# What is math? Most people in modern society have been trained in [arithmetic and algebra](math-algebra.md) from childhood, but don't often know *what* math is. ## The basis of math Math can best be described as the [science](science.md) of [structure and order](unknown.md), which represents as patterns across our [perception](image.md). Unlike other sciences, though, repetition doesn't prove a mathematical concept: the only way we can tell is with [deductive certainty](logic.md) that it can't be anything else. Structurally, math is grounded in [logic](logic.md), with zero room for [uncertainty](understanding-certainty.md). Every concept in math is a precisely parsed concept in an imaginary space. Even uncertain things are clarified on *how* uncertain they are. 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". While math itself is a type of logical science, it represents as a distinct [language](language.md) built around clearly defined [values](values.md) that represent [symbolically](symbols.md). APPLICATION: Math is a [logical](logic.md) [symbol](symbols.md), so enough inputs and outputs into a mathematical proof is guaranteed to yield itself as [offensive](morality-taboo.md), inaccurate, confusing, or inappropriate. ## The purpose and place for math Math is grouping logical values we've [interpreted](image.md), but has no [purpose](purpose.md) inherently in itself. The form of "3" may exist in some ethereal plane (like Plato had once thought), but our human structure links objects together into [patterns](symbols.md). Therefore, "3" itself isn't useful for anything, but it can be *very* useful to say there's "3 apples" or "3 cars" or "3 letters". Technically, math only exists in the mind. It's an advanced overlap of [logic](logic.md) that sits on top of what we see as [reality](reality.md) to help us [understand](understanding.md) it more clearly. APPLICATION: Since math is in our minds, it has the same fickle properties as any other [value](values.md). However, since it uses logic so intimately, we can structure it very well by comparison to any other values. Many people who [highly value](understanding-certainty.md) math have a difficult time with the mind-based location of math, simply because its reliability within [nature](reality.md) makes them believe it's [immaculate](religion.md). Within most [STEM and accounting](jobs-specialization.md), math *is* reality and not merely in our minds. Math is incredibly useful for us to find [patterns](symbols.md) in the world around us. It's the ultimate means of attaining [order](unknown.md). APPLICATION: Math is *very* useful in many parts of life, but its value is based on the accuracy of the math user's mind. Further, the signal deteriorates as it's [communicated](stories-storytellers.md), as well as whether they're [trustworthy](understanding-certainty.md) and [honest](people-lying.md), and is augmented by how they've calibrated their [feelings](mind-feelings.md). Numerical values are useful to achieve and [track outward results](results.md). We always form a *non*-numeric [purpose](purpose.md) before we start using numbers: - Someone desires to lose weight and be thin (mostly non-numeric), so they measure the 10 kg they want to lose (numeric). - A scientist wishes to [understand](understanding.md) how black holes interact with space (non-numeric), so they measure light bending from surrounding stars (numeric). - A CEO aims to know how well a company is doing (non-numeric), so they run reports that measure different parts of the company (numeric). The numbers we use are simply relational to other numbers. If someone attained 4, then later attained 6, someone would consider that an improvement, but that would change if you knew they were expected to attain 793. This is a *major* tactic for [deception](image-distortion.md), especially when we [anchor](mind-bias.md) to the first number we hear. APPLICATION: Math is useful, but it's always comparative [logic](logic.md). We must keep in mind what *any* number is comparing itself to, such as percentages or statistics. APPLICATION: To solely value money is to assign a measurement of power as [power itself](power.md). However, living [the good life](goodlife.md) requires us to *avoid* using numbers to find [purpose](purpose.md) or [meaning](meaning.md). By quantifying anything, we define our [feelings](mind-feelings.md) by an added abstraction from reality instead of [reality](reality.md) itself. In any meaningful situation, the ultimate goal of a numerical measurement will be a non-numerical purpose. Any measurement that is also a higher [purpose](purpose.md) on its own is either [performing another's non-numerical desires](groups-member.md) or veering into a form of [addiction](addiction.md). ## Using math For the sake of *working with* math, all we need is the basis that it's perfectly consistent and something we can be [certain over](understanding-certainty.md). Proving a mathematical concept is accurate 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](accounting.md), and even some [domains of art](art.md) require consistently revisiting math concepts. - All the patterns of nature, including [science](science.md) and [engineering](engineering.md), 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](education.md) often overlooks informing people on *why* math has value, and many standard college-educated math teachers will focus on [rote memorization](mind-memory.md) and formulas instead of conveying the core concepts that give mathematics any use whatsoever in the first place. This level of [ignorance](https://gainedin.site/idiot/) 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: 1. Modern Western [education](education.md), especially in the USA, has handed off most of the calculation work to computers. While some disciplines (like [accounting](money-accounting.md)) still train the old-fashioned way (which takes more work and therefore creates more [understanding](understanding.md) from the effort), [computers](computers.md) in general cheapen the required effort to build the math work we do. 2. Math is *all* cumulative, and it takes patience to work through it. Unlike other [language](language.md) (like prose) that's only *partly* cumulative, math requires [revisiting old concepts repeatedly until you understand them *entirely*](http://www.geometry.org/tex/conc/mathlearn.html). If you only [memorized the information](mind-memory.md) to repeat it back on a test, it will *not* be useful later, and you *will* be confused. 3. 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. 4. Higher-education math is composed of many math *researchers*, but many high-end university [cultures](people-culture.md) treat the professors' actual educating of students as side work in lieu of their research. Generally, unskilled [teaching](education.md) 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: 1. Use little numbers. If the problem has big, gigantic numbers, swap out that problem for little numbers and try to solve *that* one instead. 2. 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. 3. Look *beyond* the book you're reading. The textbook author may have worded it badly or is a [technical idiot](https://gainedin.site/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. 4. If you simply want the clear answer, use a calculator to find the answer, then work your way through it. 5. Use website tools as well for step-by-step walkthroughs of your exact math problem, which allows you to reproduce the process. 6. Once you *do* understand it, rework it yourself, without looking at a reference. Take your time, and *do not rush the [learning process](understanding.md)*. ## Branches/Disciplines On a highly advanced level, math has two broad [analytical](logic.md) purposes: 1. Reproducing the elements of [reality](reality.md) with numbers, which is largely the domain of most math disciplines like calculus and game theory. 2. Capturing reality itself with numbers to observe any [trends](trends.md) that may emerge, which is all derived from statistics. Math isn't really one discipline, but has technically become *hundreds* of domains. There are many classes of math, and far too many to easily specify, but they all start with the primitives of [arithmetic and algebra](math.md). The branches of math have naturally expanded proportionally to the needs of the people calculating with them. This means that math (and its [pedagogy](education.md)) branch off into many, *many* [specializations](jobs-specialization.md). 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](values.md) (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](math-geotrig.md)**, 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](math-algebra.md)**, 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](understanding-certainty.md), 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). - **Analysis**, with its base coming from **[calculus](math-calc.md)**, 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](science-physics.md). However, at the end of the 19th century, there was an issue about Euclid's fifth postulate (or parallel postulate): 1. Given a point and a line... 2. When that point is not on a line... 3. 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](trends.md) 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](data-viz.md) of [networks](https://gainedin.site/networks/). Over time, the math [pedagogical culture](education.md) has also created [their own language](glossary-math.md) 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](math-algebra.md), [geometry/trigonometry](math-geotrig.md), and sometimes [calculus](math-calc.md). - The rest are simply highly precise [thought experiments](paradoxes.md). - [Accounting](money-accounting.md) manages large piles of numbers using basic arithmetic and algebra. - Most [science](science.md) uses [statistics](math-stat.md). - Geography can often use non-euclidean math to represent planes on spherical objects (since the planet is a sphere) - Some social sciences and [investors](money-investing.md) will explore [game theory](math-gametheory.md). - [Data science](database.md) will as well as [rule](people-rules.md)-based formulas called algorithms.