James Hamblin thinks you should eat apples whole, core and all.
The core is a product of society, man. There is a thin fibrous band, smaller in diameter than a pencil and not bad to the taste. If you eat your apple vertically, it is not noticeable.
But what about cyanide! Apple seeds contain amygdalin, which can release cyanide if digested. According to John Fry:
Apple seeds contain about 700 milligrams of cyanide per kilo, so about 100 grams of apple seeds should be enough to dispatch a 70-kg adult human, but that’s an awful lot of apple cores even if you don’t eat the rest of the apple first. In addition, the seeds would have to be pretty finely crushed to let the enzymes get to the amygdalin at all.