Question

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.)

Answer 1

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.

Answer 2

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:

http://mathoverflow.net/questions/8285/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

Answer 3

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.

Answer 4

There are very many examples of the misuse of probability arguments in legal cases. See e.g. the Prosecutor's fallacy.

Answer 5

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.

Answer 6

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".

Answer 7

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...”

Answer 8

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.

Answer 9

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.

Answer 10

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.)

Answer 11

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

Answer 12

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.

Answer 13

  "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.

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.

Answer 14

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.

Answer 15

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.

Answer 16

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.

Answer 17

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.)

Answer 18

It's physics rather than math, but surely this creative paper by Alan Sokal deserves mention.

Answer 19

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.

Answer 20

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".

Answer 21

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.

Answer 22

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.

Answer 23

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.