Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research. -- Aristotle
Mathematics is a language invented by humans to describe certain objects and processes. Evolving from counting, simple arithmetic, and geometry, it has blossomed most wonderfully into set theory, boolean logic, calculus, and many odder disciplines. From the beginning, math has been fueled by smart, obsessive, and bored people who had enough leisure time to discover that universe was filled with odd regularities.
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. -- Aristotle
Math has the distinction of being the only a priori science (unless you consider philosophy to be a science). Visual aids and experimentation can certainly help, but most math is done on paper. None the less, it has great worldly benefit, from architecture to zoology. It's fun to play with -- magic squares, sudoku, the Prisoner's Dilemma, the Monty Hall Problem. It's important to our lives -- it programs our computers, launches our spaceships, and encrpts our emails.
The whole is more than the sum of its parts. -- Aristotle
In theory, math can be applied to describe any observable phenomenon that exists, and many that don't. But Godel's theorem shows that it can't actually describe every observable phenomenon, because any given system can't describe itself. This has not yet caused math to fall out of fashion.
To Thales the primary question was not what do we know, but how do we know it. -- Aristotle
There is, to my knowledge, no formal categorization of the subfields of mathematics, and no clear hierarchical structure between the 'main fields' of mathematical study and the 'sub-fields'. But here is a list of some of the main branches of mathematics.
Needless to say, you should learn more math. You should node more math. Unfortunately, while E2 is an excellent place to node math, it doesn't actually do a very good job of teaching it, unless you've gotten a good foundation elsewhere. Fortunately, your local library will have the primers to get you started.
Kurt Godel's Incompleteness Theorems (published 1931) correctly identify that any formal system (i.e. Mathematics) is inherently flawed, specifically "in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system." To restate, Mathematics is effectively a study of the consequences of the particular compromises inherent in human cognition. Caveat emptor.
Mathematics is mathematics.
Please /msg eien_meru if you have any additions, broken links, or comments. This list will probably always be incomplete, despite all efforts to the contrary.
00 - General Mathematics 01 - Biography 02 - History 03 - Books, Papers, and Essays 04 - Mathematical Logic and Foundations 05 - Combinatorics 06 - Ordered Algebraic Structures 07 - Rings and Algebras 08 - Number Theory 09 - Field Theory and Polynomials 10 - Algebraic Geometry 11 - Linear Algebra and Matrix Theory 12 - Category Theory 13 - Group Theory 14 - Real Analysis 15 - Complex Analysis 16 - Measure and Integration 17 - Ordinary Differential Equations 18 - Partial Differential Equations 19 - Dynamical Systems 20 - Sequences and Series 21 - Approximations and Expansions 22 - Fourier Analysis 23 - Harmonic Analysis 24 - Functional Analysis 25 - Calculus of Variations 26 - Geometry 27 - Differential Geometry 28 - Topology 29 - Manifolds 30 - Probability Theory and Statistics 31 - Numerical Analysis 32 - Physics, Biology, and Natural Science 33 - Computer Science 34 - Game Theory, Economics, and Social Science 99 - Other
