A real flat tax would mean everybody pays the same amount!
It is interesting to think about, or to hear, arguments as to why a tax should be anything other than a fixed quantity per person. It exposes a lot of assumptions, some of them questionable.
For instance, one might argue, "The rich should pay more because they get more in services." But if you look at "services" supported by tax dollars, don't the poor make much more use of police and welfare than the rich? So should not the poor pay more?
I seem to recall Dr. Walter Williams making an argument along this line.