Origins
Every radical is, at heart, a gardener. The word comes from Latin rādīx — 'root'. Late Latin rādīcālis meant 'of the root, fundamental'. A radical solution digs down to the root of a problem rather than trimming its branches.
The political meaning arrived in 1802 when Charles James Fox called for 'radical reform' of Parliament — reform that addressed root causes. Within decades, Radical became a political label for those who wanted to remake society from the foundations upward.
But the word had been technical long before it was political. Medieval scholars used radical to mean 'fundamental to existence'. In medicine, the radical moisture was the essential bodily fluid whose loss meant death. In grammar, the radical form of a word was its root, stripped of suffixes.
Figurative Development
Mathematics adopted the same metaphor. The radical sign (√) asks: what is the root of this number? The square root of 9 is 3 because 3, planted in the ground and squared, grows into 9. The botanical image is exact.
Chemistry followed suit. A chemical radical is a fundamental molecular fragment — a root component that persists through reactions.
The humblest member of the family is the radish — from Latin rādīx via Old English rædic. A radish is simply 'the root'. To eradicate means to 'pull up by the roots' — ē- ('out') plus rādīx. Destroy the root and the plant dies. Radical thinking aims for the same completeness.