Humanistic Mathematics Network Journal

Issue 8 Article 24

7-1-1993

A Fairy Tale: Being A Pseudo-History ofMathematics With Special Attention Given to TheEvolution of the Number SystemPeter FlusserKansas Wesleyan University

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Recommended CitationFlusser, Peter (1993) "A Fairy Tale: Being A Pseudo-History of Mathematics With Special Attention Given to The Evolution of theNumber System," Humanistic Mathematics Network Journal: Iss. 8, Article 24.Available at: http://scholarship.claremont.edu/hmnj/vol1/iss8/24

PeterFlusserKQ1Uas Wesleyan University

which humanistic professors of mathematicstell their students under the illusion

that this will turn them into humanists

A Fairy Tale:Being

A Pseudo-History of MathematicsWith Special Attention Given to

The Evolution of the Number System

good. for it was the smallest ordered field. But itwas big enough for all practical purposes. AndMOM opened her mouth and said to all peoples:"All numbers ye shall add and subrract andmultiply, and by all numbers ye shall divide but byzero ye shall not divide. And this commandment Igive unto you. and it shall be a commandment untoyou and unto your posterity. even unto the lastgeneration. And it shall be called the EleventhCommandment: 'Thou shalt not divide by zero!'For whosoever shall divide by zero shall eat of theFruit of the Tree of Infinity and of the Worm ofIndetmninacy that liveth in the Fruit of the Tree ofInfinity, And of that Worm and of that Fruit yeshall not cat lest errors and inconsistencies invadeyour work and ye shall be scorned and derided byyour colleagues and successors forever." Andthatwas the third BURDEN AND IMPEDIMENT, andMOM saw that it was good.

And it came to pass in those days that allancient mathematicians lived contentedlysomewhere in Greece. unless they lived in Egyptor Babylon. And they added and subtraeted andmultiplied and divided, but by zero they did notdivide because me Greeks did not have zero. Burthe Babylonians did perhaps have zero, but theydid not divide by zero because the Babylonians didnot divide: they multiplied by reciprocals. and thiswas indeed a sign and a mark: of distinctionbetween Babylonians and Amoebas, for whereasAmoebas multiply by dividing, Babyloniansdivided by multiplying. Now the Babyloniansmighr have invented zero. or they might haveobtained it from the Indians or the Chinese, forthere lived ancient mathematicians in ancient India

And this was the first BURDEN ANDIMPEDIMENT. And MOM saw that it was goodfor theinduction axiom caused much weeping andgnashing of lCCth among mathematics students.

And Peana defined addition andmultiplication inductively on thenatural numbers,and the nalllr3l number>could always be added andmultiplied. but they could only be subtracted anddivided some of the time, but not all of the time.So MOM invented equivalence classes of pairs ofintegers. And she called them the set of rationalnumbers. but they were really fractions. That wasthe second BURDEN AND IMPEDIMENT fornobody likes fractions.

And MOM saw that the set of rationals was

In the beginning the Mother ofMathematicians (MOM) created sets. But sets werewithout form and void, and darkness lay upon theface of the Universe of Discourse. And MOMsaid: "Let there be number>" and there were naturalnumbers. For MOM created the natural numbers:all else is the work of MAN. But the naturalnumbers were without order and harmony. SoMOM created the music of the spheres. but nobodycould hear iL And MOM saw that all was good.And that was the evening and the morning of thenewageof mathematics.

And Peano opened his mouth and said:"Let the Induction Axiom be postulated and let it bea BURDEN AND IMPEDIMENT TO LEARNINGFOR MATIIEMATICS STUDENTS who shallneither rccall nor understand it,"

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and in ancient China; but in modem Western booksancient Chinese and ancient Indian mathematiciansdo not live: except that the ancient ChineseRemainder Theorem, and the ancient Indianmathematicians arc credited with inventing themodern Arabic numerals,

At this time there lived in Crotona, on thetoe of the boot of Italy, a set of Greekmathematicians known as Pythagoreans, whomade Music and Mathematics as MOMcommanded. The Pythagoreans saw music inmathematics and mathematics in music. Forwhenthe strings of their lyres were as the ratios of smalln=~n~~~w~u~w=beco~gand

harmonious, but if the rations were notso, then thesounds came forth that were harsh and grating.When the Pythagoreans saw these things there sawthat numbers were very good and they opened theirmouths and said: "All is number!"

And it came to pass in those days thatMOM said unto Pythagoras, the Lord of thePythagoreans: "Come up to me unto the mount andbe there: and I wiD give thee a tablet of clay, whichthe Babylonians have written; that thou mayestteach it," And Pythagoras rose up and girded hisloins and went up into the mount and MOMdelivered the tablet unto him. And lo! there wereinscribed upon the tablet fifteen sets ofPythagorean triples. And the name of the tabletwas Plimpton 322. And when Pythagoras saw thetablet he rejoiced exceedingly with a great joy, andopening his mouth he said: "This shall be knownas the Theorem of Pythagoras, for verily , verily Isay to you: Unless you know that a2+b2=c2 youcannot enter Plato's academy." And this was not aBURDEN AND IMPEDIMENT, and so MOM'sanger waxed exceedingly hot,

And so it came to pass that there lived inMetaponrum a Pythagorean Blabbermouth, and hecame to Pythagoras and said: "Master, thy theoremimplies that the square root of two is not a number.Neither is it a ratio of numbers. Itis a surd" AndPythagoras' anger waxed hot and he cast Plimpton322 out of his heads and he brake it upon theground. For this was indeed the fourth BURDENAND IMPEDIMENT. And then Pythagorasopened up his mouth and said unto theBlabbermouth: "Verily thou hast said it. Butproclaim not this message to the people lest thouleadeth them into confusion, for they think that a

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surd is absurd."

But MOM darkened the Blabbermouth'smind, and he kept opening his blabber mouth andblabbered to one and all. And thus he confused thepeople. So MOM made a big fish, and the big fishcame to the Blabbermouth and opened its mouthand said "Gulp!" and swallowed theBlabbermouth. And MOM was well pleased.

And then there came Theaetetus andDemocrirus and they fixed up the number systemand called it R because it was real even thoughPlato and all Platonist mathematicians. in otherwords. almost all mathematicians. that is allmathematicians except for a set of measure zero,said that R is a mental concept and hence ideal andnot real . And Euclid wrote it all down in Book Xof his "Elements", and Book X was the fifthBURDEN AND IMPEDIMENT for nobody couldread it And MOM saw that this was good.

But the Romans did not like BURDENAND IMPEDIMENTS. So they conquered theGreeks and there fen upon the earth a thousandyears of darkness when there was no light to domathematics by. The old mathematics wasforgotten, and Euclid 1.5 was called the PONSASINORUM, the bridge of asses, because youcould notcross that bridge and understand Euclid1.5 whilesitting on your donkey.

But the Arabs had light in those days. forthey invented Algebra and brought Arabic numeralsto the Darkened West And there came OmarKhayyam, and he opened up his mouth and said:

"A book of verses underneath the sky,A loaf of bread, a jug of wine, and ICan solve the cubic. if I try!"

And Omar, the poet and algebraist, solved thecubic geometrically.

But Cardano, the sooth-sayer, Tartaglia,the stutterer, and other Italian geometers solved thecubic algebraically. For after thousand yean ofdarkness there came the light of the Renaissanceand all men could see what theAncients had done.And the light brought Fermat and he discovered theLast Theorem. And when MOM saw that Fermatmay have proved the Last Theorem sheopened hermouth and said: ''Truly ifFermat proveth the LastTheorem then woe be unto me and unto allmathematicians for there shall be naught left forusto do and we shall suffer death from boredom:'

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So MOM sent an angel to shrink the margins ofDiopbantus' book so that Fermat could not writethe proof there. And she sent guardian angels withflaming swords to guard the proof from allgenerations even unto this day so that no one mightrediscover iL And Fermat write in the margin ofDiophantus" book: "Beholdl I have discoveredsomething the ancients did not know: the marginsof Diophantus' book arc too small 10 write in."

And then came Descartes, and he stayed inbed even in the light of the Renaissance. forDescartes, the soldier of fortune, was weak andinfirm. And Descartes opened his mouth and said:"Cogito ergo suml" And then he got out of bed toteach Queen Christina Analytic Geometry andcaught cold and died. And then Newton stood onthe shoulders of giants and played with pebbles onthe shore of the unexplored ocean and discoveredthe Calculus, and then Lcibniz invented theCalculus and Newton and Lcibniz fought over whodiscovered this, and who invented that, and whocopied from whom, and what it all means, andNewton lost the argument and therefore Britishmathematicians did not know how 10 use Lcibniz'snotation. But Euler did know how to use thatnotation, and he discovered lots of mathematicsand he made lots of mistakes and so Cauchy,Bolzano and Weierstrass opened their mouths andsaid: "Mathematics is like a house without afoundation built upon sand. against which seriesand infinite simals beat vehemently. andimmediately it falls and the ruins of that house aregreaL So let us dig deep and lay a foundation formathematics on the rock of the realnumbers: thenseries and infmitesimals cannot shake it,t o

And there came a Wise but UnknownMathematician and he opened his mouth and said:"The real numbers arc infinite decimals. " And thatmade sense, and it was not a BURDEN ANDIMPEDIMENT and students did not weep nor didthey gnash their teeth. And MOM saw that thiswas not good. So MOM made Cantor and heopened his mouth and said: "Real numbers arcequivalence classes of Cauchy sequences." AndMOM also made Dedekind who, opened hismouth, said: "The real numbers are DedekindCuts." And there carne Shakespeare and he saidunto Dedekind: ''That ' s the most unkindest cut ofall." For this was the sixth BURDEN AND

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IMPEDIMENT. And MOM saw that this wasgood. for DOW mathematics students came untoMOM and said unto her: "What is a real number,rea1ly?"

And MOM said: "The real numbers arc acomplete ordered field. and all complete orderedfields arc order-isomorphicso that there is only onecomplete ordered field; Cantor and Dcdckind andthe WlSC Mathematician notwithstanding. And thatis good." For this was the seventh BURDENAND IMPEDIMENT. And it should have been thelast BURDEN AND IMPEDIMENT, for on theseventh BURDEN AND IMPEDIMENT, MOMwanted to rest.

Yea verily, this should be the end of thestory except for Gauss. who discovered thecomplex numbers. which arc ordered pairs of rea1numbers and which Gauss discovered before thereal numbers were invented. Only Wessel andArgand invented the complex numbers beforeGauss discovered them, which is strange becauseusually Gauss discovered things before otherpeople invented them. But then Gauss discoverednon-Euclidean geometry before Bolyai andLobachevski invented it, and that was importanlbut also sad. because before the discovery of non­Euclidean Geometry mathematicians had beenSEEKERS OF TRUTII, bUI after this inventionthey became hewers of wood and DRAWERS OFNECESSARY CONCLUSIONS FROMARBITRARY ASSUMPTIONS and thereforeFORMAL MANIPULA TORS OFMEANlNGlllSS SYMBOLS.

And then came Godel, and he opened hismouth and said: "You cannot proved that theaxioms of a complete ordered field arc complete orconsistent or categorical. So the foundation ofmathematics is indeed built upon sand." And thiswas a BURDEN AND IMPEDIMENT FOR ALLMA1HEMATICIANS. And when Hilbert heard ofthis BURDEN AND IMPEDIMENT he tore hisclothes, covered his head with ashes, wrappedhimself in sack-cloth and retired from mathematics.And this was not good. So MOM transformed allmathematicians intohumanists whocan overcomemost BURDEN AND IMPEDIMENTS. And thiswas very good, and so everyone lived happily everafter.

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