[pgustafs]

There are three broad subareas of mathematics: geometry, algebra, and analysis. Geometry studies space, algebra studies time, and analysis studies infinity. They are not independent -- most professional mathematicians use some mixture of the three, and virtually every mathematician understands the basics of all three.

The most important object in modern geometry is the manifold. This is a space that looks locally like n-dimensional Euclidean space -- 1-dimensional manifolds are curves, 2-dimensional manifolds are surfaces, and higher dimensional manifolds are simply called n-manifolds. All of physics takes place on manifolds. Differential equations correspond to vector fields on manifolds. The manifold hypothesis says that much of the high-dimensional data we see actually lives on much lower-dimensional manifolds (partially explaining the unreasonable effectiveness of deep learning on very high-dimensional datasets).

The most important object in algebra is the group. The collection of symmetries of any object (e.g. a Rubick's cube, a piece of paper, or three-dimensional space) forms a group under composition. A group that is also a manifold is called a Lie group. These are everywhere -- n-dimensional rotations form groups, fundamental particles correspond to representations of Lie groups, invertible matrices form a group. Spherical harmonics and Fourier series are both naturally viewed in terms of representations of Lie groups.

The most important object in analysis is the limit. Limits first appear in the construction of the real line by adjoining limits of Cauchy sequences to the rational numbers. Using the real line, one can measure volumes, probabilities, and distances in geometric spaces such as manifolds, but also in spaces of functions, sequences, and more abstract objects. The proof of the fundamental theorem of calculus (that derivatives and integrals are roughly inverse operations) requires rigorous analysis of the definitions of derivative and integral as limits.

To learn math, you should begin by understanding what a proof is. All of mathematics is based on proving theorems. A mathematical proof is a sequence of statements that explains the logical steps required to use the assumptions of the theorem to verify the result. Just as a computer program cannot "almost output" the correct answer, there is no such thing as an "almost correct" proof. A proof either describes a correct chain of logic to reach the conclusion, or it does not. The reason math is based on proofs is because more advanced math and science builds upon more basic math. An error in a mathematical theorem or an imprecise definition will lead to bigger problems down the line, so every step must be carefully validated. For an individual student as well, only through proving theorems can one deeply understand a mathematical subject, and a solid understanding of basic subjects is required to understand more advanced topics.

Fortunately, you can learn to prove theorems at the same time as learning the foundations of math. The first books you should work through are "Principles of Mathematical Analysis" by Walter Rudin, and "Linear Algebra" by Georgi Shilov. This will be hard, not for an arbitrary reason, but because assimilating new math into your brain is intrinsically difficult, especially at the beginning. If possible, try to find a teacher.