Never let a young child cry it out. not until their older like 6 months. They can do damage to their lungs crying out to you. Even if your child was older I would not let the child cry more then 15 minutes. I have worked in Early Childhood Education in a child care center for the past 5 years. Any other questions just ask.
No such thing as spoiling a 6 week old. This is when they need you the most. So if she cries...pick her up if you want. It's your baby:-) No one else's opinion matters...
You can't spoil a 6 week old by giving her too much attention. You CAN ruin one by ignoring her cries for help. Children bond and learn to trust the world when they cry and their cries are answered.
Tell your boyfriend that if he doesn't believe responding to babies is the natural order of things, to look to the animal world. A baby animal's cry will not only get the full attention of the mother, but also other adults of the same species will come running to help out.
Children who live in orphanages or who live in families where their cries aren't answered become disconnected.
If your boyfriend continues to give you trouble about your attentive mothering, have him talk to your pediatrician.