That’s because patriarchy has made you believe in a zero-sum world where somebody has to lose in order for somebody else to win. For the record, there are not now and have never been hierarchies of domination (which is what patriarchy is) where women oppressed men. Matrilineal and matrifocal societies, both in the past and in the present, believe in a balance between men and women even in cases where they have what we would consider traditional gender roles.
The largest matrilineal society in existence today is the Minangkabau of West Sumatra, Indonesia — with about 4 million people. Mothers are considered to be the most important people in society and sons leave their wives early each morning to go and have breakfast with their mothers, but there is still a true balance between men and women within this culture. It’s built around cooperation and not around the belief that one ought to have a relationship to others and the world through competition, hierarchy, domination, and control (which is what patriarchy believes).
So no, the opposite of patriarchy is not a dominance hierarchy with women instead of men at the top of the pyramid of power. It’s a partnership-oriented society.
“Hunter-gather tribes like the !Kung counter domineering behavior from senior males by forming coalitions within the group. They are intolerant of hoarding or selfishness and social order is maintained through ridicule, banishment, and sometimes even execution. What would happen in our culture if instead of approving and admiring selfish and domineering behavior, we refused to tolerate it? What would happen if we prioritized contributing to our tribe, even if only the one in our neighborhood or community? If history and current examples are any indications, we would be a culture with lower levels of violence, depression, and mental illness — a much happier society.”
