I came across the notion of duality recently to explain a physical concept.
What is duality?
Why does it occur?
How do I know if two things are dual?
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Duality between two different theories means that these two theories when applied to a problem yield the same answers. The dualities can be approximate or exact (ex: the gauge/gravity duality, string dualities etc.) depending on whether the calculations performed using the two different theories agree exactly or only up to a certain limit.
Dualities are usually a characteristic feature of the underlying mathematical logic and structure. As a vague example, just because you can explain effectively say the structure of stars using hydrodynamics and quantum mechanics separately, does not mean that hydrodynamics is dual to quantum mechanics. The duality occurs because the underlying mathematical structure between the two objects end up calculating the exact same thing or reduce to the same quantity in certain limits. Of course, there are some dualities which are well understood mathematically like mirror symmetry while some others which are still well accepted conjectures (from a mathematical framepoint) like the holographic duality. But there are dualities in physics and given their mathematical origin, they must also exist in mathematics, and they do.
Knowing two things are dual requires you to understand how to construct and calculate quantities precisely using the theories between which you want to describe a duality. However, proving two things are dual is much more mathematical.
Duality arises in physics when two different aspects of nature are apparently related to each other by a common underlying theory. There are various examples of dualities in physics. eg Wave particle duality, electric field/ magnetic field duality, gauge/ gravity duality. Usually it appears when the underlying theory is viewed from two different limits.
In order to verify if the two or more aspects of nature are dual or not, you need to know about the basic underlying theory, and see whether the aspects manifest themselves as the limits of the theory / or are sub-parts of the full theory.