What are examples of math hoaxes/interesting jokes published on April Fool's day?
For a start P=NP.
Added 2017-04-01 Anything new in 2017?
What are examples of math hoaxes/interesting jokes published on April Fool's day?
For a start P=NP.
Added 2017-04-01 Anything new in 2017?
I enjoyed the hexasphere by
A. V. Akopyan, J. Crowder, H. Edelsbrunner, R. Guseinov
from last year:
http://pub.ist.ac.at/~edels/hexasphere/
In the link, the sphere is animated, so you can look at it from all sides.
In 2009, it was discovered that the numerical value of $\pi$ has changed over time. This is a truly interdisciplinary work connecting the study of ancient cultures with string theory, cosmology and bicycle tires. Let me quote from the introduction:
Physicists have long speculated that the fundamental constants might not, in fact, be constant, but instead might vary with time. Dirac was the first to suggest this possibility, and time variation of the fundamental constants has been investigated numerous times since then. Among the various possibilities, the fine structure constant and the gravitational constant have received the greatest attention, but work has also been done, for example, on constants related to the weak and strong interactions, the electron-proton mass ratio, and several others.
It is well-known that only time variation of dimensionless fundamental constants has any physical meaning. Here we consider the time variation of a dimensionless constant not previously discussed in the literature: $\pi$. It is impossible to overstate the significance of this constant. Indeed, nearly every paper in astrophysics makes use of it. [..]
In 1975 Martin Gardner produced a map with 110 regions which he claimed required five colours:
Someone (widely believed to be Henri Darmon) circulated the following email on April Fools' Day, 1994:
There has been a really amazing development today on Fermat's Last Theorem. Noam Elkies has announced a counterexample, so that FLT is not true after all! His spoke about this at the Institute today. The solution to Fermat that he constructs involves an incredibly large prime exponent (larger that 10^20), but it is constructive. The main idea seems to be a kind of Heegner point construction, combined with an really ingenious descent for passing from the modular curves to the Fermat curve. The really difficult part of the argument seems to be to show that the field of definition of the solution (which, a priori, is some ring class field of an imgainary quadratic field) actually descends to Q. I wasn't able to get all the details, which were quite intricate...
So it seems that the Shimura Taniyama conjecture is not true after all. The experts think that it can still be salvaged, by extending the concept of automorphic representation, and introducing a notion of ``anomalous curves" that would still give rise to a ``quasi-automorphic representation".
The email reached Gian-Carlo Rota at MIT, who took it at face value and circulated it more widely. Eventually David Feldman posted it to the Usenet group sci.math. The thread is here.
This one is my favorite (especially a mixture of anyons and morons with opposite spins):
> From: Enrico Bombieri <eb@IAS.EDU> Tue, 1 Apr 1997 12:35:12 -0500
> Date: Tue, 1 Apr 1997 12:35:12 -0500 To: eb@IAS.EDU,
> zeilberg@euclid.math.temple.edu
>
> Dear Doron,
>
> There are fantastic developments to Alain Connes's lecture at IAS last
> Wednesday. Connes gave an account of how to obtain a trace formula
> involving zeroes of L-functions only on the critical line, and the
> hope was that one could obtain also Weil's explicit formula in the
> same context; this would solve the Riemann hypothesis for all
> L-functions at one stroke. Thus there cannot be even a single zeroe(1)
> off the critical line!
>
> Well, a young physicist at the lecture saw in a flash that one could
> set the whole thing in a combinatorial setting using supersymmetric
> fermionic-bosonic systems (the physics corresponds to a near absolute
> zero ensemble of a mixture of anyons and morons with opposite spins)
> and, using the C-based meta-language MISPAR, after six days of
> uninterrupted work, computed the logdet of the resolvent Laplacian,
> removed the infinities using renormalization, and, lo and behold, he
> got the required positivity of Weil's explicit formula! Wow!
>
> Regards also from Paula Cohen. Please give this the highest diffusion.
> Best,
>
> Enrico
>
>
> (1) This is the correct spelling, according to vicepresident Dan
> Quayle.
> --------------------------------------------------------------
[only borderline mathematical]
Today Ali Frolop and Douglas Scott published a paper (http://arxiv.org/abs/1603.09703) in which they found:
... there is a remarkable correspondence between each type of peculiarity in the digits of π and the anomalies in the CMB.
https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant
This concerns the number $e^{\pi\sqrt{163}}$, and says "In a 1975 April Fool article in Scientific American magazine,[7] "Mathematical Games" columnist Martin Gardner made the (hoax) claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name."
The punchline is tied into complex multiplication, though I don't know the details.
Well, there's the April 1, 1997 paper by Doron Zeilberger, The Transcendence of E plus Pi and E times Pi (the following quote is snipped a bit; full text available at the link).
The purpose of this note is to announce that Hermite's[H] celebrated result that $e$ is transcendental, combined with an amazing (but apparently overlooked) statement of Goodwin[G], imply the transcendence of both $e + \pi$ and $e \pi$.
But even more interesting than the above implication is the way by which it was arrived, via computer-generated deduction.
We first developed a C-based meta-language, MISPAR, that has built-in number-theoretical deduction capabilities, that inputs suitably formatted statements about numbers (especially targeted to handle transcendence theory), and outputs new statements. Then, using ten diligent graduate students, many results that appeared in papers on the subject were entered in the appropriate format. Then we used a genetic algorithm to deduce million of new results, most of them either trivial or uninteresting (or both!).
Then we made a long list of open problems. Whenever the computer made a new deduction, it was compared against the statements in the list, looking for possible matches.
While we sure hoped to obtain new interesting results, even in our wildest dreams we did not anticipate such a spectacular deduction.
We are sure that MISPAR would make many more interesting deductions in the future. The package itself, and implementation details, will be eventually published at the author's website (http://www.math.temple.edu/~zeilberg).
References
[G] E. J. Goodwin, Amer. Math. Monthly, 1 (1894), 246-247.
[H] C. Hermite, Comptes Rend. Acad. Sci. Paris, 77 (1873), 18-24, 74-79, 285-293.
The explanation is here; apparently, due to some careful phrasing, the statement of the paper is actually technically correct (or at least so says Zeilberger).
One year maybe about twenty years ago an April 1 story circulated on email, giving the news that TeX had been sold to Microsoft (and would therefore no longer be free). It included a pretty convincing firsthand account, complete with embarrassing technical glitches when Bill Gates took the stage at the grand public announcement.
Paul Taylor, who has often contributed to MO, once posted this to the categories mailing list; the original thread can be found here:
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From: Paul Taylor
Message-Id: <199904011210.NAA21705@wax.dcs.qmw.ac.uk>
To: categories@mta.ca
Subject: categories: Is Zermelo-Fraenkel set theory inconsistent?
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Status: O
X-Status:IS ZERMELO-FRAENKEL SET THEORY INCONSISTENT?
At the end of this message is a sketch of an argument that leads to the conclusion that Zermelo-Fraenkel set theory is inconsistent.
The impact on mathematics is not as devastating as the incautious observer might suppose. Recall that ZERMELO set theory (1908), which is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS with natural numbers and the axiom of choice, is adequate for most of the purposes of mathematics, though not, as I shall try to explain, logic (and theoretical computer science).
ZERMELO-FRAENKEL set theory is the extension of this system by the axiom-scheme of REPLACEMENT, which was first formulated by Adolf (later Abraham) Fraenkel, Nels Lennes and Thoralf Skolem in 1922, although Dimitry Mirimanoff already had something of the idea in 1917. Notice that this is some two decades after the appearance of the famous "antinomies" of set theory, so presumably the set theorists' guard had dropped by that time, and they had begun again to assert megalomaniac axioms. On the other hand, it is a decade before the second generation of paradoxical results, Godel's incompleteness theorem and Turing's unsolvability of the Halting Problem.
Whenever I see set theory books in a library or bookshop, I turn to the index to find out what they have to say about Replacement. Usually there is some trivial result, such as the existence of what categorists call image factorisation, that could have been proved from Zermelo's axioms with a little more facility in set-theoretic constructions.
The basic use of Replacement, that you will find in the better set theory books, is the recursive construction of sets (in substance -- types or objects to type-theorists and categorists -- rather than their names).
For example, Mostowski's theorem states that every well founded extensional binary relation < is isomorphic to the membership relation for a unique transitive set. This is found, recursively, by means of the formula f(x) = { f(y) | y < x }, which also provides the extensional reflection (quotient) of any well founded relation. In fact the latter result (where the quotient relation is merely another relation, rather than a membership relation) can be proved using the topos or Zermelo axioms alone, and not Replacement [1], although there are categorical generalisations of this that certainly do need Replacement.
Richard Montague [2] proved a result that should have been taken as a warning of the perilous nature of Replacement, though I suspect that Montague's personal eccentricities may have been the reason why he was ignored. ZF can prove the consistency, not only of Zermelo set theory (Z) itself, but also of Z extended by any single theorem of ZF.
Adrian Mathias has claimed [3] that Bourbaki was "ignorant" of Replacement, ie that it did not occur in "Theorie des Ensembles" [4]. Although Bourbaki is hardly very clear on this matter, it does include a version of Replacement in its axioms, indeed one that is in widespread use in category theory and other parts of mathematics, namely that one can form the UNION of any SET-INDEXED FAMILY of SETS.
One application of N-indexed unions in theoretical computer science is Scott's "D-infinity" construction of models of the untyped lambda calculus. Starting from any domain D0=D, one may form its function-space D1=(D0->D0), and similarly D2=(D1->D1), etc., linking these together with embedding- projection pairs. If D was one of the examples of L-domains, having a pair of elements with infinitely many minimal upper bounds, then one can show (classically) that D-infinity has the cardinality of a model of Zermelo set theory, so need not exist within such a model unless it also satisfies Replacement.
These two ways of seeing Replacement have a common theme: we use N-indexed or transfinite unions to unfold a free(ish) model of one logic within a model of another.
Having seen this in the context of a messy domain-theoretic construction, we may think in a more disciplined way about free models of the lambda calculus, the topos axioms, etc. In fact, there is no difficulty in constructing these models, as they are merely TERM ALGEBRAS. The problem lies in proving that the term algebra has the universal (initiality) property that qualifies it as "free":
Let S be the universe (the category of ZF-sets, for example) and F the term algebra (internal to S) for the logic L. Suppose that S itself is a model of L.
Then there is a unique interpretation functor []:F->S that takes each syntactic operation of F (eg prod(a,b)) to the semantics ([a] x [b]) in S.
It is merely unique up to unique isomorphism if the L-structure in S is defined by universal properties rather than being chosen.
This initiality property may also be expressed type-theoretically. Per Martin-Lof [5] introduced objects with such a property, called UNIVERSES, observing the analogy with Replacement. This point of view stresses that the above property is a RECURSION SCHEME.
Let me explain how I came to realise that the existence of []:F->S depends, in general, on Replacement.
There is an amazingly simple but incredibly powerful argument, due to Peter Freyd and known variously as (Artin-Wraith) glu(e)ing, sconing, the Freyd cover, logical relations and other names. It is based on some very elementary categorical investigations of a certain comma category involving F and S. This argument has been developed rather a long way (the most recent paper that I know of is [6]), and we are pretty close to having a purely categorical proof of the strong normalisation theorem for lambda calculi that, unlike the syntactic proofs, is completely generic with regard to the calculus in question.
Freyd originally showed that the terminal object (1) of the free topos (F) is projective, and more generally the "global sections functor" F(1,-) : F -> S preserves colimits. In particular, it preserves the initial object (0), which is categorical jargon for saying that S proves the consistency of F, because the S-set of F-morphisms 1->0 is the initial (empty) S-set.
I found this suspicious, because the punch-line of Andre Joyal's 1973 (but as yet unpublished and unavailable) categorical proof of Godel's incompleteness theorem is that such a functor F(1,-) : F -> S does not preserve the initial object.
The more careful amongst categorists ought also to be suspicious when I speak of "a functor F(1,-) or [] : F -> S" where F is an INTERNAL category in S. The meaning that we must give to this phrase is that it is "syntactic sugar" for a certain FIBRATION p: V -> F, where V is also an internal category and p and internal functor in S.
This brings us back to the relationship between Replacement as a recursive construction of objects and Replacement as infinitary colimits: "p: V -> F" is the colimit (in a 2-category whose objects are fibrations) of a recursively defined diagram vaguely similar to that which gives Scott's D-infinity.
I have come to the conclusion that attempts to define "colimits" such as this are inherently circular: what, after all, does it mean to have a "cocone" to test such an alleged colimit?
My categorical formulation of Replacement speaks about fibrations and smaller colimits defined internally in the style of Benabou. This is to be found in the final section (9.5) of my book [7], Section 7.7 of which also gives an account of Freyd's gluing construction.
This book is officially due to be published in mid-May, but it is already in stock (and I have my own copy in front of me), and is available direct from the publishers at 50 pounds (inclusive of overland postage and packing). Please contact Richard Knott, email: rknott@cup.cam.ac.uk fax: +44 1223 315 052 tel: +44 1223 325 916 (but other methods are preferable) snail: Cambridge University Press, The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 2RU, UK with your address and credit card number. (2.50 pounds extra for airmail.)
Having seen that Replacement provides a UNIFORM way of proving consistency of any fragment of logic, we come at last to the inconsistency argument:
Let L(0) be Zermelo set theory (or the axioms for an elementary topos).
For each n, let L(n+1) be L(n) plus as much of the axiom-scheme of replacement as is needed to justify the gluing construction that shows that
L(n+1) |- ``L(n) is consistent.''
Now let L(infinity) be the union of L(n) over n:N.
If L(infinity) |- false then L(n) |- false for some n.
But L(infinity) |- ``L(n) is consistent,''
so L(infinity) proves its OWN consistency, contradicting Godel's theorem.
However, L(infinity) has a standard non-trivial interpretation in Zermelo--Fraenkel set theory, which is therefore inconsistent.
[1] Paul Taylor, Intuitionistic Sets and Ordinals, JSL 61 (1996) 705-44
[2] Richard Montague, Fraenkel's Addition to the Axioms of Zermelo, pp 91--114 of Bar-Hillel, et al., eds., Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, 1966 (distributed by Oxford University Press).
[3] Adrian Mathias, The Ignorance of Bourbaki, Mathematical Intelligencer, 14 (1992) 4-13.
[4] Nicolas Bourbaki, Elements de Mathematique XXII: Theories des Ensembles, Livre I, Structures, Hermann, 1957 (English translation 1968).
[5] Per Martin-Lof, An Intuitionistic Theory of Types: Predicative part, pp 73--118 in Rose and Sheperdson, eds., Logic Colloquium '73, North-Holland, Studies in Logic and the Foundations of Mathematics #80, 1975
[6] Djordje Cubric, Peter Dybjer and Philip Scott, Normalisation and the Yoneda Embedding, MSCS 8 (1998) 153--192.
[7] Paul Taylor, Practical Foundations of Mathematics, Cambridge University Press, Cambridge Studies in Advanced Mathematics #59, xii+572pp, 1999.
http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/
Paul Taylor 19990401
This message may be copied elsewhere, ON CONDITION that it is quoted in its ENTIRETY.
(I did quote it in its ENTIRETY, including all the headers at the top!)
(Quite a few got the joke, but I think also quite a few missed it and took Paul at his word.)
The most recent additions to our Seeley G. Mudd Manuscript Library feature contributions from the estate of Oliver Wendell Holmes, Jr., which include letters and legal manuscripts of Pierre de Fermat (a lawyer by vocation). It is in the density of Fermat's litigation records during the period 1660-1662 that his lost mathematical proof is finally to be found.
It turns out that Fermat's proof employs what is now known as the Mason-Stothers theorem (proved independently by Stothers [2] and Mason [3] in the late 20th century). In the discovered manuscript, Fermat himself gave an elementary proof of the Mason-Stothers theorem, but his approach resembles that presented in An alternate proof of Mason's theorem by Snyder [4]. For this reason we here omit Fermat's proof of the Mason-Stothers theorem, and only reproduce the subsequent part of his proof of his last theorem, paraphrased in modern terminology.
More Physicsy than Maths, but there's Don Schneider's "discovery" of a quasar with redshift 4.1 (NB - the largest quasar red-shift known at that time was 3.7), announced at Institute of Advanced Study, Princeton on 1st April.
The number 4.1 was chosen to be a subtle hint, that this whole presentation was a prank. Few people got it right away, most others didn't and were particularly curious regarding the finer nuances of the discovery, which Schneider did happen to address convincingly on the course of his "report". It was a well-cooked up prank!
This is chronicled in Ed Regis - ``Who Got Einstein's Office'', Addition-Wesley (1987):
... But Schneider hands out his charts to the audience, and there's no disbelieving the data. Wavelength plotted against energy flux, the graph looks like a distorted view of lower Manhattan, with sharp peaks and valleys, and one very sharp spike, looking like the World Trade Center. That's the quasar, with its record-breaking redshift. The whole room is abuzz. People are talking to each other a mile a minute, and John Bahcall has the devil of a time moderating the question period. They want to know everything: Where's the object located? What's its coordinates? What's the exact time the observations were taken? But Don answers them all, every last one. . . until it's clear that the thing has gone far enough, and he brings it all to a close. There's another speaker to be heard from, poor fellow. Schneider is going to be one tough act to follow.
Indeed. Some people are even now getting the drift, an inkling of what's actually been going on here. A redshift of four point one, and today is April first. Can this be? . . . Oh, Jesus! It must be. And in fact, yes, it is! It's all . . . an April Fool's joke! Don Schneider has just pulled off the coup of the decade, getting the combined astrophysical brains of Princeton University, Bell Labs, and the Institute for Advanced Study to believe that in the space of a few hours in the morning, at an Institute with absolutely no observing facilities whatsoever, not even so much as a pair of binoculars, he's discovered the world's farthest object smack in the middle of a gravitational lens.
This list is missing one of the greatest mathematics April Fools, the Mandelbrot Monk article by Girvan from 1999. It is beautifully done and famous enough to have its own Wikipedia article.
I saw Doug Ravenel give a talk that began with him announcing a proof of the Riemann Hypothesis. It was beautifully done, I thought. I didn't even realize what date it was.
This is perhaps only borderline mathematical, but the ChessBase website has historically made a fine art of the April Fool's joke, and arguably its most successful prank was also the most mathematical: A claim that the King's Gambit had been solved. They were careful not to claim that the opening had been mathematically solved, but that a generous cutoff for the score function had been set (i.e., if the computer thought that one side was "far enough ahead" then the assumption was made that that side really did have a win) and that the game tree was exhausted under this assumption. It was clever enough that even after knowing that it might have been an April Fool's joke, I was still uncertain for a while.
Here is a counterexample to Fermat's Last Theorem, which is correct according to double precision calculations:
These agree to 10 of 44 decimal digits, but notice that simple divisibility rules show 3987 and 4365 are multiples of 3 so that a sum of their powers is also. The same rule reveals that 4472 is not divisible by 3, so that this "equation" cannot hold either.
$\endgroup$
– Greenonline
Apr 7 '16 at 16:37
A breathless announcement of a weakness in ubiquitous cryptography based on elliptic curves:
this result will require a major increase in parameter for elliptic curve cryptosystems ... we recommend increasing elliptic curve key sizes from 256 bits to 3072 bits
This 2016 claim seems almost plausible until one considers how sphere packing really relates to faster discrete logarithm computation. It deserves bonus points for using recent actual papers by Viazovska et al. to construct its argument.
The following is output from Maple:
A := 34816783:
B := 29698715047:
C := 120979604904878607889:
D := 103195600023374741883001:
isprime (A); true
isprime (B); true
isprime (C); true
isprime (D); true
AxD; 3592938812568633315821457205783
BxC; 3592938812568633315821457205783
AxD-BxC; 0
Thus AxD and BxC are two prime factorizations of 3592938812568633315821457205783.
This was widely distributed decades ago by a German mathematician. In the current version of Maple versions this no longer works,but a similar hoax with different numbers can probably be found.
There is also Daniel Schoch's article in which he give way how to obtain number of Gods in our universe using Euler characteristic.
The mathematician philosopher Hilary Putnam otherwise not known for lightheadedness revealed on april 1, 1980 that subtle logical phenomena in the context of the Lowenheim-Skolem theorem imply the impossibility of fixing an intended model of anything, including the natural numbers (!) and the reals. This actually got published the same year:
Putnam, Hilary. Models and reality. J. Symbolic Logic 45 (1980), no. 3, 464-482.
There is an "April Fools" issue of the Math Horizons, published in 2007:
http://www.maa.org/sites/default/files/pdf/horizonsarchive/Horizons-Apr07lores.pdf
https://arxiv.org/abs/1504.00108
A Farewell to Falsifiability Douglas Scott, Ali Frolop, Ali Narimani, Andrei Frolov (Submitted on 1 Apr 2015)
Some of the most obviously correct physical theories - namely string theory and the multiverse - make no testable predictions, leading many to question whether we should accept something as scientific even if it makes no testable predictions and hence is not refutable. However, some far-thinking physicists have proposed instead that we should give up on the notion of Falsifiability itself. We endorse this suggestion but think it does not go nearly far enough. We believe that we should also dispense with other outdated ideas, such as Fidelity, Frugality, Factuality and other "F" words. And we quote a lot of famous people to support this view.