Wednesday, June 27, 2012

mathematics is mining

During one of last week’s #mathchat discussions on Twitter, the topic was “What makes a question mathematical?” In the course of that conversation, I developed a few more thoughts on what constitutes mathematics (sort of a prerequisite to debating how a question can be mathematical, but on Twitter you don’t always cover the prerequisites first!), and I thought I’d share them here.

I’ve heard the question “what is mathematics?” be asked more and more often over the past 10–15 years. Specifically, people want to understand what it is that mathematicians study. Sometimes this latter question is answered by “numbers” or “patterns”. Not bad, but I think those answers reflect an underdeveloped sense of the subject. I once read an essay making an extended argument that the proper answer is “categories”, but that’s a bit inaccessible to non-mathematicians.

For a while now, my answer to “what is the object of mathematical study” has been some variation on “models”. (Actually, I first came up with this answer when one of my students in Guinea asked me the question.) This covers several kinds of activities: producing physical or conceptual models of an object or situation; determining what properties these models have; exploring how the properties can be mapped from one to another; extrapolating information from a given model beyond its initial construction. A scale replica of a boat or a building is a model (scaling is definitely a mathematical activity, as is deciding which salient features to include). A circle is a model; so is the number 5. An equation is a model. The Fibonacci sequence is a model (one which may or may not accurately represent the reproductive properties of rabbits, although it certainly reflects properties of other systems).

At this point I step back and wonder, what does it mean to do mathematics? Because, like any other scientific endeavor, it doesn’t just consist of enumerating the kinds of things it studies. Nor, as is commonly thought, does it consist of a list of results. Stating the answer to “What is the area of a triangle?” as “one-half of the product of its base and its height” is not mathematics. Finding this answer is. Applying this answer to another question is. Understanding this answer oneself or explaining it to someone else is. Can we quantify the difference between something that seems mathematical and something that actually lies within the field of mathematics?

Let’s start with the notion of a mathematical gem—a fact that can be appreciated aesthetically as beautiful. The world is full of these. The Pythagorean Theorem is one which has been known from antiquity. More recently we have uncovered the truth of Poincaré’s Conjecture (Perelman’s Theorem). We can gaze at these in awe and delight. But in and of themselves, they are insufficiently mathematical. A fact without origin or destination is sterile and inert. We do not call the collector of art an artist. In the same way, someone who simply collects theorems is not a mathematician. The practitioner is the one who produces. This does not mean there is no value in collecting, simply that a result is merely that—a product of some kind of human activity, not the activity itself.

Mathematics, then, is a kind of motion or process. If we continue the analogy of the gems, then mathematics is mining, cutting, and setting each gem so that it can be appreciated and used. Different mathematicians may be more skilled at different parts of this process (sometimes results are found in a fresh but rough form, and other labors are needed to clean them up so they can be properly appreciated). Occasionally someone will uncover an entirely new vein, or a shaft connecting two (or more) areas already being explored. This is how the history of mathematics arises. At different points in time, certain mines may be very active, and later they are shut down if productivity or interest decreases.

What is the mine in this analogy? It is a question-rich context. Questions are the motivation and the source of mathematics. The question may simply be “why?” (so that when one learns of a fact, like how to calculate the area of a triangle, one may be led to do further mathematics), or it may be “what happens if…?”, or “how can I describe…?”, or any number of other things.

To return, then, to the idea of models and what it means to study them, because “answering questions” is far too broad a description to accurately capture the essence of mathematics. In my view, mathematics is grounded at two ends in the universe as we know it, although these points of contact do not constitute its entirety. The ends may be roughly characterized as “extracting conceptual models from physical systems” and “interpreting properties of models into the physical world”. In between these ends lies the vast realm in which the models are studied for their own sake, not for any conclusions they may give about our physical world. Mining that is done closer to the ends may be described as “applied math”, and that which is carried in the middle may be called “pure”, but there is clearly a spectrum of possibilities here; for instance, often when I am seeking a physical, concrete interpretation of a mathematical model, I find myself wading through a great deal of what appears to be highly abstract and non-physical material. Thus my perspective lies somewhere between an Aristotelian stance (the existence of mathematical objects depends on physical reality) and a Platonic one (the study of these objects can be carried out independently of the reality from which they came).

Part of the glory of science is that no mine (i.e., field of inquiry) need be closed permanently, nor closed at all to any person. We can all return to what has been explored before and uncover its treasures anew, or we can explore new territory and bring back tales of a new world.

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