Copyright © 2000 Amir D. Aczel. All rights reserved.
Sometime between the fifth and sixth centuries B.C., theGreeks discovered infinity. The concept was so overwhelming,so bizarre, so contrary to every human intuition,that it confounded the ancient philosophers andmathematicians who discovered it, causing pain, insanity, andat least one murder. The consequences of the discovery wouldhave profound affects on the worlds of science, mathematics,philosophy, and religion two-and-a-half millennia later.
We have evidence that the Greeks came upon the idea ofinfinity because of haunting paradoxes attributed to thephilosopher Zeno of Elea (495-435 B.C.). The most well-knownof these paradoxes is one in which Zeno described arace between Achilles, the fastest runner of antiquity, and atortoise. Because he is much slower, the tortoise is given ahead start. Zeno reasoned that by the time Achilles reachesthe point at which the tortoise began the race, the tortoisewill have advanced some distance. Then by the time Achillestravels that new distance to the tortoise, the tortoise will haveadvanced farther yet. And the argument continues in this wayad infinitum. Therefore, concluded Zeno, the fast Achillescan never beat the slow tortoise. Zeno inferred from his paradoxthat motion is impossible under the assumption thatspace and time can be subdivided infinitely many times.
Another of Zeno's paradoxes, the dichotomy, says that youcan never leave the room in which you are right now. Firstyou walk half the distance to the door, then half the remainingdistance, then half of what still remains from where youare to the door, and so on. Even with infinitely many stepseachhalf the size of the previous oneyou can never getpast the door! Behind this paradox lies an important concept:even infinitely many steps can sometimes lead to a finitetotal distance. If each step you take measures half the size ofthe previous one, then even if you should take infinitely manysteps, the total distance traveled measures twice your firstdistance:
0 0 1 1/2 2 1+1/2 + 1/4 + 1/8 +1/16 +1/32 + 1/64+ ........ =2 ------------ 1 3/4
Zeno used this paradox to argue that under the assumptionof infinite divisibility of space and time, motion can nevereven start.
These paradoxes are the first examples in history of the useof the concept of infinity. The surprising outcome that an infinitenumber of steps could still have a finite sum is called"convergence."
One could try to resolve the paradoxes by discarding thenotion that Achilles, or the person trying to leave a room,must take smaller and smaller steps. Still, doubts remain, forif Achilles must take smaller and smaller steps, he can neverwin. These paradoxes point to disturbing properties of infinityand to the pitfalls that await us when we try to understandthe meaning of infinite processes or phenomena. But the rootsof infinity lie in the work done a century before Zeno by oneof the most important mathematicians of antiquity, Pythagoras(c. 569-500 B.C.).
Pythagoras was born on the island of Samos, off the Anatoliancoast. In his youth he traveled extensively throughoutthe ancient world. According to tradition, he visited Babylonand made a number of trips to Egypt, where he met thepriestskeepers of Egypt's historical records dating fromthe dawn of civilizationand discussed with them Egyptianstudies of number. Upon his return, he moved to Crotona, inthe Italian boot, and established a school of philosophy dedicatedto the study of numbers. Here he and his followersderived the famous Pythagorean theorem.
Before Pythagoras, mathematicians did not understandthat results, now called theorems, had to be proved. Pythagorasand his school, as well as other mathematicians of ancientGreece, introduced us to the world of rigorous mathematics,an edifice built level upon level from first principles usingaxioms and logic. Before Pythagoras, geometry had been acollection of rules derived by empirical measurement.Pythagoras discovered that a complete system of mathematicscould be constructed, where geometric elements correspondedwith numbers, and where integers and their ratioswere all that was necessary to establish an entire system oflogic and truth. But something shattered the elegant mathematicalworld built by Pythagoras and his followers. It wasthe discovery of irrational numbers.
The Pythagorean school at Crotona followed a strict codeof conduct. The members believed in metempsychosis, thetransmigration of souls. Therefore, animals could not beslaughtered for they might shelter the souls of deceasedfriends. The Pythagoreans were vegetarian and observedadditional dietary restrictions.
The Pythagoreans pursued studies of mathematics and philosophyas the basis for a moral life. Pythagoras is believedto have coined the words philosophy (love of wisdom) andmathematics ("that which is learned"). Pythagoras gave twokinds of lectures: one restricted to members of his society,and the other designed for the wider community. The disturbingfinding of the existence of irrational numbers wasgiven in the first kind of lecture, and the members were swornto complete secrecy.
The Pythagoreans had a symbola five-pointed starenclosed in a pentagon, inside of which was another pentagon,inside it another five-pointed star, and so on to infinity.In this figure, each diagonal is divided by the intersecting lineinto two unequal parts. The ratio of the larger section to thesmaller one is the golden section, the mysterious ratio thatappears in nature and in art. The golden section is the infinitelimit of the ratio of two consecutive members of the Fibonacciseries of the Middle Ages: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, ... where each number is the sum of its two predecessors.The ratio of each two successive numbersapproaches the golden section: 1.618.... This number is irrational.It has an infinite, nonrepeating decimal part. Irrationalnumbers would play a crucial role in the discovery of ordersof infinity two and a half millennia after Pythagoras.
Number mysticism did not originate with the Pythagoreans.But the Pythagoreans carried number-worship to a highlevel, both mathematically and religiously. The Pythagoreansconsidered one as the generator of all numbers. Thisassumption makes it clear that they had some understandingof the idea of infinity, since given any numberno matterhow largethey could generate a larger number by simplyadding one to it. Two was the first even number, and representedopinion. The Pythagoreans considered even numbersfemale, and odd numbers male. Three was the first true oddnumber, representing harmony. Four, the first square, wasseen as a symbol of justice and the squaring of accounts. Fiverepresented marriage: the joining of the first female and malenumbers. Six was the number of creation. The number sevenheld special awe for the Pythagoreans: it was the number ofthe seven planets, or "wandering stars."
The holiest number of all was ten, tetractys. It representedthe number of the universe and the sum of all generators ofgeometric dimensions: 10=1+2+3+4, where 1 element determinesa point (dimension 0), 2 elements determine a line(dimension 1), 3 determine a plane (dimension 2), and 4determine a tetrahedron (3 dimensions). A great tribute tothe Pythagoreans' intellectual achievements is the fact thatthey deduced the special place of the number 10 from anabstract mathematical argument rather than from countingthe fingers on two hands. Incidentally, the number 20, thesum of all fingers and toes, held no special place in theirworld, while the relics of a counting system based on 20 canstill be found in the French language. This strengthens theargument that the Pythagoreans made inferences based onabstract mathematical reasoning rather than commonanatomical features.
Ten is a triangular number. Here again we see the strongconnection the Pythagoreans saw between geometry andarithmetic. Triangular numbers are numbers whose elements,when drawn, form triangles. Smaller triangular numbersare three and six. The next triangular number after tenis fifteen.
A later Pythagorean, Philolaos (4th c. B.C.) wrote aboutthe veneration of the triangular numbers, especially thetetractys. Philolaos described the holy tetractys as all-powerful,all-producing, the beginning and the guide to divineand terrestrial life. Much of what we know about thePythagoreans comes to us from the writings of Philolaos andother scholars who lived after Pythagoras.
The Pythagoreans discovered that there are numbers thatcannot be written as the ratio of two whole numbers. Numbersthat cannot be written as the ratio of two integers arecalled irrational numbers. The Pythagoreans deduced theexistence of irrational numbers from their famous theorem,which says that the square of the hypotenuse of a right triangleis equal to the sum of the squares of the other twosides, a² + b² = c². This is demonstrated in the figure below.
When the Pythagorean theorem is applied to a triangle withtwo sides equal to one, the result is that the hypotenuse is givenby the equation c²= 1² + 1²=2, so that c=[square root of 2]. The Pythagoreansrealized that this new number could not possibly be written asthe ratio of two integers, or whole numbers. Rational numbers,which are of the form a/b where both a and b are integers,have decimals that either become zero eventually, or havea pattern that repeats itself indefinitely. For example,1/2=0.50000 ...; 2/3=0.6666666 ...; 6/11=0.54545454....Irrational numbers, on the other hand, have decimals that donot repeat the same pattern. Thus to write them exactly onewould need to write infinitely many decimals.
The irrationals were a devastating discovery for Pythagorasand his followers because numbers had become thePythagoreans' religion. God is number was the cult's motto.And by number they meant whole numbers and their ratios.The existence of the square root of two, a number that couldnot possibly be expressed as the ratio of two of God's creations,thus jeopardized the cult's entire belief system. By thetime this shattering discovery was made, the Pythagoreanshad become a well-established society dedicated to the studyof the power and mystery of numbers.
Hippasus, one of the members of the Pythagorean order, isbelieved to have committed the ultimate crime by divulgingto the outside world the secret of the existence of irrationalnumbers. A number of legends record the aftermath of theaffair. Some claim that Hippasus was expelled from the society.Others tell how he died. One story says that Pythagorashimself strangled or drowned the traitor, while anotherdescribes how the Pythagoreans dug a grave for Hippasuswhile he was still alive and then mysteriously caused him todie. Yet another legend has it that Hippasus was set afloat ona boat that was then sunk by members of the society.
In a sense, the Pythagoreans' idea of the divinity of the integersdied with Hippasus, to be replaced by the richer conceptof the continuum. For it was after the world learned the secretof the irrational numbers that Greek geometry was born.Geometry deals with lines and planes and angles, all of whichare continuous. The irrational numbers are the natural inhabitantsof the world of the continuumalthough rational numberslive in that realm as wellsince they constitute themajority of numbers in the continuum. A rational numbercan be stated in a finite number of terms, while an irrationalnumber, such as [Pi] (the ratio of the circumference of a circleto its diameter), is intrinsically infinite in its representation: toidentify it completely, one would have to specify an infinitenumber of digits. (With irrational numbers there is no possibilityof saying: "repeat the decimals 17342 forever," sinceirrational numbers have no patterns that repeat forever.)
Pythagoras died in Metapontum in southern Italy around500 B.C., but his ideas were perpetuated by many of his discipleswho dispersed throughout the ancient world. The centerat Crotona was abandoned after a rival mystical groupcalled the Sybaris mounted a surprise attack on thePythagoreans and murdered many of them. Among thosewho fled, carrying Pythagoras's flame, was a group that settledin Tarantum, farther inland in the Italian boot than Crotona.Here Philolaos was trained in the Pythagorean numbermysticism in the following century. Philolaos's writings aboutthe work of Pythagoras and his disciples brought this importantbody of work to the attention of Plato in Athens. Whilenot himself a mathematician, the great philosopher was committedto the Pythagorean veneration of number. Plato'senthusiasm for the mathematics of Pythagoras made Athensthe world's center for mathematics in the fourth century B.C.Plato became known as the "maker of mathematicians," andhis academy had at least four members considered amongthe most prominent mathematicians in the ancient world.The most important one for our story was Eudoxus(408-355 B.C.).
Plato and his students understood the power of the continuum.In keeping with number worshipnow brought toa new levelPlato wrote above the gates of his academy:"Let no one ignorant of geometry enter here." Plato's dialoguesshow that the discovery of the incommensurable magnitudestheirrational numbers such as the square roots oftwo or fivestunned the Greek mathematical communityand upset the religious basis of the Pythagoreans' numberworship. If integers and their ratios could not describe therelationship of the diagonal of a square to one of its sides,what could one say about the perfection the sect had attributedto whole numbers?
The Pythagoreans represented magnitudes by pebbles orcalculi. The words "calculus" and "calculation" come fromthe calculi of the Pythagoreans. Through the work of Plato'smathematicians and Euclid of Alexandria (c. 330-275 B.C.),author of the famous book The Elements, magnitudesbecame associated with line segments, as arithmetized geometrytook the place of the calculi. The dichotomy betweennumbers and continuous magnitudes required a newapproach to mathematicsas well as to philosophy and religion.In keeping with this new way of seeing things, Euclid'sElements discussed the solution of a quadratic equation, forexample, not algebraically but as an application of areas ofrectangles. Numbers still reigned in Plato's academy, but nowthey were viewed in the wider context of geometry.
In the Republic, Plato says "Arithmetic has a very great andelevating effect, compelling the mind to reason about abstractnumber." Timaeus, a book in which Plato writes aboutAtlantis, is named after a member of the Pythagorean order.Plato also refers to a number he calls "the lord of better andworse births," a number that through the centuries hasbecome the subject of much speculation. But Plato's greatestcontribution to the history of mathematics lies in having haddisciples who advanced the understanding of infinity.
Zeno's idea of infinity was taken up by two of the greatestmathematicians of antiquity: Eudoxus of Cnidus (408-355B.C.) and Archimedes of Syracuse (287-212 B.C.). These twoGreek mathematicians made use of infinitesimal quantitiesnumbersthat are infinitely smallin trying to find areas andvolumes. They used the idea of dividing the area of a figureinto small rectangles, then computing the areas of the rectanglesand adding these up to an approximation of theunknown desired total area.
Eudoxus was born to a poor family, but had great ambition.As a young man, he moved to Athens to attend Plato'sAcademy. Too poor to afford life in the big city, he foundlodgings in the port town of Piraeus, where the cost of livingwas low, and commuted daily to the academy in Athens.Eudoxus became Plato's star student and traveled with himto Egypt. Later in his life, Eudoxus became a physician andlegislator and even contributed to the field of astronomy.
In mathematics, Eudoxus used the idea of a limit process.He found areas and volumes of curved surfaces by dividingthe area or volume in question into a large number of rectanglesor three-dimensional objects and then calculatingtheir sum. Curvature is not easily understood, and to computeit, we need to view a curved surface as the sum of alarge number of flat surfaces. Book V of Euclid's Elementsdescribes this, Eudoxus's greatest achievement: the method ofexhaustion, devised to compute areas and volumes. Eudoxusdemonstrated that we do not have to assume the actual existenceof infinitely many, infinitely small quantities used in sucha computation of the total area or volume of a curved surface.All we have to assume is that there exist quantities "as smallas we wish" by the continued division of any given total magnitude:a brilliant introduction of the concept of a potentialinfinity. Potential infinity enabled mathematicians to developthe concept of a limit, developed in the nineteenth century toestablish the theory of calculus on a firm foundation.
The techniques first developed by Eudoxus were expandeda century later by the most famous mathematician of antiquity:Archimedes. Influenced in his work by ideas of Euclidand his school in Alexandria, Archimedes is credited withmany inventions. Among his discoveries is the famous lawdetermining how much weight an item loses when it isimmersed in a liquid. His work on catapults and othermechanical devices used to defend his beloved Syracuseenhanced his reputation in the ancient world. In mathematics,Archimedes extended the ideas of Eudoxus and made useof potential infinity in finding areas and volumes using infinitesimalquantities. By these methods, he derived the rule statingthat the volume of a cone inscribed in a sphere withmaximal base equals a third of the volume of the sphere.Archimedes thus showed how a potential infinity could beused to find the volume of a sphere and a cone, leading toactual results. After Archimedes' death at the hands of aRoman soldier, a stone mason chiseled the cone inscribed ina sphere on his gravestone to commemorate what Archimedesconsidered his most beautiful discovery.
Greek philosophers and mathematicians of the GoldenAge, from Pythagoras to Zeno to Eudoxus and Archimedes,discovered much about the concept of infinity. Surprisingly,for the next two millennia, very little was learned about themathematical properties of infinity. The concept of infinity,however, was reborn during medieval times in a new context:religion.