Most people in modern society have been trained in arithmetic and algebra from childhood, but don’t often know *what* math is.

## The Basis of Math

Every concept in math is a precisely parsed concept in an imaginary space. While some numbers represent real things (e.g., 2, 3) and are therefore called “real numbers”, others can only exist in the philosophical realm of mental ideals (e.g., √2, π) and are called “imaginary numbers”.

The actual existence of math is subject to value theory, specifically the question over whether it exists in nature around us or in the minds that perceive nature.

However, for the sake of *working with* math, it doesn’t really matter where it exists, or even if it does. All we need is the basis that it’s perfectly consistent and something we can be certain over.

Proving math requires generating proofs, and that is *most* of the work mathematicians spend time on.

Most people find math too difficult to work with, and it is *not* a trivial discipline, but it has many uses:

- Many STEM careers, accounting, and even some domains of art require consistently revisiting math concepts.
- All the patterns of nature, including science and engineering, have mathematical patterns behind them.
- Our capacity to observe and reproduce math patterns have allowed us to make
*very*efficient factories, get to the moon, have well-designed traffic circles, and build better cities.

Math education often overlooks informing people on *why* math has value, and many standard college-educated math teachers will focus on rote memorization and formulas instead of conveying the core concepts that give mathematics any use whatsoever in the first place. This level of ignorance has magnified math anxiety more than it should be.

## Math Anxiety

Math anxiety is a legitimately real problem, and most people who don’t specialize in math-based occupations simply imagine it’s too difficult for them to understand.

However, this is a self-reinforced falsehood driven by several converging factors:

- Modern Western education, especially in the USA, has handed off most of the calculation work to computers. While some disciplines (like accounting) still train the old-fashioned way (which takes more work and therefore creates more understanding from the effort), computers in general cheapen the required effort to build the math work we do.
- Math is
*all*cumulative, and it takes patience to work through it. Unlike other language (like prose) that’s only*partly*cumulative, math requires revisiting old concepts repeatedly until you understand them*entirely*. If you only memorized the information to repeat it back on a test, it will*not*be useful later, and you*will*be confused. - Some higher-level math concepts are filled with dense and sometimes confounding jargon. Integers and integrals, for example, have nothing to do with one another. The ideas are often simpler than the jargon makes them sound.
- Higher-education math is composed of many math
*researchers*, but many high-end university cultures treat the professors’ actual educating of students as side work in lieu of their research. Generally, unskilled teaching therefore contributes to students’ stereotypes that math is an impenetrable subject.

Math is very thorough, so it’s never really “easy”, and it can often be tedious, but there *are* several techniques to make it easier:

- Use little numbers. If the problem has big, gigantic numbers, swap out that problem for little numbers and try to solve
*that*one instead. - Separate out the concepts. If you see a long formula, break apart the pieces and solve those individual pieces as entirely separate concepts. If there are big numbers, apply #1 to get a firmer grasp of it.
- Look
*beyond*the book you’re reading. The textbook author may have worded it badly or is a technical idiot, and it might be too difficult for*anyone*to understand. Any legitimately useful math will have*many*videos, books, and tutorials on the subject. - If you simply want the clear answer, use a calculator to find the answer, then work your way through it.
- Use website tools as well for step-by-step walkthroughs of your exact math problem, which allows you to reproduce the process.
- Once you
*do*understand it, rework it yourself, without looking at a reference. Take your time, and*do not rush the learning process*.

## Branches/Disciplines

Math isn’t really one discipline, but has technically become *hundreds* of domains.

As late as the Renaissance, there were simply 2 branches of mathematics:

- Basic
**Arithmetic**, which is manipulating numbers, is exceptionally useful for many purposes, and is most of the math that the average person ever needs:- Addition — putting things together according to a shared value (i.e., counting) or differing values (i.e., categories).
- Subtraction — removing some things from other things in a group.
- Multiplication — addition, but compounding a fixed number of times.
- Division — evenly separating things into a desired number of portions of the original value.

**Geometry and trigonometry**, which is the study of shapes and their relationships- Euclidean geometry uses Euclidean planes (i.e., plane geometry) and the three-dimensional Euclidean space we all inhabit.

During the Renaissance, two more areas appeared:

**Algebra**, which started as mathematical notation, works with precise symbolic descriptions of mathematical ideas, and is still very useful for most people at least once a week or so.- Algebra adds a few extra elements into the calculations:
- Variables are numbers that haven’t been clearly specified, meaning the outcome of a calculation will be a second variable relative to the other variables (e.g., X+5=Y).
- Exponents essentially compound multiplication and division, similarly to how multiplication compounds addition.

- The purpose of arithmetic is relatively clear (what do these things equal?), but algebra usually needs more specific context (e.g., solve for X).

- Algebra adds a few extra elements into the calculations:
**Analysis**, with its base coming from**calculus**, which studies nonlinear relationships between different quantities (i.e., patterns for how things are related).- Differential calculus studies the rates that quantities change (e.g., the average wavelength of light).
- Integral calculus, or simply integrals, studies the accumulations or volumes of those relationships (e.g., the total sum of all light rays hitting a moving object.

For a while, the 4 domains existed independently, and most of its framework was grounded in ancient texts (e.g., Euclid’s book titled “Elements”).

- Some domains like celestial mechanics and solid mechanics arose, but they were effectively subdomains of physics.

However, at the end of the 19th century, there was an issue about Euclid’s fifth postulate (or parallel postulate):

- Given a point and a line…
- When that point is not on a line…
- Only 1 other line can pass through that point and be parallel with the existing line.

The problem was that nobody could prove the fifth postulate, and that unleashed a post-modern deconstruction movement called the foundational crisis.

After the foundational crisis, there were *many* new branches of mathematics, and most of the previous math disciplines became labeled as the “classical” discipline. As of 2020, there were 63 primary domains of mathematics.

- Many of the domains were trying to prove mathematical concepts that hadn’t been proven, meaning more explorations into stranger domains.

Most of them aren’t worth dissecting much, but some of them have had limited usefulness beyond math itself:

- Non-euclidean geometry measures things that do
*not*follow the parallel postulate (and also don’t exist in reality). - Number theory spun off from arithmetic to clarify and qualify precise definitions of what numbers even
*are*. - Graph theory intimately studies graphs, which are effectively visualizations of networks.

Over time, the math pedagogical culture has also created their own language among each other to accommodate this additional depth.

- There are
*many*adjectives that add to existing domains, and these domains become absurdly long (e.g., derived non-commutative algebraic geometry).

For all intents and purposes, 99.99% of humanity only needs the four domains of arithmetic with algebra, geometry/trigonometry, and sometimes calculus.

- The rest are simply highly precise thought experiments.
- Accounting manages large piles of numbers using basic arithmetic and algebra.
- Most science uses statistics.
- Some social sciences and investors will explore game theory.
- Data science will as well as rule-based formulas called algorithms.