Thanks to Robert H. Olley for linking the question of how al-Khayyam saw the relation between algebra and geometry to a mid-70s controversy regarding Euclid’s anticipation of algebra. Three eminent mathematicians — Hans Freudenthal, Bartel van der Waerden, and the always intimidating André Weil — took the Romanian-Israeli historian Sabetai Unguru to task for denying that Euclid had algebra. Leo Corry, who studied with Unguru, has an interesting account of this controversy which (as I should have known) represents a decisive moment in the affirmation of historical methodology, rather than mathematical reinterpretation, in the history of history of mathematics as a separate discipline.
I allude briefly to this on p. 215 of the book when I quote Ivor Grattan-Guinness on “royal road to me”-type history. In footnote 75 to Chapter 7, Grattan-Guinness identifies Weil as a primary exponent of this ahistorical (Whiggish) approach to history. Weil made a point of terrorizing historians of mathematics — notably Michael Mahoney, author of a biography of Fermat that was the subject of a memorably scathing review by Weil. Historians argued that Weil and other mathematicians did not understand the point of history; as Unguru wrote: “The history of mathematics is history not mathematics.” Weil had written a typically nasty letter to the editor of Archive for History of Exact Sciences to support Freudenthal and van der Waerden in their complaint about Unguru’s “betrayal” of Euclid. In a subsequent article in Isis, Unguru responds to Freudenthal and van der Waerden, reserving his comment on Weil for a final footnote, which contains the single best put-down of Weil I’ve ever read:
Here, I have in mind Andre Weil’s unprecedented missive to the editor of the Archivefor History of Exact Sciences, entirely repetitive in its few non ad hominem passages of the arguments of van der Waerden and Freudenthal: “Who Betrayed Euclid?” Arch. Hist. Exact Sci., 1978, 19:91-93. Concerning this letter, the less said the better. In adopting this position, I am guided by Simone Weil’s words in her sensitive and penetrating essay on the Iliad (The Iliad or the Poem of Force, Wallingford, Pa.: Pendle Hill, n.d., pp. 3, 36): “To define force-it is that x that turns anybody who is subjected to it into a thing. Exercised to the limit, it turns man into a thing in the most literal sense: it makes a corpse out of him. Somebody was here, and the next minute there is nobody here at all;” And: “The man who does not wear the armor of the lie cannot experience force without being touched by it to the very soul. Grace can prevent this touch from corrupting him, but it cannot spare him the wound.”
Unguru, it should be said, was not above nastiness himself; see slide 34 of this presentation by John B. Little of Holy Cross.