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Gulliver in the Land without One, Two, Three

Author(s): Karl MengerSource: The Mathematical Gazette, Vol. 43, No. 346 (Dec., 1959), pp. 241-250Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/3610649 .

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THEMATHEMATICALAZETTETheJournalof theMathematicalssociation

VOL. XTLTTT DECEMBER959 No. 346

GULLIVER IN THE LAND WITHOUTONE, TWO, THREEBY KARL MENGER

About the year 1700, in the course of heretofore unrecordedtravels, Gulliver met islanders who used a rather odd arithmeticalvocabulary. They counted:stix, stixpair, stixtrip, four stix, five stix, six stix, and so on;and they wrote(1) I, 1,, 41, 51, 61,and so on.The symbol I and the word stix also denoted kauri sticks-theinsular currencyand principalobject of applied mathematics.The symbols 1, I1, It and the corresponding words for the firstthree numbers had, as Gulliver learned, always been in use. Butbeyond stix, stixpair, and stixtrip, the islanders originally counted:four, five, six, and so on (and they wrote: 4, 5, 6, and so on). Thecombination of these two types of original symbols, however,created difficulties. For instance, while one could write(2) I+lP=It and 4+5=9,the symbols -+ 4 and 14+ 5 as well as the words stix plus fourand stixpair plus five appeared to be incongruous. As a remedy,mathematicians proposed a uniform symbolism; and they achieveduniformity by assimilating the higher numerals to the lowest three.After this reform,(1)became the officialsequenceof insularnumerals.Ever since, sums were indicated by(3) 1P+ It, I + 41, I - 51,41 + 51,...Stix plus four stix (in contrast to stix plus four)soundedunobjection-able and so did stixpair plus five stix. For instance, addition tables

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THE MATHEMATICAL GAZETTE

included the formulae(4) 1+ I = 1t, 1+41= 51, + 5 = 71, 41+ 51=91.But the islanders never wrote(5) 41+ 51= (4 + 5)1.Aiming at uniformity, they avoided (5) because they could notwrite(6) |4 + 51 (IP+ 5)1.As a result, even though the meaning of (5) differs from that of thelast formula (4), the contents of formulae such as (5) were rarelyexpressed at all.Difficulties arose for the islanders in their higher mathematics,that is, in multiplication. In advanced treatises, mathematicianswrote, e.g.,(7) 4.51 - 201(read: four times five stix equal twenty stix); that is to say, theydenoted the first factor of the product by the original numeral 4,and not by the official41. But products such as(8) 5. It, 6.41, 4.51,...(whose first factors exceeded stixtrip) were the only ones that couldbe designated in a direct way. Eschewing any incongruity, mathe-maticians refrainedfrom using the numerals I, Ip,and Itas the firstterms in designations of products. They never wrote, e.g.,(9) 1P.41 or 1t. or . ,nor did they ever say: stixpair times four stix or stixtrip times stixor stix times stix.Gulliver, who attributed these difficulties to lacunae in theinsular system of numerals, wrote in his diary: "Landed on anisland withoutOne, Two, Three."Of course mathematicians also discussed products with smallfirst factors,-but in an indirect way, namely, by introducing forthat very purpose (wheretheir ordinarynumeralsfor small numbersfailed) other symbols-usually the letters h and k. But even thead hoc use of such letters did not make it possible to express factsabout multiplication in simple formulae such as (5). Mathematicianswere forced to resort to implications. For instance, they wrote:(10) If h =- I, then h . hi = hi and thus h . hI= .(read: if h stix equal stix, then h times h stix equal h stix and thusstix) and, more generally,

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GULLIVER IN THE LAND WITHOUT ONE, TWO, THREE 243(11) If kl = I, then k. nI = n = n . kl for any number nl(or, as some wrote, for any number n).

Ad-hoc-symbols also played a great role in the formulation ofproblems. Pages and pages of textbooks were devoted to exercisessuch as(12) Findk2ifk 1+ It; k-=I +41; k= 41+51; k = 5.61.

Naturally under the circumstances, multiplication was a topictaught only in universities and, even there, teachers as well asstudents grumbled a good deal about the fact that few beginnersother than those majoring in mathematics acquired more than amerely mechanical facility.Not everything went smoothly in applied mathematics, especiallyin counting objects other than sticks. The islanders wrote

(13) 4 arrows, 5 arrows, ... and not 41arrows, 51arrows,...in this connection, too, reverting to the original numerals 4, 5, ...But, of course, they refrained from writing(14) 1arrow, IParrows, and Itarrowsand from saying: stix arrow, stixpair arrows, and stixtrip arrows.What they did say and write was: arrow,arrowpair,and arrowtrip.Hence, when Gulliver bought four arrows for six kauri sticks, hereceived the following bill

4 arrows... 61.But when he bought two arrows for three sticks, the bill read:

arrowpair... It.How simple would everything be, Gulliverthought, if the islandershad achieved uniformity by assimilating the lower to their highernumerals (and not the other way round); in other words, if theyhad introduced symbols, say,

(15) 1, 2, 3for the first three numbers-for the numbers themselvesand not forgroups of sticks-and kept the original numerals 4, 5, 6, ... for theothers. This minor reform would even have permitted the islandersto retain the ancient symbols i, i2, and I',to which some of them werestrongly attached. Of course, these symbols (in accordance withtheir spoken counterparts: stix, stixpair, and stixtrip) woulddescribe one stick, two sticks, and three sticks; in other words, theywould serve as synonyms of 11,21, and 31,respectively-facts thatmight be expressed in the formulae:

11= I, 21= i", 31= I.16)


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In the same spirit, the formula I + \1 = Itmight be retained as anexpression of the fact that(17) 11 +21 = 31.The statement (17) about sticks would reflect a relation betweenthree pure numbers, namely,(18) 1 +2 3.

Gulliver developed these ideas in a paper "Arithmetic withoutStix." He showed that, along the lines indicated, one would arriveat the following results:(a) Unified symbols for sums. Formulae, acceptable to seman-ticists, would include (18) and, instead of (4),

1 + 4 5, 2+5= 7, 4 + 5 9.(b) Unified symbols for products, such as 1 . 3, 2 .4, 4 . 5.The simple formula 1.1 =1 would replace the cumbersomeimplication (10).(c) The awkward implication (11) would be replaced with thetransparent law

1 . n = n nn. 1 for any number n.(d) The problems (12) could be formulated without symbolsintroduced ad hoc, namely, by saying simply:

find (2 + 3)2, (1 + 4)2, (4 + 5)2, (5 . 6)2.(e) The separation of numerals and symbols for sticks would leadto a unification of applied mathematics, satisfactory to philo-sophers of science; e.g., the afore-mentioned bills would read:

4 arrows... 61; 2 arrows... 31.There would be no reason for shunning formula (5) concerningsticks, since it would have the analogue

21+ 51= (2 + 5)1,instead of (6).The reception of these ideas on the island was mixed. Severaloutstanding mathematicians, including the discoverer of thecelebrated formula I + -1=I + 1, said Gulliver's paper wastrivial. Noted educators, on the other hand, called the criticism oftheir numerals extremely far-fetched and Gulliver's introduction ofthe symbols 1, 2, 3 (or, as they said, his new notation) a tour deforce. But the two groups concurred in one point: that the idea ofteaching terms other than stix, stixpair, and stixtrip was simply

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GULLIVER IN THE LAND WITHOUT ONE, TWO, THREE 245ridiculous. Again, some philosophers of science hailed Gulliver'sideas as a great simplificationof the existing system. Two professorsof higher mathematics ventured the prediction that, in terms of1, 2, 3, multiplication would one day be taught in secondary schools.But these voices were silenced by the powerful organization of theIsland's Major Mathematicians of Real Talent and Learning-briefly, the Immortals-whose judgment naturally carried greaterweight as being more detached, since no Immortal had seen afreshman in decades. The Immortals were reported to feel that (a)even though no one had ever taken the trouble to write dowh theideas propounded by Gulliver, all of them had been clear to allmathematicians all the time; (b) the so-called difficulties, pointedout by him, were merely didactic; they had never in the leastbothered any Immortal nor, as far as Immortals could place them-selves into the position of minor intellects, even mortal researchmathematicians.Gulliver's schedule called for his departurefrom the island beforethe dispute was settled. But the discussion aroused in him abroader interest in mathematics and, upon his return to Englandin the early 1700's,he familiarized himself with the latest progress-the theory of functions and fluents. He studied(1') x, x2,x3, /x, log x, tan x, ...(read: the functions x, x square, x cube, square root of x, ...)Among the functions, as Gulliver noticed with surprise, only themore complicated ones had self-contained symbols, such as a/,log, and tan. The small raised 3 and 2 in the symbols for cube andsquare could not occur by themselves. These two functions as wellas the first power (assuming for any number x the values x3, x2,and x, respectively) were referredto by those very values.The combination of the symbols y/, log, and tan with the symbolsx, x2, and x3 would have led to difficulties, e.g., in designating pro-ducts of functions. Mathematiciansmight have written(2') x . x2 = X as well as 6V . 3/ = ,but they could not very well write x2. 3X/ nor, for that matter,x2 + 3/ or x + log.These difficulties were overcome by a uniform notation, and uni-formity was achieved by assimilating the symbols for the morecomplicated functions to those for the simpler ones; that is to say,mathematicians also designated the functions a/, log, and tan bytheir values for x; and (1') represented the official symbols forfunctions.Sums of functions were denoted by

x + x2, x2 + \/x, log x + cos x;


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THE MATHEMATICAL GAZEJTThi

products, by(3') x2 3x. -3 ,x .3\/X, log x. tan x

V/x'(the last three dots were sometimes omitted); for instance,(4') x . x2 = X3, X . = v, 6X 3v/ = V.But mathematical treatises lacked formulae such as

log x + cos x = (log + cos)x; log x . cos x = (log . cos)x or(5') 6VX. 3Vx (6 .3V)XAiming at uniformity, mathematicians avoided (5') because theycould not write(6') (x2. \/)x = x2 . /x or (x + log)x = x + log x.As a result, even though the meaning of (5') differs from that of thelast formula (4'), the contents of formulae such as (5') were rarelyexpressed at all.Further difficulties arose in substitution. For instance, mathe-maticians wrote(7') 3V6V/ = 18V/that is to say, they denoted the function into which they substitutedanother function by 3'/ and not by its official symbol 3Vx. Butresults of substitution such as(8') 3/ log x, log tan x, tan x2were the only ones that could be designated in a direct way. Mathe-maticians could not designatethe results of substituting the functionslog x, tan x and x2 into simple functions such as x2, 2x + 1, and x,respectively, by writing(9') x2log x, (2x + 1) tan x, xx2or x2(logx), (2x + l)(tan x), x(x2).Nor could they write xx or x(x) for the result of substituting thefunction x into itself.Of course mathematicians also discussed the results of sub-stitutions into simple functions-but in an indirect way, namely,by introducing for that very purpose (wheretheir ordinary symbolsfor power functions failed) other symbols, e.g., the letters h and k.But even the ad hoc use of such letters did not make it possible toexpress facts about substitution in simple formulae such as (5').Mathematicianswere forcedto resort to implications; for instance,(10') If h(x) = x, then h(h(x))= h(x) and thus h(h(x))= x

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GULLIVER IN THE LAND WITHOUT ONE, TWO, THREE 247and, more generally,(11') If k(x) = x, then k(f(x)) = f(x) = f(k(x)) for any functionf(x)

(or, as some wrote, for any functionf).Pages and pagesof textbooks were filled with problemsformulatedin terms of ad-hoc-symbols, the letter f being a favorite; e.g.(12') Find the differential quotient off(x) if

f(x) = x + x3; f(x) - X2 log x; f(x) = log x cos x; f(x) = log tan x.Even applied mathematics was somewhat affected by the diffi-culties of the functional notation. The results of substituting

s, v, and t (the distance travelled, the volume, and the time) intothe square root, the logarithm, and the cosine (in other words, thesquare root of the distance, the logarithm of the volume, and thecosine of the time) were denoted by(13') /s, log v, cos t,respectively. But the results of substituting the time into thefunction x2, or the volume into the function 2x + 1 could not bedenoted by(14') x2t, (2x + 1)v or x2(t),(2x + 1)(v).

Yet it seemed to Gulliverthat everything would have been simpleif uniformity had been achieved by assimilating the symbols or thesimplerfunctions to those or themorecomplicatedones instead of theother way round; in other words, if mathematicians had introducedsymbols for the former, say,(15') j, j2, andjn (for any number n)as designations of the identity function (or first power), the secondpower, and the nth power, respectively-the functions themselves,not their values-and had used the symbols y/, log, cos, etc. for theothers. They might have retained x, x2, and xn as symbols for thevalues that the functions (15') assume for x; that is to say, assynonyms ofj(x), j2(x), andjn(x), respectively-a fact that might beexpressed in the formulae:(16') j(x) x, j2(x) = x2, and jn(x) = xn for any n.In the same spirit, the formulax . x2 = 3 might have been retainedas an expression of the fact that(17') j(x) .j2(x) = j3(x) for any number x.The statement (17') about numbers would reflect a relation between


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three functions, namely,(18') j .j j3

Rescuing the identity function and the power functions fromanonymity and dealing with functions themselves instead of dealingwith their values would, it seemed to Gulliver, lead to the followingresults:(a')Unifiedsymbols for sums, products,and quotients of functions,such as sinj3 + cos, j . j2, . , log . ta n, 6an . 3

(the dot for multiplication would never be omitted). Examples ofacceptable formulae would include (18') and, instead of (4')j j-! = ji and 6/ . 3/_ = orjl/6 .jl/3 jl/2

There would be no reason for shunning the formula (5') aboutnumbers, since it would have the following analogues(j2 . V)x j2X . 'x and (j + log)x jx -+ log x,

instead of (6').(b') Unified symbols for the results of substitution could be setup by mere juxtaposition of function symbols; for instance,

log tan, j2 cos, j2 log, cosj2, (2j + 1)j2.In particular, the simple formula

jj -=j,(the analogue of 0 + 0 0 and 1. 1 1) would replace thecumbersomeimplication (10').

(c') The awkward implication (11') would be replaced with thetransparent law jf = f = fj for any function f,the analogue of the laws

O + n=n=n + Oand . n = n .1 foranynumbern.(d') The problems (12') could be expressed without symbolsintroduced ad hoc, namely, by writing simply (if D stood for thedifferential quotient):

find D(j + j3), D(j2 . log), D(log . sin), D(log tan).(e') Thanks to the clean separation of symbols for functions con-necting fluents and the symbols for the fluents themselves, appli-cations of the function concept could be unified. Of course such a

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GULLIVER IN THE LAND WITHOUT ONE, TWO, THREE 249separation is possible where the connecting functions have symbols,as the sine function in the classical descriptionof a certain harmonicoscillator:(19') s =sint.This formula indicates that the position (in a certain unit) isconnected with the time (in a certain unit) by the sine function.On the other hand, a clean separation is impossible where theconnecting function is anonymous, as in the law for falling objects

s = 16t2.This formula looks simpler than(20') s 16j2(t)to him who is used to taking the identity and power functions forgranted. But only formulae such as (20') and (19'), in which func-tions are cleanly separated from fluents, make it possible to applypure mathematics to observable material automatically and byuniform proceduresfollowing articulate schemes.Consider, e.g., the basic scheme for applying the derivative of afunction to the rate of change of one fluent with regardto another.This scheme reads:dww = f(u) implies du - (Df)u for any two fluents u and w and anydifferentiable function f, where (Df)(u) is the result of substitutingthe fluent u into the function Df.Accordingto differential calculus,

Dsin =cos and D(16j2) =32j.Hence the scheme yields, by purely substitutive procedures (whichcould be easily carriedout by machines),dss = sin t implies d = cos tdtand dss = 16j2(t) mplies t- 32j(t).

Since, according to (19') and (20'), the antecedents in thesedsimplications are valid, so are the consequents. Identifying dwith the velocity v, one thus has provedv = cos t and v = 32j(t), respectively.

Gulliverintended to describehis experiences in the Land withoutOne, Two, Threein letters to Newton, to the successors of Descartes,


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THE MATHEMATICALGAZErI'IIHE MATHEMATICALGAZErI'IIto Leibniz, and to the Bernoullis. One of these great minds, rushingfrom one discovery to the next, might have paused for a minute'sreflection upon the way their own epochal ideas were expressed.It is a pity that, because of Gulliver's preparations for anothervoyage, those letters were never written.* K.M.5506 N. WayneAvenue,Chicago40, Ill., U.S. A.

ON LANGFORD'S PROBLEM (I)BY C. J. PRIDAY

For numbers a > b > 1 we shall denote by (a, b) the set ofnumbers b, b + 1, ..., a. We shall say that a set S of numbers isperfectif there exists a sequence containing just one pair of each ofthe numbers in S, satisfying the condition: for everynumberr inthe set, the two r's are separatedby exactly r places, and having nogaps (a perfectsequence).Example 1. (4, 1) is perfect: 41312432.We shall say that S is hooked f there exists a sequence containingthe same numbers and satisfying the same condition, but having agap one place from one end (a hooked equence).Example 2. (2, 1) is hooked: 121*2.Example3. (8, 2) is hooked: 8642752468357*3.We note that (by juxtaposition of the correspondingsequences)if two sets S1 and S2 without common elements are both perfectthen so is their union S, and if one is perfect and the other hookedthen S is hooked; while if both are hooked then the correspondingsequences can be "hooked together" and so S is perfect.Example 4. (8,2) and (1, 1) are hooked, so (8,1) is perfect:

8642752468357131.Langford's problem (Math. Gaz.(1958),p. 228)may be formulated* The writing of this paper is part of the work made possible by a grantfrom the Carnegie Corporation of New York for the development of theauthor's approach to mathematics.

to Leibniz, and to the Bernoullis. One of these great minds, rushingfrom one discovery to the next, might have paused for a minute'sreflection upon the way their own epochal ideas were expressed.It is a pity that, because of Gulliver's preparations for anothervoyage, those letters were never written.* K.M.5506 N. WayneAvenue,Chicago40, Ill., U.S. A.

ON LANGFORD'S PROBLEM (I)BY C. J. PRIDAY

For numbers a > b > 1 we shall denote by (a, b) the set ofnumbers b, b + 1, ..., a. We shall say that a set S of numbers isperfectif there exists a sequence containing just one pair of each ofthe numbers in S, satisfying the condition: for everynumberr inthe set, the two r's are separatedby exactly r places, and having nogaps (a perfectsequence).Example 1. (4, 1) is perfect: 41312432.We shall say that S is hooked f there exists a sequence containingthe same numbers and satisfying the same condition, but having agap one place from one end (a hooked equence).Example 2. (2, 1) is hooked: 121*2.Example3. (8, 2) is hooked: 8642752468357*3.We note that (by juxtaposition of the correspondingsequences)if two sets S1 and S2 without common elements are both perfectthen so is their union S, and if one is perfect and the other hookedthen S is hooked; while if both are hooked then the correspondingsequences can be "hooked together" and so S is perfect.Example 4. (8,2) and (1, 1) are hooked, so (8,1) is perfect:

8642752468357131.Langford's problem (Math. Gaz.(1958),p. 228)may be formulated* The writing of this paper is part of the work made possible by a grantfrom the Carnegie Corporation of New York for the development of theauthor's approach to mathematics.

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