Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.
A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra.
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics. Of concern to this discussion are several topics: the relationship of empiricism (see Penelope Maddy) with mathematics, issues related to realism, the importance of culture, necessity of application, etc.
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science, and more specifically the computer sciences, which uses advanced computing capabilities to understand and solve complex physical problems in science. While this typically extends into computational specializations, this field of study includes:
In practical use, it is typically the application of computer simulation and other forms of computation from numerical analysis and theoretical computer science to solve problems in various scientific disciplines. The field is different from theory and laboratory experiments, which are the traditional forms of science and engineering. The scientific computing approach is to gain understanding through the analysis of mathematical models implemented on computers. Scientists and engineers develop computer programs and application software that model systems being studied and run these programs with various sets of input parameters. The essence of computational science is the application of numerical algorithms and computational mathematics. In some cases, these models require massive amounts of calculations (usually floating-point) and are often executed on supercomputers or distributed computing platforms.