Tools and toys.

First off, apologies for the lack of posts recently, I've been somewhat more busy than I expected getting back into the swing of things here as school starts up again. I'm sure glad we get Good Friday off, as I could us the extra day off to get homework done.

I was musing on my way home today about the difference between applied and theoretical mathematicians, since often both kinds use very similar levels of math.  What I realized, is that applied mathematicians (which includes physicists, in my opinion) view math as a tool; theoretical mathematicians view it as a toy. Both physicists and mathematicians play around with math, but mathematicians play with it like you play with any other toy, while physicists play with it like you play with a tool. When I say 'like a tool', I mean in the same way that an accomplished carpenter might 'play around' with his tools when designing a fancy piece of furniture to give it an interesting or beautiful design, or a chef might 'play around' in decorating a cake he baked. The focus is not on the tools; it's on what can be produced with those tools. The tools may be valuable and precious objects in their own right, but they're not visible in the final product, usually.

I hope I'm not sounding unfair to theoretical mathematicians, because even though I am not one, I sometimes envy the state of mind that regards pushing back the frontiers of ignorance on the cutting edge of mathematical research as something fun. After all, a carpenter without tools would be able to produce practically nothing, and the tools of mathematics are precious and valuable in their own right. In fact, the picture as I have drawn it is a bit simplistic. Historically, theoretical and applied mathematicians have worked together, though perhaps without always being aware of it. Scientific research (and particularly physics) has often revealed a profound lack in the mathematics of the time period, prompting investigation and extension into new areas (perhaps even the discovery of a whole new field, as with the calculus). This investigation may be done initially by the scientists who are affected by it, but it usually gets absorbed by theoretical mathematicians who may have no interest whatsoever in the physical problem. Eventually they probe it even further than the scientists needed it taken, and everything seems happy again until someone discovers that taking this new math and applying it somewhere in science makes new discoveries possible. People look into this, and eventually, after a while, discover that the existing math is not enough, and the cycle begins again.
I have unfortunately forgotten whatever point I intended to make when I started writing this, so I will leave you with this marginally-useful bit of advice: if you want to know whether someone is an applied or theoretical mathematician, look at how they use math. Do they use it as a tool, or as a toy?