Well that's a misnomer but it had a certain alliteration. A maven, I'm not! The closest I come to that is that I'm a retired EE. I worked in the areas of communications, radar, microwave components, and antennas and that did entail a lot of applied math. So being engaged thus, math was a constant companion and valuable tool. But mathematics is a lot deeper than just being a tool. Mathematics is the lingua franca of science and technology and is omnipresent. But again, it is not just a servant of applied fields. At the risk of oversimplifying I think of physics as applied math to natural phenomena and engineering as applied physics. Although mathematics was conceived in earliest times as a completely practical endeavor to enhance the functioning of everyday life, it has evolved over the centuries to fill many roles. During the Renaissance mathematics suffered a big shift, going from a pragmatic science to a more mystical, philosophical discipline History of QF. A major role of modern mathematics, if not the defining role, is the pursuit of mathematical truth for its own sake.
As an engineer using math I worked at times, depending on the job at hand, with some fairly high level stuff. The usual math subjects of algebra, geometry, trigonometry, calculus and some of the more advanced including linear systems, functions of complex variables, probablity theory, differential equations, partial differential equations, some differential geometry, Fourier and LaPlace transforms, and the list goes on. I learned what I had to learn but always kind of knew that I was just scratching the surface. One can learn enough about elliptic functions to use them, e.g. in the design of some wave filters. But I knew that there was a lot lying beneath the surface that was beautiful that I couldn't quite grasp. And that was typical, you could use the stuff but you knew that you didn't know enough to really appreciate.
Now being retired and able to pursue some interests, I have been engaged in a slow, low pressure program of learning, relearning and more fully understanding and appreciating mathematics. I'm not approaching it with a view to becoming a self taught mathematician and functioning on that level. I'm sure I wouldn't be successful at it had I wanted to. I have no ambition to function as a mathematician even if able. And to confess I never really had much patience to do proofs; I wanted the results and then run.
I think mathematics is one of the highest intellectual pursuits (don't push me) and I'm blown away by what these 'magicians' have done over the course of human endeavor. The totality of it all is just amazing and brings to mind the phrase 'truth is stranger than fiction'. If you don't believe, read Fermat's Enigma.
Perspective is so much, maybe even it's almost everything. When we see the Earth and the world around us at ground level we see one thing, when we see it from the vantage point of space (as depicted in a video or even in our own imagination) it's something completely different. When we see our civilized world through the prism of societal and economic constructs we see it in almost surreal terms. But that's a subject for another day. The point I'm making here is that we see mathematics from our modern eyes. Most people not engaged in work requiring mathematics don't think about it very much after they are done with their school responsibilities. Nevertheless it does impinge on our modern lives ever more and will do so in a more accelerating manner. That's just the nature of societal evolution. We are just vaguely aware of the power of mathematics as a factor in our complex technological lives. But OK, there it is, I don't have to understand what lies under the hood of my cell phone. It's just there for me to use.
But that's not even what I'm trying to say. Forget about the high class sophisticated stuff. How about the stuff that everybody uses. Everyday ubiquitous counting. How about the perspective on that. That's always been around. Anybody could dream that up and use it even if it wasn't done before and handed to you on a platter. Well it turns up NOT! If you look at the early history of counting you'll find that there was a time before counting. Like everything in the human equation nothing came easy. Mundane counting was a long, slow, difficult evolution. It came about to better deal with the difficulties of everyday life. Even the use of our Arabic numerals was a major step forward; try multiplying in Roman numerals.
Here's a good one, Pi. Everybody knows Pi, the ratio of the circumference to the diameter of ANY plane circle. It's irrational and transcendental. It's ubiquitous in the sciences and it would be hard to imagine a world without it. Everybody was taught that its value is close to 22/7. If that was its true value it wouldn't be irrational. It is a never ending or repeating decimal. The current value has been calculated by computer to over a trillion digits. There is no calculation that would ever need the exact value of Pi; that's good because it doesn't exist. Probably even for very exacting requirement a value out to only twelve places or so would be more than enough. A commononly used value might be 3.14159. So how does one go about calculating Pi without benefit of modern computer?. Well one can try drawing a large circle and measuring the circuference and the diameter and carrying out the division. That's not going to yield high accuracy. Way back, a couple hundred years BC, a fellow by name of Archimedes of Syracuse calculated upper and lower bounds. He calculated that the value of Pi was between 223/71 and 22/7. The average of these bounds yields Pi to an accuracy of about 0.02%. His method was sophisticated and laborious. It would have been made a lot easier if he had access to trigonometry which had not been invented yet. His method consisted of inscribing and superscribing a circle with polygons of ever increasing number of sides and calculating their perimeters. As the number of sides increases their perimeters come closer and closer to the circle's circumference. By purely geometrical means (no trig) he did the calculation for 96 sides. That's a lot of number crunching by hand. So we in our 21st century self image of how smart we are have to be humbled by the fact that most people would not be able to recreate this very old truth.
A fascinating site devoted to the history of math is found here: Overview