Friday - - Mind Games -

Let’s play the num­ber game Math­e­mat­ics and word puz­zles have never been dis­tinct from each other. A prime il­lus­tra­tion of this would be the ex­ten­sive out­put of Martin Gard­ner, who wrote on math­e­mat­i­cal puz­zles for decades but was equally fas­ci­nated by words and word puz­zles. Gard­ner wrote the Math­e­mat­i­cal Games col­umn in the Sci­en­tific Amer­i­can from 1956 to 1981, which was then taken over by Dou­glas Hof­s­tadter who gave it the clever ana­gram­matic spin Me­ta­m­ag­i­cal The­mas.

In one se­ries of now-an­thol­o­gised ar­ti­cles orig­i­nally ap­pear­ing in that col­umn, Hof­s­tadter dis­cusses many va­ri­eties of self-ref­er­en­tial sen­tences. These range from the el­e­men­tary (“This sen­tence no verb”) to the more head-scratch­ing (“This sen­tence con­tains ex­actly threee erors”). Oth­ers are just plain fun: “This sen­tence is a !!!! pre­ma­ture punc­tu­a­tor”, and “You have, of course, just be­gun read­ing the sen­tence that you have just fin­ished read­ing”. A longer sub­mis­sion to the

Sci­en­tific Amer­i­can had Hof­s­tadter him­self (as edi­tor) in a snit in try­ing to ver­ify it: “Only the fool would take trou­ble to ver­ify that this sen­tence was com­posed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, six­teen f’s, four g’s, thir­teen h’s, fif­teen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, six­teen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven com­mas, twenty-three apos­tro­phes, seven hy­phens, and, last but not least, a sin­gle!”

Dutch math­e­ma­ti­cian Hans Freuden­thal sent Hof­s­tadter a story nar­rated by 18th-century Ger­man poet Chris­tian Fürchte­gott Gellert, which com­bines self-ref­er­en­tial sen­tences and a moral les­son about hon­esty. It tells of a fa­ther and son tak­ing a walk dur­ing which the son tells a big lie. His fa­ther is­sues a dire warn­ing to him about the “Liars’ Bridge”, which they are ap­proach­ing, say­ing this bridge al­ways col­lapses when a liar walks across it (there is an ac­tual bridge so named in Sibiu, Ro­ma­nia, with such a leg­end at­tached). Af­ter hear­ing this warn­ing, the boy ad­mits his lie.

When Freuden­thal told a boy this story, the lad asked him what hap­pened when they came to the bridge. Freuden­thal replied, “It col­lapsed un­der the fa­ther, who had lied, since in fact the Bridge doesn’t do that”! We’ll leave you pon­der­ing on that Catch-22 un­til next week.

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