Maybe you could do some math and deal with reality. Some people on the bottom of the scale with mouths to feed cannot afford to pay taxes and still feed their children or care for their families. That is why a sliding scale exists. A flat tax would push some people barely making it into total poverty. I would not think I would have to point that out to you. Not every problem has a simple black and white solution...not matter what FOX news says.
People pay taxes to enable the government to function. Everyone has the same obligation to the government in that the rich person enjoys the benefits of streets, parks, military, police, etc., as much as the poor man does. (There is an extremely strong argument that poor individuals use a vastly disproportionate amount of government services compared to their wealthier counterparts, but I'm not going to go into that now)
As a result, having everyone pay the same percentage is incredibly fair. Your example of people "barely making it" is irrelevant. There will always be people barely making it whether we have a 90% tax rate or no taxes at all. Furthermore, there isn't anything that says that the tax rate must be 20%. I have no doubt that if a flat tax was proposed, people under a certian income would pay no tax at all (which is very similar to what happens today).
However, pointing to a single mom who is trying to feed little Timmy and Johnny doesn't really have anything to do with the tax system. A 10% income tax will no doubt affect someone making 35k a year far more than someone making 100k a year, but thats not the point. ANY financial obligation will be harder on them. Their obligation to the government does not diminish with their salary however.
Taxing the rich more because they won't be affected by it is simply socialism. Its fine if you adhere to that, but at least come out and say it.