Both words, radish and radical, derive from the Latin word for root (radix). The vegetable we call a radish is an edible root.
Radical, functioning as both noun and adjective, is used with multiple meanings, depending upon context. Its earliest use in the context of politics and political thought and action dates from the late eighteenth century:
That the omnipotence of the state is not lodged, by the constitution, with the people, but with the whole legislative body in parliament assembled, was a radical doctrine of this obnoxious ministry.—OED citation dated 1783
“A radical doctrine” is one that would strike at the root of an established political or social norm. A radical is “a person who advocates radical or far-reaching political or social reform.”
The earliest OED citations for the noun radical are dated 1822:
Love is a great leveller; a perfect Radical.
General Scott said Archer was a Radical and inclined to be Jacobinical.
Note: As a political term, Jacobin derives from a French political club established in 1789 with the purpose of propagating the principles of extreme democracy and absolute equality. By 1800, the word Jacobin was used to refer to any political reformer.
Every society is rooted in specific institutions and conventions. At the time that radical acquired its political meanings, European society was rooted in the model of a landed elite supported by a disenfranchised working class.
In the early nineteenth century, efforts to accomplish the following were seen as radical ideas in Britain and the United States:
end the employment of children in factories and mines
extend the vote to all men
extend the vote to women
end imprisonment for debt
end the slave trade
grant full civil rights to Catholics and Jews
provide elementary schools for the children of the working classes
provide humane treatment for the mentally ill
The verb radicalize in the sense of “to make radical, especially politically; to imbue with radical principles” appears early in the nineteenth century (1825). The earliest citation for the noun radicalization—“the action or process of making or becoming radical, especially in political outlook”—is 1867.
Among the OED citations for radicalize and the noun radicalization are references to soldiers who were radicalized by witnessing the horrors of war and to “radicalized students of the late 1960s.”
These political terms have been used to describe different degrees of radicalism, as indicated in this definition of the adjective radical in the OED:
radical adjective: Advocating thorough or far-reaching political or social reform; representing or supporting an extreme section of a party; specifically (also with capital initial) (a) British belonging to, supporting, or associated with the extreme wing of the Liberal Party which called for a reform of the social and parliamentary system in the late 18th and early 19th century. (b) U.S. belonging to a faction of the Republican Party seeking extreme action against the South during the Civil War and Reconstruction. Now more generally: revolutionary, especially, left-wing.
Although in the past, radical belief was sometimes accompanied with violent behavior—e.g., John Brown, Carrie Nation, the French Revolution—it was more often contained and acted on within a framework of constitutional or parliamentary changes. A “radical” could be any person who regarded some aspect of society as unfair or undesirable and believed that the way to change it was to overturn or uproot existing norms. In that sense, suffragettes and abolitionists were radicals.
Nowadays, radical, radicalize, and radicalization have come to carry connotations of a type of extremism closely association with violence.
This is how The US Federal Bureau of Investigation (FBI) defines radicalization and radical:
At its root, radicalization takes the basic tenets of a faith or a political movement and carries them to extremes, extremes that often are drastic enough to adopt violence to intimidate others into accepting those extremes or to punish those who will not accept the extremes, and that process carries across lines of nationality or religion, from Mohammad Atta to Timothy McVeigh.
The FBI…defines radical individuals as persons who encourage, endorse, condone, justify, or support the commission of a violent act or other crimes against the U.S. government, its citizens, or its allies for political, social, or economic ends.