I had to laugh when I saw my response next to @getsat's. Both are valid viewpoints. In other words, not everyone who uses tools has to know how to make tools.
Tool makers have more insight into the assumptions behind the tools they create -- they might know how a tool was meant to be used and what its limits are. That being said, there are times when using a screwdriver as a hammer is expedient and does no harm. Of course, there are other times where using Black-Scholes to model a high-volatility market outside of the 'smoothly differentiable' market assumptions it was built on can crash a large part of the economy. It can be a dangerous game to use someone else's tools without knowing their assumptions.
2) Read this essay "A Mathematician's Lament" which points out the math learned in school is likely not really mathematics and that surprisingly math is not practical but aesthetic -- mathematics is closer to art than we are taught:
http://www.maa.org/external_archive/devlin/LockhartsLament.p...
I'm no algo trader, but I used to think the same way you do about math. I found that studying physics really gave me that first appreciation for how awesome math really is, and how it can be used to model so much of our world.
The same could be said by mathematicians about code. They look at a hash map and might think it is cool that you have O(1) lookups, but have no idea how it matters.
With experience and practice with theoretical math, you also learn how to apply it.
This really is jurisdiction dependent. At larger law firms that have to deal with paper courts still have runners that handle the docket and other details of filing (assuming the court allows it). At small firms this is the lawyer's responsibility.
Any suggestions on applied math? I suspect I just haven't been exposed to enough of it to really connect the theory to the real world.