When choosing a programming language to focus on for scientific computing or mathematical finance, it's recommended to learn three types: a compiled language, a fast interpreted language, and a full-featured interpreted language. C++ is suggested as the primary compiled language due to its stability, speed, and portability, making it suitable for low-level programming. Python is highlighted as an excellent fast interpreted language, valued for its versatility and ease of use. Matlab is noted as the industry standard for mathematical finance, despite its cost and proprietary nature, due to its powerful graphics and toolboxes. Additionally, Ruby is mentioned as a promising high-level language that reinforces object-oriented programming principles, making it a valuable complement to C++.