What is mathematics?
It’s the map, and the territory comes after
I suppose we’ll start here.
I asked my Facebook feed what they thought mathematics was. I’ll give you the answers anyway, but the question was problematic for two reasons. First, I inadvertantly primed them to give me a “less correct” answer. Second, as a friend of mine pointed out, the question was “ill-posed” because “the concept of ‘mathematics’ is too broad and ambiguous; it refers simultaneously to objective facts of knowledge as well as our subjective experiences and interpretations regarding these facts.” Regardless, some answers I received involved “numbers,” “numbers and a couple of x’s and y’s.” Even answers that pointed more in an abstract direction like “making sense of an infinite amount of numbers that some how fit together in a nice puzzle” still involved numbers.
(Interestingly enough, one can bait incorrect answers for “What is mathematics?” on Facebook, but were I to post “Who here is illiterate?” that would go over much more poorly.)
A friend of mine, a lecturer in the philosophy of mathematics at Clayton State in Georgia, once (may have) said that “mathematics is the rules by which reality works, where the rules themselves are interesting.” We drew a map and realized, looking at it afterward, that the map was worth studying for its own sake, not just for the territory it charted.
That sentence contains the two main points that I cannot and will not belabor enough. First, that we created mathematics: it’s not the language of the gods and it’s not the capital-u Universe speaking to us. Second, that it’s the map in which we’re interested, not the territory: that the territory tends to follow along is a happy accident.
“God created the whole numbers, all else is the work of man.” — Leopold Kronecker
Calculus grew out of physics, but the first could be defined without any reference to the second. The exponential function grew out of studying compound accumulation, but students can see it long after and long before they start doing interest problems. The reason for the apparent idiosyncrasy is that school-taught mathematics is the highlights.
As an example, most students are taught to draw a pair of axes on gridded paper and plot points to do “analytic geometry,” using algebra to understand geometric figures. (For instance, a line whose movement up is three times larger than its movement to the right, intercepting the vertical axis four units above the intersection of the axes, is given by the equation y = 3x + 4, as an automotive worker in Boston’s Chinatown was all too happy to recount to me.) These axes are given the name “the Cartesian coordinate system” (or something like this), after René Descartes, the “father of analytic geometry.”
But Descartes never drew a grid. Most agree that the first instance of using a coordinate system was in La Géométrie, when Descartes tackled (and solved) the classical “problem of Pappus,” which dealt with creating conic sections (circles, ellipses, parabolae, hyperbolae, for those of you not still in high school) from three given lines. His method began with fixing one of the given lines in place, calling it “the origin,” and only looking at the other lines in position relative to this one.
Which, if you think about it, is exactly what the Cartesian coordinate system does: we fix an “origin” and define all points by their distance relative to this point. However, this is still the highlight reel, and it takes some small effort to move from Descartes’ approach to the coordinate system with which the student is familiar.
So we start with a problem, like Pappus’ or like figuring out how bodies move over time, and draw a sketchy, unrefined map that gets us to the answer. But the maps are still functions of the territory. It’s axiomitization that breaks us away from the territory. Let’s return to geometry, because it’s an easy example and I think we can all remember enough to talk about it. If I define a triangle as three distinct points in the plane, or three line segments, or three vectors added “tip-to-tail,” or the union of three topological spaces homeomorphic to [0,1] where the endpoints of each have been identified in the natural way, and all those words were already clear to the reader—would it matter if a triangle actually existed?
If I clearly define all the properties of a unicorn, then we could all have a very lengthy academic discussion about unicorns, publishing papers and being awarded tenure; that there is no such thing is irrelevant. (Draw parallels to a mythical being of your choice as you like.)
Triangles actually exist (from a certain point of view), but plenty of very interesting things don’t (a series of posts on projective geometry may be in order.) As an example going in the other direction, consider the bridges of Königsberg.
Is it possible to walk through the city and cross each of the seven bridges exactly once? Leonhard Euler’s solution—it’s “no,” by the way—to this problem was to abstract each land mass into a vertex and each bridge into an edge. Who cares if they’re bridges and buildings?
Asks the skeptic: who cares about any of this? The problem of Königsberg led to the development of topology, the branch of mathematics dealing with the abstract study of space. (If you aren’t familiar with the term, don’t worry. Neither are most sophomores in mathematics programs.) A sub-branch of topology is knot theory, which is exactly what it sounds like. Knot theory is used for, among other things, studying DNA. (Lord Kelvin once thought, incorrectly, that the “prime knots”—those that could not be decomposed into a combination of other knots, like prime numbers cannot be factored into other numbers—corresponded exactly with the table of the elements.)
“Math is just a bunch of LEGO bricks.” — Alex Reinhart
In an utterly fantastic article on the mis-focus in mathematics education, Alex Reinhart lays out the answer to the oft-asked question “where will I use any of this?” Try to describe the accumulation of interest or the flight of a plane in English or any other natural language, he argues, and you will run into ambiguity. Mathematics is thought-bricks that we can stack together to produce new thoughts that are communicated clearly and precisely.
It isn’t numbers, it isn’t formulae, and it isn’t shapes. Mathematics is the study of precision and thought. It’s a science and an art. It’s the map that doesn’t need any territory. Most importantly, it’s a hell of a lot of fun.
p.s. This is the correct answer to the ill-posed question.
p.p.s. All of my references to “the map and the territory” were stolen from Lesswrong, who in turn adopted it from Alfred Korzybski.
