In this final volume, we take a brief look at the various ways in which mathematics is used by mathematicians as well as any number of other professionals. The hope is that this is a guide for those to find interesting directions to pursue.
The characterizations here are my own, based on my own subjective viewpoints, experiences, and research. The idea that this captures all of the math goings-ons is absurd and this should not be taken as definitive in any fashion. It is a bit like a movie trailer, designed to inspire pursuit rather than to give the full story.
While there are a number of professions covered here, many are left out. A notable category left out are those of statisticians. This is in part because there is a whole book in this series dedicated to that field. But it is also difficult to cover a field which is used across so many disparate fields and used by many different people.
Along with statisticians, I also failed to figure out where to put data analysts. They are a bit statistical, a bit computer sciency, and a bit artist. I do mention them under the visualizations section in the artist chapter, but it is a field which is very multidisciplinary and in flux.
The chapters are roughly organized with the more pure mathematical fields coming first.
We start with algebraists which covers both manipulating integers as well as abstracting out the properties of objects and studying the abstracted version. The most applied area listed in that section is that of cryptography, a field strongly linked to the use of prime numbers. Definitions are complicated leading to simple and widely applicable theorems.
Analysts take their cues from the real numbers and calculus. They love to generalize notions to find something that they can call a solution to an equation. The most practical part of analysts that we cover is that of numerical analysis which is the art of getting reliable numbers when exact methods fail to answer a question. Definitions are generally simple with very complicated proofs supporting messy theorems.
Geometers refer to those who pursue the ideas of curved spaces. It covers analyzing the curvature of ordinary surfaces and soap bubbles as well as abstract spaces such as the space giving the configuration of everything in the universe. Relativity is founded on the ideas coming from differential geometry. The most practical area discussed here is that of control theory, a field which attempts to figure out how to find the optimal paths under various constraints. This is a mixture of definitions and theorems with a large portion of story telling.
What I have termed the booleans covers those who love to count. This covers discrete areas of mathematics. Here, questions are often very easy to state and almost impossible to answer. There is a good deal of small number explorations, but the problems often change in nature as the numbers get large and out of the reach of computers. I have included logicians here as I have often felt that there core is counting beyond the finite; it would also be reasonable to group them with algebraists. In terms of understanding what is doable in the world, this is the area of mathematics that provides the most tools for figuring all of this out. Computer scientists, included here, are often concerned with whether an algorithm will scale with larger numbers.
Scientists use mathematics to investigate reality. Science is very wide ranging and we sample some of the biggest areas. Physics and mathematics has a long history of being codeveloped. Biology has been pushing the limits of the combinatorial aspects of mathematics. Ecology is a great example of modeling intertwining statistics and differential equations.
Engineers are those who use science and mathematics to shape reality. In my classification, this includes fields called engineering, but it also includes other professional fields such as meterology and actuaries. All of these fields use a variety of different parts of mathematics, much of which has been incoorporated into computer systems.
Our final section is that of artists. Mathematics is as much an art as any field; most professional mathematicians pursue mathematics because of its beauty. We explore here some of the ways that mathematics intersects with areas that are more commonly understood to be artistic such as music, writing, and the visual arts.