31 lines
6.4 KiB
Markdown
31 lines
6.4 KiB
Markdown
# Mathematics
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Mathematics (also math or maths) is the best [science](science.md) (yes, it is a formal science), which deals with numbers, abstract structures and [logic](logic.md) in as rigorous and objective way as possible. In fact it's the only true science that can actually prove things. It is immensely important in [programming](programming.md) and [computer science](compsci.md).
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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 that's 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), 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. In this sense mathematics is more than a science.
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[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 and they think everyone just if-then-elses everything. In reality, this is like thinking that merely being able to read and write makes you someone being capable of being a great writer or that being able to drive a car makes you able to design and construct cars. Such people will be able to get jobs and do some repetitive tasks such as web 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.
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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.
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**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).
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## Overview
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Following are some areas and topics which a programmer should be familiar with:
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- **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), ...
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- **advanced notation**: ability to understand the notation that's often used in papers etc. (the big sigma for sum, calculus notation etc.)
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- **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.
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- **[proofs](proof.md)**: core of high level mathematics, one should know the basic proof techniques (direct, [contradiction](contradiction.md), [induction](induction.md), ...).
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- **[linear algebra](linear_algebra.md)**: aka "vectors and matrices", essential in almost every field ([graphics](graphics.md), [machine learning](machine_learning.md), ...).
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- **[calculus](calculus.md) and differential equations**: just essential for advanced math and many fields (graphics, machine learning, electricity, physics, any optimization, ...).
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- **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), ...), ...
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- **[graph theory](graph.md)**: generally useful tools, especially important e.g. in networks or indexing structures in [databases](database.md).
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- **[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), ...), ...
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- **discrete math**: basic structures such as [groups](group.md) and [fields](field.md), [abstract algebras](abstract_algebra.md) and the properties of these structures.
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- **[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, ...
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- **[numerical methods](numerical.md)**: for simulations and [approximations](approximation.md) of solutions to problems we can't solve exactly.
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- **[probability](probability.md)/statistics**: encountered practically everywhere but very important e.g. in [cryptography](cryptography.md).
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- **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), ...
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