Radical did not originally mean extreme or rebellious. It came from the Latin word radix, which means root.
That older meaning helps explain several modern uses. In mathematics, the symbol for a square root is called the radical sign. For example, √9 is read as the square root of 9, and the symbol itself is a radical.
In older political and social writing, radical could describe change that goes to the root of a problem, not just change that is shocking or severe. The idea was about getting down to the basic source.
You can also see the same Latin root in related words:
- radix, a technical word for root or base
- radish, a plant named for its edible root
- eradicate, literally to pull out by the roots
So when you see radical, the oldest image behind it is not intensity first. It is a root. That historical meaning still survives most clearly in math and in the idea of addressing something at its source.