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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3$\begingroup$ Do you want examples where they use the theorem correctly, but the real-world context violates one of the assumptions (e.g., ignoring that the Earth is not thermodynamically a closed system), or that they just misunderstand the theorem itself? $\endgroup$– Scott McKuenCommented Apr 12, 2011 at 14:59
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54$\begingroup$ Does "applying the Banach-Tarski paradox to an orange" qualify? $\endgroup$– SomeoneCommented Apr 12, 2011 at 15:14
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11$\begingroup$ 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... $\endgroup$– godelianCommented Apr 12, 2011 at 15:19
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21$\begingroup$ I feel like most people misapply Godel's incompleteness theorem. $\endgroup$– Sean TilsonCommented Apr 12, 2011 at 15:25
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10$\begingroup$ 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! $\endgroup$– F ZaldivarCommented Apr 13, 2011 at 0:45
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33 Answers
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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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I just came across the paper BAK-SNEPPEN MODELS FOR THE EVOLUTION OF STRUCTURED KNOWLEDGE in SOCIETY. INTEGRATION. EDUCATION Proceedings of the International Scientific Conference.
The abstract states:
Models of biological evolution can help to understand many social and economical phenomena where the search for optimality is hindered by voluntary or random competition. Bak-Sneppen is one of the most significant models because it balances at best explication power and simplicity. Unlike cellular automata models, Bak-Sneppen models join locality and globality. The authors try to re-read these models in the framework of mathematics, where, despite its high developped structure, knowledge waves can hinder comprehension both of pupils and of scholars.
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[Copying an answer I posted to another question over here. I think these are actually nice examples of scientists taking pains not to misapply science to theology, although Mendeleev still argues for the immortality of the soul by analogy from conservation laws in physics.]
Once, browsing in the local history section of the Central Library in Rochester, NY, I stumbled upon a curious book:
The book (a full scan of which is available here) is a collection of responses by various thoughtful people to the following request:
Dear Sir:
The Author of this letter, inspired by the untimely decease of a dear friend, and in contemplation of the numerous philosophical and logical theories leading to a belief in the continued existence of the soul, or personal identity after death, begs of you the great favor of a letter, setting out as briefly, or at such length as may be convenient, what you consider to be the strongest reason, or argument, advanced by science or philosophy, or by common sense, in favor of an affirmative answer to this mighty question; or preferably, a statement of your own deductions thereon.
It is our desire to obtain from thinkers and educators of the world, an expression—a twentieth century bulletin, on this subject.
Our request will impress you doubtless as an unusual one, but none the less will you see the force of it, and its possibilities. Who can measure the impetus such a compilation may have upon the inquiring human mind?
May I not have your co-operation in this matter?
Thanking you now in advance for the courtesy of a reply, I am
Fraternally yours, ROBERT J. THOMPSON.
Wellington Ave., Chicago, U. S. A. October, 1901.
I had fun skimming through the first collection of responses, from "The Scientists". The variety of responses is interesting. Although there's the usual bloviation on the cosmological and teleological arguments, as well as some self-citations to studies confirming the existence of telepathy and other paranormal phenomena, there's also some healthy scientific modesty. E.g., E. Ducleaux writes,
Excuse me for not being able to help you in your investigation. I have no scientific opinion regarding the questions you put. I mean, no opinion that rests on anything but personal beliefs. Besides, I think that everybody is in about the same position and that any reasons that may be brought forth in favor of one's opinion are only good for the person that brings them forth, and that they cannot impress the listener; they are therefore not scientific reasons.
D. I. Mendélieff's response begins similarly, although he ends by arguing for the immortality of the soul by analogy with the laws of conservation of mass and energy (an analogy I was surprised to see repeated very often in other responses!):
The question as to the continuance of the existence of the soul or personal identity after death, mentioned in your letter of August, 1901, I, as a natural philosopher, consider to be an hypothesis which cannot be proved by evidence of real facts. But as a man educated in a religious sense, I prefer to remain in the belief of the immortality of the soul. It is my opinion that the philosophical side of the question consists in the relation between the soul, the natural forces, and matter; and if it were possible to clear up to some extent this feature of the problem---the relation between force and matter---then also the relation between the soul and natural forces would be forwarded to a great extent.
The unquestionable existence of reason, will and consciousness compels us to acknowledge the existence of a special world of relations of this kind, and any rational conclusion in relation to this special world cannot be accepted as proved quite in the same manner. Knowledge of physics and mechanics does not give anything in relation to chemistry or in relation to the existence of celestial bodies.
We must simply confess that it is impossible to comprehend this question in a general way, but it would also be sheer nonsense to ignore the physical world; and as matter and natural forces must be acknowledged as eternal, it is also probable that the soul is eternal.
But my favorite response was by James R. Nichols, marveling at the modern-day wonders of technology and anticipating the next version of the iPhone:
Do we not every day converse with unseen friends long distances away; do we not recognize their familiar voices, in homes separated from us by rivers, woods, and mountains? These voices come out of the darkness, guided by a frail wire which science provides as a pathway.
[...]
If our friends in this life, dead to us (hidden as they are by the shroud of space), can be seen, and we can hear their voices, their shouts of laughter, the words of the hymns they sing, the cries of the little ones in the mother's arms, is it very absurd to anticipate a time when those dead to us by the dissolution of the body may, by some unknown telephony, send to us voices from a realm close at hand, but hidden from mortal vision?