Well, you can "scratch" with just a turntable, but you can't "cut". You'd be able to rub the record back and forth, you just couldn't affect the volume. It's kind of hard to explain without going into a tutorial on scratching... If you just want to get that "wikki-wikki" sound or the sound of a DJ scratching a bass kick, you can do it without a mixer, but that's about all you'll be able to do. Er... actually there's a whole slew of "faderless scratches" but you have to be pretty serious about scratching to do those with any competence, so if you could scratch that way you'd already be a pretty good turntablist and definitely already have a mixer.
Sorry if this is kind of confusing, it's sort of a loaded question...