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Miloslav Ciz 2024-03-18 14:04:06 +01:00
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@ -10,7 +10,9 @@ Mathematics as a whole is constructed with [logic](logic.md) from some basic sys
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 and to innovate in big ways, 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** because the power of [logic](logic.md) itself 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).
**The power of math is limited** because the power of [logic](logic.md) itself is limited. In 1930s this actually caused a big crisis in mathematics, connected to so called Hilbert's program which aimed to establish a completely "bulletproof" system to be the foundation of mathematics, however 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 of course killed Hilbert's program; since then we simply know we will never have a logically perfect system. 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).
**What is mathematics really about?** Elementary school dropouts think math is about calculations and [numbers](number.md) -- sure, these are a big part of it but mathematicians mostly give a different answer. The core and [art](art.md) of high mathematics is constructing **[proofs](proof.md)**, but it also involves exploration, a common theme is e.g. **[generalization](generalization.md)**: mathematicians love to take already existing knowledge and patterns and extend them into other domains, find more general rules of which currently known rules are only a [special case](special_case.md). By this they are discovering universal laws and find that even seemingly unrelated concepts may have a lot in common.
## Overview