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What is the difference between trivially and obviously in a sentence like "This condition obviously/trivially holds"?

What about using them as adjectives? For example, "It is obvious/trivial that this condition holds"?

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  • Do you actually see people use "This condition is obviously/trivially held"? Should it not be "This condition obviously/trivially holds"?
    – Bence Mélykúti
    Commented May 6, 2018 at 11:40
  • I changed it. But not sure if the previous one was correct or not.
    – Shayan
    Commented May 6, 2018 at 13:52

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Consider first the adjectives trivial and obvious, while they have related meanings they are not synonyms. Informally people do on occasions use the words interchangeably but they do have distinct meanings.

We can speak of trivial problems. Such problems have obvious solutions. Obvious relates to a thing we find, the etymology is of something that is "in the way", something we stumble over or just find in our path.

How shall we get to the zoo?

It's obvious, get on this bus that is labelled "London Zoo"!

In contrast, when we say

We need to return the gorilla to the zoo. The obvious problem is: how do we catch him?

We are not saying that the problem is trivial (easy), we are saying that the problem is clear to everybody, and is probably quite difficult.

We can speak of trivial solutions and obvious solutions. The equation

Fermat's last theorem

is true for any value of n if a, b and c are all zero; this is a trivial solution, a solution of no interest. The interesting problem concerns the case when a, b and c are not zero.

This leads us to

The condition is trivially held.

covers cases such as "a, b, c are all zero" above; easy to see it's correct, but of no great interest.

The condition is obviously held

here there is some reason why the condition can be seen to be true. Sometimes a supposedly obvious solution, is only obvious once a particular insight is achieved.

Is it obvious?

Yes look ...

Oh yes! Completely obvious

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  • These are both good points. What I would highlight from the usage in mathematics is that obvious and trivial are informally used interchangeably/as synonyms meaning easy, but trivial has the extra meaning for cases when e.g. everything is zero and something is true but it's an uninteresting border case.
    – Bence Mélykúti
    Commented May 2, 2018 at 10:20
  • I do think that informally, away from maths, trivial tends to be used to describe problems whereas obvious more usually is applied to the solution.
    – djna
    Commented May 2, 2018 at 12:01
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I don't see any difference between the meaning of trivially and obviously as used in your first question, but trivial can also mean unimportant. You can use both as adjectives just how you suggest. Obvious is heard frequently in everyday speech, while trivially is more like scientific jargon. The Cambridge English Dictionary classifies obvious at the B1 level, trivial at the B2 level.

obvious
adjective

[B1] easy to see, recognize, or understand
Cambridge English Dictionary

obviously
adverb

[B1] in a way that is easy to understand or see
Cambridge English Dictionary

trivial
adjective

[B2] having little value or importance
A trivial problem is easy to solve
Cambridge English Dictionary
Examples for trivially

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