The pagan algebraists

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“I dispute the availability, and thus the value, of that reason which is cultivated in any especial form other than the abstractly logical. I dispute, in particular, the reason educed by mathematical study. The mathematics are the science of form and quantity; mathematical reasoning is merely logic applied to observation upon form and quantity. The great error lies in supposing that even the truths of what is called pure algebra are abstract or general truths. And this error is so egregious that I am confounded at the universality with which it has been received. Mathematical axioms are not axioms of general truth. What is truth of relation – of form and quantity – is often grossly false in regard to morals, for example. In this latter science it is very usually untrue that the aggregated parts are equal to the whole. In chemistry also the axiom fails. In the consideration of motive it fails; for two motives, each of a given value, have not, necessarily, a value when united, equal to the sum of their values apart. There are numerous other mathematical truths which are only truths within the limits of relation. But the mathematician argues from his finite truths, through habit, as if they were of an absolutely general applicability – as the world indeed imagines them to be. Bryant, in his very learned “Mythology”, mentions an analogous source of error, when he says that “although the pagan fables are not believed, yet we forget ourselves continually, and make inference from them as existing realities.” With the algebraists, however, who are pagan themselves, the “pagan fables” are believed, and the inferences are made, not so much through lapse of memory as through an unaccountable addling of the brains. In short, I never yet encountered the mere mathematician who would be trusted out of equal roots, or one who did not clandestinely hold it as a point of his faith that x^{2}+px was absolutely and unconditionally equal to q. Say to one of these gentlemen, by way of experiment, if you please, that you believe occasions may occur where x^{2}+px is not altogether equal to q, and, having made him understand what you mean, get out of his reach as speedily as convenient, for, beyond doubt, he will endeavor to knock you down.”

From “The Purloined Letter” (1844), Edgar Allan Poe (1809 – 1849)

C. Auguste Dupin, the central figure of this short story by the great Edgar Allan Poe, argues against the “addling of the brain” induced by studying the algebraic “pagan fables“. The passage could be read as a comment on mathematical education, which often places emphasis on problems detached from reality and on various special problems having as a starting point some rather comfortable assumption. As an example, Poe seems to be referring to polynomial equations the solution of which is greatly facilitated by assuming that they possess at least two equal roots. A text of the time referring to exactly that topic is the paper “A new method for finding the equal roots of an equation, by division“, by the Rev. John Hellins, Curate of Constantine, in Cornwall (Phil. Trans. R. Soc. Lond. 1782 72, 417-425, published 1 January 1782).