I have been thinking for a long time about the way mathematicians use
terms like “obvious”, “straightforward”, “trivial”, and so forth, and
the different shades of meaning these communicate. Someday I will
publish a longer and more complete discussion.
Meantime, here's a thought. Discussing the Petersen graph
recently, I said:
The standard presentation, above, demonstrates that the Petersen
graph is nonplanar, since it obviously contracts to !!K_5!!.
To someone not versed in graph theory, this not only isn't obvious,
it's unintelligible. In fact, it's indistinguishable from a
meaningless parody:
The Cosell configuration, shown above, is semispatulated, since
it obviously extends to a !!\zeta!!complete net.
But I also think this is an exactly correct use of “obvious”:
I said it obviously contracts to !!K_5!!. If you know what a
contraction is, and what !!K_5!! is, this is obvious. In fact the
first thing you might notice, if you were seeing the Petersen graph
for the first time, is how much it resembles !!K_5!!:


 Petersen   !!K_5!!

But it isn't meant to suggest that the meanings of “contract” or
“!!K_5!!” are themselves obvious. Compare:
Obviously, a fly ball that leaves the field ouside of fair
territory is never a home run.
If you don't know at least the approximate definitions of the
technical terms there, you will be in the dark. But that doesn't
make this an inappropriate application of the term “obvious”.
The Petersen graph also contracts to !!K_{3,3}!!, but I doubt
anyone would say that it was obvious, at least not from seeing this
presentation.
I didn't say that the graph was obviously nonplanar. The
contractibility is obvious, but the nonplanarity follows from that
by Kuratowski's
theorem, which
nobody claims is obvious. (Quite the opposite!)
Contrast this with:
The Petersen graph is nonplanar, since it *trivially contracts to !!K_5!!.
I think “trivially” here is wrong, and people might object. That
would suggest that no actual contractions need to occur. !!K_5!!
trivially contracts to !!K_5!!, but the Petersen graph does not.
