[Math] Most harmful heuristic

big-listsoft-question

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?

Best Answer

Not the most harmful, but a fun example (credit due to Tony Varilly):

"You can't add apples and oranges."

False. You can in the free abelian group generated by an apple and an orange. As Patrick Barrow says, "A failure of imagination is not an insight into necessity."