Michael Kinyon reminded me of Edgar Allan Poe’s comments on algebraists in his story “The Purloined Letter”. Here they are in full.

“But is this really a poet?” I asked. “There are two brothers, I know; and both have attained reputation in letters. The minister I believe has written learnedly on the Differential Calculus. He is a mathematician, and no poet.”

“You are mistaken; I know him well; he is both. As poet *and* mathematician, he could 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 à parièr,’*” replied Dupin, quoting from Chamfort, “‘*que toute idée publique, toute convention reçue, est une sottise, car elle a convenue au plus grand nombre.’* 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 of 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 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. Mathematial axioms are *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 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 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, of 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 when *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 endeavour to knock you down.”

I must admit I would quite like to knock him down for this amazing farrago. But it is more interesting to refute him point by point. A good exercise for students?

And yet there is something in what he says. Julian Jaynes gives an example of valid poetic reasoning, which goes something like this: “Man dies; grass dies; so man is grass”. Could it be that this kind of reasoning is closer to what makes Dupin, Holmes, and their ilk great detectives than Boolean logic? Indeed, is it closer to how we *do* mathematics, rather than how we write it up afterwards?

I actually don’t think that mathematicians are much better than other people in applying logical reasoning to things outside mathematics; and I think that we teachers should feel a bit ashamed of that fact.

That is what Aristotle called “apagoge”, usually rendered “abduction”. Peirce wrote a lot about it. See the following essay of mine:

http://intersci.ss.uci.edu/wiki/index.php/Functional_Logic_:_Inquiry_and_Analogy

And this is precisely why Plato, with a hearty and Pythagorean ‘Nevermore!’, voted poets off his island.

On subsequent reflection, I think the common element is imagery, imagination.