Parodies of abstruse mathematical writing

Question

Perhaps under the influence of a recent question on perverse sheaves, in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53), I recalled a long-ago parody of abstruse mathematical language that I can no longer remember in detail nor find by searching.

I am not seeking merely "examples of colorful language," as in that earlier MO question, but rather parodies almost in the Alan Sokal Fashionable Nonsense sense (although I don't think he parodied abstract mathematics directly).

I am partly motivated by the possible educational advantage of self-mockery (or self-awareness), tangentially related to an MESE question, "Wonder as Motivation." But I ask here to tap into the likely greater density of mathematicians working in abstract fields ripe for parody.

Q. Can you provide examples of (or pointers to) intentionally comic parodies of abstruse mathematical language, written by knowledgeable mathematicians so that they could (in another universe) make mathematical sense.

Answer 1

Is this what you're looking for?

http://thatsmathematics.com/mathgen/

Mathgen is an random math paper generator, based on SCIgen which does the same for computer science papers. It will provide you with an unlimited supply of abstruse nonsense: definitions, theorems, proofs, references, and all.

Here is a sample title and abstract.

"Some Reducibility Results for Ultra-Universally Nonnegative Arrows"

Assume we are given a contra-intrinsic subring $\mathscr{{W}}$. Recently, there has been much interest in the description of manifolds. We show that $\| t' \| > \mathbf{{b}}$. So is it possible to describe compactly ultra-prime systems? Hence recent interest in finitely Huygens--Hilbert, closed, meager groups has centered on describing canonical homomorphisms.

(Disclosure: edited by Nate Eldredge, author of Mathgen, to include additional details.)

Answer 2

The online version of the closing entry of Reports of the Midwest Category Seminar IV (1970, Springer LNM 137) costs $29.95 so I decided to place a transcript here.

CATEGORICALLY, THE FINAL EXAMINATION    
-------------  ---------------------
FOR THE    

SUMMER INSTITUTE AT BOWDOIN COLLEGE (Maine) 1969



               'I thought I saw a garden door that opened with a key,
                I looked again and found it was a Double Rule of Three,
                And all its mysteries, I said, are plain as day to me.'

                                         (Verse by the true founder of
                                           Category Theory)


Important Instruction:  This is a take-home exam:
---------------------    Do not bring it back!


Answer as many as possible at a time.



 1. Are foundations necessary? To put it another way, given a
    chance, wouldn't Mathematics float?

 2. Describe the category of foundations. Is this a concrete cate-
    gory? A re-enforced concrete category?

 3. Discuss the relations and limitations of the foundations set
    forth by:
       a) Frege-Russell
       b) Bernays-Gödel
       c) Playtex.

 4. (Mac Lane's Theorem) Prove that every diagram commutes.

 5. Considering a left-adjoint as male and a right adjoint as female,
    give the correct term for a contravariant functor self-adjoint
    on the right.

 6. Considering a left-adjoint as husband and right-adjoint as
    wife, give a precise definition of "marital relations". Do the
    same for the pre-adjoint situation.

 7. Discuss the Freudian significance of exact sequences. (Hint:
    consider the fulfillment by one arrow of the kernel of the next.)

 8. Find two new errors in Freyd's "Abelian Categories".
         --- ===

 9. Trace the origin of the Monads-Triads-Triples controversy to the
    important paper of St. Augustine.

10. Using theorems from both Freyd and Mitchell, prove that every
    reflective category is co-reflective. Dualize.

11. Give your opinion of the following exercises:
       a) Ten pushouts
       b) Twenty laps around an adjoint triangle
       c) Two supernatural transformations.

12. Write out at least one verse of
       a) "Little Arrows"
       b) "Doing What Comes Naturally"
       c) "Hom on the Range"

13. Why is the identity functor on 2 called the "Mother Functor"?
                                   -

14. Write down the evident diagram, apply the obvious argument, and
    obtain the usual result. (If you can't do it, you're not
    looking at it hard enough, or, perhaps, too hard.)


                                                      Phreilambud

PS After some controversy in comments I just googled for "who is phreilambud" and found this:

Date: Mon, 3 Oct 2005 11:58:38 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: categories: Re: Phreilambud at Bowdoin 1969

``Phreilambud'' was written by me, a young student named Lambert who
disappeared, I think, from mathematics and David Eisenbud, now paying
for his sins as head of MSRI (Berkeley).
   Peter

PPS Another thing that came to my mind, although not exactly what the question asks for but closely related to the above. Jack Duskin once told me that after one of his talks on simplicial sets, with the blackboard full of dozens of parallel bunches of arrows sticking in all directions behind him, somebody in the audience warned him of the high risk of sharing the fate of St. Sebastian.

Answer 3

In a comment to one of the answers here Marius Kempe mentioned a similar case described in Mikhail Gromov's autobiographic text A Few Recollections; I liked it so much I decided to put it in a separate answer. Recalling how he was trying for a year to reconcile his previous views with what he learned from the work of Tony Phillips on submersions, Gromov then continues

Something else written by Tony, a private letter to me, also kept me puzzled for quite awhile. This letter contained a couple of pages of incomprehensible mathematics, starting with something like:

... an involutive gromomorphism $G ∶ SU → US$ of admissible type... $T$ transforms $MG → SB$...

I could not understand a single sentence in it. But when I showed this to my friend, an analyst Volodia Eidlin, he asked me: ”What is a gromomorphism?”

”You mean homomorphism” – I replied – ”There is no such thing as gromomorphism”. (”Homomorphism” is spelled and pronounced as ”gomomorphism” in Russian.)

”Do you ever read anything as it is written?” – he was annoyed – ”This is ”gromomorphism”, black on white.”

”Must be a misspel...” – I mumbled, but then it dawned on me. Tony’s was an encoded message. He was suggesting I would immigrate from the Soviet Union to US and invited me to SUNY at Stony Brook where he worked. (We met with Tony when he visited to Russia a year earlier. His visit was brief, but long enough to learn the basic conspiracy survival rules in Soviet Russia.)

Answer 4

Well there is C. E. Linderholm's Mathematics made difficult ("available on the internet")...

Also, if I remember well, D. Nordon's Les mathématiques pures n'existent pas! has a pretty biting parody of a Bourbaki-era seminar and/or thesis defense.

Third, K. Meyer: An application of Poincaré's recurrence theorem to academic administration (lifted from another question here).

Fourth, the definition of left- and right-circular cows in P. Jordan and R. de L. Kronig: Movements of the Lower Jaw of Cattle during Mastication.

Answer 5

In this example, a parody of mathematical writing serves a purpose which is definitely not comic, but it is so good that it deserves a mention. In 1982, during the martial law in Poland, Stanislaw Hartman, a professor of mathematics in Wroclaw, was put in an internment camp by the authorities (for being an ``extremist"). The news could not be circulated because of censorship of mail and phone calls, so a sample of abstruse ( pseudo- )mathematical language was used by his friends to communicate the fact to the outside world, in particular to mathematicians abroad. Wroclaw mathematicians continued the tradition of the Scottish Book from Lvov by establishing the New Scottish Book and publishing some of its problems in the journal Colloquium Mathematicum, so in Colloquium Mathematicum 44 (1981), the following problem (P 1217) appeared:

S. Manhart (Sany) P 1217 (Q). Consider a random walk of extreme element Hint =$ H(t)$ of the solid category $S$. The process develops within a rectilinear 3-cell $N$ whose boundary $\partial N$ is connected and closed. Estimate the expectation of $T_\varepsilon =\inf \{t > 0: H(t) \notin N\}$.
Letter of January 4, 1982: P 1217 (Q), R1. In the Manhart case, $T_\varepsilon$ turned to be $2^5+1$ (letter of February 6, 1982). In other cases the problem is still open."

Here is the explanation from an article by Roman Duda on the New Scottish Book (whose English translation can be found here: http://kielich.amu.edu.pl/Stefan_Banach/e-duda.html)

The alleged S. Manhart (Sany) is S. Hartman (Nysa) whose supposed letter of 4 January reminds the reader that since that day he is on 'a random walk (...) inside a rectilinear 3-dimensional cell $N$, whose boundary $\partial N$ is connected and closed' in the internment camp in Nysa . The time of his internment was to be deduced from $T_\varepsilon =\inf \{t > 0: H(t) \notin N\}$.
In an update it could be noted that in his case the time was $2^5+1$ (= 33 days) but `in other cases the problem is still open'.

Answer 6

There is the truly wonderful Mustard watches: an integrated approach to time and food by "Y.-J. Ringard" (Jean-Yves Girard).

http://girard.perso.math.cnrs.fr/mustard/article.html

Answer 7

Topologische Differentialalgebra. [Update: Site seems to be down. Drat. See link in Martin Brandenburg's comment for an archived version.] This is a script about a (hitherto) non-existing field of mathematics which six grad students at the University of Münster wrote in the 1990s, apparently during "seminar" meetings at a local pub. They managed to get an announcement of the lecture into the official course catalogue of the University for two semesters in a row. It has quite a legendary status at the institute in Münster, but since recently a physicist from Bonn mentioned it to me, I realised its fame has spread.

One chapter opens up new areas of graph theory, starting with the definition of trees, then forests, as well as lightnings, termites, and the crucial invariant of "beavericity". Important theorem: "After finitely many autumns every forest is defoliated". -- Many jokes rely on puns, some of which would work in English too (e.g. a section on "pope numbers"), some of which rely on German math terminology and sometimes even local knowledge. E.g. there is an elaborate discussion of "autos" (automorphisms, but in German "Auto"="car") switching "orbits" (the German word "Bahnen" also meaning "lanes") on certain rings, especially the famous Rishon-Le-Zion-Ring (a ring road with heavy traffic in the city of Münster), which culminates in the definition of "accidents" ...

The true highlight and reason for its notoriety is the chapter about "rings with 17", a property which (annoyingly and on purpose) is not exactly the same as having characteristic $\neq 17$. Accordingly, they manage to go into a case distinction about whether or not some parameter is $17$ in almost every proof, sometimes admitting that the cases can be handled quite similarly, sometimes leaving the case $n=17$ as an exercise, or often just excluding the case $n=17$ in statements. This has a nasty Trojan effect: Of course you know it's silly, but once you've read it, whenever you think about some ring theoretic statement, a voice will come up: "Does this proof work even if the ring has no 17 ...?"

Answer 8

N. J. Wildberger: Let H be a load of hogwash.

Answer 9

A note on piffles. I am not sure where it was first published; according to this page it was in the Mathematical Gazette 1967: jstor link.

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.

[... answered by] defining a Wuffle to be a reducible Biffle and he was then able to show that all Wuffles were reducible.

[...]

[...] defined a Piffle to be an infinite multi-variable sub-polynormal Woffle which does not satisfy the lower regular Q-property. He states, but was unable to prove, that there was at least a finite number of Piffles.

Answer 10

I remember picking up Whitehead and Russell's "Principia Mathemematica" as an undergraduate and finding it about as interesting as a telephone book.

You know you have something special in your hands when page 367 is about the number $1$ and looks like this:

I think that pretty much sets the bar for how abstruse mathematical writting can be.

Answer 11

The following is somehow a parody of "proof by contradiction" with an obvious educational purpose taken from the book "The Foundation of Mathematics" written by Ian Stewart and David Tall:

COMEDIAN: You're not here.

STRAIGHT MAN: Don't be silly, of course I am.

COMEDIAN: You're not, and I'll prove it to you...Look, you're not in Timbuktu.

STRAIGHT MAN: No.

COMEDIAN: You're not at the South Pole.

STRAIGHT MAN: Of course, I'm not.

COMEDIAN: If you're not in Timbuktu or at the South Pole, you must be somewhere else!

STRAIGHT MAN: Of course I'm somewhere else!

COMEDIAN: Well, if you're somewhere else, you can't be here!

Answer 12

Physicists are way ahead of mathematicians here, see here. The Stuperspace article is a classic.

Answer 13

The m-Lab is a randomly generated parody of n-Lab.

Answer 14

Might as well add this gem.

Group theory for homotopy theorists (Expository note), Krause and Nikolaus.

available on Nikolaus' website, direct download

Abstract: We demonstrate how to effectively work with the theory of groups using Quillen model structures avoiding the overly abstract definition of a group as a set with a binary operation. Our approach is highly inspired by the modern point set approaches to the category of spectra. One major advantage is that in our approach it is easy to write down examples of free groups and colimits of groups. We also use it to define the tensor product of abelian groups.

---

The great thing about this one is that in some sense the joke is really quite interesting mathematically!

Answer 15

Of course, there's this old classic: http://bjornsmaths.blogspot.com/2005/11/how-to-catch-lion-in-sahara-desert.html

Answer 16

Love and Tensor Algebra from The Cyberiad by Stanislaw Lem (translation by Michael Kandel)

Come, let us hasten to a higher plane
Where dyads tread the fairy fields of Venn,
Their indices bedecked from one to n
Commingled in an endless Markov chain!

It continues: http://www.aleph.se/Trans/Cultural/Art/tensor.html

Answer 17

There was a parody of mathematical research in the Math Monthly many years ago. I'm going to have to paraphrase, since I don't have the reference, but it went something like:

Definition: A zipple is a commutative zapple.

Theorem: The existence of non-commutative zapples would imply a non-zipple.

Etc, I'm sure the idea is clear. But the actual note was much cleverer that this. (With any luck, someone with a better memory than mine knows the exact reference.)

There's also Lehrer's "There's a delta for every epsilon", which moves into parody as it attempts to find a delta for those poor negative epsilons that are so neglected in calculus courses.

Answer 18

In the 80s I attended a conference titled ``Rigid Bodies with Flexible Attachments''. The sad part is that nobody, me included, saw anything strange about that title. It took my wife and her friends, looking at me with incredulous humor, to catch on that there could possibly be anything ... biological...meant to be construed from the title.

I could not find the conference proceedings .. maybe they changed the name: but here is a title from that period: ``Hamiltonian Structures and Stability for Rigid Bodies with Flexible Attachments, Arch, Rat. Mech. Anal., vol. 98, no. 1, pp. 71-93. ''

Answer 19

I guess John Walsh's "lost scroll", an Asterix-inspired parody of the "Séminaires de Probabilités" should appear in this list. The letter purporting to explain how the scroll was found is particularly hilarious.

Answer 20

I recommend the book A Random Walk In Science. That should keep you busy for a while.

Answer 21

Yet another entry, for what it's worth: C. Adams and S.G. Krantz, The cohomology of proofs, Math. Intelligencer 28 (2006), N3, 29-30.

Answer 22

Colin Adams often inserts mathematical doubletalk in his humorous short stories. Here's an example, from the story, A Proof of God, from the book, Riot at the Calc Exam:

Well, then we factor by the kernel of the homomorphism, yielding an abstract subvariety determined by the maximal ideal. The definition of this subvariety can be analytically continued and then completed to yield a simplicial complex in a fundamental domain for the action of the cusp subgroup of a hyperbolic orbifold commensurable with a Bianchi group of arbitrarily large discriminant. The trace field generates a dilogarithmic map that lifts to the universal cover. Quotienting out by the orientation-reversing isometries yields a manifold of Hausdorff dimension 3/2. The cohomological sheaf of this manifold allows us to prove the existence of a bilocal diffeomorphism onto the generators for the fundamental group of a CR-manifold of dimension 12. The primary obstruction to a lifting of the associated Steenrod algebra affords a means to define a weakly contractible map to the commutator. Suspending this map yields a cofibration of the associated Eilenberg-MacLane space. Taking the one-point compactification under the Zariski topology generates a moduli space that parametrizes the finitely generated quasi-Fuchsian groups of rank one. If we restrict to codimension three, we obtain an excellent ring, the localization of which is a factor field. Projecting to the generic fiber yields a Lipshitz map from the set of names to the set of all wives. When the range is restricted to just my wives, the commutativity of the map forces my first wife to have the name Gladys. And that is a contradiction.

Answer 23

It is a presentation, but I think it should count. Graduate students at Carnegie Mellon setup a "telephone game" presentation: $n $ people write $n $ beamer slides, but person $k $ only sees slide $k-1$. A separate person delivers the presentation without seeing the slides beforehand.

The result is available on youtube. http://youtu.be/XIz1XcPpcx4

Answer 24

D. Knuth's "The complexity of songs" is definitely in this category. The article contains a few gems such as

However, the advent of modern drugs has led to demands for still less memory, and the ultimate improvement of Theorem 1 has consequently just been announced: THEOREM 2. There exist arbitrarily long songs of complexity O(1).

and

It remains an open problem to study the complexity of nondeterministic songs.

Answer 25

Does mathematical physics count? http://www.landsburg.com/rasputin.pdf

Answer 26

It's not quite in the spirit for which you ask, but it's always a good time to mention Serre's How to write mathematics badly. (I'm sorry for the abysmal video quality, but you can still get much of the sense of it. Did anyone ever make a transcript?)

Answer 27

The Qualifying Exam by Richard Roth (Mathematics Magazine 38(3):166–167 (May, 1965)) mentioned in Zhen Lin's comment seems worth posting as an answer. Excerpt:

ALPHA: The candidate will please define what is meant by a continuous denominator.

CANDIDATE: Consider the set of all doubly evocative singly homologous functions on the unit sphere. Introducing a continuous group structure in the usual way we may define the Skolem uniformity of automorphic cycles to be the theta relation on all sets of measure zero and the zeta function on left ideals whose valuation is Gaussian, uniformly on compacta. Then given any cardinal predicate, the continuous denominator is the corresponding normal quaternion for which the problem vanishes almost everywhere.

BETA: Could the candidate please give an example of a non-Skolem uniformity?

CANDIDATE: I believe the inversion of the reals under countable intersections is non-Skolem… at least almost everywhere.

BETA: That’s correct. Now could you…

OMICRON: (Interrupting) I wish to contradict. It isn’t a non-Skolem uniformity since the third axiom concerning the density of the seventh roots of unity is not in fact satisfied.

BETA: Ah, yes, but you see, in my paper on toxic algebras… 1957… Journal of Refined Mathematics and Statistical Dynamics of the University of Lompoc… I showed that the third axiom need not be satisfied if the basis is countably finite and the metric is Noetherian, hence…

Answer 28

"De statu corruptionis" is something along such lines. It is available here: https://www.amazon.de/corruptionis-Entscheidungslogische-Ein%C3%BCbungen-H%C3%B6here-Amoralit%C3%A4t/dp/3922305016

The parody is not about mathematics as such, but rather about its applications to economics. The book presents a "similar" application to religion.

Answer 29

Found this wonderful specimen few days ago:

A non-judgmental reconstruction of drunken logic by Robert J. Simmons

This work builds on and extends a seemingly equally important paper which unfortunately I was not able to locate: in the references it is cited as

6. Neel Krishnaswami, Rob Simmons, and Carsten Varming. Handwaving logic. Journal of the Eighth Floor Whiteboard, December 8, 2006. Possibly erased.

My favorite quote:

A connection to linear logic would also allow us to investigate connections between the consumption of resources and the consumption of alcohol, perhaps giving a satisfactory logical justification for the truth of the “Three Tequila Proposition”:$$\textsf{tequila}\otimes\textsf{tequila}\otimes\textsf{tequila}\multimap\bot$$

The paper is accompanied by slides of some presentation, in shockwave flash. Seems to have been presented at the SIGBOVIK2007 conference organized by ACH SIGBOVIK, the Assocation for Computational Heresy Special Interest Group on Harry Quaint-costumed Bovik. At least the paper is listed in its Proceedings under the heading "Papers not yet rejected, Track I -- Psychopathology and Logic" (in fact I could present the whole proceedings here but...)

Answer 30

There is an xkcd comic with the following parody of "weirdly abstract" unsolved mathematical problems:

Is the Euler Field Manifold Hypergroup Isomorphic to a Gödel-Klein Meta-Algebreic ε<0 Quasimonoid Conjection under Sondheim Calculus? Or is the question ill-formed?

Some other xkcd comics might qualify, e.g., Fairy Tales ("the grasshopper contracted to a point on a manifold that was not a 3-sphere"), Proofs ("let's assume the correct answer will eventually be written on this board at the coordinates $(x,y)$"), and Math Paper ("I use an extension of the divisor function over the Gaussian integers to generalize the so-called “friendly numbers” into the complex plane").

Answer 31

In a spirit similar to MathGen, there is The proof is trivial!, a small website which randomly generates short snippets along the lines of:

The proof is trivial! Just biject it to a

Lebesgue-measurable variety

whose elements are

orientable generating functions

Answer 32

Meta variant: Does the Euler-Diderot incident count? E.g. here Because it is made-up?

Answer 33

A lovely example of this genre is Burritos for the hungry mathematician by Ed Morehouse, which includes such lines as "To wit, a burrito is just a strong monad in the symmetric monoidal category of food."