I'd personally say the symbols and their manipulation are a good representation of what's going on in math, but math is overall more general. (By the way, arith

I'd personally say the symbols and their manipulation are a good representation of what's going on in math, but math is overall more general. (By the way, arithmetic is about numbers, maybe you meant algebraic.) Sometimes manipulating symbols towards an answer is great, but sometimes taking a step back, and looking at a problem through a different lens (e.g. at what you are trying to do intuitively) is vastly superior, and the symbol-manipulating, rigorous formalization (and verification) part comes afterwards. A few examples (out of very many): * In signal processing (both digital and analog) it can often be much more insightful to play with visualizations of time domain, spectra, and convolution and multiplication thereof. * Related but more general: Thinking about the complex exponential as spinning in a circle, or tracing out a corkscrew in 3 dimensions, is a way easier method to grasp it than to look at the equations, which for someone getting into it will look like abstract nonsense[1]. * Topology is about "shape" and "deformation" of objects. * Discrete Structures is about trees, graphs, and so on. In all of those, you can hit paths where an intuitive understanding may stay out of reach, and symbolic manipulation through e.g. algebra might remain the only way to work with it, but that is often not generally true for the whole field. [1] Funnily, that's an actual term used by mathematicians, but usually in another field: https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense