William E. Emba.|
"This is the third and last C. Auguste Dupin mystery. The
Prefect of Paris police explains a very delicate situation
to Dupin, involving a royal letter whose possession grants
its bearer great powers, and whose rightful owner is unable
to publicly seek redress. The thief is known, and it is
known he keeps the letter handy, yet eighteen months of
the most assiduous searching of the thief's residence has
failed to locate it. The Prefect then mentions that the
reward has gone up considerably, but it's not the money
that interests him, it's the principle. Dupin asks for the
money, for which he will produce the letter. The Prefect
writes Dupin a check, and right then and there, Dupin hands
the Prefect the letter.
"Before explaining his solution, Dupin launches into a long
spiel about mathematics and the nature of reasoning. In
particular, he criticizes the Prefect for attacking the
problem along purely mathematical angles, when the thief in
question is both mathematician and poet."
Here is the discussion of mathematics. (The rest of the story is also available on-line...reach it by following the link above or below.) I must say that I find it very wrong-headed. In particular, I agree with the very notion that he claims to argue against: it is true that mathematicians must have exceptional skills of reasoning, and these skills extend to topic other than mathematics. Though it is true that individual mathematical topics (algebra, for instance) are limited in application outside of mathematics, the logic involved, understanding implication, the need to be very explicit about assumptions, being able to critically analyze one's own beliefs to determine whether they can be proven with a rigorous argument, and being willing to give them up if such an argument can be constructed against them, these are not to be found only among mathematicians, but any mathematician must have these useful skills. In any case, this is what Poe has to say on the topic:
|(quoted from The Purloined Letter)|
"But is this really the 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 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.' 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 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 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.
A lenghty and deep discussion of the mathematical themes in "The Purloined Letter" can be found in John T. Irwin's "The Mystery to a Solution - Poe, Borges and the Analytic Detective Story" (Baltimore, The Johns Hopkins University Press, 1994). Irwin dissects all the numerical and geometrical ideas behind Poe's story, and discusses in detail the text quoted above, from Chapter 34 on. He also examines the influence of French mathematicians on Poe's own studies (West Point was considerably influenced by French mathematics, through the Ã‰cole Polytechnique). It is an exhausting but ultimately rewarding analysis, if you happen to be a mathematically-oriented Edgar Allan Poe fan.
As far as I can recall, I read this but never quite "got it". I was perhaps 13. So, I enjoyed some of it but was a but frustrated. It looks like there isn't as much to "get" as I had hoped, since the argument seems silly now.
N.B. Another of Poe's mysteries featuring Chevalier C. Auguste Dupin, The Mystery of Marie Roget, contains further evidence that Poe did not really understand math. I have debated whether to give this story its own entry, but aside from being barely mathematical, it is also barely fictional. (Although reset in Paris, it is actually Poe's account of an actual murder which took place outside New York.) It begins with some rather nice praise for probability:
|(quoted from The Mystery of Marie Roget)|
There are few persons, even among the calmest thinkers, who have not occasionally been startled into a vague yet thrilling half-credence in the supernatural, by coincidences of so seemingly marvellous a character that, as mere coincidences, the intellect has been unable to receive them. Such sentiments --
for the half-credences of which I speak have never the full force of thought -- such sentiments are seldom thoroughly stifled unless by reference to the doctrine of chance, or, as it is technically termed, the Calculus of Probabilities. Now this Calculus is, in its essence, purely mathematical; and thus we have the anomaly of the most rigidly exact in science applied to the shadow and spirituality of the most intangible in speculation.
However, at the end he quite mistakenly suggests that according to the theory of probability the outcome of a roll of the die depends on prior rolls, that it is only uneducated common people who think otherwise, and that there are sophisticated reasons which guarantee that this is not so:
|(quoted from The Mystery of Marie Roget)|
For, in respect to the latter branch of the supposition, it should be considered that the most trifling variation in the facts of the two cases might give rise to the most important miscalculations, by diverting thoroughly the two courses of events; very much as, in arithmetic, an error which, in its own individuality, may be inappreciable, produces at length, by dint of multiplication at all points of the process, a result enormously at variance with truth. And, in regard to the former branch, we must not fail to hold in view that the very Calculus of Probabilities to which I have referred, forbids all idea of the extension of the parallel, -- forbids it with a positiveness strong and decided just in proportion as this parallel has been already long-drawn and exact. This is one of those anomalous propositions which, seemingly, appealing to thought altogether apart from the mathematical, is yet one which only the mathematician can fully entertain. Nothing, for example, is more difficult than to convince the merely general reader
that the fact of sixes having been thrown twice in succession by a player at dice, is sufficient cause for betting the largest odds that sixes will not be thrown in the third attempt. A suggestion to this effect is usually rejected by the intellect at once. It does not appear that the two throws which have been completed, and which lie now absolutely in the Past, can have influence upon the throw which exists only in the Future. The chance for throwing sixes seems to be precisely as it was at any ordinary time -- that is to say, subject only to the influence of the various other throws which may be made by the dice. And this is a reflection which appears so exceedingly obvious that attempts to controvert it are received more frequently with a derisive smile than with anything like respectful attention. The error here involved -- a gross error redolent of mischief -- I cannot pretend to expose within the limits assigned me at present; and with the philosophical it needs no exposure. It may be sufficient here to say that it forms one of an infinite series of mistakes which arise in the path of Reason through her propensity for seeking truth in detail.
Poe's misunderstanding is an instance of The Gambler's Fallacy.
Good site! I too was struck by the Gambler's fallacy appearing after otherwise valid points on probability-e.g., the multiplication of a chain of probabilities.