Wednesday, 27 October 2010
The Piffle Paradox - or how pure mathematicians have fun
Ever wondered how pure mathematicians have fun? The following is from the 1967 paper Modern Research in Mathematics by A. K. Austin, from the Department of Pure Mathematics at the University of Sheffield. It's a send-up, by the way...
A note on piffles by A. B. Smith
A. C. Jones in his paper "A Note on the Theory of Boffles," Proceedings of the National Society, 13, first defined a Biffle to be a non-definite Boffle and asked if every Biffle was reducible.
C. D. Brown in "On a paper by A. C. Jones," Biffle, 24, answered in part this question by defining a Wuffle to be a reducible Biffle and he was then able to show that all Wuffles were reducible.
H. Green, P. Smith, and D. Jones in their review of Brown’s paper, "Wuffle Review, 48", suggested the name Woffle for any Wuffle other than the non-trivial Wuffle and conjectured that the total number of Woffles would be at least as great as the number so far known to exist. They asked if this conjecture was the strongest possible.
T. Brown, "A collection of 250 papers on Woffle Theory dedicated to R. S. Green on his 23rd Birthday" defined a Piffle to be an infinite multi-variable sub-polynormal Woffle which does not satisfy the lower regular Q-property. He stated, but was unable to prove, that there were at least a finite number of Piffles.
T. Smith, L. Jones, R. Brown, and A. Green in their collected works "A short introduction to the classical theory of the Piffle," Piffle Press, 6 gns., showed that all bi-universal Piffles were strictly descending and conjectured that to prove a stronger result would be harder.
It is this conjecture which motivated the present paper.
Not to be outdone, S. J. Farlow from the Department of Mathematics, University of Maine, wrote in the seminal A rebuke of A. B. Smith's paper, 'A Note on Piffles':
In A. B. Smith's recent paper, 'A Note on Piffles', The American Mathematical Monthly, 84, p. 566 he completely fails to mention one of the most significant results yet discovered in Piffle Theory, namely A. K. Puddle's paper, 'Products of Planar Piffles'.
In this short but succinct note Puddle proves that a denumerable product of Pi Piffles is in fact a P-Pi Piffle (assuming of course pairwise permutation of the Piffles). That Puddle's condition was only necessary and not sufficient did of course not detract from this significant work—but did in fact open the door to the well-known Piffle Paradox (of which I'm afraid Professor Smith is completely unaware).
Readers interested in obtaining a complete up-to-date history of the Piffle should consult P.U. Piper's comprehensive review, The Piffle: 1840-1978 (Pauper Press). Here Piper describes some modern approaches taken by American Mathematicians during the last fifteen years. I am sorry to say that the classical treatment of Piffles taken by most English Mathematicians, notably the work of author Smith, is, by American standards, obsolete even before it hits the printing press. In particular the classic theorem of Smith, Jones and Brown on Polynomial Piffles would be only a simple corollary to Puddle's basic result on Homological Piffles. In fact it is fairly safe to say that all the English results so far on Piffle Theory can be subsumed in Piper's short note, 'Spectral Decompositions of Partial Piffles', American Piffle Review, 27, pp. 1-2.
Hat-tip to Let ε < 0 where I first saw this lovely work. I believe the original paper came out of discussions between mathematicians and educators regarding good (and presumably bad and confusing) forms of mathematics education. I dare say that had I seen this treatise in undergraduate maths, or had Homological Piffles been mentioned at least once, I wouldn't have transferred from Metric Spaces to Astronomy....
Austin, A. (1967). 3183. Modern Research in Mathematics The Mathematical Gazette, 51 (376) DOI: 10.2307/3614400
Farlow, S. (1980). Three Mathematical Satires A rebuke of A. B. Smith's paper, 'A Note on Piffles' International Journal of Mathematical Education in Science and Technology, 11 (2), 285-304 DOI: 10.1080/0020739800110222