On 1/10/09, David Wolfe <davidgameswolfe@gmail.com> wrote:

On Sat, Jan 10, 2009 at 3:52 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:

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Did you have any particularly (counter-) instructive instances in mind? [Present company excepted, naturally!] WFL

Actually, no. My main reason for such a textbook would be to encourage students to read through all proofs carefully and skeptically, and not just the ones in the textbook. I envision a book that is still focused on one topic (such as Number Theory), not a textbook on reading proofs, per se.

[At this point I must make a determined attampt to avoid lapsing into riding various hobby-horses --- but (mixing the metaphors) it'll be sailing close to the wind.] Anyone who has attempted seriously to program mathematical algorithms for computer [an activity almost co-extensive with theorem- proving] will be aware of the pleasurable momentary surge of astonishment when a progam of --- say --- ten lines or so apparently works at the first attempt; not to mention the familiar incredulous frustration when closer investigation establishes that it in practice still fails to do after the tenth. It follows that any proof published at the research level is almost certainly technically incorrect --- even discounting failures at the "typographical" level, which might be dismissed as amenable to an informed reader's error-correction facility. A less obvious consequence involves the observable inherent reluctance on the part of most readers to accept that a purported proof is fallacious, particularly once it has been digested and (unconsciously) accepted. As a result, there's certainly plenty of raw material out there --- I'll go out on a limb (metaphorosis permitting) and bore on about a couple of my rather obscure personal favourites. The Padé block theorem, which says that zeros in a Padé C-table or number wall occur only clumped together as disjoint square "blocks" or "windows" [the rectangular property is almost trivial]. I had seen around half-a-dozen attempted proofs of this long-accepted result in various published papers and books, every single one seriously flawed. [Incidentally, the roll-call of shame includes my own first attempt, further dissemination of which was happily intercepted by an astute colleague.] Much earlier I had not been (quite) so lucky, when I inadvertantly circulated a misproof of the frame theorems --- the basis of a recurrence algorihm circumventing such zeros --- to a handful of colleagues. Although I had soon realised it was a dud, a couple of years later the identical argument appeared in a book by another author --- fortunately, without any complicity therein being ascribed to myself. This specimen is noteworthy for the occurrence of an elaborate, plausible but irredeemably fallacious diagram at its head, preceding the very first line of text --- the reasoning which follows it being straightforward computation, and taken in isolation, surely, essentially correct. There is a moral here --- if you're going to perpetrate a really outrageous swindle, stick it to the punter right at the start of the performance, before he has a chance to find his bearings at put his guard up. Enough of these self-indulgent tropes! Fred Lunnon