I blogged earlier about the "new age of empirical math", and how simulation complements deductive analysis as a math tool. For a recent client I've been doing queueing theory simulations with a Python library called SimPy. There a great tutorial on SimPy at http://simpy.sourceforge.net/SimPyDocs/Manual.html, but basically it's a discrete event simulation (DES) package. In the past I've only used DES to back up analytical results. For example, if you set the distribution of times between events to be exponential, then you get a Poisson process, which you can prove a lot of theorems about. But you can easily have the times between events be non-exponential. In that case you've got a snowball's chance in hell of proving anything, but it's perfectly easy to simulate it.
Until recently I was involved in a discussion on LinkedIn about which area of math was most fundamental (http://www.linkedin.com/groupItem?view=&gid=2074892&type=member&item=181517933&qid=7833e132-755b-469e-8f4a-8649dd36e5ac&trk=group_most_popular-0-b-ttl&goback=%2Egde_2074892_member_181517933%2Egmp_2074892). It started off as a poll about which area people thought was most fundamental, but eventually descended into mostly theoretical tail-chasing.
It's fascinating from a sociological perspective to see how the conversation evolved. This is an area where neither proofs nor experiments nor applications can be arbiters of what is true or relevant. And in this situation, it becomes largely a game of personalities, with no real resolution ever happening. Politics comes to mind...
Back in the day, before the Greeks, math was a more-or-less empirical subject. People did derivations and algebra and all that, but the basic rules were treated as empirical truths rather than platonic absolutes. For example, the Egyptians calculated the volume of a sphere in a way that used the wrong value of pi; the results were technically wrong, but they were close enough for everybody's needs. This all changed though when the Greeks (most notably Euclid) established deductive logic as the tool of choice for math. They played the role of Prometheus, bringing formal rigor to an ignorant world, and in so doing they transformed math from a toolkit of heuristics into a pursuit of higher truths. Math was done deductively, by hand, or it wasn't done at all.
But since the advent of computers, empirical math has come back into vogue. Computations that you'd never tackle with pencil and paper are calculated numerically, and much of the deductive work has moved over to studying the properties of numerical algorithms, rather than the actual problems those algos are solving.
The honest fact of the matter is that most math problems can't be solved analytically. Higher-order roots, most differential equations, and many integration problems are intractable in principle. And even if some of them happen to be exactly solvable, it's a moot point; we can solve them numerically without a problem, so there's no need to waste time using deduction. Some people lament the lack of theoretical rigor, but I think we should embrace a wider view of mathematics. Deduction is a hammer, and numerical methods are a screwdriver; you use them both to build a house. Mathematics is not limited to what's theoretically tractable; it's just that, until recently, *mathematicians* were so limited. When you're calculating the strange attractor of a dynamical system, or inverting a large matrix, you aren't quibbling about which axioms are satisfied; you're hacking out approximate truths, just like the Egyptians did long ago.
This blog features a series of mini-essays about the history and nature of math, and how it fits into the spectrum of human activity. I'm considering putting them together into a book, so comments are encouraged!