Unsolved problems in arithmetic

ide syelpnantie saan nf di sword pe gnishing correct Argument Isar incort A ZENO OF ELEA me sure of all natural numbers (1, 2, 3 an so on). A number was understood as ga multitude composed of units. The writhnetical unit was thuught In be it is olivions that the unit is Ihe common incommensurables, Commensurate is having a camuon measure, and n at it can pose more problems tha Ir ts the nature of arithmetic th mphs of mathematie al logic is it can solve. Indeed, one of the triu re are problems chat can never be solved dl cal logie, ©- sue whe iu nat, EUCLID by Ber suusebvabiltty in this drawing PHILOSOPHERS AND MATHEMATICIANS pende in order to deal with auch issues 46 nada Bry te Actetotle (304-322 8.6.2 founded lo ten. 300: s Gen. 580-500 acd and Zena bee, 4430 80,1; Enelid dvaueeil by Pythug Mathematies, and specifically geameoprovided au ever ronger motive to logie: the Pythagareol hud discovered the existence tbe demonstration thar the DSi DE Loner. \ 3 bere ave arithmetiv problems that a child of 19 can understand Lun that Juve nevertheless remained ‚tens, hundreds and ev ved For a long Un wonsands nf years. insafficicat ingennily stood ia the way uf lem of arith. as Hhoneht that if a prob ed, only mette voukl Le accurately stat was its solution, and that if a problem mately be moved, a solntion would ulti a new found trough the application of method ar the more ingenious applica. \ linn of aa old one, Metulogic, an area stem]? veal that the elic probof the u ay be due not to lack of ingenuity ther te hhereut limitations of the thaws tough u refinement of lies pf both wen and machines. E This pussibility gives metaloge philo8 iphicul as well us mathematical iaterest. ort begins sith Aristotle, ais that mathemabcal Greek i me into being in the tate 1h id the varly 20th century. Hime our vented lice. He did so in ersponse to tein demands, ene philosophical and one mathematical, The philosophical demund arose in response to the extreme riety of arguments with which he d 10 deal, Thates had argued that Ue: basic Heraclitus had Anaximandec stuf of the world is that itis not one thing but un ineletertaimale something or other; are relative, Socrates that they are uraned dut all things are iu mation, Parmenides that no things ave; Protagaces had argued that ethical judgments aml so on. Aristotle also had tu be prepared to deal with paradoxes, such as Zeus paradox of Achilles and the tor- 50 toise or quasilegal arguments such as the one alleged to have taken place between Vrolagoras and Enathlus {sce illestrations an pages 52 and 53]. There was u need w lay down general principles that \ s 1/2 or 2/3 were understood not as being a part of + init but always as being ane unit out of two, two ont of three and so où, On the basis of this arithmetic it was probably obvions that given any Gvo lengths there had to be a geometric unit .o small that either length would be a multiple of it. .- [t would then follow thee every two lengths would he in a definite fixed proportion lo each other: first length is to length and y is Ihe number of units in second length as x is to y, where « is the munber uf veomotric unils in the first the secund, Civen the Pythagoreans’ quasireligious belief in number as the unifying principle nf arithinetic, geomelry, cone syllogism, Logie was understood ta he a only one validly drawn conclusion: “Some cancer victims are not men.” The that the premises are truc bat rather chat if the premises are true, then the concluthat an argument is valid is to assert not and conlusies. ‘Phat is, to assert three propositions together constitute à and the tartoise examples of proofs that clashed with strong beliefs held on other grounds—inchiding some fallacious “proofs” such us stady of the connections between premthe argument of Achilles raised the question of the exact nature and reliability of proof, Aristotelian Logie and considered what he called a demtion, a valid syllogisen yv true premises He tha gave an answer ta the questions of when sore» thing is proved in ge: aby. À proposi. is proved in ge ‚thee consion is also true, Aristotle went further and mathenatical üdernde invented log . He concentrated his ulleation où four Bene eral types ol proposition: the nuiverzal affirmative, the universal negative, the particular affirmative and the partic Hence it was in response Lo hoth philis white,” » “No man is ry onan uegative. Examples af ech in tur thin which Bertrand Russell (4873-1970) dii parad al theorems BER Se ERTRAND shortly alter Avistotle’s tine, Euclid bee is Euclidl’s Element, whieh dates Crom proceed svllagi:ticallv to cı ions, du conception lay the grim of the tras ditional axionatie method. The classic example of that method ile, nat legge i- detuonstrutive, Que inst start with sd vident truths Gvhich ave not chen able) und "Some man is white” and “Some KURT ce DEL m. Nor « mple, s white” und "Some vot white” there is mology anc philowphy, their discovery ure evel ed set theury, GEORG CANTOR victor af incommensurals Jengths mest have iu the recy nl white.” Aristotle considered whut arguments are valid if two such Propositions sharin ler are axsamed andr ligion: been n shock, the Best of many elushes between seienee West. The disvovery ciune in the form of u proof that the side aud the diagonal of a square have nu common measure [se bottom illustration vn page 54). Further H KARTFRIEDRIC GAUSS e method to geemetiy, Kart Fried. BC) applied Aristotle's ax rich Gauss (1777-1855) and D IL Lobache+sky 61793-1456) d le Sped nomEucliden geometry, Georg Cantor (18-15-1918! fuunds

Stand famous one. Greek geometers tailed ghtedge and ZENO’S PARADOX suggests that the 52 Like B gap Oils Lu aT sD i DR} sato D 2) à sat 5 a OD suite à AR C D BRYSON A) wo Av a. Sii Rose B Cc demonstration; the geometer’s job was ta find logical eunnections between geanetric proposit Ins, not to determine if the starting assumptions are true, It was asioms be important, however, that the co stent, because un inconsistent colAF statements cannot possibly be lect in the 19th centrae. Muthematicia tury developed means of showing that if geomeler as providing an Aristotelian led to a rejection nf the conception of a ties this attitude became widespread, Tt and with the discovery of mure grumenot hold. They ‘decided that only by empirical observation and experiment could one determine which geametry was true, try different from Enclid's, a geometry in which the Pythagorean relation does ers, independently discovered a geomegeometry. Karl Friedrich Gauss and Nikplai Ivanovich Lobachevsky, among othe quite sure what to make of them. Let us first consider non-Euclidean maticians and philosophers were not sume dietunste, 337 Achilles reaches B the tortoise will have moved ze will be at D, and so ote les gets to C, tbe to to Ci when Ar de oan Chita A ÈPon OH Wea + mot Achilles ile By Ul n tertoise, Suppose te: tortoise is given 4 beed AB, A line riga à 0 OÙ: rd $DRDO ee RDA «Or ever sons A A logic for which neither intuition nor Ar istutelian logic had any answers, Mathea A > nl ew and troublesome paradoxe: of systematic und fundamental problems Al about the same tine the deficienies in Aristotelian logic became apparent. They became so because the developent af non-Enclidean geametry and set theory and the appearance of some Challenges to Lugic naLick uf ingenuity ar tu the intrinsic was ture of the proble: themselves. It not until the 19th century that such con. structions were shown te he impossi with only a st ghiedge anid pompuss, r was left lo posterity lo decide whethe unthe failure to solve these problems der the limiting condition was due to It impossibility might be open to proof. is known, conceived of the iden that the the stuted condition, but no one, as far ooo GDS ob UO sens oft Sun. ZI Ÿ to solve any ol these problems under compil iS à tion with the aid of a st von cproblem was to make au exact constru and ng cube, squaring the cirele and wiseeti Ihe à (asbitrary) angle. In each case were the problems of dinplicating tv: selves could not solve. The most in geometry that was important in the development ot mathematical logie: they themproposed geome «- problems they so defieient, but Ihe deficiencies were t subtle ns to escape the notice of almos everyone [or 2,0U0 years. The ancient Greeks did something ele 35]. Euclid’s method and practice were method [sce illustration on pages proofs that Euclid gains his importance in the development of the axiomatic s and tions, postulates, common notion of many theorems) and in the exhibition problems were impossible to solve under edistin number of theorems. It is în the notions, and from these he proves a large common tions, five postulates and five (definitions among these four categories that limiting condition, although they x did solve them by using, more comple methods, Some nf the ancient Greeks the seem to have been convinced that gins his book w h a series of 23 defini 3 - sets does not into one to ane corre that he is un inhabitant of the village; Such a postin cun exist {ur if one does, not really a paradox bevause we can easily resolve it, notably Ly denying that sell’s formulation, The pupularizalion is the Paradox of the postman [see illustration on page 57], but it is important to distinguish tl popularization from Rus trand Russell in 1901 having to do with sets of all sels that are noi members of themselves. Its spirit can be conveyed by cal contradi. tions, The most simple and famous is the paradox proposed by Berwas shaken by the discovery that within set theory there were a ımmnber of logi common notion that the whole is greater than the part, Set theory seemed so fundamental that for a time it appenred that all matlıematics coukl be based ou it. This belief spondence (1-2, 2-4, 3-6, ..., n~2n, ...) This euntradiets Euclid's by putting thea show that there are just as many even numbers as there are natural oumbers necessarily apply. For instance, one can logic that holds for finite ple, just as the set of state capitals is equal in size lo the set of states but smaller than the set of U.S, se Cantor showed, the set of natural minnbers is equal in size to the set ot rational numbers (quotients of integer: but smaller than the set ol real numbers (decimals), In reasoning about infinite sets one has to be careful, because Ihe finite sets with differeut sizes, For exam tremely fruitful, Cantor was able te: slow that the concept of the inlinite is a struetred ane, aud that just as fuit sets may have diflurent sizes, so there may be inof dishes. Sut theory proved to be exse kinds sa set of chessmen or the set of natural numbers or the set ol all sets ncepto the growth of math tical logie. „Georg Cantor defined a set as any colieugon constituting a whole cf definite ane distinguishable objects of our intuition the truth of the statements. Set theory provided another impetus, ment of mathematical logic in at Joust two ways. I focused attention on the mathemalician’s job as a discoverer of logical connections, and it raised the “problem of how to prove the con istency „of a collection of statements apart from “by, was thus important to the ‘levelopsystem (say Euclidean geometry) is “consistent, then another (sav some nonÉEuclidean geometry) is also consistent. Yet the question of how one can absozlutely prove the consistency of à system remained open. Nou-Euctidean gromeFOR IE HE WINS Tris | potent een HS CASE THER BY case mE ME, BET IS 1S ‘ABSUR FORD, TF in whieh top illustration an next tico pages}. The argument Aristotle had applied to prop» Silions also applies to definitions: Just as (on pain of infinite regress) there are with paradoxes and the question of proof in mathematics, Non-Enclidean geomeand in matters of explicituis: with regard to symbols, grammar aud logie [sce from Euclid's in matters of definition hols for these entities, sa that new rekte tions among mathematical entities can be discovered by manipulation of their symbols in accordance with certain roles. The secoud was the axtomatic methit= not Euelid'i but a revision of that method based on various insights and discoveries made since his lime. To distinguish it from Euelid's method, it is often called the formal axiomatic method. The formal axiomatic method 4 relations among mathematical entities are reflecte:! in relations among the symfirst was the algebraic method, lashed with current belief. Aristotelian logic was inude quate to re solve the issues and a new approach was needed, The new approach was by way of mathenitienl methods and it resulted ins new discipline: mathematical logic. Two methods of mathematics tr: formed logic into a new discipline. ‘The try and set Cheary both provided state hh Basaldits ebecto tual beds s stica for lis feu. Buch bawyer armıed as shown, ! sukE about Aristotelian logic: the need tn deal Hence the rebirth of logieul inquiry that took place in the 19th and the curly 20th century was brought about by considerations similar to those that brought Mathematical Logic over, Russell's puradox, along with mauy other paradoxes, indicated the need for careful inquiry into the source if the Lgical errors. essarily sound for infinite ones. Moreis sound for finite collections is nol wecdevelopment of matbewalical logie in two ways, ft regularly provided urguments having to do with infinite collections and showed that the “logic” that Set theory, then, was important to the tions of which appear In be necessarily tru: and all the ways of escape from which appear to have Ingieally unclesirable consequences, with Russell's paruchox, all the assımp- ) version, This is not the cuse or if he is, that he receives mail und so on). In short, there are many ways to escape the paradoxical consequences of the pos PAY meed half oi bis fee only after Euathlus bud wor his first ense, When E OT HAVE TO Pay; à WIN OR ZURTS JUNG MENT. 2 LOSES" THEN, BY Trief) KEY OUR AGREE MEN „To DONT HAVE te Bay Mot. Tr, ON HE OTHE * HAND, MALL HOT HAGE wor. Pay FIRST pise Toy OLG THAT T MUST Dry

ions that eanuot be praved, some proposit some terms that so there must also be to define be defined. (By uying annet ted terms Euclid was all his geometric his defini. ty, for example in proved propusitinus ‚o Ai into ereulari tar which lies Lian of a straight line The puis on itch} eveulv with the Just as all proposte analogy gues farther. in terms of un tions must be proved teens must undefined test cally thecelore theoreti Definitions are RI _ x ey es uo reference to proving theorems mak efincd ible meanings of the nad the poss sense the axionis are terms. In another cts that make the abent any set of obje nis of far example, the axio axioms true; tem ight be true a particular formal sys natural numbers. when applied to the defined terms in The Fuet that the è dillerent things axioms appiy 30 my ct ta represent it dete might be thou he es methenl. listoad in Uae formal aromat hod's gren strength it represents the met e it makes it pos1 versatile, becaus : uel fav alla huge sible to work aut ance [hat ae true Tor suy body ol Uteorens ashes ME 244 dye which the set of a cul, and h player wies une, g for this feature, con To get a feelin eh there are nine sider il gene in whi re and 2 wenegls I OS pl viag Sa e between two Iving face np ona tabl picking The plavers take nus players. 10 be the vico ve DESCRIPTION . A TRADITIONAL AXIOMATIC MEN: _-Â . -’ {as APPLIED TO GEOMETRY) EXAMPLES DESCRIPTION sitional « ” O) > ¢ (xr {x + Q=0 yay If vand y aree in individual „ar vis à formal sentence. apes, then EXAMPLES FORMAL AXXIOMATIC METHOD (AS APPLIED TO ARITHMETIC) ‘ (Befinitions.) Definitions are eliminated Since rom a theoretical point of view ‘hey serve no function. th only us is a praclical apbreviative one rue ' L (that is, can stand for any eea N List of symbols, Bols. MMust include all te be,“se ia ing stem, For eae a individual veriables {Ihal SEE an Ae chine moden A point is that which has no part ‚Aline is breadihless length, * Aste ht line isa line that lies „evenly wilh the points on itsell, Formation : ¢ ules. Indicate i wi ays in whic Symbols may be legally combuied ion en. duce fon al sentences. For example, "x = x" ornial sentence whereas = = is5 notnot. were intended to be Delinitlons. Definitinns were 10 include all objective and true and Sasic geornelen: terms, fences with ueh we starl, These. azione are nen ed to be rterpreted as anthmelic se pie ces, although na reference is made sin me Gevelopment of Ihe system (Arithmetic) axi ams, A list of forme seit-endent Postulates. Intended lo be apoly specif. ee tat truths or constructions ically to geometry. (p >{q > nn ttris i a formal senlence conta: individual varabie, infer lhe formal sen o ic ve nh the individual lence i variable veplaced lems, they are essentially similar, so thal ıyone familiar with long division hus an algorithm by which he can solve any represents an infinite number of probpredicate is an open sentence: one that can be completed by assigning nantes to its variables.) Although this predicate and y can be any natural numbers, (A represents a decision procedure for th : predicate “x is divisible by 4,” wher ex familiar in everyday mathematics For example, the technique of long divisie u dures—sametimes called algarithias—ave exists a method whereby every probl N expressible in A+ can be solved in ak nite number of steps. Decision proce. has a “decision procedure,” that is, ther v vugeometra pst amd esdi nuples of euch d element systems ace descrilied ure cited desert: le eme are of geo application of à transfo:matien rule. n proof mus ıxion or follow from an noi enere omen by the Theovems. The last lines in proofs, Sach a (Logica!) èaxioms. » Whi Wher ıalerpri logical asgumplons. The ALAN become Hi gas: “Up is true, then if q is true. 9 I wen 23S rue. to the same shing ident truths that È Things that are equet Corman aatians. Seil r sciences, for exam. È ase.equal to one another. Itequats be added ta equals, the wholes aright also supply to othe \" equal, The whole ts greater than ine part. ple to anıthetic. — Transtormation rules, Rutes by which we # manıpufa sentences to produce ie te formal hev For instance, from “ i l1[ PV presumably inciuded and if p imp es q. ences, such as "If p, then a." ll to troths ar constructio fl3 03 finile straight ne il is possible Theorems. Nonewdeat in sucha way as lo sconsuuci an equilateras triangle. he square l hypolenuse of a right tiiequaofl the intended to be proveddoubt {by aol atlawind za note al to the sm of thi squares remove all legilimate È proof escent e laies, common NONO theorems}. ok applied iumatie method A+) set up with the intention of formal. negation of any other theorem, For in. stance, the proof ensures that if" + 1 = numbers, It is possible to prove that such izing the cules for the udilition of natural se i to the syinbols in such a wav that a systeut is “consistent,” that is that ic is impossible for any theorem to he the the ni na ne time ol the elements of then eat M jects, This set of objects is Ingen tts model fer the system, The to arithatet tatetic develop dev ed out of the traditional ra any assumptions lo entedefin itions. postu: ),cal the sides. nose contained in the NS and oreviously prove sentences METHOD, along FORMAL AXIOMATIC od, wanstorimed lag with the algebraic meth e seywence of formal je a finit sentence is ML, h that euch formal suc an cartier formal | axivm or follows from ion of a trans” i utence by the applicat f| The last line uf the proo with w bieb it is compared here. The elements ent: of sl Di explicitly siated but! ’. {Rules of inference.] Not ordinary intuitive inlets ght angles are equal to one another. I A straight line can be drawn from any | pom) to any point, 4 A citcle can be drawn ith any center ‚and.any distance, VE o . explicitly stated; [Formation rules] Netthe ordinary geammatica presumaoly included uage. niles of the natural lang es ly fisted: presumably [Symbols.}Not expticit s from the natural included ordinary symbol language. eo rds whase sum is first ta obtain three € ally similar 10 15. The gine is atructar mol abvions tly ie Linkiacktoe, The sia the cards in à atb one imagines puting , instead of pickviektacktiur grid so that cin wank then ing a i d, the players On (see hes alternativelv with pr en it is easy to ration on page 581. ‘Th me of ticktacktoe and gamer there is « see that for euch rave re- C ta apply anyone who the strategy of one RM erg te ge id euubles ; That, in elfeel, di it to the other game. n strings a forwhut the mathematiciv studies the strc: mal system clues. He and derives sluleture ul both © games” structure that are ments deseribing the true of either “game.” some analogy A Loval system has symbols corwith a natural fangnage. Lis the alphabet, prince respond 10 letters ol erals and so forth. tution snierks, num s co espund to the The lumation rule res sfursation rules cor ral linguag ge ammalical rules ol a natu pond to The trau speaker can pere varinus operations iy h as chauging form su the language, suc to passive, The a sentence trom active able correoms have fewer identifi 2° is a theorem, then.“1 + 1» 2” (“does not equal 2") will not be a theorem r ou also be shown that A+ is" correct : hat js ver i \ PA i al numbers, nuore, A+ can be shown to be complete,” that is, every truth about Finally, it cou be shown that the system the anturat numbers expre ble in the symbolism af the system is a theorem, I ox li ox be defined in terms of rile e univ the per not necessary; they siv pose vi abbreviation. al system are al The avionis of a form undefined sven fowed tn enaliin dute ane sense the axioms ut hols. Thus in : the mi nol about aayehing a e à common measure. Gupease s end d hav Tren A x fellows numbers here rand y af natural in no common dhvtsorIf vre gouare both sides. VIE ef = 28% By the Pythagorean thearen, geste s _ sì _ 1 _ è est 2 tas we have 54 axi vale. Fence. metalen “i en he system is the t be odd. cal language, formation on rules that y is even. 39 « muscommos disor. sponding items in the nutu m. The transformati object ‚sung i he formal system is the since 1 andy nave ao sidesed com- is u theore merely a mechini ‘ af oare = 22 and ahhongh Wey might be con gage, because it gust be such that it is is ity seven, MEN y the object ver ine whether oF | U such sentences as “Winde erm Lo det ble to para 4 ure ced N alk. For example, if we say, “Tu Te tz? a 2 a 50 that=7:3 even, Since A ave of course ine cal pro2 furmal sentences : trom which it follows number lo be both . is, is” or “A is A.” There System S wo e Cal i not a given sequence of its impossible for a erences between natural Jane the formal senà diff e ant hav „move = not port 1 can + I lence A - is a proof. we nre making a cod and even, s and d is, they represent formal systems, but the ane) and ges Sta gau tem ent . in the common Measure. {hat A metalangnage about when ford tems incommensurate targins. ogy is dose enough so that le lag” Praperties of Formal Sys Image hi a Statement in the object they are often was discovacrd i systems WE interpreted fa + That is why the analysis of the INCOMMENSURABILITY n. of objects that are ij artificial languages. here in modern Sorte ed call Zah proper ete ties pertic theougit a prool, given of systenis has come lied Consider a set app are s of a square have e formal system iu j be knawn as metidogic. n transformation sule som Whe that the side and diagonal orem. dependent of | the me prouf utilized tation 19 axions, the result is a the to ~ ;Consi no common measure. The . We give an interpre sid,er now now a à tormat system {call i t ication of the question rem: the square of g members of th: the Pythagureun theo The exhibition of the appl the system by assignin triangle is ed e explicitly, u proof the hypotenuse of a right rules 15 a proof. Mor of the two sides. to the sum of the squares yt = 227. from wnich I

PROBLEMS UNSOLVED ANCIENT GREEKS UNSOLVED PROBLEMS QF ARITHMETIC — Duplication of the cube a cube with twice the va gwen cube, A 8 at A) = volume c' B 2: (volume Problem of perfect numbers. A periect number is à natus) of tis number qual to the sum Grass fescluding its a'e 26, 436, 812%, 235 order LEBRICIGHE, 137438591929. Praùtern: Determine wheiher there is a finite of ai number of them, | | — . 8 Squaring the circle: constructing a square with an area equal lo that ol a given circle. A area of A= area cf 8 Goldbach's conjecture. In 1742 that every Goldbach zor ject umber greät- than en natu 78 the sum ol toro primes fa prune isa Natural Number greäler thar 1 that :s divisible only by teelt and 9). For example, M= 7 + 13, 59 = 54-88, 7000 = 3 + 6907 Problem: Prove this ar find ar natural number thal cannot be so represented. Trisecting an (arbitrary) angle: ES" . ing an arbbary given angle into three equal paris, Y d a caos EE Fermat's last thearem. Fermat slammed to have a prool (which was never found) toe the statement that for alt natural num. bers x.y. 2.9, aes taz’! All Fermat's other claims about mathematics Lieve zu been confirmed or ‘eluted. Hence ihe name “last theorem.” proved suntniy Ihe conete sastliy these limited memi—weer impossibile Phe arithnutie problems, on the other li el, nay be ilve them, inade:puate Le id available n Uribee bee a Formal system, namely thot the set a that makes Ihe axioms true be idene with the set of provable sentences, grin, like the Cheshire cat's, reils metalanguagr) could be Let us now consider Gödel's argument in a little more detail, Let C stand for the sentence “For any natural number % ta say that Gidel assigne:l a unique nunder to each symbol, seutence and praol so that they can be ordered and one can speak, for example, of symbol Na. Vor proof No. 15, tivic AMERICAN, June, 1956]. Sulfice il Nagel and James R Newman; SCIENreflected in the system (in its object language). Familinrity with the details of the code is not important for our purposes (see "Gödels Proat,” by Ernest system that statements auc rexionings abvat a Gidel did this by developing a code (analogous to the ticktickloe grid) by ineans of whieh he could demonstrate are structurally identical, so, Gödel was able ta show, one willanetical zene lences are structurally identical with P. as licktacktoe aud the related card game mul systems for arithmetic there are formul sentences aualogens le P, that is, either the system is incorrect (proves falselioads) or i is incomplete (coninins truths nat provable in the systeci), Just (roughly) that for any af the kun Tore Gödel's incompleteness theoren states but his tical sentences Irue under any inlerpretatheorem put farward by Alonzo Church in 1936, These theorems and some athers like them are sometimes called limiis destroyed. The bur las disappecred The detailed statement of Gödel; incompleteness theorem is difficult, and a great deul of bac kground knowledge is necessary to understand the full proof, the cozy relation between truth and provability that one atten pts to achieve P is true if and only if P is nor provable. Hence we conclude that if we have a system rich enough to express P, then dees produce something disconcerting: hot make the system inconsistent, but it which it is expressible ís inconsistent, Now suppose the expressive powers of were such that some formal syst “provable” or “nuprovalile” could he expressed but not “true” or “Eslse.” The analogue of the ar sentence would then he “This sentence ts nut provable,” Call this sentence £. The existence of P does tenve is ve: “This senlonen is not true.” A contradiction vests Front assuming that the liar simience Is either (ne ov nut mul system in true, aul heuce any ing whether or not the follawing sen be roughly conveyed [see Musization on page 591. Consider the “Jar yoondax” it Greeks, which farmulsted by the can be restated as the problem of decid- The spirit of th: proof van nenetheless Giuots Theevem lstive theorems, since they suggest limitatiors on man's abilities, SOME UNSOLVED PROBLEMS of geometry and arithmetie Noted. Por al the ce division problem given him in amount of lime without any sew ine out, Innes, The exiswhe only a fuite number of e etsy been car zal interest sights, The existence of a deci ion procedure deeveases the theoretical iuteresi in su aren. Alt ried long divise prohioms wre prac tence of a decision procedure for a tore resolves the theomal system velienl prubl ms far all models for Ihe ssively pow systen:, Suceess with a number of systems led by 1930 te the hape, if val the expectolins, Wot more es]: artil systems thar A-+ odght be proved to be consistent, correct mel cumplote and ta possess a decision procedure. was hoperl that a fuif Jo partieulur, it syster har aritbunetic=cali A A~could li proved io have dire quidities. (By “oil systems” is mit the addition and mule plication of natural numbers med all ope erations definable in terms uf addition ciaus to solve the theoretical problems of and multiplication, such us division.) Such a proof would enable mathematinritlimetio: it vas even concervable that all the problems of matheraties might be solved itt à similar way, It was one of the great achievements of mathematical logicians to show that these hopes cannot be fulfilled, As examples of how this ‘vas done I shall discovered by Kurt Gödel in 1930 and a cuss the incompleteness throrem dis- 56 } ;® 187 ) # ~ proof No, x is vol a proof in A of sentence No. a.” We shall assmine what Gödel showed: that n can be so chosen that it is the number of the sentence G itself. IF we examine the situatinn closely, we come across the eurions phenomenon of wincompleteness [see illustration on page 59). A formal system is e-incomplete if it contains a Beneralization of which it is passible to prove every nu merical instance but it is not possible to prove the generalization. And it is just with regard to o-iscompleteness that mathematical logie may throw light ou some of the unsolved problems of arith. in general to trisect an augle requires à precise definition of all possible construc. tions with a straightedge and campass. May Sn, a definition they were never ir position to prove the impossibility of trisection. Similarly, decision procedures have existecl fac thousands of yeurs, but us the 1930’s no one gave im exact mathe. ed such Because the ancient Greeks difficulties we may be fuced with in regard to the unsolved problems of arithmetic. It states (roughly) that there is na decision procedure far any tortua l svstem ol arithmetie, An analogy may ‘he helpful. To biseet an arbitrary angle in matie I analysis of the concept. With Hi, vsis Church was ahle ra plane geometry is easy; one does not have to know everything that con be doue with a straightedge aud Compass. All arie has to do is construct a bisecto r, Ta contrast, to show unt it i impossible „IM PEOPLE 40 0e ZN PGE Y UP TAEID5 eck NONE Mar EIS a =~ metic. For example, in the case of Goldhach's BEEN process to deter. VPPOSEDHE àmail CK UP D OLE PARADOX OF THE POSTMAN is stucthar te 1 Bertra nd Russells garadux about set: that du Unlike Russell's parad Sy hewever, this poputarized version van ‘aposed of eusily. Far sample, une can deny that such à postman dee sen indeed ed exist, exist a net cuntaiti theiuselves, conjecture [see illustration on opposite pagel, for ny given even number it is u completely mechani mine if the number is the sum of twe primes. Suppose, huwever, Coldbach’s conjecture, like G, iMustrutes wien. pleteuess in A. We could then prove each and every one ol the following without being able to prove the general statement: “4 is the sum of twa primes, 6 is the sum of two primes, 8 is the sum of two primes...” In oti words, the reason the conjecture remains undecileit may be that, from the Assumptions math ematicians have actually made when teying to decide it, there is no proof or ref. que in m thematics ulation, Looking at it another way, a that of dividing a proof into eases. standard tech ple of a probiem that breaks down ints Galdbuch's conjecture is an iuste t of incompleteness, we have au examie an infinite number of casest On the other hand, if we could metamathematic dy prove that Caltbach 's Conjecture ix au illustration of meincompleteness in A, we would prove (as in the dr wwe dua i i une of the natura) » trates incu Beten ewan shaw that it is in possi pe sibiletoto find hbo. f nd auwow even " er greater than 2 that is unt the gn Nd bro prie, and this is equivalen t “greater vt 8 ! hi Malti eve ununber an 2 sum of two primes, What such a proof would mean is that those en zoen nod: transcending method, a N et ie, for example the mente coni ue catene Similar state. theoren, tae > me e about Fermat's last pue} tes see rusiration on opposite Ch 0 know they may, the tie Inet mt pe enphavizedl that we da hue on Fe é ree Goldbach s conjecsince ema s lust theorem illustrutes current e ness in A, but frou what we urch’s theorem also ilustrates

show that there can be no algorithm to solve the predicate “Sentence x expresses a th of arithmetic’ in the way (hat there is a long-division algorithm to solve the predicate “x is divisible by y.” Church proved his theorem by an argumeut thet hes similarities both ta zadel's argument and to Cantor's argumeut showing that there are mare real numbers than natural numbers, Chnrelvs that there is u decision pravedure for prov smonstrated that the assumption arithmetic leads 10 an absurdity, This reaus that there is uo algorithm that will always work for finding arithmetical fruth—aat merely that we have not vet found such an algoritlun. In other words, as there is 16 method in ticktacktow jv that will guarantee à win against all strategies, so there is no method in avithmetic that will guarantee a proof for all truths. Tt follows that there will always Hanetical baths far which present be SB um vv vxv À A methods will not work and that the creanecessary. tion of new methads will forever be The limitative theorems of Gödel and Church appear to have far-reaching phil. osophical consequences, and yet someone might object: “I deny that any philosnphical conclusions can be drawn from these Iwo theorems. Gödel and Church have shown that all the known formal systens for arithmetic are ipcomplete and lack a derision procedure, but this is hardly y more signiticang than the fact that one cannot square the circle or trisect an angle. It is perfectly possibie to do these things, but not with merely a sitaightedge and compass. Sinsilarly, all that the theorems af Gödel and Church come to is that given the means they huve selected just as Enelid selected a straightedge and compuss), their ‚onelusiuns follow, There is no philo- | TT ng, a [vv saphical import to their theorems, which Y d game STRATEGY for a wen to be identical with the strategy for ticktucktoe, Similurly, math matie adding to 17 dille clans Joule for identici] structures in aevas of mathematica taat ure apparently Vu || < «© € > 1 simply suggest that we must look for other means.” sion was arrived al through the follow. This objection would have a great deel of sting if it were not for one cireumstance: There do not appear lo be any other means. This profound concluing reasoning, Euclid had studied ideal triangles in an effort to understand the concept of space, In an analogous way Emil Post in the 1930’s undertook an analysis of what au ideal human compntprocedure, Independently and at about er does when he computes, in an eflort ta understand the concept of « decision the same time A. M. Turing andertouk a for the same per vis, corresponding pose, of an ideal computing machine, The two analyses were shovn i be equivalent. Church suggested that any schu} computation by men or machines could be duplicated by the ideal man or ideal machine. Church's thesis is an empirical one, but the evidence lor it is overwhelming. IF we accept it, the linitutive aspects of Güdel's and Chureh's theorems can be made explicit (see il. hestration où page 60]. fn fact, Engine MH. Gulanter has suggested that the limitive theorems of Gödel and Church e “the only known psychological la vs cunuparuble în exactitude with the af physics.” Unanswerable Questions precise questions do not have precise an- The central change that the limititive throrenss required of alt previous theovies of the nature of mathematics was the ecognition that there are unansw ible Earlier it had questions in the subject. bern thought that if a question could be ade precise, that question had an auswer. Now it was seen Ihat perhaps some ULI For any naluzal number x, proof : is vol a prcot in A of sentence a. GÖDEL'S ARGUMENT (IN MORE DETAIL} This sentence is not provable. Lel G be the name of the above inte sa chosen that senlence n is G sel GÖOEL'S ARGUMENT Let P be the name of the above sentence. LIAR PARADOX This sentence is not true since it states that itis not alur € ‘a e (218 a natural number Let L be the name of the above sentence. Suppose P is provabie. Then, Suppose L is true. Then what it says is correct proci 1 is not a proof in A of se-lence n. roof 2 is npt a proof in Avi sentence n. . Proof 3 is not a proof in A of sentence n. It can be shown that i each is al so provable. But "For any nat. number x, proof x is nol a proof in A of sentence niis rot en prov able, Hence ıf A is consistent, each and every numeri inslan cat of G is ce provable but G itse:f se: is nat. . Any formal system sys é conlarıs that a (formal penlenc such that each and every numerical instance 18 arava, fut net e gene: al statement, is called u-ncomplete. Hence if A is consistent G is noi provahie anti A is w- incompiete. ‘ A {nol-G . == il is fais, E m ce we know that each acd every sun S \ ol -G is orovabie hen, uPpose inf system in the formal I Syster numerical instance of G is pi rovabte, we have a s exists a (formal) sentence each and every instance wei ee piable ul {he negation of whch is also provable,ofSince each i e is true (provided that A is consistent), of not: G is a proof of a falsehaod. Hence, if A is Corel non die proof not provable, ' Therelore,i, G 1s Irue of the nat wal numbers if and oniy # ever, numerical instance of 6 is provabie in A bul G ıs nol Brovabie in A ‘ Conct:sion;: The syste system in which "N G is expressible (that is, A) i erlher incorrect (that is, proves falsehoods) Ke corner dk tis ct or incompleta with trish, What he alov ed iis that contains teuths not provabie in the system). ca al systean for wthaaeties maant hi with provalaility rash scope), so our abstract conceptions are sults reach still further, In order to perceive this a distinction must be drawn between the undecidable and Ihe undo. There does nut seem to be any way around these conclusions; indeed, the reour powers of perceptual discrimination intended to extend them (fo example limited and the methods of mathematics The indeterminacy is due to a refusal to mathematical induetion) are ai best pur our assumplions so that they ure ademike additional assumptions, The indeterminucy in arilinnetic, in coudrust, is due to u systematic inability to enlarge tid ju their effect. Our powers of conceptual discrimination have limils just as apleteness Hee n is soggested by of the neient Greeks. Gide) deslt plete (that is, contains truths not pravabie in the system). Conclusion: The system in which P is expressibie is either incorrect (that is, proves falsehoods) or ıncomonly if Pis rol provable, Therefore, P is true if and Suppose P is nol provable, Then, since il states that it is noi Provable. it is true. So if P 15 nat provable, i is bue, provable, ils nol true. So it A is provable, it +8 not true. Suppose G is provable p “able in th e format system A, Assuming i 15 correct a expresses is ve, However, G 8 «presses that G 3 A not provanie sistency, Henceif A ıs consistent,G is noi ' provabie. This means that each of the following is true: snot and ¢ says that it is nat true. So if Lis true, it is net true. Suppose L Is not true, Then what L says is incorrect and i. musi be true, So if £ 15 not true, itis true, Therefore, ¢. is true if and only if Lis not true. Conclusion: The system 1 ar pivades”™ U his 5 in which L 1s exprassible is inconsistent, with dhe GÖDEL'S PROOF anutugy tn certain ways. Just as indeterminatehess, previously considered peculiar to Imaginativo creations, was found in the swers. By way of avalogy, think of au ib jeet, say a tight bulb. 1£ you then ask, "Is it nade partly of glass?” the answer will probably be yes; if you ask the question, Quate to prove alt the truths of arith. metic. From the point uf view of any particular human being there must exist accidental truths of arithmetic, that is, statements that are true but ave such that nothing he assumes would enable him to prove them to be true; they would be neither necessary nor impossible, What the for which it is impossible lor aay human techuiques {for example by a micro being to make systematically complete and correct assumptions. Just as our sensory perceptions hive limits that can be extended but nut eliminuted by certain 5? cidable in another. For example, G és. unA but is, as we have seen, decidabi decidable {and true) in the larger systern is provable in the system, The concept if undeciduble is relative to a system; what is undecidable in one system may be detheorems represent, then, solvable. A sentener is undeciduble in a on vil is thelimilative discovery of an abstract structure given system il neither it aor its negation ite nee of such a square is accidentali either necessary vor impossible. With cul to that of a given circle. È ist we conceptual limitations the limited to Euclide it is not possible to \ A the existence of a square Drove quare whose wl Pula geni din” philosophical impact can be in- Mental philosophical conceptions. that of the Pythagoreans’ discovery of the incommensurabit y of the side andl the diagonut of a square, lt upsets funda physical world with the discovery of the quantum: theory, su indetenmiratenoss “Is it made partly uf cork?” the answer Was also found in mathematics with the discovery of the imitative theorems. The impact of this discovery is comparable to will probably be no. If, huwever, you ask, “Does it weigh exactly 3.1 ounces?” the question is probably unanswerable, The reality toward whieh the question is directed is indeterminate in some ways. Such indeterminateness is characteristic of prodnets af the imagination, including artistic creations. (How often did Juliet sneeze during the year before she met Romeo?) In these areas it is pointless to ask questions about things that are not determined by evidence, Compured with imaginative creations, physical reality is determinate, and yet the resulis of quautum theory suggest that physical reality is also indeterminate

THEOREM THEOREM The the human comguter capabic of LANGUAGE OF PSYCHOLOGY no consist Thees exists à sel of problems of ar cunsssient buman computer can 501 and Cla hb le plesity. LANGUAGE OF PHYSICS the true sentences of arithmetic. san There is no con tent conguting machine that the true and only ri' be programmed Thare exists a set of problems of anthmenc that mmed progra no consistent comp. tra machine can te to salve. heen ce Ariihmelical wuth thus has an “ideal” suggested by Join Myhill) The or “prospective” nature, (The terms have analogy of a game is useful in explaining this uotion, Hoften happens that a game is juvented and rules are laid down that define it but that later a civcunstince arises for which the cules give no guide uuce, At this point a decision has to be made as to what will henceforth be the rele concerning that cireumstaace. The decision might be made on the ground of tairsess or of what procedure makes for a better spectator sport or increases the danger or has some other effect. It cannot be made on the basis of the rules of the game, however, beesuse the rules discover mittametical truth not only are are incompletely defined. Now part of the iupact of the Huilative theorens is that the rules by which we define and but must he incompletely defined. We are therefore forced to define the notion of aritbnetical nutb historically; it canuot be esplicated once and fur all but must he redefined cuntinually, We have seen how both Gauss and Lobachevsky me to the conclusive that the problems of unth in non-Enclilean geometry required them to go beyond the data of pure geometry. In an analogous way we must go beyond the data of arithmetic in arder to define arithmetical truth, Maa has invented a game of arithmetic that is incomplete, apparently because of the incommensurability of man’s ideals and his abilities, Yet the temptation is strong to say that although arithnetic may be a game, it is a game in which we leam about realityabstract reality. There is, then, an incomand our ability to understand it commensurability between abstract reality pletely. JE the predicate “Sentence x expresses a truth of acithuselic” were solvable, arithmetic would become theoretically boring, It is not solvable, and so classical mathematics] about abstract enthat arithmetic is consistent! As Gödel noted, the question of consistency tums on aur ability "to replace [the axioms of aritlanetie will continue to absorb our interest precisely because its investigaour mind.” ZA tion demands human creativity, tities of an objective Pfatanic realm by insights about the given operations of pressing consisteucy. In order to be sure that the formal sentence expresses consistency, however, we have to assume can indeed prove a formal sentence ex plicated than arithmutic itself, then we fienit ourselves to methuds no more come the systems. On the other hand, if we and no, There arc proofs for consisteney based on methods so complicated as lo Le more in doubt than the consisteney of are consistent?” So fur the answer is yes decision procedure, Grant further that there may be serious problems of pre» tical undecidabilily and practical unsolvability for urithinetie, Can we at least prove that these systems of arithmetic systems of withmetic and Want there is no nut prove Li might be asked: “Grant thee we canthe completeness of formal solutions of andy the most extreme cunibeing sure whecher or not vere of the well-known problems of arithmetic have complicated. We have as yel no way of problems whose solutions are arbitrarily imited, whereas avithmetie presents pacity of any conceivable human being or computer. The ability of both men nd machines tn deal with complications sated Ihat they would be beyond the cacate is practically unsolvable if it is thearetically solvable but the techniques of solution are physically impossihle to car. ry ont. For example, the tecluiques for sulution of the predicate “x is a divisor of a perfect number” may be sp complinevertheless be faise, Or wgain, a prediras been checked into the niiilinus without being shown lo be false, but it may is virtually nil. Actnally the conjeetne confevture is falso, the Gest even number or te the effect thot any actual computation by a man mucbine. chine can he duplica ted by an ites? man oe na ideal rete stat na carted out, would produce a proe am that, alsentences INCOMPLETENESS. | tarmciateg ch arithene Wud Ang © y the GÖDEL'S URCH'S is that is not the sum of tivo primus may be we that the chance of discovering it whieh ys shown here, in the light of Chu Ar’ thesis, LIMITATIVE ASPECT of the sennons of Güdel comes exil (IS EOCIN Ivable if it is solvable by 1 of natal nmube t dual i Judes A and ¢ ù the argreuen that G is sudecil Me in AL In vont st, the conrept cf unsolvability i absolute. 5 sively solvable, at arithe ide! Laman enmprler {or ident compt ing macliitic) mentioned above; ott cecursively ansolvahle. Pus wise it is instance, the niinerical 4 redicate “x is ie, able, This does net menti that for some is that there given vaine nf the predivate we cur e, not salve the problem. Fur exwupl UL 4 km 2" is sentence 2,467 in A, then “Sentence 2,467 expresses a truth le. of arithmetic” is both true and provab What unsolvahitity means no technique whilsouver that will at ways w ork, as long division always socks, The question valurally arises of n whether or not some well-know probJem of elementary arithiuetio is recursively unsolvable, There seems to be no case. reason why (hit could not be the For example, te predicate “x is a divisor of a perfect number” might very well be recursively vurolvable and ther fore, sut Church’s thesis, uusolvon the alte in the ordinary sense, that fi, une solvable by any man ov computer. Practical Problems As il this were not enough, there ace further possible difficulties. [ have been describing what might be called the theoreticaliy undecidable aad the theoreti cally unsolvable. Phere also exist the phenomena nf practical undevidability aud practical unsolvability. A formal sentence is practically undecidable if it is thearetivally deciduble but the proof is so lung as to be physically impossible to curry oul, For example, it might he that Goldbach's conjeeture is true but that the shortest proof in a given system would require more pouuds of ink than there are atoms in the universe. Or if the