Recently, Stu Denenberg got me to reread The Purloined Letter by E.A. Poe. If you recall, the story centers around the search for a stolen letter secreted in the home of a court schemer. The chief of police has failed to find the letter, yet it must be there. He comes for help to Dupin, an amateur sleuth. Dupin discusses the matter with the narrator (his "Dr. Watson.")
The particular part relevant to Ira's presentation is as follows below, in which Dupin speaks about the court schemer and why the chief of police's (the Préfet) disdain for poets has caused him to underestimate the schemer's reasoning power.
"You are mistaken; I know him well; he is both. As poet and mathematician, he would reason well; as mere mathematician, he could not have reasoned at all, and thus would have been at the mercy of the Prefect."
"You surprise me," I said, "by these opinions, which have been contradicted by the voice of the world. You do not mean to set at naught the well-digested idea of centuries. The mathematical reason has long been regarded as the reason par excellence.
"'Il y a a parier,'" replied Dupin, quoting from Chamfort, "'que toute idee publique, toute convention recue, est une sottise, car elle a convenu au plus grand nombre.' (trans: I would wager that every popular notion, every convention, is a bit of stupidity that exists only because it is useful to the majority.) The mathematicians, I grant you, have done their best to promulgate the popular error to which you allude, and which is none the less an error for its promulgation as truth. With an art worthy a better cause, for example, they have insinuated the term 'analysis' into application to algebra. The French are the originators of this particular deception; but if a term is of any importance — if words derive any value from applicability — then 'analysis' conveys 'algebra' about as much as, in Latin, 'ambitus' implies 'ambition,' 'religio' religion or 'homines honesti,' a set of honorable men."
"You have a quarrel on hand, I see," said I, "with some of the algebraists of Paris; but proceed."
"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 uponform 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 axiomsare not axioms of general truth. What is true 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 falls. In the consideration of motive it falls; 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 inferences from them as existing realities.' With the algebraists, however, who are Pagans 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 could be trusted out of equal roots, or one who did not clandestinely hold it as a point of his faith that x squared + 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 squared + 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."
Before we get too upset with what seems like an unfair attack upon mathematicians, we should remember the Pythagoreans who made trigonometry into a religion. As Ira has demonstrated, it's necessary to set up a problem "correctly" before one can use mathematics to find a solution.
With respect -Joel