Why should men be "chivalrous" toward women (opening the door, paying for dinner, giving up a seat, etc)?

Chivalry is about kindness and courtesy. If a man holds open a door for me, it's no different than a woman doing it -- it's just an act of kindness and good manners. If I give up my subway seat for a pregnant woman or a man on crutches (and I've done both), it's just a matter of being kind and considerate. Kindness is not an antiquated virtue, you know?