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@ -8,18 +8,18 @@ Some see math not as a science but rather a discipline that develops formal tool
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 completely logical truths which however math itself can never prove, and that math itself cannot prove its own consistency (which killed so called Hilbert's program which seeked 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).
**The power of math is limited.** In 1932 [Kurt Godel](godel.md) mathematically proved, with his [incompleteness theorems](incompleteness.md), that (basically) there are truths which math itself can never prove, and that 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 areas and topics which a programmer should be familiar with:
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), analythical geometry (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), ...
- **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)), formal logic is also just generally used in formal texts, one should know about predicate vs propositional logic 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**: just essential for advanced math and many fields (graphics, machine learning, electricity, physics, any optimization, ...).
- **[calculus](calculus.md) and differential equations**: 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), ...), ...