## Sunday, September 07, 2008

### Trivia (no pun intended)

Writing about rhetoric the other day reminded me of a recent exchange I had with a Melbourne philosopher and historian of science. Other logicians and myself were bemoaning the fact there is an inflated use of the word 'trivial' in logical and mathematical writing. These days, just about any proof can be called trivial; when unpacked (however, routinely), they might take pages!

We were then gently reminded that 'trivial' in fact derives from the medieval Latin 'trivium', a scholastic term for the three disciplines logic, rhetoric, and grammar. So, by a cheap argument-from-etymology, we reach the conclusion that indeed any truth in the area of logic, rhetoric, and grammar is trivial. Now, that's inflation.

Andreas said...

There is of course an interesting question here about the relation between how people use 'trivial' and another notion, namely uninformativeness. Suppose we agree that statements like 'John is John' or 'If Mars is a planet, then it's a planet' are uninformative. We might then think that all analytic truths (if we may so call them) are uninformative. However, obviously, some analytic truths seem to be highly informative. So why are these statements uninformative?

I think what usually goes on is that people use 'trivial' to mean something like 'easy to see is true'. More precisely, we could say that something is trivial if it requires no (or only a little) cognitive work in order to achieve knowledge of its truth. This of course makes triviality a relative notion, but I suspect that this is really how people use it.

Of course, whether it requires none or a lot of work to see that some claim is true has no bearing on whether or not that claim is informative or not. Although in a derived sense it migh in that when something requires a lot of work, we very often learn a substantial amount while attempting to carry out that work. Think of how much mathematicians over the centuries learned from trying to prove Fermat's Last Theorem, for instance.

Paal said...

You remember Walicki's terminology? It's also called "the Polish taxonomy". All proofs in logic (and mathematics) are either:

1) trivial,
2) not so trivial
3) obvious
4) not so ovious

Where the level of complexity goes from 1) to 4). I still use it, and 3) means that I can't do it, and 4) means that I will never be able to do it.

Catarina said...

'Trivial', 'surprising', 'paradoxical' all seem to be relative, epistemological notions anyway, not directly related to objective factors such as e.g. the length of a proof. And yes, it looks like we simply can't do without these 'subjective' notions when doing philosophy of logic and mathematics... Personally, I'm particularly interested in the notion of 'surprises' in logic.
A small historical addendum (which I'm sure you are aware of): 'trivial' came to mean what it does because the disciplines composing the 'trivium' were the first ones taught according to the medieval curriculum.