The puz­zling world of Ray­mond Smullyan

Cosmos - - Spectrum -

His work ranged from the aca­demic study of math­e­mat­ics to some of the world’s clas­sic logic brain­teasers.

RAY­MOND SMULLYAN is a name that will be fa­mil­iar to most math­e­mat­ics en­thu­si­asts. Born in 1919 in the small town of Far Rockaway, New York, Smullyan was fas­ci­nated by mu­sic, maths and magic from a very young age. Mov­ing with his fam­ily to Man­hat­tan at 13, he en­tered Theodore Roo­sevelt High School.

While the school pro­vided an ad­e­quate mu­sic cur­ricu­lum, Ray­mond was dis­sat­is­fied with its math­e­mat­ics of­fer­ings and left soon af­ter. He con­tin­ued his stud­ies on his own and even­tu­ally en­rolled at Pa­cific Col­lege in Ore­gon. He moved around from one univer­sity to an­other, some­times as a stu­dent and other times as a mu­sic or maths teacher for the next twenty-odd years. At one point, the Univer­sity of Chicago even gave him credit for a cal­cu­lus course that he had never ac­tu­ally taken, but that he had been teach­ing for some time. Even­tu­ally he re­ceived his PHD in math­e­mat­ics from Princeton in 1959.

Smullyan pub­lished 30 books in his life­time. While his aca­demic works had in­tim­i­dat­ing ti­tles such as Gödel’s In­com­plete­ness The­o­rems and Re­cur­sive The­ory for Me­ta­math­e­mat­ics, his of­fer­ings to the pub­lic were more re­flec­tive of his true, whim­si­cal na­ture. Alice in Puz­zleLand, The Lady and The Tiger, and King Arthur in Search of His Dog are all typ­i­cally Smullyan ti­tles. Ad­di­tion­ally they are con­sid­ered clas­sics in the field of logic puz­zles – one of Smullyan’s spe­cial­ties.

Two of my all-time favourite brain­teasers come from his first book of logic puz­zles, What is the Name of This Book?. The first puzzle in­tro­duces “Liars and Truth-tell­ers”. Liars can only tell lies and truth-tell­ers can only tell the truth. Liars and truth-tell­ers are iden­ti­cal in ap­pear­ance. In some prob­lems you must ques­tion them un­til you fig­ure out which is which. In oth­ers, you must get them to re­veal a lo­ca­tion or spe­cific path­way.

With that in mind, imag­ine you are on a de­serted is­land whose only in­hab­i­tants are a sin­gle liar and a sin­gle truth-teller. You come to a fork in the road, guarded by these two men. You know one of them is a liar and the other a truth-teller, but you do not know which is which. You wish to know which way leads to the beach. You may turn to ei­ther of the two men but may ask only one ques­tion. The one you ask will only an­swer with “Yes” or “No.” Is there a ques­tion you can ask that will en­sure you are di­rected down the ap­pro­pri­ate fork? (An­swer at the end of the ar­ti­cle.)

A much sim­pler brain­teaser wasn’t de­vised by Smullyan but was one of his fa­vorites. A man is look­ing at a photograph and says: “Broth­ers and sis­ters I have none, but this man’s fa­ther is my fa­ther’s son.” Whose photograph is he hold­ing? (An­swer be­low.)

Smullyan had other in­ter­ests apart from logic puz­zles. He was a lead­ing Taoist philoso­pher and also an am­a­teur as­tronomer who even ground his own six-inch tele­scope mir­ror. His many ad­ven­tures in­clude ap­pear­ing on The Tonight Show Star­ring Johnny Car­son in 1982 and in 2001 be­ing the sub­ject of a bi­o­graph­i­cal doc­u­men­tary en­ti­tled This Film Needs No Ti­tle: A Por­trait of Ray­mond Smullyan. Seem­ingly in­de­fati­ga­ble, he con­tin­ued to give talks on math­e­mat­ics and mu­sic un­til well into his 90s. Ray­mond Smullyan passed away on 7 Fe­bru­ary, 2017, at the age of 97.

I’ll close with a fiendishly dif­fi­cult puzzle of Smullyan’s. It is pre­pos­ter­ously hard. In 1979, two friends, Arthur and Robert, were cu­ra­tors at the Mu­seum of Amer­i­can His­tory. Both were born in the month of May, one in 1932 and the other a year later. Each was in charge of a beau­ti­ful an­tique clock. Both clocks worked pretty well, con­sid­er­ing their ages, but one clock lost 10 sec­onds an hour while the other gained 10 sec­onds an hour. On one bright day in Jan­uary, the two friends set both clocks right at ex­actly 12 noon. “You re­alise,” said Arthur, “that the clocks will start drift­ing apart, and they won’t be to­gether again un­til – let’s see – why, on the very day you will be 47 years old. Am I right?” Robert then made a short cal­cu­la­tion. “That’s right!” he said. Who is older, Arthur or Robert? Good luck.

AN­SWER TO FIRST PUZZLE: The se­cret is to ask a ques­tion that forces both liars and truth-tell­ers to re­spond the same way. For in­stance, turn to ei­ther in­hab­i­tant while point­ing down one of the roads and ask, “Would he tell me this road leads to the beach?” (“He” refers to the one you are not talk­ing to). A liar will an­swer dis­hon­estly “yes” if it’s the wrong road and “no” if it is the cor­rect road. But a truth-teller would give ex­actly the same an­swers! He will say “yes” if you’re point­ing down the wrong road and “no” if you’re point­ing down the cor­rect road. AN­SWER TO SEC­OND PUZZLE: The man is look­ing at a pic­ture of his son.

His pub­lic of­fer­ings were re­flec­tive of his true, whim­si­cal na­ture.

There is noth­ing like crack­ing a tough math­e­mat­i­cal nut.

YOU MIGHT BE FOR­GIVEN FOR sup­pos­ing that a hum­drum num­ber like 4 would be of lit­tle in­ter­est to math­e­ma­ti­cians or sci­en­tists. But 4 turns out to be deeply sig­nif­i­cant, crop­ping up across sci­ence and maths. Part of the num­ber’s sig­nif­i­cance can be traced to an­tiq­uity. Greek philoso­phers as­so­ci­ated var­i­ous qual­i­ties with whole num­bers, and 4 was cho­sen to sig­nify jus­tice. Even to­day we say that a fair bargain is “a square deal”. Other an­cient uses in­clude ref­er­ences to the four winds, the four cor­ners of the Earth and the four horses of the Apoc­a­lypse.

In physics, 4 is the num­ber of di­men­sions of space­time – at least, those di­men­sions we per­ceive. The lo­ca­tion of an ob­ject in space re­quires three num­bers to spec­ify it – lon­gi­tude, lat­i­tude, al­ti­tude – and time re­quires just one to say when (e.g. 6 pm on Thurs­day).

The num­ber of fun­da­men­tal forces of na­ture is, so far as we know, also 4. They are grav­i­ta­tion, elec­tro­mag­netism, the weak force re­spon­si­ble for some forms of ra­dioac­tiv­ity and the strong force that holds atomic nu­clei to­gether. Even the fun­da­men­tal par­ti­cles of mat­ter the uni­verse is made from clus­ter nat­u­rally into fam­i­lies of 4.

Pro­fes­sional math­e­ma­ti­cians also get ex­cited about the num­ber 4. In 1852, an English­man named Fran­cis Guthrie was busy colour­ing a map show­ing the coun­ties of Eng­land. Nat­u­rally he wanted to avoid hav­ing two ad­ja­cent coun­ties the same colour, and he won­dered how many dif­fer­ent types of crayon that would re­quire, for all pos­si­ble ar­range­ments of coun­ties. ( Cos­mos 65, p122) From com­mon ex­pe­ri­ence the min­i­mum num­ber seems to be four, but could that an­swer be rig­or­ously proved?

In 1879 a math­e­ma­ti­cian named Al­fred Kempe claimed to have solved the prob­lem, but his proof was flawed. A sub­se­quent at­tempt by Peter Tait in 1880 was also in er­ror. Over time, the four­colour prob­lem be­came some­thing of a cause célêbre; there is noth­ing like crack­ing a tough math­e­mat­i­cal nut to mo­ti­vate ob­ses­sive math­e­ma­ti­cians. In 1900 the math­e­mat­i­cal su­per­star David Hil­bert listed solv­ing the four-colour prob­lem as one of the 23 great out­stand­ing chal­lenges in math­e­mat­ics.

The is­sue re­mained un­re­solved when I was a child. I vividly re­mem­ber draw­ing imaginary maps show­ing coun­ties with in­cred­i­bly con­vo­luted shapes and ar­range­ments in a fu­tile at­tempt to dis­prove the four-colour con­jec­ture. (I also tried to dis­prove Pythago­ras’s the­o­rem by analysing a com­plex mess of in­ter­sect­ing tri­an­gles, a lost cause if ever there was one.)

In 1976 Ken­neth Ap­pel and Wolf­gang Haken at the Univer­sity of Illi­nois Ur­banaCham­paign fi­nally came up with a proof, but one with a dif­fer­ence. Rather than the usual pages of al­ge­bra, Ap­pel and Haken re­sorted to 1,200 hours of com­puter time, re­sult­ing in thou­sands pages of out­put. A lively de­bate en­sued over whether this con­struc­tion con­sti­tuted a real proof.

Most math­e­ma­ti­cians who study the proofs of oth­ers in de­tail ex­pect to ar­rive at an “Aha!” mo­ment when they fi­nally “get it”. Glimps­ing the log­i­cal, and of­ten el­e­gant, rea­son­ing of a pre­ced­ing math­e­ma­ti­cian is what adds to the charm of the sub­ject. But no hu­man can “glimpse the mind of a com­puter” to ap­pre­ci­ate how or even whether it has got the proof right.

This raises a pro­found is­sue. There is clearly a limit to the ca­pa­bil­i­ties of even the clever­est hu­man math­e­ma­ti­cian, yet there is no limit to the dif­fi­culty and sub­tlety of math­e­mat­i­cal prob­lems. As com­put­ers be­come more pow­er­ful, there is a grow­ing temp­ta­tion to do the math­e­mat­i­cal equiv­a­lent of ‘ask­ing Siri’ in or­der to set­tle out­stand­ing con­jec­tures. But this strikes at the very heart of what it is to know some­thing. In many hu­man af­fairs, to pro­fess knowl­edge it is suf­fi­cient to be con­fi­dent of the facts. But in the realm of pure math­e­mat­ics, founded as it is on the ground of un­shake­able logic, proofs are sup­posed to rep­re­sent the most se­cure form of knowl­edge and un­der­stand­ing. To out­source this supreme in­tel­lec­tual re­spon­si­bil­ity to a ma­chine rep­re­sents a shat­ter­ing de­mo­tion of hu­man priv­i­lege.

Ap­pel and Haken’s com­pu­ta­tions were the math­e­mat­i­cal equiv­a­lent of Coper­ni­cus de­mot­ing Earth from the cen­tre of the uni­verse. It was a sign of things to come. Com­put­ers are now rou­tinely used to ex­plore the math­e­mat­i­cal land­scape.

In Dou­glas Adams’ A Hitch­hiker’s Guide to the Galaxy, a com­puter is fa­mously asked to solve the rid­dle of “life, the uni­verse and ev­ery­thing”. The an­swer it de­liv­ers is 42 – at once ba­nal yet pro­foundly mean­ing­less to mere mor­tals. Why did Adams pick that num­ber? Well, 42 = 43 – 42 – 4 – √4.

As the an­cient Greeks might have com­mented: “Fair enough.”

There is clearly a limit to the ca­pa­bil­i­ties of even the clever­est hu­man math­e­ma­ti­cian.

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