While imagining how awesome of a maths and philosophy teacher I am going to be for my daughter, whose patience occasionally expires in expressions such as "dad, you don't have to teach me all the time", I recalled my own experience with elementary maths and how wastefully and long I fought against both the formalism and, more significantly, the threat and charge of inadequacy for miscomprehension or substandard performance--so much for meritocracy.
The first real arrival a student experiences in the primary education of mathematics in the US is elementary algebra, which is something of a sterilized and frustrating scope from the wonderfully weird and bizarre natural scape of algebra. The subject of elementary algebra is necessarily constructed from two theories: the theory of equations and the theory of functions, and typically features a myopic fixation on polynomial functions. Neither command of nor felicity in the topic can be attained without a fluent proficiency in both theories. For me, it was not until I took foundations of analysis, well into a maths BS, taught by a practicing mathematician at a research university that my understanding of these two theories was finally made clear. How much more advantage that my daughter would have then I, I thought, to learn from someone who knows the rigor and loves the subject.
There are other mathematical lexicons so fundamental to their domain, and all those that derive therefrom, similar to the theories of equations and functions, that I wish I either had had the intelligence to recognize or the fortune in a teacher wise enough to make it clear when I formally began learning them. The first comes from the theory of differential equations and the second from the theory of probability. Fluency in elementary algebra and calculus is not sufficient to be successful in constructing and applying differential equations and their solutions or appropriating their models. The same can be said about elementary algebra, some of whose ancient practitioners conjured wild scenarios to solve as if by magic. But since we are to learn that elementary algebra only has merit in the service of other topics, specifically those that form the mathematical prerequisites for a proper engineering education, no such interest or wonder was ever afforded by me to the magic of long form word problems in elementary algebra and how the application thereof is artful or captivating. A proper engineer's pursuit, however is the creative and discriminating application of differential equations, both ordinary and partial, and their solutions. Again, it was not until my second semester of graduate quantum mechanics that I think I really started to see such maths in nature. Once you learn the ways of differential equations, you never look at things in the same way again. You see them as boundary conditions, resonances, and decays.
The last item comes from the theory of probability, is more of a concept than a dialect, and is called a random variable. Like differential equations, which can be seen as a generalization of the theory of equations to which the solutions are functions rather than variables (sets), random variables generalize the meaning of a variable in ways that are profoundly insightful to any field that applies maths to reality, but a full comprehension for me only came together near the end of my first semester in undergraduate probability.
My relative travails in each topic I think demonstrates my own daft beginnings more than anything else. We know so little about ourselves in the beginning and how we learn. Mathematics has never been easy for me and despite the unrelenting contrary accusations. It is a fulfilling competence that was refined only through years of intellectual agony and determination. I think I was improbably lucky to have obtained and retained a handle on its innate, sublime beauty from when I was very young.
Maths is a language but they are also a curious collection of disparate locales with translations and orthonormal purposes as needed. Seductive formulas such as the complex exponential and elliptic curves belie an awesome variety and utility underneath their deceptively simple forms and we conjecture about yet more ambitious, unified truth from simple forms. I always return to Hilbert's defiant declaration:
and wonder what weird meaning is wrought from raw logic in the human mind against its eclectic Condition and whether that construct really does sing the music of the universe, or the mad phantasms and sprites that spring into consciousness or both.