Why there is "no such thing as a fish"

YouTube link.

I have been listening to podcasts of QI's "No Such Thing as a Fish," each of which is introduced with the statements "No, seriously.  It's in The Oxford Dictionary* of Underwater Life.  It says it right there in the first paragraph: there's no such thing as a fish."

I wondered about the logic of the statement, which maddeningly was never discussed in any of the podcasts I had heard.  Finally a web search led me to the QI episode above, during which Stephen Fry quotes the eminent Stephen Jay Gould's assertion that there is no such thing as a fish. (q.v. for explanation) (addendum: or read Sylvia's notes in the Comment section).

While I'm on the subject of podcasts, I can also recommend QI's series of "International Factballs" produced to coincide with the World Cup.  Lots of tidbits of "things you wouldn't know" in those broadcasts.

*There is actually no book with that title.  Oxford does, however, publish an encyclopedia of underwater life (see the comment thread below).