Symbolic mathematical computing systems have served as a canary in the coal mine of software systems for more than sixty years. They have introduced or have been early adopters of programming language ideas such as dynamic memory management, arbitrary precision arithmetic and dependent types. These systems have the property of being highly complex while at the same time operating in a domain where results are well-defined and clearly verifiable. These software systems span multiple layers of abstraction with concerns ranging from instruction scheduling and cache pressure up to algorithmic complexity of constructions in algebraic geometry. All of the major symbolic mathematical computing systems include low-level code for arbitrary precision arithmetic, memory management and other primitives, a compiler or interpreter for a bespoke programming language, a library of high-level mathematical algorithms, and some form of user interface. Each of these parts invokes multiple deep issues. We present some lessons learned from this environment and free flowing opinions on topics including: The key message is that the real world is often messier than presentation in papers and we need to be able to cross between very low and very high levels of abstraction to deal with this, as Alan Mycroft has done in his work.