Mathematics (also math or maths) is the best science (yes, it is a formal science), which deals with numbers, abstract structures and logic 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 and computer science.
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, but that's a limited view as scientific methods 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, however this is not considered good enough in math as it actually has the superior tool of proof that is considered the ultimate goal of math. I.e. math relies on deductive reasoning (proof) rather than less reliable inductive reasoning (scientific method) -- in this sense mathematics is more than a science.
Soydevs, 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 "coding" of baby programs or point-and-click "coding" in gigantic GUI frameworks such as Unity 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 game development or system administration, but they will never create anything innovative and all they will ever make will be ugly, bloated spaghetti solution that will likely do more harm than good.
On the other hand, one does not have to be a math PhD 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 mathematically proved, with his incompleteness theorems, 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 due to undecidability (there are problems a computer can never decide), proven by Alan Turing.
Following are some math areas and topics which a programmer should be familiar with:
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