# Mathematics

Mathematics (also math or maths, from Greek *mathematicos*, *learned*) is the best [science](science.md) (yes, it is a formal science), which deductively deals with [numbers](number.md) and other [abstract](abstraction.md) structures with the use of pure [logic](logic.md), in as rigorous and objective way as possible. In fact it's the only true science that can actually prove things thanks to its tool of mathematical proof (other sciences may only disprove or show something to be very likely). It is immensely important in [programming](programming.md) and [computer science](compsci.md). Mathematics is possibly the intellectually most difficult field to study in depth, meant for the smartest people; the difficulty, as some mathematicians themselves say, comes especially from the extremely deep abstraction (pure mathematics often examines subjects that have no known connection to reality and only exist as a quirk of logic itself). It is said that mathematics is the only **universal [language](language.md)** in our universe -- if we ever get in contact with an intelligent alien civilization, mathematics is likely to be used for communication.

Some see math not as a science but rather a discipline that develops formal tools for "true sciences". The reasoning is usually that a science has to use [scientific method](scientific_method.md), but that's a limited view as scientific method is not the only way of obtaining reliable knowledge. Besides that math can and does use the principles of scientific method -- mathematicians first perform "experiments" with numbers and generalize into [conjectures](conjecture.md) and later "strong beliefs", however this is not considered [good enough](good_enough.md) in math as it actually has the superior tool of [proof](proof.md) that is considered the ultimate goal of math. I.e. math relies on [deductive](deduction.md) reasoning (proof) rather than less reliable [inductive](induction.md) reasoning (scientific method) -- in this sense mathematics is more than a science.

[Soydevs](soydev.md), coding monkeys (such as webdevs) and just retards in general hate math because they can't understand it. They think they can do programming without math, which is just ridiculous. This delusion stems mostly from these people being highly incompetent and without proper education -- all they've ever seen was a shallow if-then-else [python](python.md) "[coding](coding.md)" of baby programs or point-and-click "coding" in gigantic [GUI](gui.md) frameworks such as [Unity](unity.md) where everything is already preprogrammed for them. By Dunning–Kruger they can't even see how incompetent they are and what real programming is about. In reality, this is like thinking that being able to operate a calculator makes you a capable mathematician or being able to drive a car makes you a capable car engineer. Such people will be able to get jobs and do some repetitive tasks such as web development, [Unity](unity.md) game development or system administration, but they will never create anything innovative and all they will ever make will be ugly, [bloated](bloat.md) [spaghetti](spaghetti.md) solution that will likely do more harm than good.

On the other hand, one does not have to be a math [PhD](phd.md) in order to be a good programmer in most fields. Sure, knowledge and overview of advanced mathematics is needed to excel, to be able to spot and sense elegant solutions, but beyond these essentials that anyone can learn with a bit of will it's really more about just not being afraid of math, accepting and embracing the fact that it permeates what we do and studying it when the study of a new topic is needed.

**The power of math is limited.** In 1932 [Kurt Godel](godel.md) mathematically proved, with his [incompleteness theorems](incompleteness.md), that (basically) there are logical truths which math itself can never prove, and that, put in a simplified way, "math itself cannot prove its own consistency" (which killed so called Hilbert's program which sought to do exactly that). This is related to the limited power of [computers](computer.md) due to [undecidability](undecidability.md) (there are problems a computer can never decide), proven by [Alan Turing](turing.md).

## Overview

Following are some math areas and topics which a programmer should be familiar with:

- **basics** (high-school level math): arithmetic, algebra, expressions, [functions](function.md), [equations](equation.md), geometry, [trigonometry](trigonometry.md)/goniometry, systems of linear equations, quadratic equations, [complex numbers](complex_number.md), [logarithms](lofarithm.md), [analytic geometry](analytic_geometry.md) (many problems are equivalent to relationships of shapes in N dimensional spaces), [polynomials](polynomial.md) (used in many areas, e.g. error correction codes in networking), ...
- **advanced notation**: ability to understand the notation that's often used in papers etc. (the big sigma for sum, calculus notation etc.)
- **formal [logic](logic.md)**: computers are based on [Boolean](boolean.md) logic, knowing basic formulas and theorems here is crucial (e.g. the completeness of [NAND](nand.md) or [De Morgan's laws](de_morgan.md)), formal logic is also just generally used in formal texts, one should know about predicate vs propositional logic etc.
- **[proofs](proof.md)**: core of high level mathematics, one should know the basic proof techniques (direct, [contradiction](contradiction.md), [induction](induction.md), ...).
- **[linear algebra](linear_algebra.md)**: aka "vectors and matrices", essential in almost every field ([graphics](graphics.md), [machine learning](machine_learning.md), ...).
- **[calculus](calculus.md) and [differential equations](differential_equation.md)**: just essential for advanced math and many fields (graphics, machine learning, [electronics](electronics.md), physics, any optimization, ...).
- **theoretical [computer science](compsci.md)**: [computational complexity](computational_complexity.md) (very important), [computability](computability.md), [formal languages](formal_language.md), computational models ([automata](automaton.md), [Turing machines](turing_machine.md), ...), ...
- **[graph theory](graph.md)**: generally useful tools, especially important e.g. in networks or indexing structures in [databases](database.md).
- **[number](number.md) and [set](set.md) theory**: sets of numbers ([natural](natural_number.md), [rational](rational_number.md), [real](real_number.md), [complex](complex_number.md), ...), [prime numbers](prime.md) (important e.g. for [cryptography](cryptography.md), [quantum computing](quantum.md), ...), ...
- **discrete math**: basic structures such as [groups](group.md) and [fields](field.md), [abstract algebras](abstract_algebra.md) and the properties of these structures.
- **[signal processing](signal_processing.md)**: [Fourier transform](fourier_transform.md) and other integral transforms (important e.g. for [compression](compression.md) and analysis of signals), [aliasing](aliasing.md), filter theory, ...
- **[numerical methods](numerical.md)**: for simulations and [approximations](approximation.md) of solutions to problems we can't solve exactly.
- **[probability](probability.md)/statistics**: encountered practically everywhere but very important e.g. in [cryptography](cryptography.md).
- **other**: things important in specific fields and/or other weird stuff, e.g. [topology](topology.md), [quaternions](quaternion.md) (graphics, physics), [lambda calculus](lambda_calculus.md), [game theory](game_theory.md), [fractal geometry](fractal.md), ...

## See Also

- [thrembo](thrembo.md)