Show mathematically how increasing the carbon dioxide content of air contributes to Global Warming?

The CO2 system is non-linear. A small change in CO2 can have a large effect. Here's why.

There is a natural "carbon cycle" which recycles CO2. Many sources emit it, many sinks (like plants) absorb it.

But it's a delicate balance. We're messing up the balance by digging up a whole lot of carbon the natural cycle buried over thousands of years and burning it real fast.

You can see it clearly in this graph. The little teeth are the natural cycle; plants reducing it in summer, it going up a little in winter. The huge move upwards is us, burning fossil fuels.

http://scrippsco2.ucsd.edu/graphics_gallery/mauna_...

We're messing up nature big time. And that threatens very bad things for coastal flooding and serious damage to our agriculture. We need to fix it.

How much do we need to cut back? This site discusses that:

http://www.realclimate.org/index.php/archives/2006...

To everyone that has answered so far: The asker expressely said to show mathematically. So far all you have done is make verbal expressions. One would think the least you could do was show some type of correlation between increasing CO2 levels and increasing temperatures. Oh, I forgot, the data doesn't correlate very well. My bad.

umm.... I don't know