Romantic Mathematical Art: Part II
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Jim Henle, Editor
Romantic Mathematical Art: Part II JIM HENLE
This is a column about the mathematical structures that give us pleasure. Usefulness is irrelevant. Significance, depth, even truth are optional. If something appears in
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. —Albert Einstein.
T
he mysterious is indeed beautiful. And the most beautiful mysteries are those surrounding The Impossible.
Part I of ‘‘Romantic Mathematical Art,’’ which appeared in the previous issue of this journal, focused on the infinite. Part II focuses on impossibility, though as noted in Part I, the two are closely related. Some of the topics covered in Part I (Penrose tiling, the hydra, hypergame, astrology, and M. C. Escher) dipped noticeably into impossibility. The art of impossibility is singular for its history, for its philosophical interest, and for its political connections. It’s all here. Well, some of it is here.
this column, it’s because it’s intriguing, or lovely, or just fun. Moreover, it is so intended.
Truth It all began, probably, with the liar paradox. The Liar This sentence is false.
â Jim Henle, Department of Mathematics and Statistics, Burton Hall, Smith College, Northampton, MA 01063, USA. e-mail: [email protected] 1
If the sentence is true, then it’s false. And if it’s false, then it’s true. Impossible! But it could be art. Versions of the paradox date to the ancient Greek philosopher–poet Epimenides. Was the paradox intended to please (hence, by our criteria, art)? That’s a question that can’t be answered. Instead, consider the paradox of Euathlus, which came along about a century later. It seems like a lot of fun. It could be art. The story is that Euathlus studied law under Protagoras with the understanding that payment for his education would be made as soon as he won his first case. But after his studies were completed, Euathlus didn’t practice law. Protagoras grew impatient and sued his student for payment. Euathlus undertook his own defense. You see the difficulty. If the judgment is that Euathlus must pay, it would violate the agreement, since he would have lost the case. On the other hand, if the judgment is that he doesn’t have to pay, then surely he does have to pay, having won the case!1 Let me say now that by ‘‘paradox’’ I mean a self-contradictory situation in which there appear to be good arguments on both sides. A paradox can be considered
The pleasure of this paradox continues today. A look on the web yields a variety of modern perspectives.
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resolved only if a flaw is found in one of the arguments or if the situation can be shown not to arise. One answer to the liar is to argue that not every sentence has a truth value. Self-referential statements are especially prone to paradox. We could resolve these paradoxes simply by denying truth values to such statements. Miniac My favorite self-referential parado
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