I haven't read them myself yet, but I've heard good things about Structure and Interpretation of Classical Mechanics and Functional Differential Geometry. I he

I haven't read them myself yet, but I've heard good things about Structure and Interpretation of Classical Mechanics and Functional Differential Geometry. I hear there is a reasonable alternative to classical notation in these books, and with Prof. Sussman's insistence on good names for things, I'm sure single-letter variables do not happen that often. About partial derivatives being rewritten as functions: it's common for ∂/∂xᵢ to be thought of as a linear operator from the vector space of functions of some number of variables to itself (let's just assume the functions are smooth). The derivative of a function between smooth manifolds is a function (linear transformation), and it contains the idea of all the partial derivatives of a function, with the benefit of being able to reason without using explicit coordinates: bases are arbitrary, after all. > is there a way to “break up” long functions into shorter expressions (like sub-functions) without losing their relationships? I think that's already somewhat happening by using things like divergence or the Laplacian. But mathematicians could do a lot better with naming subexpressions (or giving better structure to proofs...). Though, thinking in public, perhaps the equation itself isn't actually central, and instead the equation is just a mnemonic for all of the explanation of what the equation represents, and so the subexpressions are named by their explanation. About abstractions: The thing about math is that the material of the subject is new ways of understanding phenomena (physical or nonphysical). A big difficulty with the subject is that you have to somehow learn completely new metaphiers (metaphor illuminates a metaphrand with a metaphier) -- for instance linear algebra has linear combination, linear independence, vector spaces, linear transformations, bases, inner products, and so on, and while their are many examples of these concepts, it takes significant effort to get to the point where the concepts themselves become primary. Teaching the subject, it feels like the most I can do is point out the moon in as many ways as I can, so to speak, and hope the student can follow my finger and recognize the moon from my descriptions -- I can't look at the moon for them. From the assumption that complexity is proportional to the number of new introduced metaphors, SQL and OpenGL are not nearly as complicated. Effective SQL needs some understanding of a relation and how to use the relation metaphor to model data, and OpenGL sure has a lot going on, but it seems to all be standard systems-type programming (but with too much global state, in my opinion). OpenGL gets a lot more difficult when you then have to bring in the linear algebra to actually make some shaders, but that's not intrinsic to OpenGL itself, I don't think. It could be that there is some minimal set of metaphors, or even a new easier-to-internalize metaphor, which would make partial differential equations transparent to anybody, or at least make a single partial differential equation transparent. But this hasn't happened yet, and people have been thinking about them for about a couple hundred years, much longer than either SQL or OpenGL.