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For something I'm writing -- I'm interested in examples of bad arguments which involve the application of mathematical theorems in non-mathematical contexts. E.G. folks who make theological arguments based on (what they take to be) Godel's theorem, or Bayesian arguments for creationism. (If necessary I'm willing to extend the net to physics, to include bad applications of the second law of thermodynamics or the Uncertainty Principle, if you know any really amusing ones.)

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Does "applying the Banach-Tarski paradox to an orange" qualify? –  Someone Apr 12 '11 at 15:14
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Rather than Gödel's incompleteness theorem applied to theological arguments, there is Gödel's ontological proof of the existence of God (en.wikipedia.org/wiki/Gödel's_ontological_proof), which is more likely to be misapplied... –  godelian Apr 12 '11 at 15:19
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I feel like most people misapply Godel's incompleteness theorem. –  Sean Tilson Apr 12 '11 at 15:25
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Gödel's work, together with that of Church and Turing and others, does show, however, that much philosophy and theology (even stuff which is supposed to be respectable) misconceives the relationship between truth, provability, possibility, rationality, knowledge and reality. –  Mark Bennet Apr 12 '11 at 22:02
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Perhaps it was my being ignorant of algebraic topology as a kid, but splitting my sandwich with my brother did not seem to be fair! –  F Zaldivar Apr 13 '11 at 0:45
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28 Answers 28

A tragic example of this is the case People v. Collins, in which a prosecutor asked a mathematician (as an expert witness) a question of the form, "assuming these events are independent, what is the probability that...". The events were obviously not independent, things like "drives a convertible", "has a caucasian girlfriend", "girlfriend has blond hair", and some others. The mathematician answered the misleading question correctly (assuming independence), and the defendant went to jail. The California Supreme Court later overturned the verdict, in a decision that shows a surprisingly solid understanding of probability.

This case could be required reading (the supreme court decision, anyway) in any introduction to probability course. It has counting, independence, and conditional probability all involved in a fundamental way.

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And one of the judges dissented?! –  Mariano Suárez-Alvarez Apr 13 '11 at 18:57
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Very interesting! I'd heard of the case before, but I never knew the verdict was overturned. Arguably, the best-known (and most awful) example of this kind of thing is the Sally Clark case. The Royal Statistical Society wrote a very good public statement about it. The statement is posted on the RSS web site, but unfortunately the link seems to be broken. –  Vectornaut Apr 13 '11 at 23:03
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From the Supreme Court's decision: "Mathematics, a veritable sorcerer in our computerized society, while assisting the trier of fact in the search for truth, must not cast a spell over him." –  Beren Sanders May 31 '11 at 19:42
    
Eric, maybe you mean 1-(1-1/10000)^20000 ? –  Kevin H. Lin Jun 1 '11 at 23:49
    
It is worth looking at the multiple testing example under the Prosecutor's Fallacy: en.wikipedia.org/wiki/… This explains another way that data in the court room can be misinterpreted. –  Eric Naslund Jun 2 '11 at 1:11
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Here are some examples, ranging from the comical to the debatable.

Comical: Pretty much any mention of mathematics in Jacques Lacan. To give you an idea, here is a typical passage:

This diagram [the Möbius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. [Lacan (1970), pp. 192-193]

And here's another one:

Thus, by calculating that signification according to the algebraic method used here, namely $$\frac{S(\text{Signifier})}{s(\text{signified})} = s(\text{the statement})$$ with $S=(-1)$ produces $s=\sqrt{-1}$[...]Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the form of an image, but as a part lacking in the desired image: that is why it is equivalent to the of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of the lack of signifier -1. [Lacan (1971); seminar held in 1960.]

Interesting/Rigorous but still quite a stretch: The work of Alain Badiou on set theory, although more rigorous and advanced, also provides a very good resource for misapplications of formal mathematics in order to draw non-mathematical conclusions, cf. especially Being and Event which is his magnum opus, in which he uses set theory to support the tagline that 'Mathematics is Ontology'. Unlike Lacan, Badiou at least knows his stuff when it comes to the statement and development of formal results. That said, his interpretations and conclusions are often huge stretches.

Here's a related MO post on Badiou:

Badiou and Mathematics

Interesting/Philosophy: I don't know if you'd call these misapplications, but they are certainly attempts to use formal results to draw philosophical conclusions that are not in any formal way entailed by those results. Here are some examples:

  • Michael Dummett on how Godel Incompleteness might/might not threaten the thesis that meaning is use (philosophical anti-realism):

The philosophical significance of Gödel's theorem, M Dummett - Ratio, 1963

  • Hilary Putnam on how the Lowenheim-Skolem Theorem proves that reference is underdetermined by all possible theoretical or operation constraints (i.e. that the meaning of our mathematical vocabulary can never be accurately understood in order to fix an intended model):

http://www.jstor.org/stable/2273415

Pretty much anything philosophical that has been written about the so-called Skolem Paradox involves formal-to-informal entailments.

  • Roger Penrose in The Emperor's New Mind again using Godel to draw conclusions about consciousness and mechanism
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Great answer -- and your mention of Lacan reminds me that, while it's not quite about a THOEOREM, the Lang-Huntington affair is certainly a good example of what I'm looking for! –  JSE Apr 12 '11 at 15:54
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Lacan featured prominently in Sokal's hoax. –  Steve Huntsman Apr 12 '11 at 18:04
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... Given how frequent misunderstanding of words and even whole passages of text are, I would not be too surprised if one day somebody manages to misunderstand a whole language. –  darij grinberg Apr 13 '11 at 8:02
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I've looked at Badiou's work and I'd put it in your third category of "Interesting/Philosophy." These don't strike me as misapplications. It sounds to me that you're tacitly assuming that it is always illegitimate to use formal results to support a philosophical thesis unless the formal results formally entail the thesis, but this is itself a controversial philosophical stance. I'd reserve the term "misapplication" for situations where the applier misunderstands the mathematical result, or gets the technical details wrong, or claims that something follows formally when it doesn't. –  Timothy Chow Apr 13 '11 at 21:55
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One of my collegues, Lucien Guillou, told me that he was asked to give lessons in topology, especially knot theory, to psychanalysts of the Lacanian sort. One of the reason for their interest was the Borromean rings which they took for a illustration of the link between body, spirit and soul. Take one out and the remaining two fall apart. –  Roland Bacher Jun 1 '11 at 6:15
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My favourite in this direction is an application of Noether's theorem to public relations: Sha, "Noether's Theorem: The Science of Symmetry and the Law of Conservation", J. Public Relations Research, 16 (2004) 391-416.

I quote from the abstract:

Noether's Theorem shows that symmetry-or change-can only exist simultaneously with conservation or invariance. For public relations, the implication is that an organization can behave "symmetrically" while maintaining certain beliefs, principles, or purposes that will never be relinquished. A case study of the Democratic Progressive Party (DPP) on Taiwan using participant observation (13 months), qualitative interviews (n = 22), and a quantitative survey (n = 166; response rate = 28.77%) showed that the organization exhibited symmetry by reaching out to external publics, engaging in dialogue with them, and expressing openness regarding Taiwan independence. Simultaneously, the party conserved its interests in gaining power and establishing an independent Taiwan. Recent electoral victories of the DPP suggest the effectiveness of symmetry-conservation for public relations practice.

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This is amazing. I find it hard to believe it's not a joke, but the paper actually seems to have been cited a few times, with no indication that it is being read as anything other than at least a serious analogy with physics (for example, in "New media and public relations" by Sandra Duhé; see books.google.com/books?id=n6hyFnSRkEwC&pg=PA8). –  Henry Cohn Apr 12 '11 at 21:08
    
I agree, this "application" is extremely laughable. I was expecting a million answers about Godel's theorems, but Noether's theorem applied to public relations? grooooaaaaaan..... I guess the problem is in thinking that our definition of "symmetry" is the same as their definition of "symmetry" –  William Feb 24 '12 at 2:28
    
From Bey-Ling Sha's biography at San Diego State University: "Dr. Sha's primary research program combines theories of mathematical physics with public relations scholarship. Her other research areas include international public relations, activism, cultural identity, gender, and health communication. Her research has been published in Journal of Public Relations Research, Public Relations Review, and Journal of Promotion Management, as well as various book chapters." Needless to say, she has a PhD in mass communication, not mathematical physics. –  Tom LaGatta Apr 29 '12 at 6:36
    
This is actually very typical for social sciences courses. I've seen a number of required books with titles like "quantum leadership" that start by misquoting physics or math and then pretend to apply the misquoted concepts. –  Michael Jun 12 at 14:38
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This is not an answer. Just a very long comment. Mostly I am stunned by the answers given.

(1) I'm surprised to see Lacan featured as the main example. What I see in these quotes is an attempt to formalise human condition. Is it laughable? Yes! But no more that 16th century physics and widely taken as such. I'm pretty sure 99,9% of the human population never heard of Lacan and was never affected by his thoughts on maths in any way.

(2) If I was in the audience for a talk on "Theorems misapplied to non-mathematical contexts" I'd selfishly want to see examples that affected me or someone I know. Amazingly, none of the answers given until now mentionned the field of ECONOMICS. Some people in this field are passing opinions (often political) for mathematical facts every day and this translates into policies that have influence on the lives of millions (if not billions) of people.

Just an example. When the subprime mortgage buble exploded, we heard most banks and insurance companies were shocked because "their experts(*) said the price of houses couldn't go down everywhere in the US at the same time". In fancier terms, it was widely believed that the use of Collateralized Debt Obligations (CDO) and Credit Default Swaps (CDS) were minimizing the risk of default while it was actually just spreading and increasing it. I am very ignorant in mathematical finance but I'd like someone to try and explain to me which theorems that was based on. I'm pretty sure this should go straight to the top of the list.

(*) I used the word "experts" as a generic word for "economists and mathematicians employed by financial institutions".

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The "experts" also more or less assumed that mortgage failures were events independent from each other... –  Thierry Zell Jun 1 '11 at 0:14
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Not sure where you heard that from, but I've never heard of someone in credit products assuming defaults are uncorrelated. But I feel this is getting off-topic. People build models based on assumptions that approximate reality, sometimes the assumptions are good and sometimes not. It would be unfair to say that is misapplying math--if anything, math is the science of making conclusions from assumptions! I don't think people claimed to have theorems saying mortgage delinquencies couldn't double, but they might have had models assigning 0 probability to that due to the assumptions. –  Luke Gustafson Jun 1 '11 at 5:58
    
@Luke: I suggest you try to explain to your Greek friends the subtle distinction you drew in the last sentence of your comment. You know, like, in terms of actual consequences for their real lifes... –  Did Jun 1 '11 at 6:46
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But Didier Piau, while it might be true that the distinction Luke Gustafson makes does not have much practical influence, I think it does make quite a difference for the question at hand. Consider something else and made up: If a bridge colapses because a mathematician used the wrong PDE or an illsuited solver or whatever to compute the static it is on-topic here, if the math. was told the max load will be 1000 tons and s/he should compute with a marging of safety for 1200 t. but then for some reason there where 1500 t. on that bridge and it collapsed then it seems off-topic here. –  quid Jun 2 '11 at 13:20
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A couple of misapplications of physics come to mind:

Conservation of angular momentum does not mean what people think it means. If you have an object spinning on a flat surface, it can't turn around without outside forces, right? Wrong, the rattleback toy does this (video).

The Coriolis effect is real, but the idea that this has something to do with the direction water spins down the drain is a false urban legend.

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That's neat. Thanks for the link to the rattleback. –  Willie Wong Apr 12 '11 at 20:46
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Evidently I don't understand conservation of angular momentum. I am very confused. –  Harry Altman Apr 12 '11 at 23:29
    
Well, perhaps my presentation was misleading. Rolling is complicated, and there is a transfer of angular momentum from the rattleback to the surface. –  Douglas Zare Apr 13 '11 at 1:30
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On a related note, this reminds of R. Montgomery's paper "On the Gauge Theory of a Falling Cat", which use some fairly high powered mathematics to show how it is possible for a falling object (ie a cat) to spin around and reorient itself without acting on any outside forces. The same technique is actually used in satellites to reposition themselves in outer space! (Link to PDF: count.ucsc.edu/~rmont/papers/cat.PDF ) –  Mikola Apr 13 '11 at 18:41
    
I have been taught that friction and normal reaction are both outside force and that the sum of one vertical vector and one horizontal is just anything you want, so if applied together away from the center of gravity, these two can create any torque you fancy. Of course, the difficulty is in making them to coordinate in an interesting way, but that has little to do with any conservation laws. –  fedja Jun 24 at 15:08
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There are very many examples of the misuse of probability arguments in legal cases. See e.g. the Prosecutor's fallacy.

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The link is broken. –  John Bentin Apr 12 '11 at 17:29
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I fixed the link. –  Nate Eldredge Apr 12 '11 at 17:55
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So there is no more chance than one in a million that person has these characteristics - and a person of this kind committed the crime. So it must be the person in front of you, how could it possibly be anyone else? But the population of the nation is 60 million. What distinguishes this person from the other 59 (on average) who have the same profile? –  Mark Bennet Apr 12 '11 at 21:56
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As you mentioned, an often misapplied mathematical statement is Heisenberg's uncertainty principle, which for me, as a reader of Chriss-Ginzburg, is the purely mathematical statement that any subvariety of classical phase space ($\mathrm{Specm}(\mathrm{gr}A)$) that arises from a noncommutative system of equations (an ideal in A) is coisotropic. The Encyclopedia of Science and Religion states:

There has also been an interest in using quantum uncertainty, and the breakdown of rigid determinism that it ensures, to defend the concept of free will and to provide a channel for divine action in the world in the face of unbreakable laws of nature.

I've come across this often in religious discourse- the claim that the uncertainty principle states that "everything is uncertain" and that therefore the laws of nature are subject to the decisions of G-d. I've heard it freely confused with the "law of relativity", which apparently states that "everything is relative". Moreover, some anthropologists cite Heisenberg's uncertainty principle as follows:

In social situations, too, the simple presence of an observer - an anthropologist at a tribal ceremony, a news reporter at a schoolboard meeting, or a TV camera in a courtroom - generally influences the course of events to some uncertain degree as they are recorded. The distortion that results from measurement or observation is called the Heisenberg Effect as in “No one does or can do the same thing on stage that he does unobserved...”
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This answer does not make it obvious, to me, that the uncertainty principle cannot be applied to religion in the way suggested. I would guess that, at least, most attempts to make this application are flawed, but the surely the physically interpreted version, not the abstract mathematical one, is the uncertainty principle in question? –  Charles Staats Apr 13 '11 at 20:32
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I don't see why $\Delta x \Delta p \geq \frac{\hbar}{2}$ would have any more application to religion that the formulation which I gave... I would doubt that most non-scientists are familiar with either formulation, nor do they mean either formulation when they cite it (although it would be entertaining if they did). Rather, it's turned into "everything is uncertain" or "the presence of an observer influences what is being observed". –  Daniel Moskovich Apr 13 '11 at 22:39
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Daniel, I voted down this answer for two reasons: (1) As Charles tried to tell you, your first example is, at best, an example of misuse of a physical, not mathematical result, and therefore does n;t answer the question as asked. (2) But actually, this is not a misuse at all, since the question of determinism is relevant to the old philosophical/theological debate on free will. Indeed, for centuries, the main argument against free will was based on the syllogism "if the world is deterministic, free will is impossible", which was roughly justified as follows: "if the present state... –  Joël Oct 10 '11 at 22:45
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...of the world determines the future state, there is nothing the will can change about the future". This line of reasoning was used by both scientific and religious people, with the determinism of Newton's law of physics as one of the way to justify that the world is indeed determinism. Now the fact that the formulation of Quantum Mechanics is not deterministic surely undermines this argument. –  Joël Oct 10 '11 at 22:49
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The uncertainty principle doesn't have anything to do with determinism (or the lack thereof)! It just implies our inability to measure with arbitrary accuracy two quantities at the same time. Quantum mechanics of course "non deterministic", but non-determinism is built into the theory: it is not a consequence of the uncertainty principle. –  Qfwfq Dec 24 '11 at 22:42
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The original question, and several of the answers, refer to misuse of Godel's work, but with very few specific citations. For these, I would suggest Torkel Franzen's book, Godel's Theorem: An Incomplete Guide to its Use and Abuse.

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Oh yeah! Should have said I have Franzen's book at hand. I recommend it. –  JSE Apr 13 '11 at 1:28
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Franzen's book is excellent and a quick read: it is not an exhaustive list of the abuses in questions (or it would be longer!) but it does cover some common misconceptions and high profile cases (e.g. Penrose). More importantly, it does an excellent job of going straight to the problem, and makes the subtleties of the technique quite accessible to a non-logician but mathematically sophisticated reader. –  Thierry Zell Apr 13 '11 at 1:34
    
I was more referring to misuse in conversation with artsy people who are in love with "What the Bleep do we know?" –  Sean Tilson Apr 14 '11 at 4:33
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Is the book incomplete as a consequence of Godel's theorem? –  Asaf Karagila Jan 7 '12 at 14:10
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Arrow's theorem is often glossed as "there is no good voting system".

Press' paper Strong profiling is not mathematically optimal for discovering rare malfeasors has been misinterpreted by the popular press as a mathematical endorsement of certain politics, though that's perhaps due in part to the intentional framing of the problem by Press.

Goedel's theorem is misapplied arguably more than it is used properly.

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How exactly is that a missapplication of Arrow's theorem? It is certainly a valid interpretation. –  Michael Greinecker Feb 24 '12 at 3:14
    
@Michael: I haven't time to go into specifics at the moment, but a much better gloss of Arrow's theorem is "IIA is an unreasonable condition in ordinal voting systems"; consider the 3-voter case and any majoritarian system. –  Charles Feb 24 '12 at 4:31
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  "Therefore, socialist economy is impossible, in every sense of the word."

Robert Murphy comes to this conclusion in Cantor’s Diagonal Argument: An Extension to the Socialist Calculation Debate.$^1$

The debate is over whether a Central Planning Board can, even in theory, correctly price goods and services, as it is assumed a market economy can. Socialists such as Dickinson argued that a market economy can, in principle, be simulated by the Board, even if it means solving a large system of simultaneous equations. Hayek, on behalf of the Austrians, agreed, yet maintained the number of equations—presumably one for each product and potential product—is clearly too large in practice. Both sides claimed victory.

In the cited article, the author takes the ball from Hayek and carries it across the goal line: after a decent three-page explanation of the diagonal argument, Murphy concludes the Planning Board’s task would not merely be impractical, but fully impossible because of the requirement to publish an uncountably infinite list of prices.

I suppose if one started with the assumption there are (at least) countably infinite number of products/services $p_1, p_2, \dots$ and also agreed that any possible subset of these products is again a product itself, the price of which is not necessarily the sum of the component prices (let’s ignore issues of convergence!), then one could conclude using Cantor’s Theorem ($2^S>S$) there are an uncountable number of products the Board must “list”. But I’m not sure why, if we take the listing process literally, it matters how large the infinity is.

$^1$THE QUARTERLY JOURNAL OF AUSTRIAN ECONOMICS VOL. 9, NO. 2 (SUMMER 2006): 3–11

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The "No free lunch" (NFL) theorem from mathematical optimization was used by William Dembski to disprove Darwinian theory of evolution. (The relevance of NFL's theorem to evolution was proposed earlier by Stuart Kauffman.)

Olle Haggstrom wrote a paper debunking Dembski's argument. (Here is an early version with stronger rhetorics.)

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Alan Sokal's Book deserves some mention if we are talking about misuse of theorems.

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This isn't exactly what you asked for, but I find it so amusing I could not resist.

The Indiana $\pi$ bill, when they almost passed a bill claiming that $\pi=3.2$, in order to be able to square the circle.

Unbelievable.

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The bill did not say pi is 3.2, it was actually far too incomprehensible to infer any specific value of pi. –  Michael Renardy Apr 12 '11 at 18:11
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One interpretation I read was pi = 9. –  Allen Knutson Apr 13 '11 at 0:44
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One interesting thing about this bill is that it was not introduced to legislate on what the value of $\pi$ should be (an easy way to misunderstand the story), but rather in order to copyright of a method to square the circle for exclusive use free of charge by the State of Indiana. What the legislature thought they could use this for escapes me. –  Thierry Zell Apr 13 '11 at 1:44
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Apparently (according to Wikipedia) Goodwin had also proven such "truths" as trisecting a given angle, and had them published in American Mathematical Monthly, with the disclaimer 'published by request of the author.' Although this happened in the late 1800s, it makes me skeptical of ALL published mathematical results... –  William Feb 24 '12 at 2:42
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In order to baffle the uninitiated, some authors interpret Banach-Tarski paradox (stating that "it is possible to decompose a ball into five pieces which can be reassembled by rigid motions to form two balls of the same size as the original.", cf. http://mathworld.wolfram.com/Banach-TarskiParadox.html) in an obviously false way as if it could be applied to physical objects. E.g. Reuben Hersh writes (Reuben Hersh: "What Is Mathematics, Really?" p.255):

"Stefan Banach and Alfred Tarski proved, using the axiom of choice, that it's possible to divide a pea (or a grape or a marshmallow) into 5 pieces such that the pieces can be moved around (translated and rotated) to have volume greater than the sun."

Clearly, this formulation is very much misleading, since it suggests that the paradox can be applied to a physical objects, which is obviously false. Indeed, the construction is such that the ball is divided into non-measurable parts and, clearly, there is no physical objects corresponding to non-measurable sets.

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Why is it clear that there are no physical objects corresponding to non-measurable sets? For the same reason that "Hilbert Space" is a purely abstract mathematical construct with no utility in physics? Oh, wait... –  Igor Rivin May 31 '11 at 19:51
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The quotation from Hirsch also mixes two versions of the Banach-Taski theorem. The number 5 of pieces is, if I remember correctly, for making two balls the same size as the original. To get from a pea to the sun, more pieces would be needed (but still only finitely many). –  Andreas Blass May 31 '11 at 20:42
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I'm really bothered by the wording "applying the theorem to physical objects": a theorem is a mathematical statement, it can be applied to a mathematical object in the course of a proof, but talking about applying it to physical objects does not even begin to make sense. –  Thierry Zell Jun 1 '11 at 0:01
    
@IgorRivin: Because there is no constructive way of building a non-measurable set. –  Martin Hairer Jun 25 at 20:58
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This could be an unfair example, since I don't know the text myself. All I can say is that my skepticism is aroused just by the title of

  • Guerino Mazzola, The Topos of Music: Geometric Logic of Concepts, Theory, and Performance (Birkhäuser, 2002)

(in other words, topos theory applied to music theory). At least one MO participant at MO (Mikael Vejdemo Johansson) has tried to read this book and came away feeling skeptical, according to his remarks here. I'd be interested in hearing other reactions from people who have taken a stab at it.

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I waited a while to say something on this as I do not really feel qualified, but as nobody else said anything so far: in view of other, traditional, mathematical work of the author, it seems highly likely to me that if this deserves to be on the list at all then only in the category 'math is solid, but for some reason inapplicable/not relevant to the application.' Now, whether the latter is the case or not is perhaps hard to tell as (I assume) 'theory of music' is not a 'hard' subject with a clear right or wrong. Based on a talk I heard years ago I remember that using this theory one... –  quid Apr 13 '11 at 11:46
    
...can make concrete assertions. Specifically (the details I remember are vague and my general musical is insufficient) there was some investigation carried out whether a certain sequence A of notes constitutes the motif (in the musical sense of a certain well-known piece of music) or whether it is a sequence A'. Using this theory an anwer was given; somebody with a music background in the audience disagreed with this answer, but it was my understanding that regarding this question there is debate in the music comm., I guess somebody else might have agreed. So, not sure what this tells. –  quid Apr 13 '11 at 11:49
    
Thanks, unknown. I would love to get my hands on the book, even if my knowledge of music is not up to the task of deciding whether this is a worthwhile investigation. I am not challenging the mathematical competence of the author, by the way. Hopefully Mikael will see this sometime and share some of his thoughts on the subject (he wrote a review that was rejected for being overly harsh, even if admired within the publishing office). –  Todd Trimble Apr 13 '11 at 12:45
    
You might find the book on the internet... –  Michael Bächtold May 31 '11 at 19:40
    
Thanks, Michael. In fact, someone I know sent me an internet copy. –  Todd Trimble May 31 '11 at 21:57
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The whole "transformation" and "network centric warfare" push in the US Department of Defense last decade under Cebrowski and Rumsfeld invoked a heap of dubious interpretations and purported applications of nonlinear phenomena (perhaps most notably when 9/11 was referred to as a "system perturbation"). See here for an introductory overview.

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This is a wonderful and fascinating still life by Juan Sanchez Cotán: http://www.friendsofart.net/static/images/art1/juan-sanchez-cotan-still-life-with-quince-cabbage-melon-and-cucumber.jpg

It is thought by many art historians that Cotán used a mathematical formula to determine the heights at which the various items would appear. For all I know this may be the case -- it would seem only appropriate given the name of the artist -- but I once read part of a book by a very respectable art historian (whose name I have maddeningly forgotten but I'm working on it) who said what the formula was. His evidence was just the picture itself and not any surviving record of how it was painted. But of course, given that the heights of the items are not precisely determined (anything like), it is clear that any number of curves could be declared to fit. This is not exactly misuse of a theorem but it was certainly misuse of mathematics, similar to finding the golden ratio everywhere but a bit more sophisticated.

Added: I've tracked it down now. The critic is Norman Bryson and he says this: "In relation to the quince, the cabbage appears to come forward slightly; the melon is further forward than the quince, the melon slice projects out beyond the ledge, and the cucumber overhangs it still further. The arc is therefore not on the same plane as its co-ordinates, it curves in three dimensions: it is a true hyperbola, of the type produced when a cone is viewed in oblique section." I haven't found more of the quotation, but I seem to remember that it was quite important to Bryson that it really was a hyperbola and not, say, an exponential decay. (As a matter of fact, looking at the picture again I am not convinced that the items form a nice curve of any kind: the cabbage is too far to the left and too near to being directly under the apple. And the relationship of the string of the cabbage with the leaves of the apple leads me to doubt whether the curve lies in an oblique plane, or indeed any plane, as he suggests.)

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Heard in high-school History class, no reference unfortunately: in the early 20th century, someone published a monograph on the dimensions of a certain small building; derived many important constants from basic operations on said dimensions, showing the intent of the architects. The building in question was... ...a public urinal! –  Thierry Zell Jun 1 '11 at 0:12
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It's physics rather than math, but surely this creative paper by Alan Sokal deserves mention.

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I submit, to your consideration, this paper by Frank Tipler, Professor at Tulane University. The paper was published in the peer-reviewed Reports on Progress in Physics, volume 68 (2005), pages 897-964. Tipler's book "The Physics of Christianity" is based on this paper.

Tipler invokes Gödel's theorem (see p. 905 onwards), Presburger arithmetic, Löwenheim-Skolem, Hales' proof of the Kepler conjecture (the latter only as an example, I believe), and various other mathematical results.

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Wow, do you think he's making a bid for the Templeton Prize? –  gowers Jun 2 '11 at 7:38
    
If he does, it's a slam dunk. –  Alon Amit Jun 3 '11 at 3:50
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Sokal once again, with Brown and Friedman, wrote this paper: The complex dynamics of wishful thinking: The critical positivity ratio (arXiv version). The story behind this is that Nick Brown, "who began a part-time psychology course in his 50s – and ended up taking on America's academic establishment" according to Andrew Anthony in the guardian http://www.theguardian.com/science/2014/jan/19/mathematics-of-happiness-debunked-nick-brown.

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Not really a theorem but amusing non-sense. Somebody (it was perhaps Sokal) told me about a psychanalytical book based on set theory. The author wrote it in English and translated the french terminology "th\'eorie des ensembles" as "Theory of the (w)hole". The book was later translated into French with the title "Th\'eorie des t(r)ous".

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A rare instance of Gödel-abuse in a published paper is "Bacterial wisdom, Gödel's theorem and creative genomic webs" by Eshel Ben-Jacob. Here, Gödel's theorem is used to prove that "a system cannot self-design another system which is more advanced than itself", with application to genomics.

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Russian media provide a lot of amusing examples. Let me mention two:

1) (Perelman's proof of) the Poincaré conjecture leads to understanding the shape of the Universe;

2) (this is maybe what you mean in the post) it follows from the Godel's theorem that God does not exist.

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I'm not altogether convinced by your first example: see mathoverflow.net/questions/9708/… –  Daniel Moskovich Apr 14 '11 at 0:24
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In his book Everybody for Everybody, Samual A. Nigro argues that Gödel's theorems not only cast doubt on the theory of evolution, but prove the doctrine of original sin, the need for sacrament and penance, and that there is a future eternity.

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In the same vein as the bayesian argument for creationism and misapplications of Gödel's incompleteness theorems, there are misapplications of the second law of thermodynamics against evolution of life ("undesigned", e.g. darwinian or lamarckian).

The second law is a mathematical consequence of Hamilton and Schrödinger equations for reasonable hamiltonians, in particular of fundamental physical evolution equations, and also of simple statistical models (statistical ensembles). See Wikipedia.

The argument is that life is complex and evolution implies a decrease in entropy/increase in complexity contradicting the second law. See for instance here.

The flaw is that the Earth, where evolution occurs, is not an isolated system. If we consider rather the solar (or just Sun-Earth) system there is loss of entropy on Earth but a compensating gain on the Sun.

For a recent anecdote (and a nice blog to add to your blogroll) see Retraction Watch.

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This reference is an excellent parody of the so-called application of mathematics to economics and other social sciences (it purports to apply mathematics to theology): http://www.amazon.de/corruptionis-Entscheidungslogische-Ein%C3%BCbungen-H%C3%B6here-Amoralit%C3%A4t/dp/3922305016

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This is my favourite example.

From the text:

The Mandelbrot set provides a fractal representation of how these unique individuals provide self-similarity within the larger intrinsic case. This theme, in particular, focuses on how these individuals’ experiences with change contribute to the overall stress within the larger far-from-equilibrium system

Authors try to analyze how librarians work, by making an analogy with fractals. Also, the obligatory reference to Heisenbergs uncertainty principle.

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I call your attention to http://www.abarim-publications.com where you will find the book, Quantum Mechanics for Beginners; an Introduction with the blurb,

Quantum Mechanics studies the peculiar world of the "ones"; those things in nature that can not be divided. Since God is a One, and the Body of Christ as well, it shouldn't be surprising that the Bible discusses the "ones" at length, and this a few millennia before the emergence of Quantum Mechanics in the scientific arena. To appreciate this unexpected dimension of the Bible, Abarim Publication's fun-filled crash course in Quantum Mechanics should be mandatory at every seminary.

Also, Chaos Theory for Beginners; an Introduction:

Chaos Theory looks at patterns and their reoccurrence in nature. Since Moses built the tabernacle - which would turn into the temple, and later still in the Body of Christ - after patterns he saw in heaven, Chaos Theory is a must for every serious student of the Bible.

One of the chapters is entitled, Agape and Gravity Live Together in Perfect Harmony. Fans of Stevie Wonder may see a pun there. There is also Scripture Theory for Beginners; an Introduction:

What Chaos Theory does with nature, Scripture Theory does with Scriptures: the identification of reoccurring patterns and their meanings. Especially interesting are those Biblical patterns that are identical to those found in high-energy physics.

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