Mathematics

Mathematics is a language commonly used in science, but it is not in itself a science since it is not directly tested against experience. As with the sciences themselves, you are invites to explore mathematics as a fascinating human treaure.

Here is a partial list of the different areas of mathematics: algebra, analysis, geometry, topology, logic, foundations of mathematics, computation theory, number theory, probability theory.

Here is a partial list of applications: statistics, computer science, numerical analysis, accounting, science, engineering.

Mathematics deals with the general theory of forms and patterns. It is a collection of deductive systems, each with axioms, rules of inferences, and theorems derived from the axioms using the rules of inference. Algebra deals with sets of discrete objects and transformations on those sets. Analysis deals with the study of continuous curves and transformations on functions of those curves. Geometry deals will points, lines, surfaces, and shapes. Topology deals with contiguity under different transformations. Logic deals with rules of inference. Foundations of mathematics deals with the overall structure of mathematics and mathematical reasoning. Computation theory deals with the theory of algorithms and recursive functions and with whether certain problems can be solved mechanically or not. Number theory deals with different number systems. And probability theory formalizes the concepts of chance and randomness.

Mathematics provides a collection of ready made theories. If the axioms of the mathematical system fit the application area, then you have a ready made set of relationships and deductions that can be applied. This has been put to very good use in the sciences, and in fact many aspects of mathematics arose in the course of developing new areas of science and only become formalized later.

There are a few mathematical platonists who believe that mathematics refers to a real world of forms. Others believe that mathematics refers to forms in the human mind. But most people today treat mathematics as a language. It is the most definite discipline, the one place where you can prove your theorem and as long as the axioms are correct and the rules of inference are correct and you did not make a mistake in applying them, people will almost always agree that you are right. This definiteness though comes from the fact that you do not need to test mathematical statements against future experience. They are proved in a purely deductive way. Mathematical systems are judged by their coherence, usefulness, elegance, and simplicity, not by whether they allow us to predict and control experience. And since mathematical statements are not tested against experience, they are not scientific. On the other hand, since mathematics is the model of coherence, which is the other essential characteristic of science, mathematics is still sometimes given the honorific, science, and we speak of the mathematical sciences.