Saturday, July 14, 2007

What is Math?

Apparently, math is within the purview of the Phiolosophia Naturalis blog carnival (*nudge, poke*, suggest articles!), which has got me thinking again of a discussion I had a couple of months ago with a group of MIT alum I didn't know very well. One of the guys in the group was trying to promote the view that math is a science, but the scientists at the table didn't like this idea very much. There are actually three distinct questions at play here. At the most fundamental level, what does "mathematics" mean and encompass, how do we approach and go about learning and discovering math, and how should math be marketed so as to not scare people away from it?

So, is math a science? Webster defines "mathematics" as "the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations." But any real mathematician (except perhaps a number theorist?) would be quick to say that mathematics is about mere numbers. Keith Devlin claims that mathematics is "the science of patterns." I grimaced when I first heard this proposed definition, in part because it shifts the definition from "mathematics" to "science" and "patterns." I think what sets math and science apart is a certain degree of allowed imagination: for example, it is possible to create a self-consistent theory in which gravity goes like r-3 instead of like r-2, and the only problem with this theory would be that it doesn't describe the world we actually live in. One has to go and look at our universe to realize this, however. With math, however, one can theoretically sit in a closed room with a good brain and an endless supply for paper and pencils and derive and prove all of math—in math, something either is or is not true, and there is no way to even self-consistently describe the stuff that isn't true. That is, math is, at its core, universal truth.

From a marketing perspective, though, would it be better to treat math as a science? If the idea is that "numbers are scary" but "patterns are fun," then treating math like something to be explored and investigated instead of memorized might help fewer people get turned off by it. Such an educational approach, however, is the kind that is expensive and difficult to test the results of... though an approach that teaches how there are patterns in the multiplication table rather than insistance on memorization might have benefits. I can't really speak to this, however, since I've never been an educator, and moreso, I can't imagine what it's like to not grasp elementary level math as intuitive.

I had a friend who majored in the "philosophy of math" at Harvard. He said they didn't do math; they thought about doing math. When people actually "do math," i.e., prove new theorems and such, I think the process is rather scientific. You have some idea, a "hypothesis" perhaps, and you start with your assumptions and poke around until you prove or disprove the idea, or decide it's too difficult and the idea needs simplification. Science has a lot of trying new things to see what kinds of results can be produced; in this functional way, perhaps math is much more like an experimental science than a theoretical one, but where the "data" are trains of logical thought rather than measured values.

Of course, there are those who claim that math is a construct of the human mind, and, of course, we have no way of disproving this theory until we can communicate with other species. So you behavioral biologists and neuroscientists better get working on that.


Jacob said...

It's odd that you should equate advertising math as a science with giving the impression that it's not memorization. Because I was turned off of both physics and chemistry (you don't get more sciency than that) exactly because, for me, they were pure memorization of equations that I had no intuition for. Which made them both difficult and uninteresting.

Of course, for me, math is exactly *not* about numbers. If you're doing calculus (yuck!), you're a scientist. If you're developing calculus as a theory, you're doing math. Like you said, math is philosophy -- it's logic, it's no coincidence that philosophers like Descartes and Leibniz are also important in the history of math -- you start with axioms and you see what you can deduce from them.

Of course, I never excelled in real mathematics either -- I never took analysis, abstract algebra, etc. When I did have to do proofs, for cryptography and complexity, I could understand what other people came up with, but it was hard to come up with things on my own. I think I might be a mathematical engineer...

Vincent said...

You've hit upon part of the key difference between mathematics and the sciences. Mathematics is about deriving true statements from a set of axioms that are specified but can in principle be arbitrary. For the sciences, the only true axiom is the universe (although one can argue that secondary axioms, such as repeatability, exist).

Another key difference is in the logic used. With minor exceptions, mathematics is all about deductive logic. Most of modern mathematics is concerned with proving "true" statements (or disproving "false" ones) rigorously by applying deductive logic to certain axioms. The sciences, however, primarily involve inductive logic. The axioms are unknown a priori and must be deduced from experiment or observation. Given inherent experimental or observational errors, it is rare that a single test disproves a theory. And given the nature of inductive logic, no experiment can "prove" a theory, although it can increase one's credence in it, at least within some range of parameter space.

It may not be entirely so clear cut. Consider the physics of gravity. Newton developed a consistent theory that was able to make predictive statements (using deductive logic taking Newtonian laws of gravitations as their axioms) that were supported in experiment after experiment, thereby supporting the theory (using inductive logic). However, the theory did not hold under extrapolation, especially in the strong gravity regime. Enter general relativity, which changed the axioms. General relativity has been a spectacular success, with its deductive predictions being inductively supported by a litany of observations. Observational inconsistencies have led many to speculate that non-baryonic dark matter may exist and others to speculate that general relativity may not apply in the low-acceleration regime, leading to the theory of MOND. Each of these theories offers its own testable predictions. Thus, while both deductive and inductive logic are important to the sciences, the main difference between mathematics and the sciences is that in the latter the axioms are not known a priori and must be inferred.