Like a river draws all who ride it toward a common sea...

I crave a new philosophical method, one with a built-in tendency toward convergence, analogously to how science and math each slowly converge toward consensus by their own methods.

Perhaps this simple method?:

Place before you a blank sheet of paper. With your interlocutor, discuss each other's views. On the paper, write only those statements to which you both heartily agree. Aim to fill the page with valuable truths. Share the best of them.