting any more time with this e-mail.
The incident highlights an important fact: very few people actually understand what I do. (Most days I don't even understand what I do, which is why we call it "research.") To me this is rather unfortunate, because I actually think what I do has a tremendous philosophical contribution to make to the sciences. Too often the contribution of mathematics is seen in purely utilitarian terms: we can model a "real-world" process in mathematical terms, thereby understanding it in rigorous quantitative terms. I think there's a lot more to it than that.
What I do is actually a little strange. First, I start with a mathematical model, that is, an equation or a system of equations. The equations I study are often called evolution equations because they are supposed to describe how a system evolves at each (usually infinitesimal) increment of time. In other words, the system of equations constitutes a set of rules which describe the behavior of certain components of a system. (The technical term for this concept is dynamical system.) As a very simple example, think of a ball being dropped from a few feet above the floor. The rule governing its motion is given by Newton's law: the weight of the ball is a force, and this implies a certain acceleration. So since I know the weight of the ball, at each infinitesimal interval of time, I can describe how its position with respect to the floor is changing. This gives me an evolution equation (or differential equation).
Now here's the strange part. Instead of trying to observe what actually happens and then coming up with a way of describing it, I go the other way around: I take the system of equations I'm studying, and I see what the resulting behavior should be. For instance, the fundamental question I usually ask is whether this dynamical system is well posed. By that I mean, does it pass the following criteria: for each initial condition (how the system looks at the beginning), is there a solution? and is that solution unique? and if I change the initial condition slightly, does the solution change only slightly? In other words, does it behave the way we intuitively expect things to happen in real life? Usually we can expect that if I drop a ball from a certain point above the ground, it won't suddenly explode, it won't suddenly fall upward, and it probably won't fall in a significantly different way if I happen to hold it up slightly higher from the ground.
Once this fundamental question has been asked about the system being well posed, I can then ask other questions. How does this system behave in the long run? What if I change the system of equations slightly? What if I add some component to the equations, representing some sort of force on the system? All of these questions can be addressed mathematically.
What is going on philosophically? There are two fundamentally different notions of causality that I see at work in the sciences. One is what I'll call linear causality. This is the view that life is a series of unfolding events, and in order to explain one kind of event, you must find some other kind of event that caused it. For example, if you want to explain why someone has lung cancer, perhaps you will explain it by saying that he smoked too much. However, from this point of view there are many things which are surprisingly difficult to describe. For instance, what caused the recent financial collapse in 2008? What caused the Industrial Revolution? What causes the weather? Such questions are very difficult to understand in terms of linear sequences of events.
The second notion of causality is what I'll call systemic causality. From this point of view, instead of viewing the world as a series of events which unfold in a progression, we see things as components of a system governed by overarching rules. Mathematics justifies this notion of causality by showing how it might logically operate. What we can show is that intricate structures or patterns can come into existence through the repeated application of a rule, rather than, say, through a predetermined structure imposed on the system. Of course, one cannot explain this without referring to much more complex example than dropping a ball.
Let's consider, for example, Newton's explanation of the motion of the planets (we should also give credit to Copernicus, Galileo, and Kepler, Hooke, et al). Instead of resorting to the classical explanation that planets and stars had fixed spheres in which to move, Newton saw the whole thing as a dynamical system. That is, he could show that all of the complex orbits of the planets are natural consequences of simple rules. Actually, you really only need one simple rule: Newton's law of universal gravitation. From this rule, if you want to understand what the resulting planetary motion will be, all you have to do is solve the "N-body problem." Thus the beautiful and complex order we see in the heavens can be explained using a systemic approach. A linear explanation never could have given us the understanding we now have, because motion is not simply a sequence of events, but an abstract pattern. Fortunately, such abstract patterns can be shown to be logically tied to rules, thanks to mathematics.
There are other examples of systemic causality where mathematics was originally not part of the explanation. Charles Darwin's explanation of how species came to be so diverse relied not on a linear, but a systemic explanation. He proposed a simple rule, natural selection, which could be repeatedly applied to result in biological diversity. Like Newton's law of universal gravitation, natural selection is simply a rule that all components of a system--namely living organisms--seem to follow. Darwin's burden was to show that there was indeed a logical connection between this rule and the kind of general pattern we see in nature.
Another good example, before Darwin and after Newton, would be Adam Smith, attempting to explain how wealthy nations had acquired their prosperity. Instead of seeking a linear explanation, which would have made Wealth of Nations into a history book, he sought a systemic explanation, thus helping to usher in economics as a science in its own right. He showed that the principle of market exchange could explain the general order we observe. If Newton's and Darwin's theories may have been religiously subversive, explaining nature without God (though Newton himself didn't see it that way), Smith's theory was political subversive, explaining how economic flourishing could come about without any particular guidance from the state.
I have often noticed, however, this is not the kind of explanation usually desired by most people. Linear explanations are still the most frequent in our daily lives. This is probably because, in order to accomplish our particular goals, we have to plan things sequentially. We do not often need an abstract theory to explain the concrete events of our own lives. I imagine this is the case even for many scientists, or at least for engineers, medical researchers, and the like, whose main focus is on solving concrete problems. However, I am willing to wager that certain fields, particularly medicine, could benefit greatly from a more systemic rather than linear approach. It's fine that doctors are able to link symptoms with diseases, and thereby prescribe a correct treatment. This has certainly made a lot of sick people well. But if we are concerned, say, with curing cancer, I highly doubt any progress can be made without systemic explanations, and I bet mathematics might be able to help with that.
On a political level, I would say our politicians are overwhelmingly biased toward ad hoc solutions rather than considering long term principles. This is simultaneously a moral failing and an intellectual error. The idea that some particular jobs program will in fact create the kind of economy that we want (as if anyone knows what that may be) completely ignores the scientific insight of Adam Smith. If we're concerned about the economy, we ought to be focused on the general rules which make prosperity possible, rather than on the particular goals we think are important at the time.
Finally, on a theological level, I find it interesting that today we have such a controversy over Darwin's theory of evolution, when in fact this theory is a natural consequence of the general way of thinking I have outlined here. I wonder if Christians who understand the implications of Newton's laws find them to be subversive to the faith. My sense is that they do not. But why, then, is Darwin so demonized? If God has so structured the universe that simple rules can result in beautiful complexity, what are we to say against that? Do we subvert his authority by using mathematics? It should be abundantly clear from what I have said that evolution is not "random." It need not be without purpose, either. It is written in our scriptures that God shows no partiality, and in a way, that is all the theory of evolution says: all things in heaven and on earth are subject to the same laws, which wondrously result in the beautiful and complex order which we get to experience.
I suspect all of this raises more questions than it answers, but I hope it raises the right questions. That is, after all, how we make progress.