TITLE: What does the adjective "natural" actually mean?
QUESTION [35 upvotes]: Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The word "natural" seems to be one of the most ambiguous terms used in formal mathematics. I have never seen anyone actually define it. People just use it and expect others to understand it.
What exactly is meant by "natural" in mathematical writing?

REPLY [5 votes]: One issue I have with all of the answers is that mathematical writing is writing by humans. Much of the language in a mathematical paper written by a human is plain language, intended to impart understanding from one human to another human, rather than formal language intended to codify a proof. In proving many existence theorems, one has to make a rather wild guess at the definition of the object whose existence is asserted, long before one is able to carry out the actual proof that the object satisfies all of the required properties of the theorem. Such wild guesses have many different motivations in humans.  I often use the word "natural" to let my reader know what motivated me to guess at the right object. Sometimes the motivation was background knowledge or experience, sometimes it was just blind intuition which I cannot account for, and sometimes it is motivated by what felt "natural". I would not assert that there is any formal mathematical definition for this use of "natural". I suppose we'll have to wait to see what happens with this term once mathematical papers are written by computers to impart understanding from one computer to another.