Content-Type: text/shitpost

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”:

1. 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!!

2. 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”.

3. 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.

4. 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.