Math in the Modern World (C)
Math in the Modern World (C) MATH 1319
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What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/29/15
Hindu Beliefs Author Anonymous It is difficult to assign a dogmatic orthodoxy to Hinduism Many variations have developed from Hinduism over the years and many nonHindu cults and eligious movements gained their inspiration from Hinduism Even in India today the most orthodox divisions of Hinduism have changed signi cantly over the last three thousand years One of the oldest aspects of Hinduism is as much social as religious and that is the caste system It is important to understand the caste system before delving into Hindu religious beliefs According to Hindu teaching there are four basic castes or social classes Each caste has its own rules and obligation for living The elite caste is the Brahman or priest caste Second are the Kshatriyas or warriors and rulers Third are the Vaisyas or merchants and farmers Finally the fourth caste is the Shudras or laborers Outside the caste system are the untouchables The untouchables are the outcasts of Hindu society Though outlawed in India in the 1940s the untouchables are still a very real part of Indian society One does not get decide his or her caste 7 that matter is decided when one is born into a particular caste As previously stated there is not a strict orthodoxy in Hinduism There are however several principles that share a commonality among the various sects Virtually all Hindus believe in The threeinone god known as Brahman which is composed of Brahma the creator Vishnu the Preserver and Shiva the Destroyer The Caste System Karma The law that good begets good and bad begets bad Every action thought or decision one makes has consequences 7 good or bad 7 that will return to each person in the present life or in one yet to come Reincarnation Also known as transmigration of souls or samsara This is a journey on the circle of life where each person experiences as series of physical births deaths and rebirths With good karma a person can be reborn into a higher caste or even to godhood Bad karma can relegate one to a lower caste or even to life as an animal in their next life Nirvana This is the goal of the Hindu Nirvana is the release of the soul from the seemingly endless cycle of rebirths Hinduism is both polytheistic and pantheistic There are three gods that compose Brahman 7 Brahma Vishnu and Shiva Hindus also worship the wives of Shiva such as Kali or one of Vishnu s ten incarnations avatars This is only the beginning There are literally millions of Hindu gods and goddesses 7 by some counts as many as 330 million Mathematical mysteries The Ba rber39s Paradox by Helen Joyce A close shave for set theory Suppose vou Wak past a barber s shop ohe dav and see a srgh that saw Do vou Shave voursertv Ithot come h and m shave vou Ishave ahvohe who does hot Shave hrrhsert aho ho ohe e se Thrs seerhs tarr ehough aho tarrh srrhpre unt a httre ater the toHoWhg ouestroh occurs to vou a does the aroer Shave hrrhsertv It he does then he mustn t because he doesn t Shave rheh W o s ave therhsehes out then he doesn t so he must because he Shaves everv man who oesn t Shave hrrhsert and so on Both possrorhtres ead to a cohtraorctroh Thrs rs the Barber s Paradox orscovereo ov rhatherhatrcrah ph osopher and q t e set theorv the Barber s paradox exposed a huge prob em and changed the ehtrre orrectroh ottwehtreth centurv rhatherhatrcs 1h hawe set theorv a set Js Just a coHectJoh orobJects that Satrstv some cohthJoh Ahv dearw ohrased cohthJoh Js thouoht to de ne a set a hamew those thhgs that Satrstv the cohthJoh Here are some sets The set oraH red motorcvdes The set oraH hteoers oreater than zero The set oraH bTue bahahas 7 thch Js Just the emptv Set Thrs Set 5 not a member of Ttse t me sets are hot members or themsewes a tor examp e the set orau red motorcvdes a and some sets are a tor examp e the set orau nOnT motorcvdes Now what about the set oraH se s thch are hot members or t a er o t and Th Jsh t theh t s Just We the barber who Shaves thseTr but mustn t ahd thererore doesn t and so must W We reahze that RusseH S Barber s Paradox means that there 5 a cohtradJctJoh att orhaJ e set the v That Js herer a statement 8 su h that both Jtserrahd Jts hegatJ hots aretue Th o rtJ r st ehthereJ t ora et thch areh m orthems wes em cohtahs Jtsew But once vou have a cohtradJctJoh vou can prove anvthmg vou Me Just ushg the rures or rochar deductroh ThJs Js how Jt goes 1 Its Js true and o s ahv other statemeht then s or o Js dearw true 2 shoe hot s s arso true so s s o Q hd hot 8 3 Therefore Q Js true ho matter what Jt Js The Daradox raJses the rrrohtemho prospect that the Whoe or mathemaths Js based oh Shaw rouhdatJohs and that ho broor can be trusted In essence the brobrem was that h hawe set theorv t was assumed that ahv cohereht cohthJoh courd be used to determhe a set In the Barber s Paradox the cohthJoh Js Shaves thsew but the set oraH men who Shave themsewes FRACTAL CHAOS Crashes in the Wall Between Science and Religion By Ralph Losey At any given moment life is completely senseless But viewed over a period it seems to reveal itself as an organism existing in time having a purpose trending in a certain direction Aldous Huxley 18941963 The rapidly accelerating discoveries of Chaos are overtaking our worldview They teach us that Newton and indeed almost all of the prechaos scientists were dead wrong in their basic view of the Universe They thought that there was a predictable cause and effect for everything and that everything happened according to fixed physical laws They believed in certainties not probabilities Theirfundamental image of the Universe was a big clock The presence of a divine being was only necessary to make the clock and wind it up After He created the Universe all God had to do was sit back and watch The laws would operate in a predictable causal fashion Old science actually used to think that if you only knew all of the initial conditions how the clock worked you could predict what would happen at any point in time Science assumed that everything could be known and eventually predicted The Universe was ruled by a detailed system of unchanging laws Cosmos and causality reigned supreme There was no room for chaos and so it was conveniently swept under the rug The inevitable outcome of the ordered machine view was the complete winding down of the clock the end of time in complete entropy the second law of thermodynamics where everything tends to breakdown to dissipate This big picture of science naturally spawned the quotGod is deadquot philosophies nihilism the life nausea of existentialism behavioralism communism and the like Now with the Chaos theories this paradigm is itself dead A whole new scientific view has been born one much more in accord with an organic view the common law and philosophies of hope and spirit The cosmic clock image of establishment science first began to crumble at the turn of the century when physicists found that at the nuclear level the causal laws of physics didn39t hold true The behavior of the atom and individual electron could not be predicted Still even in the face of incontrovertible evidence of quantum physics old ideas die hard The static civil law mind set would not die easily Even Einstein could not believe that God would play dice with the Universe He searched in vain for a unified field theory that would explain away the chance and unpredictability so obvious in the subatomic world Science struggled to maintain its centuries old view The belief in a causal cosmos was now on shaky ground because it lacked a subatomic foundation Still it prevailed because the rest of the world of physics seemed to follow linear orderly and predictable clock like processes Besides no one had articulated a different view to replace it The subatomic world was considered an insignificant anomaly an exception that proved the rule Then along came the Science of Chaos in the last part of this century to show that causality did not apply everywhere else as thought In fact close measurements revealed that the unpredictable appeared in what was previously believed to be the most ordered and predictable of systems the swinging of a simple pendulum the very heart of a clock As James Gleick39s book Chaos shows the brave early explorers of Chaos found that Science had been fooling itself for centuries by ignoring tiny deviations in its data and experiments If a number was slightly off what the causal laws predicted the prechaos scientists simply assumed there was an error in measurement in order to uphold the sanctity of the law itself In order to preserve their pseudocosmos scientists limited their investigation to closed and artificial systems avoiding the turbulence of open systems like the plague Causality was the prime assumption behind all prechaos science and it never occurred to anyone to question it This conceptual bias created a blind spot of enormous proportions But the reality of open systems the Chaos lurking behind all order would not be denied The charade of perfect order and fudged experimental data could not last forever By the nineteen seventies it began to crumble the conceptual blinders were falling from the eyes of more and more scientists By the nineteen eighties the fly in the ointment the unpredictable results in what should have been perfect predictability could no longer be denied The Science of Chaos was born Our understanding of the world will never be the same After nearly two decades now of work by Chaoticians made up of the leading scientists and mathematicians in a wide variety of fields the evidence is overwhelming The world is not a gigantic clock where everything happens in an ordered and predictable manner The real world is fundamentally disordered free Chaos reigns over predictability Simple linear systems which are causal and predictable are the exception in the Universe not the rule Most of the Universe works in jumps in a nonlinearfashion that can not be exactly predicted It is infinitely complex Freedom and free will the Strange Attractors prevail over rules and determinacy Yet Chaos is no enemy and destroyer of Cosmos for from out of Chaos a higher order always appears but this order comes 39 and p quot quot It is quot if a 39 quot39 The creation of the Universe is an ongoing process not just a one time event at the beginning All and everything and everyone is part of this creative process Over time all systems from molecules to life to galactic clusters are continually creating new organizations and patterns from out of featurelessness and chaos The world is not a Clock it is a Game a Game of Chance and Choice In the game random processes chance and serendipity allow room for free will individuality and unpredictable creativity The Universe is governed by laws but the laws are of a different kind than previously thought Like the common law system the Laws of Wisdom are inherently flexible They are not written in stone they are general They leave infinite room for creativity within certain general parameters A few fundamental principals exist to establish the parameters but the Law governs much more loosely than previously thought The Laws are subject to changes and modifications over time and depend upon the particular facts Like the common law the Laws of Nature appear to have flexibility many things are decided on a case by case basis Self organization is the rule not the exception Everything is not predetermined by a rigid and David Hilbert s Grand Hotel Paradox Courtesy Wikipedia Situation In a hotel with a finite number of rooms it is clear that once it is full no more guests can be accommodated Now imagine a hotel with an infinite number of rooms One might assume that the same problem will arise when an infinite number of guests come along and all the rooms are occupied However in an infinite hotel the situations quotevery room is occupiedquot and quotno more guests can be accommodatedquot do not turn out to be equivalent There is a way to solve the problem if you move the guest occupying room 1 to room 2 the guest occupying room 2 to room 3 etc you can fit the newcomer into room 1 It is also possible to make room for a countably infinite number of new clients just move the person occupying room 1 to room 2 occupying room 2 to room 4 occupying room 3 to room 6 and so on and all the oddnumbered rooms will be free for the new guests Now imagine a countably infinite number of coaches arrive each with a countably infinite number of passengers Still the hotel can accommodate them first empty the odd numbered rooms as above then put the first coach39s load in rooms 3n for n 1 2 3 the second coach39s load in rooms 5n for n 1 2 and so on for coach number i we use the rooms pn where p is the i1th prime number You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers if the seats are not numbered number them Regard the hotel as coach 0 lnterleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests The guest in room number 1729 moves to room 01070209 ie room 1070209 Leading zero added to clarify we take the first digit of the coach number first The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel Some find this state of affairs profoundly counterintultlve The properties of infinite quotcollections of thingsquot are quite different from those of ordinary quotcollections of thingsquot In an ordinary hotel with more than one room the number of oddnumbered rooms is obviously smaller than the total number of rooms However in Hilbert39s aptly named Grand Hotel the quotnumberquot of oddnumbered rooms is as quotlargequot as the total quotnumberquot of rooms ln mathematical terms this would be expressed as follows the cardinality of the subset containing the oddnumbered rooms is the same as the cardinality of the set of all rooms In fact infinite sets are characterized as sets that have proper subsets of the same cardinality For countable sets this cardinality is called alephnull An even stranger story regarding this hotel shows that mathematical induction only works from an induction basis No cigars may be brought into the hotel Yet each of the guests all rooms had guests at the time got a cigar while in the hotel How is this The guest in Room 1 got a cigar from the guest in Room 2 The guest in Room 2 had previously received two cigars from the guest in Room 3 The guest in Room 3 had previously received three clgars from the guest in Room 4 etc Each guest kept one cigar and passed the remainder to the guest in the nextlowernumbered room Chinese Characters fur class discussinn Z 4mg wrrenrnerns enne whenrnerns torrents w poovammmnon m wartime 011 a 1mm tundof because afsneezmg dragon mnght madly Chinese Philnsnphy By DAVID L HALL amp ROGER T AMZES e nt u and H V V m H r assurnedrn atternptrng to rntroduee Asran thrnkrng to Western readers Unal the rst Jesurt rgnoranee of one another thrnkers L Contrasted wrth tradrtrons the Lhmkmg of the Chrnese rs far more eonerete thxsrworldly played sueh arnrnorro1e m the development of Chrnese rnteueetua1 eulture andthat as a eonsequenee Chrnese eyes were focused not upon rssues of eosrnre order but upon rnore soeral nexus stayr atrhome39 chrna and unaruculated By contrast m the West these norms had to be abstracted andratsed to the m the eon uenee of Greek Hebrew and Latrn ervrhzatrons the prominence of less formal uses of analogical parabolic and literary discourse The Chinese are largely indifferent to abstract analyses that seek to maintain an objective perspective and are decidedly anthropocentric in their motivations for the acquisition organization and transmission of knowledge The disinterest in dispassionate speculations upon the nature of things and a passionate commitment to the goal of social harmony was dominant throughout most of Chinese history Indeed the interest in logical speculations on the part of groups such as the sophists and the later Mohists was shortlived in classical China The concrete orientation of the Chinese toward the aim of communal harmony conditioned their approach toward philosophical differences Ideological con icts were seen not only by the politicians but by the intellectuals themselves to threaten societal wellbeing Harmonious interaction was finally more important to these thinkers than abstract issues of who had arrived at the truth Perhaps the most obvious illustration of the way the Chinese handled their theoretical con icts is to be found in mutual accommodation of the three emergent traditions of Chinese culture Confucianism Daoism and Buddhism Beginning in the Han dynasty 206 BC7AD 220 the diverse themes inherited from the competing hundred schools of preimperial China were harmonized within Confucianism as it ascended to become the state ideology From the Han synthesis until approximately the tenth century AD strong Buddhist and religious Daoist in uences continued to compete with persistent Confucian themes while from the eleventh century to the modern period Neoconfucianism 7 a Chinese neoclassicism 7 absorbed into itself these existing tensions and those that would emerge as China like it or not confronted Western civilization In the development of modern China when Western in uence at last seemed a permanent part of Chinese culture the values of traditional China have remained dominant For a brief period intellectual activity surrounding the May Fourth movement in 1919 seemed to be leading the Chinese into directions of Western philosophic interest Visits by Bertrand Russell and John Dewey coupled with a large number of Chinese students seeking education in Europe Great Britain and the USA promised a new epoch in China s relations with the rest of the world Chinese thinking as ars contextualis In the Western tradition thinking about the order of things began with questions such as What kinds of things are there and What is the nature physis of things This inquiry which later came to be called metaphysics took on two principal forms One which the scholastics later termed ontologia generalis general ontology is the investigation of the most essential features of things 7 the being of beings A slightly less abstract mode of metaphysical thinking scientia universalis universal science involves the attempt to construct a science of the sciences a way of knowing which organizes the various ways of knowing the world about us Both general ontology and universal science interpret the order of the cosmos Both suppose that there are general characteristics 7the being of beings or universal principles 7 which tell us how things are ordered Neither of these forms of metaphysical thinking were in uential in classical China One reason for their unimportance is re ected in the character of the Chinese language Simply put the classical Chinese language does not employ a copulative verb The Chinese terms usually used to translate being and notbeing are you and wu see Youiwu The Chinese you means not that something is esse in Latin in the sense that it exists in some essential way it means rather that something is present To be is to be available to be around Likewise wu as to not be means not to be around Thus the Chinese sense of being overlaps having A familiar line from the Daoist classic the Laozz39 or Daodejz39ng which is often translated Notbeing is superior to Being should more responsibly be translated as Nothaving is superior to having or as a Marxistinspired translator has rendered it Not owning private property is superior to owning private property The Chinese language disposes those who employ the notions of you and wu to concern themselves with the presence or absence of concrete particular things and the effect of having or not having them at hand Even in recent centuries when the in uence of translating IndoEuropean culture required the Chinese language to designate a term to do the work of the copula the choice was shi meaning this thus indicating proximity and availability rather than existence One must assume that the concrete disposition of Chinese thinkers is both cause and consequence of this characteristic feature of linguistic usage Perhaps the best designation for the most general science of order in the Chinese tradition would be ars contextualz39s The art of contextualizing contrasts with both scientia universalis and ontologia generalis Chinese thinkers sought the understanding of order through the artful disposition of things a participatory process which does not presume that there are essential features or antecedentdetermining principles serving as transcendent sources of order The art of contextualizing seeks to understand and appreciate the manner in which particular things presenttohand are or may be most harmoniously correlated Classical Chinese thinkers located the energy of transformation and change within a world that is ziran autogenerative or literally soofitself and found the more or less harmonious interrelations among the particular things around them to be the natural condition of things requiring no appeal to an ordering principle or agency for explanation The dominant Chinese understanding of the order of things advertises an important ambiguity in the notion of order itself The most familiar understanding of order in the West is associated with uniformity and pattern regularity This logical or rational ordering is an implication of the cosmological assumptions which characterize the logos of a cosmos in terms of causal laws and formal patterns A second sense of order is characterized by concrete particularities whose uniqueness is essential to the order itself No final unity is possible in this view since were this so the order of the whole would dominate the order of the parts cancelling the uniqueness of its constituent particulars Thus aesthetic order is ultimately acosmological in the sense that no single order dominates The crucial difference between these two senses of order is that in the one case there is the presumption of an objective standard which one perforce must instantiate in the other there is no source of order other than the agency of the elements comprising the order In the West mathematical order has been thought the purest In China by contrast any notion of order which abstracts from the concrete details of thisworldly existence has been seen as moving in a direction of decreasing relevance Rational order depends upon the belief in a singleordered world a cosmos aesthetic order speaks of the world in much less unitary terms In China the cosmos is simply the ten thousand things The belief that the things of nature may be ordered in any number of ways is the basis of philosophical thinking as am contextualz39s Euclid of Alexandria Born about325 BC Died about 265 BC in Alexandria Egypt V o w 9 V A F 0 Dr M AD Wrote see 1 or 9 ormany other sources r predecessors Thxs man vaed m the me afthe rst mewfar Arehrrnedes whoallowed askzd hxm xfthere were a shorted way to smdy geomevy Lhan the Elements m whxch he rephed says he 15 and 1th2 whole quot51mmst canstmcnzm afthe sarca ed Plamnxc gmes rehable T m a bom m Tyre merely invented by the authors on the who was a phrlosopher who lrved about 100 years before the mathematician Euehd ofAlexandna It Tu references to numerous rnen called Euclxdm the meracure owns penod n r nan exactly what reference to Euclid in Archimedes work Proclus is referring to in what has come down to us there is only one reference to Euclid and this occurs in On the sphere anal the cylinder The obvious conclusion therefore is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev in 48 He argued that the reference to Euclid was added to Archimedes book at a later stage and indeed it is a rather surprising reference It was not the tradition of the time to give such references moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference Despite Hj elmslev39s claims that the passage has been added later BulmerThomas writes in 1 Although it is no longer possible to rely on this reference a general consideration of Euclid 39s works still shows that he must have written after such pupils of Plato as Eudoxus anal before Archimedes For further discussion on dating Euclid see for example 8 This is far from an end to the arguments about Euclid the mathematician The situation is best summed up by Itard l 1 who gives three possible hypotheses i Euclid was an historical character who wrote the Elements and the other works attributed to him ii Euclid was the leader of a team of mathematicians working at Alexandria They all contributed to writing the 39complete works of Euclid even continuing to write books under Euclid s name after his death iii Euclid was not an historical character The 39complete works of Euclid were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier It is worth remarking that Itard who accepts Hjelmslev s claims that the passage about Euclid was added to Archimedes favours the second of the three possibilities that we listed above We should however make some comments on the three possibilities which it is fair to say sum up pretty well all possible current theories There is some strong evidence to accept i It was accepted without question by everyone for over 2000 years and there is little evidence which is inconsistent with this hypothesis It is true that there are differences in style between some of the books of the Elements yet many authors vary their style Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author remained Even if we accept i then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria He therefore would have had some able pupils who may have helped out in writing the books However hypothesis ii goes much further than this and would suggest that different books were written by different mathematicians Other than the differences in style referred to above there is little direct evidence of this Although on the face of it iii might seem the most fanciful of the three suggestions nevertheless the 20Lh century example of Bourbaki shows that it is far from impossible Henri Cartan Andre Weil Jean Dieudonne Claude Chevalley and Alexander Grothendieck wrote collectively under the name of Bourbaki and Bourbaki s Elements de mathe39matiques contains more than 30 volumes Of course if iii were the correct hypothesis then Apollonius who studied with the pupils of Euclid in Alexandria must have known there was no person 39Euclid39 but the fact that he wrote Euclid did not work out the syntheses of the locus with respect to three and four lines but only a chance portion ofit certainly does not prove that Euclid was an historical character since there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious Nevertheless the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis iii in that the Euclid team would have to have consisted of outstanding mathematicians So who were they We shall assume in this article that hypothesis i is true but having no knowledge of Euclid we must concentrate on his works after making a few comments on possible historical events Euclid must have studied in Plato s Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar None of Euclid39s works have a preface at least none has come down to us so it is highly unlikely that any ever existed so we cannot see any of his character as we can of some other Greek mathematicians from the nature of their prefaces Pappus writes see for example 1 that Euclid was most fair and well disposed towards all who were able in any measure to advance mathematics careful in no way to give o quotence and although an exact scholar not vaunting himself Some claim these words have been added to Pappus and certainly the point of the passage in a continuation which we have not quoted is to speak harshly and almost certainly unfairly of Apollonius The picture of Euclid drawn by Pappus is however certainly in line with the evidence from his mathematical texts Another story told by Stobaeus 9 is the following someone who had begun to learn geometry with Euclid when he had learnt the first theorem asked Euclid quotWhat shall I get by learning these things quotEuclid called his slave and said quotGive him threepence since he must make gain out of what he learns quot Euclid s most famous work is his treatise on mathematics The Elements The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years Probably no results in The Elements were rst proved by Euclid but the organization of the material and its exposition are certainly due to him In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong a rhombus and a rhomboid The Elements begins with de nitions and ve postulates The rst three postulates are postulates of construction for example the rst postulate states that it is possible to draw a straight line between any two points These postulates also implicitly assume the existence of points lines and circles and then the existence of other geometric objects are deduced from the fact that these exist There are other assumptions in the postulates which are not explicit For example it is assumed that there is a unique line joining any two points Similarly postulates two and three on producing straight lines and drawing circles respectively assume the uniqueness of the objects the possibility of whose construction is being postulated The fourth and fth postulates are of a different nature Postulate four states that all right angles are equal This may seem quotobviousquot but it actually assumes that space in homogeneous by this we mean that a gure will be independent of the position in space in which it is placed The famous fth or parallel postulate states that one and only one line can be drawn through a point parallel to a given line Euclid s decision to make this a postulate led to Euclidean geometry It was not until the 19Lh century that this postulate was dropped and noneuclidean geometries were studied There are also axioms which Euclid calls common notions These are not speci c geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science For example Things which are equal to the same thing are equal to each other Zeno of Sidon about 250 years after Euclid wrote the Elements seems to have been the rst to show that Euclid s propositions were not deduced from the postulates and axioms alone and Euclid does make other subtle assumptions The Elements is divided into 13 books Books one to six deal with plane geometry In particular books one and two set out basic properties of triangles parallels parallelograms rectangles and squares Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras Book ve lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes Heath says 9 Greek mathematics can boast no finer discovery than this theory which put on a sound footing so much of geometry as depended on the use of proportion Book six looks at applications of the results of book ve to plane geometry Books seven to nine deal with number theory In particular book seven is a selfcontained introduction to number theory and contains the Euclidean algorithm for nding the greatest common divisor of two numbers Book eight looks at numbers in geometrical progression but van der Waerden writes in 2 that it contains cumbersome enunciations needless repetitions and even logical fallacies Apparently Euclid39s exposition excelled only in those parts in which he had excellent sources at his disposal Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus Euclid changed the proofs of several theorems in this book so that they tted the new de nition of proportion given by Eudoxus Books eleven to thirteen deal with threedimensional geometry In book eleven the basic de nitions needed for the three books together are given The theorems then follow a fairly similar pattern to the t w 139 39 39 39 p ii 4 1 given in books one and four The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters These results are certainly due to Eudoxus Euclid proves these theorems using the quotmethod of exhaustionquot as invented by Eudoxus The Elements ends with book thirteen which discusses the properties of the ve regular polyhedra and gives a proof that there are precisely ve This book appears to be based largely on an earlier treatise by Theaetetus Euclid s Elements is remarkable for the clarity with which the theorems are stated and proved The standard of rigour was to become a goal for the inventors of the calculus centuries later As Heath writes in 9 This wonderful book with all its imperfections which are indeed slight enough when account is taken of the date it appeared is and will doubtless remain the greatest mathematical textbook of all time Even in Greek times the most accomplished mathematicians occupied themselves with it Heron Pappus Porphyry Proclus and Simplicius wrote commentaries Theon ofAlexandria re edited it altering the language here and there mostly with a view to greater clearness and consistency It is a fascinating story how the Elements has survived from Euclid s time and this is told well by Fowler in 7 He describes the earliest material relating to the Elements which has survived Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years six fragmentary ostraca containing text and a figure found on Elephantine Island in 190607 and 190708 These texts are early though still more than 100 years after the death of Plato they are dated on palaeographic grounds to the third quarter of the third century BC advanced they deal with the results found in the quotElementsquot book thirteen on the pentagon hexagon decagon and icosahedron and they do not follow the text of the Elements So they give evidence of someone in the third century BC located more than 500 miles south of Alexandria working through this di icult material this may be an attempt to understand the mathematics and not a slavish copying The next fragment that we have dates from 75 125 AD and again appears to be notes by someone trying to understand the material of the Elements More than one thousand editions of The Elements have been published since it was rst printed in 1482 Heath 9 discusses many of the editions and describes the likely changes to the text over the years B L van der Waerden assesses the importance of the Elements in 2 Almost from the time of its writing and lasting almost to the present the Elements has exerted a continuous and major in uence on human a airs It was the primary source of geometric reasoning theorems and methods at least until the advent of non Euclidean geometry in the 19th century It is sometimes said that next to the Bible the quotElementsquot may be the most translated published and studied of all the books produced in the Western world Euclid also wrote the following books which have survived Data with 94 propositions which looks at what properties of gures can be deduced when other properties are given On Divisions which looks at constructions to divide a gure into two parts with areas of given ratio Optics which is the first Greek work on perspective and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set Euclid s following books have all been lost Surface Loci two books Porisms a three book work with according to Pappus 171 theorems and 38 lemmas Conics four books Book of Fallacies and Elements ofMusic The Book of Fallacies is described by Proclus 1 Since many things seem to conform with the truth and to follow from scientific principles but lead astray from the principles and deceive the more superficial Euclid has handed down methods for the clear sighted understanding of these matters also The treatise in which he gave this machinery to us is entitled Fallacies enumerating in order the various kinds exercising our intelligence in each case by theorems of all sorts setting the true side by side with the false and combining the refutation of the error with practical illustration Elements of M usic is a work which is attributed to Euclid by Proclus We have two treatises on music which have survived and have by some authors attributed to Euclid but it is now thought that they are not the work on music referred to by Proclus Euclid may not have been a first class mathematician but the long lasting nature of The Elements must make him the leading mathematics teacher of antiquity or perhaps of all time As a final personal note let me add that my EFR own introduction to mathematics at school in the 1950s was from an edition of part of Euclid s Elements and the work provided a logical basis for mathematics and the concept of proof which seem to be lacking in school mathematics today The Elements Book I of XIII De nitions De nition 1 A point is that which has no part De nition 2 A line is breadthless length De nition 3 The ends of a line are points De nition 4 A straight line is a line which lies evenly with the points on itself De nition 5 A surface is that which has length and breadth only De nition 6 The edges ofa surface are lines De nition 7 A plane surface is a surface which lies evenly with the straight lines on itself De nition 8 A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line De nition 9 And when the lines containing the angle are straight the angle is called rectilinear De nition 10 When a straight line standing on a straight line makes the adjacent angles equal to one another each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands De nition 11 An obtuse angle is an angle greater than a right angle De nition 12 An acute angle is an angle less than a right angle De nition 13 A boundary is that which is an extremity of anything De nition 14 A figure is that which is contained by any boundary or boundaries De nition 15 A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another De nition 16 And the point is called the center of the circle De nition 17 A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle and such a straight line also bisects the circle De nition 18 A semicircle is the figure contained by the diameter and the circumference cut off by it And the center of the semicircle is the same as that of the circle De nition 19 Rectilinearfigures are those which are contained by straight lines trilateral figures being those contained by three quadrilateral those contained by four and multilateral those contained by more than four straight lines De nition 20 Of trilateral figures an equilateral triangle is that which has its three sides equal an isosceles triangle that which has two of its sides alone equal and a scalene triangle that which has its three sides unequal De nition 21 Further of trilateral gures a right angled triangle is that which has a right angle an obtuse angled triangle that which has an obtuse angle and an acute angled triangle that which has its three angles acute De nition 22 Of quadrilateral gures a square is that which is both equilateral and rightangled an oblong that which is rightangled but not equilateral a rhombus that which is equilateral but not rightangled and a rhomboial that which has its opposite sides and angles equal to one another but is neither equilateral nor rightangled And let quadrilaterals other than these be called trapezia De nition 23 Parallel straight lines are straight lines which being in the same plane and being produced inde nitely in both directions do not meet one another in either direction Postulates Let the following be postulated Postulate 1 To draw a straight line from any point to any point Postulate 2 To produce a nite straight line continuously in a straight line Postulate 3 To describe a circle with any center and radius Postulate 4 That all right angles equal one another Postulate 5 That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles the two straight lines if produced inde nitely meet on that side on which are the angles less than the two right angles Common Notions Common notion 1 Things which equal the same thing also equal one another Common notion 2 If equals are added to equals then the wholes are equal Common notion 3 If equals are subtracted from equals then the remainders are equal Common notion 4 Things which coincide with one another equal one another Common notion 5 The whole is greater than the part Propositions Proposition 1 To construct an equilateral triangle on a given nite straight line Proposition 2 To place a straight line equal to a given straight line with one end at a given point Proposition 3 Mathematics The Most Misunderstood Subject Dr Robert H Lewis Professor of Mathematics Fordham University For more than two thousand years mathematics has been a part of the human search for understanding Mathematical discoveries have come both from the attempt to describe the natural world and from the desire to arrive at a form of inescapable truth from careful reasoning These remain fruitful and important motivations for mathematical thinking but in the last century mathematics has been successfully applied to many other aspects of the human world voting trends in politics the dating of ancient artifacts the analysis of automobile tra ic patterns and long term strategies for the sustainable harvest of deciduous forests to mention a few Today mathematics as a mode of thought and expression is more valuable than ever before Learning to think in mathematical terms is an essential part of becoming a liberally educated person Kenyon College Math Department Web Page quotAn essential part of becoming a liberally educated personquot Sadly many people in America indeed I would have to say very many people in America would nd that a difficult and puzzling concept The majority of educated Americans do not think of Mathematics when they think of a liberal education Mathematics as essential for science yes For business and accounting sure But for a liberal education Why do so many people have such misconceptions about Mathematics The great misconception about mathematics and it sti es and thwarts more students than any other single thing is the notion that mathematics is about formulas and cranking out computations It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why and the student39s duty is to memorize all this stuff Such students seem to feel that sometime in the future their boss will walk into the office and demand quotQuick what39s the quadratic formulaquot Or quotHurry I need to know the derivative of 3XA2 6X 1quot There are no such employers What is mathematics really like Mathematics is not about answers it s about processes Let me give a series of parables to try to get to the root of the misconceptions and to try to illuminate what mathematics IS all about None of these analogies is perfect but all provide insight nScaffolding When a new building is made a skeleton of steel struts called the scaffolding is put up first The workers walk on the scaffolding and use it to hold equipment as they begin the real task of constructing the building The scaffolding has no use by itself It would be absurd to just build the scaffolding and then walk away thinking that something of value has been accomplished Yet this is what seems to occur in all too many mathematics classes in high schools Students learn formulas and how to plug into them They learn mechanical techniques for solving certain equations or taking derivatives But all of these things are just the scaffolding They are necessary and useful sure but by themselves they are useless Doing only the superficial and then thinking that something important has happened is like building only the scaffolding The real quotbuildingquot in the mathematics sense is the true mathematical understanding the true ability to think perceive and analyze mathematically oReady for the big play Professional athletes spend hours in gyms working out on equipment of all sorts Special trainers are hired to advise them on workout schedules They spend hours running on treadmills Why do they do that Are they learning skills necessary for playing their sport say basketball Imagine there re three seconds left in the seventh game of the NBA championship The score is tied Time out The pressure is intense The coach is huddling with his star players He says to one quotOK Michael this is it You know what to doquot And Michael says quotRight coach Bring in my treadmillquot Duh Of course not But then what was all that treadmill time for If the treadmill is not seen during the actual game was it just a waste to use it Were all those trainers wasting their time Of course not It produced if it was done right something of value namely stamina and aerobic capacity Those capacities are of enormous value even if they cannot be seen in any immediate sense So too does mathematics education produce something of value true mental capacity and the ability to think lThe hostile partygoer When I was in first grade we read a series of books about Dick and Jane There were a lot of sentences like quotsee Dick runquot and so forth Dick and Jane also had a dog called Spot What does that have to do with mathematics education Well when I occasionally meet people at parties who lea1n that I am a mathematician and professor they sometimes show a bit of repressed hostility One man once said something to me like quotYou know I had to memorize the quadratic formula in school and I ve never once done anything with it I ve since forgotten it What a waste Have YOU ever had to use it aside from teaching itquot I was tempted to say quotNo of course not So whatquot Actually though as a mathematician and computer programmer I do use it but rarely Nonetheless the best answer is indeed quotNo of course not So whatquot And that is not a cynical answer After all if I had been the man39s first grade teacher would he have said quotYou know I can t remember anymore what the name of Dick and Jane39s dog was I ve never used the fact that their names were Dick and Jane Therefore you wasted my time when I was siX years oldquot How absurd Of course people would never say that Why Because they understand intuitively that the details of the story were not the point The point was to learn to read Learning to read opens vast new vistas of understanding and leads to all sorts of other competencies The same thing is true of mathematics Had the man39s mathematics education been a good one he would have seen intuitively what the real point of it all was oThe considerate Piano Teacher Imagine a piano teacher who gets the bright idea that she will make learning the piano quotsimplerquot by plugging up the student39s ears with cotton The student can hear nothing No distractions that way The poor student sits down in front of the piano and is told to press certain keys in a certain order There is endless memorizing of quotnotesquot A B C etc The student has to memorize strange symbols on paper and rules of writing them And all the while the students hear nothing No music The teacher thinks she is doing the student a favor by eliminating the unnecessary distraction of the sound Of course the above scenario is preposterous Such quotinstructionquot would be torture No teacher would ever dream of such a thing of removing the heart and soul of the whole experience of removing the music And yet that is exactly what has happened in most high school mathematics classes over the last 25 years For whatever misguided reason mathematics students have been deprived of the heart and soul of the course and been left with a torturous outer shell The prime example is the gutting of geometry courses where proofs have been removed or deemphasized Apparently some teachers think that this is quotdoing the students a favorquot Or is it that many teachers do not really understand the mathematics at all oStep high A long time ago when I was in graduate school the physical fitness craze was starting A doctor named Cooper wrote a book on Aerobics in which he outlined programs one could follow to build up aerobic capacity and therefore cardiovascular health You could do it via running walking swimming stair climbing or stationary running In each case he outlined a week by week schedule The goal was to work up to what he called 30 quotpointsquot per weeks of exercise during a twelve week program Since it was winter and I lived in a snowy place I decided to do stationary running I built a foam padded platform to jog in place Day after day I would follow the schedule jogging in place while watching television I dreamed of the spring when I would joyfully demonstrate my new health by running a mile in 8 minutes which was said to be equivalent to 30pointsperweek cardiovascular health The great day came I started running at what Ithought was a moderate pace But within a minute I was feeling winded The other people with me started getting far ahead Itried to keep up but soon I was panting gasping for breath I had to give up after half a mile I was crushed What could have gone wrong I cursed that darn Dr Cooper and his book I eventually figured it out In the description of stationary running it said that every part of one s foot must be lifted a certain distance from the oor maybe it was 10 inches In all those weeks I never really paid attention to that Someone then checked me and I wasn39t even close to 10 inches No wonder it had failed I was so discouraged it was years before I tried exercising again What does that have to do with mathematics education Unfortunately a great deal In the absence of a real test for me actually running on a track it is easy to think one is progressing if one follows well intentioned but basically artificial guidelines It is all too easy to slip in some way as I did by not stepping high enough and be lulled into false con dence Then when the real test nally comes and the illusion of competence is painfully shattered it is all too easy to feel betrayed or to quotblame the messengerquot The quotreal testquot I am speaking ofis notjust what happens to so many high school graduates when they meet freshman mathematics courses It is that we in the U S are falling farther and farther behind most other countries in the world not just the well known ones like China India and Japan The bar must be raised yes but not in arti cial ways in true authentic ones nCargo cult education During World War II in the Paci c Ocean American forces hopped from island to island relentlessly pushing westward toward Japan Many of these islands in the south Paci c were inhabited by people who had never seen Westerners maybe their ancestors years before had left legends of large wooden ships We can only imagine their surprise and shock when large naval vessels arrived and troops set up communication bases and runways Airplanes and those who ew them seemed like gods It seemed to the natives that the men in the radio buildings with their microphones radios and large antennas had the power to call in the gods All of the things brought by the navy radios buildings food weapons furniture etc were collectively referred to as quotcargoquot Then suddenly the war ended and the Westerners left No more ships No more airplanes All that was left were some abandoned buildings and rusting furniture But a curious thing happened The natives on some islands gured that they too could call in the gods They would simply do what the Americans had done They entered the abandoned buildings erected a large bamboo pole to be the quotantennaquot found some old boxes to be the quotradioquot used a coconut shell to be the quotmicrophonequot They spoke into the quotmicrophonequot and implored the airplanes to land But of course nothing came except eventually some anthropologists The practice came to be known as a quotCargo Cultquot The story may seem sad amusing or pathetic but what does that have to do with mathematics education Unfortunately a great deal The south Paci c natives were unable to discern between the super cial outer appearance of what was happening and the deeper reality They had no understanding that there even exists such a thing as electricity much less radio waves or aerodynamic theory They imitated what they saw and they saw only the super cial Sadly the same thing has happened in far too many high schools in the United States in the last twenty ve years or so in mathematics education Well meaning quoteducatorsquot who have no conception of the true nature of mathematics see only its outer shell and imitate it The result is cargo cult mathematics They call for the gods but nothing happens The cure is not louder calling it is not more bamboo antennas ie glossy ten pound text books and fancy calculators The only cure is genuine understanding of authentic mathematics 1Confusion of Education with Training Training is what you do when you learn to operate a lathe or ll out a tax form It means you learn how to use or operate some kind of machine or system that was produced by people in order to accomplish speci c tasks People often go to training institutes to become certi ed to operate a machine or perform certain skills Then they can get jobs that directly involve those speci c skills Education is very different Education is not about any particular machine system skill or job Education is both broader and deeper than training An education is a deep complex and organic representation of reality in the student39s mind It is an image of reality made of concepts not facts Concepts that relate to each other reinforce each other and illuminate each other Yet the education is more even than that because it is organic it will live evolve and adapt throughout life Education is built up with facts as a house is with stones But a collection of facts is no more an education than a heap of stones is a house An educated guess is an accurate conclusion that educated people can often quotjump toquot by synthesizing and extrapolating from their knowledge base People who are good at the game quotJeopardyquot do it all the time when they come up with the right question by piecing together little clues in the answer But there is no such thing as a quottrained guessquot No subject is more essential nor can contribute more to becoming a liberally educated person than mathematics Become a math major and nd out So What Good Is It Some people may understand all that I ve said above but still feel a bit uneasy After all there are bills to pay If mathematics is as I ve described it then perhaps it is no more helpful in establishing a career then say philosophy Here we mathematicians have the best of both worlds as there are many careers that open up to people who have studied mathematics Real Mathematics the kind I discussed above That brings up one more misconception and one more parable which I call