What is considered trivial in mathematics? What makes a statement trivial?

Question

I'm a graduate student in a sort of Computer Science major. I'm not from US so is kind difficult to explain, but my course is more focused in the practical area of CS, is a course devoted to put professionals in the industry rather than the academical area. But we do have some theory and mathematical disciplines, like calculus, linear algebra, discrete mathematics.

Sometimes my mathematics professor says: "we have this and that, but the former is trivial". But I don't find it trivial at all. My friends also don't. I need that "trivial" statement to be either explained to me with detail or I've to read and think for sometime to understand that "obvious" statement. If it's trivial I think I should be able to understand right away. Maybe because I'm not a mathematics student or I lack talent with math I'm not able to consider some statements trivial like my professor do?

I also find a lot of trivialities in mathematics books that I need to read several times to understand.

What is considered trivial in mathematics? What make a statement trivial? Are some statements more trivial than others? Am I just not smart enough or untalented?

Answer 1

I can only really speak for myself (I'm a professor of mathematics and regularly publish research articles), of course, but when I refer to statements as being trivial (or obvious, or clear) what I really mean is that they can be proven using ideas and arguments with which I assume my audience is well acquainted. So it is a very relative notion. For instance, in an undergraduate group theory class I would, in theory, feel comfortable saying something like "It is trivial to see that there are infinitely many groups whose order is a prime number" because I would assume that all of my students would know, off the top of their heads, that

1. For every prime number $p$ there is a group of order $p$ (namely the cyclic group of order $p$), and
2. There are infinitely many prime numbers.

When I am writing a research article, if I want to make use of a fact that I am absolutely certain that my audience will be able to prove because its proof makes use of only standard arguments assembled in the obvious manner, I might refer to such-and-such as being trivial. (Example: I'm writing a paper for an audience of algebraic number theorists and want to refer to the fact that infinitely many primes split in a quadratic field.) Note that this definitely does not mean that every professional mathematician, regardless of the field they work in, would be able to prove the assertion in question. In a similar spirit, if an author wants to give their audience a tad more information they might say something like "Statement A is a trivial consequence of Theorem X."

Having said all of this, I should add that I try very hard not to refer to things as being trivial or obvious or clear when I write papers, and certainly would never do so out loud in front of students in a class. My personal opinion has always been that if something is truly trivial then it should be very easy to prove and you might as well just give the proof. In fact, I suspect that when a paper is found to contain an error it is often the case that some "trivial" statement turned out to be incorrect.

Maybe I'll end with an amusing anecdote from Boas' Lion Hunting and Other Mathematical Pursuits:

"The story is told of G. H. Hardy (and of other people) that during a lecture he said 'It is obvious...Is it obvious?' left the room, and returned fifteen minutes later, saying 'Yes, it's obvious.' I was present once when Rogosinski asked Hardy whether the story were true. Hardy would admit only that he might have said 'It is obvious...Is it obvious?' (brief pause) 'Yes, it's obvious.'"

Answer 2

In a formal sense of the word, trivial can mean two things: a possible answer that will always be the case regardless of other variables. This does not mean that the trivial answer will be an easy or simple answer. Or a valid answer of little practical use. Examples:

X = X is a trivial answer [more in the first part of my definition above] In applied mathematics using operations research [0,0,0] will often be a trivial answer because we would never want an instance of nothing happening.Yet mathematically [0,0,0] is possible. Getting an answer with a negative physical amount is often trivial. Such as a negative area. Even though it's mathematically possible.

This might not be the best explanation or examples. But it's a start.