I exist only when shared. I grow when divided. I die when hoarded. I weigh nothing, yet entire worlds are built on me. What am I?

I am not ignorance; I am choice. I see the door and swear it isn’t there. I survive only while you look away. The moment you name me, I begin to die. What am I?

I guard your treasures, yet have no eyes, I click and I turn where your secret lies. A twist of numbers, a key to my heart, open me wrong, and I won’t part.

Echoes of a quartet, multiplied force, a leap from pair to foursome, fourfold strength in a single course.

I'm spoken through problems, some yet to be solved. Through logic and science, I'm always involved. Most everyone knows me, some animals too. I help you determine what's many, what's few.