What Do Mathematicians Do?
So, do you crunch numbers all day?
Actually, pure mathematical research is mostly a pen-and-paper (alternately, chalk-and-chalkboard or marker-and-whiteboard) affair. In fact, only a small fraction of the research thought process goes into the numbers one generally thinks of in everyday life, such as the integers or the real numbers. After all, such things are mostly, though not entirely, very well understood. Most of the work involves more interesting and/or abstract objects, such as algebraic structures, topological spaces, graphs, function spaces, differential equations, and so on. (There is, of course, another discussion to be had on why we look at such abstract objects.) Although a fair amount of calculations—"crunching"—must inevitably happen, this is certainly not the focus of research efforts, only the means to a much less boring end.
Mathematical Truth and Proofs
At the most basic level, mathematicians are curious people who wish to solve problems and answer questions, like in other disciplines, such as science, philosophy, and religion. For example, one might wonder what the area of the inside of a unit circle is, or whether a given differential equation has a unique solution given certain initial conditions. This is analogous to, say, a scientist asking how a human cell reproduces, or a philosopher inquiring how the moral evaluation of a person should be affected by that person's life circumstances.
What separates mathematics from science, philosophy, and other areas of study is how its questions are answered, which in turn determines what kinds of questions can be posed. In science, one cannot simply fabricate any theory and deem it credible. Proposals of a scientific theory must be confirmed repeatedly through experimentation and empirical observations. Analogously, one establishes the credibility of a mathematical claim by "proving it". In this process, one begins with a set of assumptions (the hypotheses) and then deduces additional facts about the system (the conclusions) through formal logical reasoning. This is the fundamental line in mathematics separating a statement of substance from mere hand waving.
As a most basic example of proofs, assume two integers \(x\) and \(y\) are even. We might think it is quite obvious by just tinkering around with some small numbers that \(x + y\) is also even, but in order for this "fact" to carry any true mathematical weight, we must prove it, that is, we must deduce this from existing knowledge. To proceed with this process, we can recall from the definition of "evenness" that there are integers \(x^\prime\) and \(y^\prime\) such that \(x = 2 x^\prime\) and \(y = 2 y^\prime\). Recalling from basic algebra the distributive property of real numbers, then \(x + y = 2 x^\prime + 2 y^\prime = 2 (x^\prime + y^\prime)\). Since \(x + y\) is expressed as \(2\) times something, we have deduced that \(x + y\) is also an even number.
While the above example may seem overly simple, the problems and proofs encountered in research mathematics are, of course, much more complex and difficult. Proving results often requires a significant amount of intuition and creativity. In many cases, one must either develop new ideas or discover unexpected connections between existing ideas. However, at the end of the day, the basic philosophical idea is the same—one wishes to establish facts about objects of interest and accomplishes this mathematically by proving these facts to be true. (One should also ask why mathematicians insist on proofs, even for simple statements, but that is a separate discussion—see [Why Prove Things?].)
A Wealth of Mathematical Topics
Objects of interest to mathematicians can vary over a wide range of subjects, each of which can be inspired from other disciplines or from more abstract curiosities. Many partial differential equations that are actively studied model phenomena in physics. Examples include the Navier-Stokes equations, which model fluid dynamics, and the Einstein field equations, which model the universe under the theory of general relativity. Computer science also drives a significant amount of research in discrete mathematics. One famous unsolved problem in discrete mathematics is P versus NP, which asks whether a certain important class of problems can be solved in a tractable amount of time by a computer. A resolution of this question could have tremendous impact in algorithms, cryptography, and many other fields.
In some cases, interesting problems can be of a more frivolous nature. For example, recently computer scientist Jonathan Schaeffer mathematically solved the game of checkers: he proved that if both players adopt optimal strategies, the outcome is guaranteed to be a draw. This effort is also an example of a more recent phenomenon—the use of the vast number-crunching power of computers to aid in proofs.
On the other hand, problems such as the Poincaré conjecture, which deals roughly with determining the possible geometries available to certain three-dimensional structures, have more purely mathematical motivations independently of outside applications. However, research efforts in this direction, leading to the recent resolution of the conjecture, have resulted in the development of new ideas and tools (e.g., the Ricci flow) that are widely applicable elsewhere throughout mathematics. Furthermore, for many abstract problems of interest, their potential applications may be initially unknown but far-reaching over time. For instance, many topics in number theory, the area of mathematics that studies the integers, have in the past century become fundamental to the development of cryptography and cryptanalysis.
None of the preceding descriptions thus far has mentioned the roles of applied mathematicians, who may reside at various points along the interface between mathematics and other fields. Applied mathematicians may be fully involved in rigorous proofs within some theory, or their expertise could also be focused on applying the theory to other scientific, engineering, or economic disciplines, or they could lie anywhere between these two extremes. This is, of course, a tremendously diverse area that has many exciting developments in its own right.
All in all, mathematics—both the problem solving and the proof crafting—is a fascinating endeavor that requires much insight and ingenuity—quite different from its unfortunate stereotype as robotic number crunching.