On triviality

As a math student, one of my favorite mathematical jargon is trivial. In mathematics, a given problem or structure is trivial if it is relatively simple or well-understood.

Some trivial structures in mathematics are semi-formally defined. Empty set, singleton group, and singleton ring are considered trivial, and most mathematicians refer to them by trivial set, trivial group, and trivial ring respectively.

Informally, mathematicians say that a problem is trivial if it is well understood. This is very subjective, because it really depends on one's mathematical knowledge. "Union of two finite sets is finite" may be trivial to almost everyone. "Differentiable functions are continuous" is a trivial fact for senior math students, but it's a totally reasonable question to appear in a first year calculus exam. Algebraists say that finite abelian groups are trivial. They're all isomorphic to direct products of cyclic groups, which are relatively well understood structures.

I started to use the term trivial in my every day conversation. When someone told me that they did well on their exam, I replied "Well that's trivial, you normally get over 80 in your tests anyway". If someone complained about problem they're having with one's girlfriend, I would've replied "You know what, I think your problem is trivial. There are so many people who already went through this problem".

Then I came back to reality, and realized that the word trivial has more negative, non-mathematical meaning. Oxford English Dictionary says:

Of small account, little esteemed, paltry, poor; trifling, inconsiderable, unimportant, slight.

 

All the problems that my friends are having - they were not inconsiderable and unimportant. They were all very grave matter to me. But for me, I understood their graveness, but the problems themselves were within the predictable boundary of my expectations, thus trivial.

In some sense, mathematicians are always addicted to new unsolved problems. Their job is to turn mysteries into trivialities. I personally think that's the biggest philosophical difference between mathematicians and engineers. Whenever mathematicians discover a pattern in structure, they are awed by its beauty, and move onto the next problem, bored by its triviality. Engineers, however, when they discover a pattern in nature, they want to understand it, exploit its predictability, and integrate it to their work. Final example of this is regular expressions. Regular languages are well understood within the hierarchy of formal languages. Regular languages themselves are not very interesting in theoretical computer scientist's perspective, but regular expressions are one of the most useful things in computer programming and software development. They're simple to understand, and they're very powerful tools. Engineers love them.

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