(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISCHAPTER - I NUMBERS & THEIR CLASSIFICATIONSINTRODUCTION :The word number has no generally agreed uponmathematical meaning nor does the work number system. Thereis no rule to say that what is a number and what is not. Ancientas well modern Philosophers cum Mathematicians have madeuntiring eforts to fnd out a suitable defnition of number but aperfect defnition has not been possible as yet.The primitive conception of number seems to befundamental with human thoughts and it is well known thatevery living being has the idea of number which is expressed bythe method of counting. In the process of describing theuniverse, philosophers came across numbers not in the abstractform but in the concrete form. Use of numbers up to Parardhahas also been made in the Vedas as early as six thousand B.C.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISThe religious texts of the Vedic period provide evidence for theuse of numbers and large numbers. In the classical period ofIndian mathematics (400 A.D to 1200 A.D.) importantcontributions in numbers were made by scholars likeAryabhata, Brahmagupt&Bhaskar-II.Pythagoras ( 531-510 B.C. ) profounded the Doctrine thatnumber is the nature of things and an essential character of theuniverse. He believed that all Mathematical Sciences can bereduced to numbers. The Pythagoreans made a remarkablespeculations on the natural number and treated unit as the frstorder of things and believed that all numbers arise out of unity.Even axiomatic characterizations of number system have notbeen adequate so Leopold Kronecker, the father of modernintuitionism, has rightly said that (natural) numbers are thecreation of God himself and rest is the creation of man. Thus, heconsiderednumber as primary object and rest as secondaryobjects, Accepting natural numbers as the basic stuf,(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISconstruction of various classes of numbers such as integers,rationals, irrationals real numbers and complex numbers etc.have been possible such that each subsequentconstructioncontains an isomorphic model of the previous one, some of themore interesting examples of abstraction that can be considerednumbers include the quaternions, the octonions ordinalnumbers and the transfnite numbers.Guiseppepeano (1889), a famous Italian mathematician bymeans of his famous fve axioms claimed to reduce the wholemathematics to the theory of natural numbers. He observedthat the entire theory of natural numbers can be derived fromthree primitive ideas such as Zero, Successor and number.He defned numbers and his defnition has been improved byRussell.Again it is well known that real number system is completeand needs no further extension in Classical mathematics. Butthere are situations which carry for the use of infnitesimals and(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISinfnities which are beyond the scope of the real number systemand hence an extension of real number system is essential suchsystem may be called ultra-real number system or hyper realnumber system whose arithemetic and analysis can beeloquently developed. Similar extension of natural numbers,complex numbers and Rn may also be possible. The calculus ofthese ultra real numbers are called Nonstandard Analysis orInfnitesimal calculus which was introduced and developed byAbraham Robinson in 1966.Classifcation of numbers:Numbers can be classifed into sets, called numbersystems.1.Natural numbers:The most familiar numbers are the natural numbers orcounting numbers: one , two, three, and so on. Traditionally, thesequence of nautral numbers started with 1 (0 was not evenconsidered as number for the Ancient Greeks). However, in the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS19th Century, set theorists and other mathematicians startedincluding 0 (cardinality of the empty set, i.e. 0 elements, where 0isthus the smallest cardinal number) in the set of naturalnumber s. Today, diferent mathematicians use the term todescribe both sets, including zero or not. The mathematicalsymbol for the set of all natural numbers is N, also written N.In the base ten numeral system, in almost universal usetoday for mathematical operations, thesymbols for naturalnumbers are written using ten digits: 0,1,2,3,4,5,6,7,8, and 9.In this base ten system, the rightmost digit ofa natural numberhas a place value of one, and every other digit has a place valueten times that of the place value of the digit to its right.In set theory, which is capable of acting as an axiomaticfoundation for modern mathematics, natural numbers can berepresented by classes of equivalent sets. For instance, thenumber 3 can be represented as the class of all sets that haveexactly three elements. Alternatively, in Peano Arithmetic, the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnumber 3 is represented as sss0, where s the successorfunction ( i.e.3 is the third successor of 0). Many diferentrepresentations are possible; all that is needed to formallyrepresent 3 is to inscribe a certain symbol or pattern of symbolsthree times.2.IntegersThe negative of a positive integer is defned as a numberthat produces zero when it is added to the correspondingpositive integer. Negative numbers are usually written with anegative sign (a minus sign). As an example, the negative of 7 iswritten -7, and 7+ (-7) = 0. When the set of negative numbers iscombined with the set of natural numbers (which includes zero),the result is defned as the set of integer numbers, also calledintegers, Z also written Z. Here the letter Z comes from GermanZahl, meaning number.The set of integers forms a ring with operations additionand multiplication.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS3.Rational NumbersA rational number is a number that can be expressed as afraction with an integer numerator and a non-zero naturalnumber denominator. Fractions are written as two numbers, thenumerator and the denominator, with a dividing bar betweenthem. In the fraction writtenorm representsequal parts, where n equal parts of that size makeup one whole. Two diferent fractions may correspond to thesame rational number; for example 21and 42are equal, that is:If the absolute value of m is greater than n, then theabsolute value of the fraction is greater than 1. Fractions can begreater than, less than or equal to 1 and can also be positive,negative or zero. The set of all rational numbers includes theintegers, since every integer can be written as a fraction with(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISdenominator 1. For example: -7 can be written as . Thesymbol for the rational numbers is Q (for quotient), also written.4.Real NumbersThe real numbers include all of the measuring numbers.Real numbers are usually written using decimal numberals, inwhich a decimal point is placed to the right of the digit withplace value one. Each digit to the right of the decimal point hasa place value one-tenth of the place value of the digit to its left.Thus,represents 1 hundred, 2 tens, 3 ones, 4tenths, 5 hundredths,and 6 thousandths. In the US and UK and a number of othercountries, the decimal point is represented by a period, whereasin continental Europe and certain other countries the decimalpoint is represented by a comma. Zero is often written as 0.0(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISwhen it must be treated as a real number rather than aninteger. In the US and UK a number between -1 and 1 is alwayswritten with a leading zero to emphasize the decimal. Negativereal numbers are written with a preceding minus sign:Every rational number is also a real number. It is not thecase, however, that every real number is rational. If a realnumber cannot be written as a fraction of two integers, it iscalled irrational. A decimal that can be written as a fractioneither ends (terminates) or forever repeats, because it is theanswer to a problem in division. Thus the real number 0.5 canbe written asand the real number 0.333..(forever repeatingthrees, otherwise written ) can be written as . On the other(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIShand, the real number (pi), the ratio of the circumference ofany circle to its diameter, isSince the decimal neither ends nor forever repeats, it cannot bewritten as a fraction, and is an example of an irrational number.Other irrational numbers include(the square root of 2, that is, the positivenumber whose square is 2).Thus 1.0 and 0.999 are two diferent decimal numeralsrepresenting the natural number 1. There are infnitely manyother ways of representing the number 1, for example and so on.Every real number is either rational or irrational. Everyreal number corresponds to a point on the number line. The real(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnumbers also have an important but highly technical propertycalled the least upper bound property. The symbol for the realnumbers is R, also written as R.When a real number represents a measurement, there isalways a margin of error. This is often indicated by rounding ortruncating a decimal, so that digits that suggest a greateraccuracy than the measurement itself are removed. Theremaining digits are called signifcant digits. For example,measurements with a ruler can seldom be made without amargin of error of at least 0.001 meters. If the sides of arectangle are measured as 1.23 meters and 4.56 meters, thenmultiplication gives an area for the rectangle of 5.6088 squaremeters. Since only the frst two digits after the decimal place aresignifcant, this is usually rounded to 5.61.In abstract algebra, it can be shown that any completeordered feld is isomorphic to the real numbers. The realnumbers are not, however, an algebraically closed feld.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS5.Complex numbers.Moving to a greater level of abstraction, the real numberscan be extended to the complex numbers. This set of numbersarose, historically, from trying to fnd closed formulas for theroots of cubic and quartic polynomials. This led to expressionsinvolving the square root of negative numbers, and eventually tothe defnition of a new number : the square root of negative one,denoted by i, a symbol assigned by Leonhard Euler, and calledthe imaginary unit. The complex number consists of allnumbers of the formwhere a and b are real numbers. In the expression a+bi, the realnumber a is called real part and b is called the imaginary part.If the real part of a complex number is zero, then the number iscalled an imaginary number or is referred to as purelyimaginary; if the imaginary part is zero, then the number is areal number. Thus the real number are a subset of the complex(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnumbers. If the real and imaginary parts of a complex numberare both integers, then the number is called a Gaussian integer.The symbol for the complex number is C or .In abstract algebra, the complex numbers are anexample of an algebraically closed feld, meaning that everypolynomial with complex coefcients can be factored into linearfactors. Like the real number system, the complex number is afeld and is complete, but unlike the real numbers it is notordered. That is, there is no meaning in saying that i is greaterthan 1, nor is there any meaning in saying that i is less than 1.In technical terms, the complex numbers lack the trichotomyproperty.Complex numbers correspond to points on the complexplane,sometimes called the Argand plane.Each of the number systems mentioned above is a propersubset of the next number system. Symbolically,.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS6.Computable numbersMoving to problems of computation, the computablenumbers are determined in the set of the real numbers. Thecomputable numbers, also known as the recursive numbers orthe computable reals, are the real numbers than can becomputed to within any desired precision by a fnite, terminatingalgorithm. Equivalent defnitions can be given using recursivefunctions, Turing machines or -calculus as the formalrepresentation of algorithms. The computable numbers form areal closed feld and can be used in the place of real numbers formany, but not all, mathematical purposes.7.Other types ofnumbers :Algebraic numbers are those that can be expressed as thesolution to a polynomial equation with integer coefcients. Thecomplement of the algebraic numbers are the transcendentalnumbers.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISHyperreal numbers are used in the non-standardanalysis. The hyperreals, or nonstandard reals (usually denotedas R*).denote an ordered feld that is a proper extension of theordered feld of real numbers R and satisifes the transferprinciple. This principle allows true frst order statements aboutR to be reinterpreted as true frst order statements about R*.Superreal and surreal numbers extend the real numbersby adding infnitesimally small numbers and infnitely largenumbers, but still form felds.The p-adic numbers may have infnitely long expansions tothe left of the decimal point, in the same way that real numbersmay have infnitely long expansions to the right. The numbersystem that results depends on what base is used for the digits:any base is possible, but a prime number base provides the bestmathematical properties.For dealing with infnite collections, the natural numbershave been generalized to the ordinal numbers and to the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScardindal numbers. The former gives the ordering of thecollection, while the latter gives its size. For the fnite set, theordinal and cardinal numbers are equivalent, but they difer inthe infnite case.A relation number is defned as the class of relationsconsisting of all those relations that are similar to one memberof the class.Sets of numbers that are not subsets of the complexnumbers are sometimes called hypercomplex numbers. Theyinclude the quaternions H, invented by Sir William RowanHamilton, in which multiplication is not commutative, and theoctonions, in which multiplication is not associative. Elements offunction felds of non-zero characteristicbehave in some wayslike numbers and are often regarded as numbers by numbertheorists.8.Specifc use of numbers (80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISThere are also other sets of numbers with specialized uses.Some are subsets of the complex numbers. For example,algebraic numbers are the roots of polynomials with rationalcoefcients. Complex numbers that are not algebraic are calledtranscendental numbers.An even number is an integer that is evenly divisible by2, i.e., divisible by 2 without remainder; an old number is aninteger that is not evenly divisible by 2. A formal defnition of anodd number is that it is an integer of the formwhereis an integer. An even number has the formwhere is aninteger.A perfect number is a positive integer that is the sum ofits proper positive divisors- the sum of the positive divisors notincluding the number itself. Equivalently, a perfect number is anumber that is half the sum of all of its positive divisors, or. The frst perfect number is 6, because 1,2 and 3 are(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISits proper positive divisors and . The next perfectnumber is 28= 1+2+4+7+14. The next perfect numbers are 496and 8128. These frst four perfect numbers were the only onesknown to early Greek mathematics.A fgurate number is a number that can be represented asa regular and discrete geometric pattern. If the pattern ispolytopic, the fgurate is labeled a polytopic number, and may bea polygonal number or a polyhedral number. Polytopic numbersfor r = 2, 3 and 4 are:(triangular numbers).(tetrahederal numbers).(Pentatopicnumbers ).9.Numerals Numbers should be distinguished from numerals, thesymbols used to represent numbers. Boyer showed thatEgyptians created the frst ciphered numerals system. Greeksfollowed by mapping their counting numbers onto Ionian and(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISDoric alphabets. The number fve can be represented by boththe base ten numeral 5, by the Roman numeral V andciphered letters. An important development in the history ofnumerals was the development of a positional system, likemodern decimals, which can represent very large numbers. TheRoman numerals require extra symbols for larger numbers. CHAPTERIIHistory of Numbers1First use of numbers : Bones and other artifacts have been discoveredwith marks cut into them that many believe are tallymarks. These tally marks may have been used for countingelapsed time, such as numbers of days, lunar cycles orkeeping records of quantities, such as of animals.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISA tallying system has no concept of place valuewhich limits its representation of large numbers.Nonetheless tallying systems are considered the frst kindof abstract numeral system.2. Zero The use of zero as a number should bedistinguished from its use as a placeholder numeral inplace - value systems. Many ancient texts used zero.Babylonian (Modern Iraq) and Egyptian texts used it.Egyptians used the word nfr to denote zero balance indouble entry accounting entries. Indian texts used aSanskrit word Shunye to refer to the concept of void. Inmathematics texts this word often refers to the numberzero.Records show that the Ancient Greeks seemedunsure about the status of zero as anumber: theyaskedthemselves how can nothing be something?leading to interesting philosophical and, by the Medieval(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISperiod, religious arguments about the nature and existenceof zero and the vacuum. The paradoxes of Zeno of Eleadepend in large part on the uncertain interpretation ofzero.The late Olmec people of south-central Mexicobegan to use a true zero in the New World possibly by the4th century BC but certainly by 40 BC, which became anintegral part of Maya numberals and the Maya Calendar.Mayan arithmetic used base 4 and base 5 written as base20. Sanchez in 1961 reported a base 4, base 5 fngerabacus. By 130 AD, Ptolemy, infuenced by Hipparchusand the Babylonians, was using a symbol for zero (a smallcircle with a long overbar) within a sexagesimal numeralsystem otherwise using alphabetic Greek numberals.Because it was used alone, not as just a placeholder, thisHellenistic zero was the frst documented use of a true zeroin the Old World. In later Byzantine manuscripts of hisSyntaxisMathematica (Almagest), the Hellenistic zero had(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISmorphed into the Greek letter omicron (otherwise meaning70).Another true zero was used in tables alongside Romannumerals by 525 (frst known use by Dionysius Exiguus),but as a word, nulla meaning nothing, not as a symbol.When division produced zero as a remainder, nihil, alsomeaning nothing, was used. These medieval zeros wereused by all future medieval computists (calculators ofEaster). An isolated use of their initial, N, was used in atable of Roman numerals by Bede or a colleague about725, a true zero symbol. An early documented use of thezero by Brahmagupta (in the Brahma-Sphuta-Siddhanta)dates to 628. He treated zero as a number and discussedoperations involving it, including division. By this time (the7th century) the concept had clearly reached Cambodia asKhmer numerals, and documentation shows the idea laterspreading to China and the Islamic world.3. Negative Numbers(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS The abstract concept of negative numbers wasrecognized as early as 100 BC - 50 BC. The Chinese NineChapters on the Mathematical Art (Chinese:Jiu-ZhangSuanshu) contains methods for fnding the areas of fgures;red rods were used to denote positive coefcients, black fornegative. This is the earliest known mention of negativenumbers in the East; the frst reference in a Western workwas in the 3rd century in Greece. Diophantus referred tothe equation equivalent to 4x+20 = 0 (the solution isnegative) in Arithmetica, saying that the equation gave anabsurd result.During the 6th century B.C., negative numberswere in use in India to represent debts. Diophantusprevious reference was discussed more explicitly by IndianmathematicianBrahmagupta, inBrahma-Sphuta-Siddhanta 628, who used negative numbers to produce thegeneral form quadratic formula that remains in use today.However, in the 12th century in India, Bhaskara gives(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnegative roots for quadratic equations but says the negativevalue is in this case notto be taken, for it is inadequate;people do not approve of negative roots. European mathematicians, for the most part,resisted the concept of negative numbers until the 17thcentury, although Fibonacci allowed negative solutions infnancial problems where they could be interpreted asdebts (chapter 13 of Liber Abaci, 1202) and later as losses(in Flos). At the same time, the Chinese were indicatingnegative numbers either by drawing a diagonal strokethrough the right most nonzero digit of the correspondingpositive numbers numerals. The frst use of negativenumbers in a European work was by Chuquet during the15th century. He used them as exponents, but referred tothem as absurd numbers.As recently as the 18th century, it was commonpractice to ignore any negative results returned byequations on the assumption that they were meaningless,(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISjust as Rene Descartes did with negative solutions in aCartesian coordinate system.4. Rational numbers It is likely that the concept of fractional numbers dates toprehistoric times. The Ancient Egyptians used theirEgyptian fraction notation for rational numbers inmathematical texts such as the Rhind MathematicalPapyrus and the Kahun Papyrus. Classical Greek andIndian mathematicians made studies of the theory ofrational numbers, as part of the general study of numbertheory. The best known of these is Eculids Elements,dating to roughly 300 B.C. Of the Indian texts, the mostrelevant is the Sthananga Sutra, which also covers numbertheory as part of a general study of mathematics. The concept of decimal fractions is closelylinked with decimal place - value notation; the two seem tohave developed in tandem. For example, it is common forthe Jain math sutras to include calculations of decimal-fraction approximations to pi or the square root of two.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISSimiarly, Babylonian math texts had always usedsexagesimal(base 60) fractions with great frequency.5.Irrational numbers The earliest known use of irrational numbers was in theIndian Sulba Sutras composed between 800-500 B.C. Thefrst existence proofs of irrational numbers is usuallyattributed to Pythagoras, more specifcally to thePythagorean Hippasus of Metapontum, who produced a(most likely geometrical) proof of the irrationality of thesquare root of 2. The story goes that Hippasus discoveredirrational numbers when trying to represent the squareroot of 2 as a fraction. However Pythagoras believed in theabsoluteness of numbers and could not accept theexistence of irrational numbers. He could not disprove theirexistence through logic, but he could not accept irrationalnumbers, so he sentenced Hippasus to death bydrowning.\ The sixteenth century brought fnal Europeanacceptance of negative integral and fractional numbers. By(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISthe seventeenth century, mathematicians generally useddecimal fractions with modern notation. It was not,however, until the nineteenth century that mathematiciansseparated irrationals into algebraic and transcendentalparts, and once more undertook scientifc study ofirrationals. It had remained almost dormant since Euclid.1872 brought publication of the theories of KarlWeierstrass (by his pupil Kossak), Heine, Georg Cantor andRichard Dedekind. In 1869, Meray had taken the samepoint of departure as Heine, butthe theory is generallyreferred to the year 1872. Weierstrass method wascompletely set forth by Salvatore Pincherle (1880), andDedekinds has received additional prominence throughthe authors later work (1888) and endorsement by PaulTannery (1894). Weierstrass, Cantor, and Heine base theirtheories on infnite series, while Dedekind founds his onthe idea of a cut in the system of real numbers, separatingall rational numbers into two groups having certain(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScharacteristic properties. The subject has received latercontributions at the hands of Weierstrass, Kronecker andMeray. Continued fractions, closely related to irrationalnumbers received attention at the hands of Euler, and atthe opening of the nineteenth century were brought intoprominence through the writings of Joseph LouisLagrange. Other noteworthy contributions have been madeby Druckenmuller (1837), Kunze(1857), Lemke (1870), andGunther (1872). Ramus (1855) frst connected the subjectwith determinants, resulting with the subsequentcontributions of Heine, Mobius and Gunther, in the theoryof Kettenbruchdeterminanten. Dirichlet also added to thegeneral theory, as have numerous contributors to theapplications of the subject.6.Transcendental numbers and reals The frst results concerning transcendentalnumbers were Lamberts 1761 proof that cannot berational, and also that en is irrational if n is rational(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS(unless n = 0).(The constant e was frst referred to inNapiers 1618 work on logarithms). Legendre extended thisproof to show that is not the square root of a rationalnumber. The search for roots of quintic and higher degreeequations was an important development, the Able-Rufnitheorem(Rufni 1799, Abel 1824) showed that theycould not be solved by radicals (formula involving onlyarithmetical operations and roots). Hence it was necessaryto consider the wider set of algebraic numbers (allsolutions to polynomial equations). Galois (1832) linkedpolynomial equations to group theory giving rise to thefeld of Galoistheory. The existence of transcendental numbers wasfrst established by Liouville (1844, 1851). Hermite provedin 1873 that e is transcendental and Lindemannproved in1882 that is transcendental. Finally Cantor showsthat the set of all real numbers is uncountably infnite but(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISthe set of all algebraic numbers is countably infnite, sothere is an uncountably infnite number of transcendentalnumbers.7.Infnity and Infnitesimals a.History of Infnity The earliest known conception of mathematicalinfnity appears in the Yajur-Veda, an ancient Indian script,which at one point states, If you remove a part from infnityor add a part to infnity, still what remains is infnity.Infnity was a popular topic of philosophical study amongthe Jain mathematicians in.400 BC. They distinguishedbetween fve types of infnity: infnite in one and twodirections, infnite in area, infnite everywhere, and infniteperpetually. Aristotle defned the traditional Western notionof mathematical infnity. He distinguished between actualinfnity and potential infnity - the general consensus beingthat only the latter had true value. Galileos Two NewSciences discussed the idea of one-to-one correspondences(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISbetween infnite sets. But the next major advance in thetheory was made by Georg Cantor; in 1895 he published abook about his new set theory, introducing, among otherthings, transfnite numbers and formulating the continuumhypothesis. This was the frst mathematical model thatrepresented infnity by numbers and gave rules foroperating with these infnite numbers.b.History of Infnitesimal An infnitesimal is a number that is smaller than everypositive real number and is larger than every negative realnumber, or, equivalently, in absolute value it is smaller thanfor all . Zero is the only the real numberthat at the same is an infnitesimal, so that the nonzeroinfnitesimals do not occur in classical mathematics. Yet,(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISthey can be treated in as much the same way as can theclassical numbers. For example, each non zero infnitesimal can be inverted and the result is the number . Itfollows thatfor allfor which reasonis called(positive or negative) hyper large (or infnitely large). Hyperlarge number too do not occur in classical mathematics butnevertheless can be treated like classical number. The notion of infnitely small quantities wasdiscussed by the Eleatic School. The Greek mathematicianArchimedes (287 B.C. - 212 B.C.) in The Method ofMechanical Theorems , was the frst to propose a logicallyrigorous defnition of infnitesimals. His Archimedeanproperty defnes a number x as infnite if it satisfes theconditions andinfnitesimal ifand a similar set of conditions hold forand the recriprocals of the positive integers. A number(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISsystem is said to be Archimedian if it contains no infnite orinfnitesimal numbers. The Indian mathematicianBhaskara II (1114-1185) described a geometric techniquefor expressing the change in Sin in terms of Cos times achange in . Prior to the invention of calculusmathematicians were able to calculate tangent lines by themethod Pierre de Fermats method of adequality and ReneDescartes method of normals. There is debate amongscholars as to whether the method was infnitesimals oralgebraice in nature. When Newton and Leibniz invented thecalculus, they made the use of infnitesimals. The useinfnitesimalswas attacked as incorrect by Bishop Berkeleyin his work, The Analyst . Mathematicians, scientists andengineers continued to use infnisitimals to produce correctresults. In the second half of the nineteenth century, thecalculus was reformulated by Augustin-Louis Cauchy,Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISothers, using the- defnition of limit and set theory.While infnitesimals eventually disappeared fromthecalculus, their mathematical study continued through thework of Levi-Civita and others, throughout the latenineteenth and the twentieth centuries as documented byPhilip Ehrlich (2006). In the 20th Century, it was found thatinfnitesimals could serve as a basis for calculus andanalysis. In the 1960,Abraham Robinson showed howinfnitely large andinfnitesimal numbers can be rigorouslydefned and used to develop the feld of nonstandardanalysis. The system of hyperreal numbers representsrigorous method of treatingthe ideas aboutinfnite andinfnitesimal numbers that had been used casually bymathematicians, scientists, and engineers ever since theinvention ofinfnitesimal calculus by Newton and Leibniz.8.Complex numbers :The earliest feeting reference to square rootsof negative numbers occurred in the work of the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISmathematician and inventor Heron of Alaxendria in the 1stcentury A.D, when he considered the volume of animpossible frustum of a pyramid. They became moreprominent when in the 16thcentury closed formulas for theroots of third and forth degree polynomials were discoveredby Italian mathematicians such as Niccolo FontanaTartaglia and GerolamoCardano. It was soon realized thatthese formulas, even if one was only interested in realsolutions, sometimes required the manipulation of squareroots of negative numbers. This was doubly unsettling since they did noteven consider negative numbers to be on frm ground atthe time. When Rene Descartes coined the termimaginary for these quantities in 1637, he intended it asderogatory. A further source of confusion was that theequationseemed capriciously inconsistent with the algebraicidentity.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISWhich is valid for positive real numbers a and b, and wasalso used in complex number calculations with one of a, bpositive and other negative. The incorrect use of thisidentity, and the related identityin the case when both a and b are negative even bedelivedEuler. This difculty eventually led him to the conventionof using the special symboliin place ofto guard againstthis mistake.The 18th century saw the work of Abraham de Moivre andLeonhard Euler, deMoivres formula (1730) states:and toEuler(1748) Eulers formula of complex analysis :The existence of complex numbers was not completelyaccepted until Caspar Wessel described the geometricalinterpretation in 1799. Carl Friedrich Gauss rediscoveredand popularized it several years later, and as a result thetheory of complex numbers received a notable expansion.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISThe idea of the graphic representation of complex numbershad appeared, however, as early as 1685, in Walliss DeAlgebra tractatus.Also in 1799, Gauss provided the frst generallyaccepted proof of the fundamental theorem of algebra,showing that every polynomial over the complex numbershas a full set of solutions in that realm. The generalacceptance of the theory of complex numbers is due to thelabors of Augustin Louis Cauchy and Niels Henrik Abel,and especially the latter, who was the frst to boldly usecomplex numbers with a success that is well known.Gauss studied complex numbers of the form a+bi, where aand b are integral, or rational (and i is one of the two rootsofx2+1=0). His student, Gotthold Eisenstein, studied thetype b a + , where is a complex root of x3-1=0. Othersuch classes (called cyclotomatic felds) of complexnumbers derive from the roots of unity xk-1=0 for highervalues of k. This generalization is largely due to ErnstKummer, who also invented ideal numbers, which were(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISexpressed as geometrical entities by Felix Klein in 1893.The general theory of felds was created by Evariste Galoiswho studied the felds generated by the roots of anypolynomial equationf(x) = 0. In 1850 Victor Alexandre Puiseux took the keystep of distinguishing between poles and branch points,and introduced the concept of essential singular points.This eventually led to the concept of the extended complexplane.9.Prime numbers Prime numbers have been studied throughoutrecorded history. Euclid devoted one book of the Elementsto the theory of primes; in it he proved the infnitude of theprimes and the fundamental theorem of arithmetic, andpresented the Euclidean algorathim for fnding the greatestcommon divisor of two numbers. In 240 B.C, Eratoshtenes used the Sieve ofEratosthenes to quickly isolate prime numbers. Butmostfurther development of the theory of primes in Europedates to the Renaissance and later eras.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS In 1796, Adrien Marie Legendre conjecturedthe prime number theorem, describing the asymptoticdistribution of primes. Other results concerning thedistribution of the primes include Eulers proof that thesum of the reciprocals of the primes diverges, and theGoldbach conjecture, which claims that any sufcientlylarge even number is the sum of two primes. Yet anotherconjecture related to the distribution of prime numbers isthe Riemann hypothesis, formulated by Bernhard Riemannin 1859. The prime number theorem was fnally proved byJacques Hadamard and Charles de la Vallee-Poussin in1896. Goldbach and Riemanns conjectures remainunproven and unrefuted.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISCHAPTER - IIINUMBER CONCEPTS IN GREEK AND INDIANPHILOSOPHYIn this chapter, we discuss the concept of number in Greekand Indian Philosophy. The Pythagoreans madeveryremarkable speculations on the nature of number. Thebelieved thatthe conceptof number is fundamental and allmathematical sciences can be reduced to number. Theytreated unit as the frst order of things and believed that allnumbers arise out of unity. According to Indian Philosophynumber is the root cause of the formation of other extensivebodies. Thus just like Pythagoreans, the Indian Philosopherstoo have accepted number as the world principle.1. Number concepts in Greek Philosophy In Greek Philosophy, Number has been given thetopmost position in the description of the universe. Pythagoras,(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISarenowned mathematician and philosopher of Greece duringthe sixth century B.C.(531-510 B.C.) propounded the doctrinethat number is the nature of things and an essential characterof the universe; it is the world ground and the stuf of which theuniverse is made.[53].Just as Thales (6th century B.C.)considered water to be the ultimate reality and the frst principleof which things are composed so also the Pythagoreansconsidered number as the frst principle of things. They couldnot conceive of the universe without number. Proportion, orderand harmony are the dominant notes of the universe but theideas of all these are closely connected with number. They alsodiscovered that the musical harmony was founded uponnumbers. They were convinced that the universe is made up ofnumber but number is opposed to matter and distinguishedfrom it, although closely connected with it, something whichlimits it and gives it a shape. They believed that the concept ofnumber is fundamental and all mathematical sciences can bereduced to numbers.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS The Pythagoreans recognized the formal cause butthey reduced it to the material cause by declaring that numberis the stuf of which things are made. They abstracted from thequality of things but were left with the quantity as the ultimatereality. But the universe of the Pythagoreans was not ouruniverse, which transcends the immediate sense perception,which manifests itself so richly, even if mysteriously in thenumerous inventions which make up the essential part of ourdaily life. The universe of the Greeks was limited to things moreimmediately accessible to senses.[12] The Pythagoreans made remarkable speculations on thenature of number. They treated unit as the frst order of thingsand believed that all numbers arise out of unity. They furtherbelieved that the universe is composed of pair of opposites andcontradictions whose fundamental character is that they arecomposed of the odd and the even, which they identifed withthe limited and the unlimited respectively, because the odd(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnumber cannot be bifurcated while the even number can bebifurcated. The limited and the unlimited become the ultimateprinciples of the universe. The limit is identifed with unit; itproceeds to draw more and more of the unlimited and to limit it,and in this way the formation of the world proceeds. Later on they made distinction between unity andduality and received two further pairs of opposites whosenumber was later arbitrarily made up to ten. They attachedarbitrary meanings to numbers and treated some of them assacred. They worshipped the Tetractys of the Decad[53] andused to swear by this sacred number. Sorbonne introduced themagical properties of the number three into psychologybeginning with the Trinity of Thought, Will and Felling. Allsquared numbers were treated as Justice. It was argued that thequality of justice was to return equal for equal and the squarednumber was the product of two equals; hence it was justice.[49]The Pythagoreans believed that unit was the central fre or the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIShearth of the universe round which revolve ten bodies : counterearth, earth, sun, moon, the fve planets and the heaven of thefxed stars. The Pythagorean doctrine looks for the secret ofthe universe in number. It is a crude philosophy and theapplication of the number theory issues in a barren and a futilearithmetical mysticism. Hegel says that the association of thesefrst numbers with defnite thoughts must be purely external,these numbers no doubt conceal a profound meaning and aresuggestive of various thoughts but in philosophy the point is notwhat one may think but what one does think and the thoughtshould be thought in thought itself and not in symbols [20]. InHegels doctrine Reason or Unity was the source of all and it wasalso theAbsolute, alias the Deity, alias the orders of the Leader[21].2.Number concepts in Indian Philosophy(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS According to the Indian philosophers also, numberplays a very important role in the formation of the universe. Theminutest and the irreducible substance of this universe hasbeen called Parmanu i.e., atom. Two Parmanus make onedwayanuka and three dwayanukas make one trayanuka.Trayanuka is the minutest object whose existence can be seen;for example, the smallest visible objects appearing through thesingle ray of the Sun. Existence of dwayanuka and parmanu isproved by inference. Dwayanuka is more extensively thanparmanu and Trayanuka is more extensive than Dwayanuka.Other more extensive bodies are composed of severaltrayanukas; of course the entire universe is composed oftrayanukas and ultimately of parmanus. The philosophers triedto fnd out the object which causes increase in the extension ormagnitude, i.e., pariman. Pariman (magnitude) of parmanu isnot the cause; for as the increase in wealth makes a rich manricher and increase in the poverty of a poor man makes him(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISpoorer, so also the pariman of the parmanu, the object withoutany parts,would make dwayanuka anutar (minuter) still andnot extensive [50]. They came to the conclusion that it is due tothe number inherent in the parmanu that a dwayanuk hasmore pariman than a parmanu. Similarly number is the rootcause of the formation of other extensive bodies also. Thus justlike the Pythagoreans the Indian philosopher too have acceptednumber as the world principle. They have also tried to defnenumber and characterize it. We proceed to deal with thedefnition of number in the next chapter.In the process of describing the universe,philosophers came across Number, not in the abstract form butin the concrete form one, two, ..... many when theydistinguished between one moon, two eyes, fve fngers or manystars. Use of numbers upto parardha has been made in theVedas as early as six thousand B.C. In Lalitvistara, a Budhistwork of the frst century B.C. numbers up to Tallakshan (1053)(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISoccur in Pali Grammar of Kaccayan number as largeasamkhyeya (10140) have been used [42]. Use of very largenumbers like googol (10100) and googolplexandSkewes number [2] has also been made by thescientists.Hence it is clear that number has been an essential objectfor the creation of universe. Philosophers have been thinkingwhether to put number in the class of individuals or ofattributes and there has not been an unanimous opinion onthis issue. But the universally accepted fact is that the conceptof number is fundamental and most important for the existenceofthe universe.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS CHAPTER - VIDEFINITION OF NUMBER IN ININDIAN PHILOSOPHYThough it is very difcult to give an exact defnitionof number, yet several Indian philosophers, especially MaharashiKanad (3rd century B.C.) and followers of his Vaisesika school ofthought, have tried to explain it to a large extent. The sanskritequivalent of number is samkhyan which is composed of twowords viz. samyak meaning real and khyapan meaningknowledge. So samkhyan literally means real knowledge [4]. Italso means the cause of the real knowledge. The real knowledgeof any object is possible only after the knowledge of itssamkhyan. The Vaisesikasbelieve that samkhyan is anattribute which inheres in all substances. According to the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISMimansa school of thought, founded by Maharshi Jemini,samkhyan inheres in every category of the padarthas and isdiferent from it. Prasasta Pada, in his commentary of theaphorisms of Kanad, has characterized number by thefollowing aphorism EkadiVyavharhetuSamkhyan. [15]Sridharacarya in his NyayaKandali explains theabove as Number is an attribute by means of which the conceptof this is one, these are two, etc. and the use of such sentencesis possible. [15]. Shankar Misra has interpreted the above aphorismin his KanadRahasya. According to him the aphorism meansthat The basisof the genus unity inhering in the attributeunity whichresides in the singleindividuality and the genusduality or triplicity etc. inherning in the attribute which residesin many objects and is diferent from individuality is Number.[15](80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISUdayanacarya in his Kiranavali explains it as Numbermay be defned to be that attribute which causes counting. [15]As per the specifc intuition of Dandi (a person carrying astaf) results from the presence of the staf, so also the intuitionof one, two, three etc. is specifc and takes place due to thenumber unity, duality, triplicity, etc. But the attribute number isdiferent from other attributes like colour, taste, smell and touchetc. [13], for, the intuition of number is produced without thelatter attributes. Number cannot be a substance though it inheres inall substances. The proof is very simple and based on logicalgrounds. Suppose some person is asked to bring the numberone. Can he do so? He can bring one book or one chair butnot the number one. On the other hand if some person is askedto bring books and he is nottold how many books he has tobring, then also he cannot be sure of the number of books heshould bring. Thus number must be diferent from susbtance.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISThis is further supported by the following argument ofSridharacarya : In a dense forest collection of trees is cognizedin its real from i.e. clear intuttion of substance is obtained atonce but the exact idea of the number of the trees cannot beobtained forthwith. Hence number is diferent from substance.[15].According to the Budhist philosophy number is notan independent attribute and the basis of the cognition ofone,two, etc. is the intuition of the attribute colour etc. But this viewis not proper since due to the principle of momentarinesss [7] ofthe similarity between two attributes no attribute can produceany other attribute and more so number. Further it is seen thatif a blue coloured door be repainted with green colour, its colouris changed but its number unity does not change. Hence theexistence of number must be real and diferent from colour. Bhushan holds that unity is nondiference fromitself and the diference from itself is duality. Udayanacarya has(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScriticized Bhushans view in the following manner [15]: Wherethe self identity of a jar its unity, then there would be nointuition of unity in case of cloth and nondiference from unityitself being common to two, three, four etc. There would be nodiference between duality, triplicity, fourness etc. according toBhushan. Hence Bhushans view is not valid.Number may reside in single substratum as well as inmany substrate. Unity is the number which resides in singlesubstratum; it may be eternal as well as noneternal according asthe substratum is eternal or noneternal. Other numbers duality,triplicity etc. reside in many substances and are noneternal; forall of them may be produced and also may be destroyed with theproduction and destruction of relative understanding. Thefollowers of the Mimansa and the Nyaya schools of thought donot agree with the above view held by the Vaisesikas. Accordingto the Mimansa school numbers are not produced by relativeunderstanding; all numbers unity, duality, triplicity etc. are(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISproduced in each substance just after its formation in the sameway as its shape, size or colour is produced; due to diference inrelative understanding diference in these numbers inherent inthe substance is cognized. So according to them relativeunderstanding is not the generator but the exhibitor ofnumbers. The Naiyayikasmostly agree with the views of theVaisesikas but they do not challenge the view of theMimansakas too .[36] According to the Budhists, who do notaccept number as an attribute, diference in the number of aparticular substance may be caused by relative understanding;to the same person diferent numbers may appear to dwell inthe same substance under diferent circumstances and diferentmoments and also at the same moment diferent persons maycognize diferent numbers in the same substance.Though number resides in all substances, yet it cannot bea genus like existence, for its denotation is neither less nor morethan that of existence; nor can it be a genus confned to the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISsubstance only; for it is neither less nor more extensive thansubstanceness.[15]Sometimes expressions like one colour, two tastesetc are used. From such use of numbers one may be tempted toaccept that number is inherent not only in substances but alsoin attributes; so it may be an independent object which inheresin all the seven categories of the padarthas. But this argumentis fallacious. Firstly, the use of number in such expression mustbe secondary and not basic, since the cognition of numbers inattributes and actions is erroneous, [13] and it is nondiferencefrom itself which constitutes the derivation. But number is not,nothing but self identity. The above use of number may bejustifed in the following manner. When one colour is used inreference to some fower, the number unity as well as the colourof the fower is inherentin the substances ofthe fower soclosely that its unity iscognized also with its colour; thus in theabove use of number with attribute, the latter represents the(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISsubstance itself. It may be argued that just as the above use ofnumber, its use is always secondary even when it is used with asubstance. But this argument is fallacious; for, the secondarycannot exist without the primary.[13] Hence the real numberunity exists. Number is not universal, as it does not exist inattributes and actions but we fnd the use of numbers withboth, e.g., twenty four attributes, fve actions etc. Due to suchpeculiar use of number one may take it for an independentcategory of the padarthas. But then there would be the eightcategory viz. number and the number inhering in this eightcategory will be the ninth category of the padarthas. Thus therewill be no limit to the number of the categories and the wholeuniverse would be a mess, without a discipline [24]. So numbercannot be an independent category of the padarthas. It must bean attributes with all the specialties mentioned above.1.Production and Destruction of Number :As already mentioned, the Vaisesikas hold that unity is(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScognised absolutely and all other numbers are produced byrelative understanding and are also destroyed with thedestruction of the latter. The production and destruction ofduality take place in the following way. When two homogeneousor heterogenous substances are in contact with the eye, therelatve understanding, i.e., this is one, this is one and thesetogether are two is producd and this relative understandingproduces duality. Beginning from the sensual contact andending with samskar or impression there are eight moments [45]which occur in this order :(i) contact of the sense with the substratum ofduality in the frst moment, (ii)cognition of the genus inherent in the attributeunity in the second moment, (iii)the relative understanding in the form ofcognizance of the many along with the attributeunity as qualifed with the generic notion or(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScharacteristic of unity in the third moment,(iv) production of the attribute duality in the fourthmoment, (v)cognition of the genus inherentin duality in theffth moment, (vi)cognition of the attribute duality as qualifed withthat genus in the sixth moment, (vii) cognition of substances as qualifed with theattribute duality in the seventh moment and then(viii) samskar is produced in the eight moment.The order of destruction of duality is as follows : (i) destruction of the characteristic of unity from relativeunderstanding in the frst moment, (ii) destruction of relative understanding from thecognition of the characteristic of duality in the second moment, (iii) destruction of the characteristic of duality from(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScognition of the attribute duality in the third moment, (iv) destruction of the cognition of the attributes dualityfrom cognition of substances as qualifed with the possession ofduality in the fourthmoment and (v) destruction of the latter from samskar or fromcognition of other objects in the ffth moment [45]. In somecases duality is destroyed also from the destruction of thesubstratum instead of relative understanding and in other casesthe destruction takes place from both the destruction of thesubstratum as well as the relative understanding when there issimulateneity of action in the constituent parts of duality andrelative understanding.Duality is produced by pure relative understanding andtrplicity by relative understanding accompanied by duality.Similarly for the production of fourness etc. Hence numbers areproduced through the process of induction. According toSridharacarya multiplicity is also a number which may(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISsometimes be produced independently and without the formerproduction of duality as in case of army or forest etc. inconsequence of nonexistence of constant relative understanding.2.Remarks on the Vaisesika Defnition of Number :We fnd that someof the ancient Indian Philsophers,especially the Vaisesikas, have attempted to defne andcharacterize members unity, duality, triplicity, etc. and that theyhave accepted numbers as objectively real quantities inhering inall substance. For them unity, like God, inheres in all substance and is a basic number with which all higher numbers areformed by means of relative understanding. Modern westernphilosophers like Hilbert and Brouwer also agree with their viewthat the concept of one is fundamental in mathematics. But theVaisesikas have not defned zero, which has its use as a numberin the Indian Mathematical literature since about 200 B.C.[14]or even earlier in the vedic literature. It was Brahmagupta (628A.D.), the prince of Indian mathematics, who correctly defned(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISzero. He gave a relational rather than half - scientifc and half-intuitive defnition of zero. Since zero is the number smallerthan any one of the numbers unity, duality, etc. which are usedfor counting purposes, it is desirable and proper to defne zerofrst and then to defne other numbers with the help of zero. TheVaisesikas had no method to defne zero frst and then thenumbers unity, duality etc. (or to defne zero with the knowledgeof unity ,duality etc. )by means of relative understanding. Weshall see later that it was Frege-Russell defnition of numberwhich could enable one to defne zero frst and then other highernumbers. Further we see that we can get only fnite numbers,however large, in their system and as such the concept ofinfnity cannot be congised. But Cantor (1845-1918) has shownhow to deal with the infnite, and hence it is both desirable andpossible to deal with the fundamental properties of numbers insome way which is applicable to fnite as well as infnite(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnumbers. We shall also see later how Russells defnition wascapable to handle the fnite as well as infnite numbers. After the attempt of the Pythagoreans in Greece andthe Vaisesikas in India towards explaining the nature and use ofnumber we do not fnd any work in this direction unitl towardsthe close of the nineteenth century when mathematician cum-logicians like Peano, Frege and Russell etc. took up the workand tried to give a logical defnition of number.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISCHAPTER - VSOME REMARKS ON PEANOS DIFFINITION OFNUMBER & FREGE-RUSSELLS DEFINITION OFNUMBERIn this paper we discuss Peanos defnition ofnumber andFrege-Russellsdefnition of number. We haveseen that Peanos defnition of number is not satisfactory andFrege-Russells defnition of number too is far from beingsatisfactory. Thus we see that inspite of the valuable attemptsmade by ancientaswell asmodern thinkers a clear cutdefnition of number has notbeen possible.1.Peanos defnition of NumberGuiseppePeano (1889), a famous mathematician(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISof Italy, believed that like Pythagoras that the whole ofmathematics could be deducted from numbers. Havingreduced all traditional pure mathematics to the theory ofnatural numbers, the next step in the logical analysis wasto reduce the theory itself to the minimum number ofpremises and undefned terms from which it could bederived. Peano showed that the entire theory of naturalnumbers could be derived fromthe three primitive ideasviz. zero, successor and number and the following fvepropositions:1.zero is a number2.successorof every number is also a number3.no two number have the same successor4.zero is not a successor5.If a property is possessed by zero and whenever itispossessed by any number, it is also possessed by its(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISsuccessor, then the property is possessed by allnumbers. A thorough and critical study of Peanosaxioms of natural number is proposed to be made in thenext chapter. For the present we shall simply discuss howhe has defned numbers and how his defnition has beenimproved by Russell.Peano gave the defnition of number byabstraction, accepting that numbers are applicableessentially to classes (i.e collections) he defned number asa property of classes. Two classes have the same numberwhen one-one relation exists between their numbers. Twosuch classes are said to be similar and the similarityrelation has three properties viz. refexivity, symmety andtransitivity. Peano held that when similarity relation holdsbetween two terms, the two terms have a common property(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIScalled their number.[40]But this defnition is not satisfactory. For, all similarclasses have the same number as a common property, sothere is a many one relation which every class has to itsnumber and to nothing else. In other words, we have a set ofentities such that any class, and all other classes similar toit, have a certain many one relation to one and only one ofthe entities of the set. But there may be many such sets ofentities and hence the defnition of Peano by abstractionfails to defne the number of a class. The axioms of Peanodo not enable us to know whether there exists any set ofterms verifying the axioms. We want our numbers to besuch that can be used for counting and this requires thatour numbers should have a defnite meaning, not merelythat they have certain formal properties only. This defnitemeaning is defned only by the logical theory of arithmetic.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS2.FREGERUSSELLSDEFINITION OF NUMBER: It was Frege (1884) who frst of all attempted togive a logical defnition of numbers but none paid anyattention to his views until Russell independently gave thesame defnition of number in 1901. He did not defnenumber by means of relative understanding (like theVaisesiksa) but he used two concepts viz the similarity ofsets (which Peano also used) and the class of classes todefne all numbers including zero Russell also introducedinfnity through an axiom known as the Axiom of infnity. One might confuse the concept of number of acollection with its plurality. But this is not proper. Pluralityis not a number but an instance of some particular number.For example, the trio of men, say, Ram, Shyam and Hari isan example of the number three, but the latter is notidentical with the trio consisting of Ram, Shyam and Hari.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS According to Frege and Russell, number is anattribute (unlike colour, taste etc. as it is according to theVsaisesikas) it is a defning property of a collection. Thereare two ways of defning a collection. One way is byextension. i.e.,enumeration and the other way is byintension. i.e., characterization. According to the extensionmethod number of a collection is to be determined bycounting the members of the collection. But this systemsufers from two defects. The frst defect is that counting isnothing else than labelling each member of a collection witha number and thus it sufers from the defect of circularity.Secondly, numbers themselves form an infnite class andhence it is not always possible to enumerate or count thenumbers of an infnite collection. Hence Frege and Russellattempted to defne number by intension. Russell defnes itin two steps. First he defnes the concept of similarity of twosets and then he formulates the concept of number in terms(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISof this similarity concept, He defnes the number of a set tobe the class of all sets similar to it. [40] Similarity can be defned as in case of Peano,without the help of the number concept . For example, in acountry where polygamy or polyandry is prohibited, one cansay without counting that the set of husbands is similar tothe set of wives; for, every husband has a wife and everywife has a husband uniquely. We need not count the actualnumber of the husbands or the wives. The second part of this defnition describesnumber as a class of classes or more clearly, a class ofsimilar classes . At the frst sight it appears paradoxical toaccept number as class of classes and Peano neveraccepted this defnition. But if we treat class as a conceptand not as a collection, then a number is really defned as acommon property of a set of similar classes and nothing(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISelse. This view removes the appearance of the paradox to agreat extent. Further it may be seen that this defnitionallows the deduction of all the usual properties of numbers,fnite as well as infnite, and is the only one which ispossible in terms of the fundamental concepts of generallogic.[40]3.Axiom of Infnity :Russell further claimed that not only fnite but infnitecollections also exist in this universe. For, if it be assumedthat there exists only fnite number of objects, say 1000only, in the whole universe, then there would be no class of1001 or 1002 individuals and hence 1001 and 1002 willeach belong to the empty class and thus will be identical.This means that two diferent numbers, viz .1000 and 1001,have the same successor. But this contradicts the thirdaxiom of Peano. To remove this difculty Russell introduced(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISthe axiom of infnity which assumes that, there exists inthe universe a class consisting of infnitely manyindividuals. We can never verify the truth of this axiom, yetall the scientists and philosophers have accepted this andnow it has become the foundation of the structure ofmathematical as well as other scientifc systems. With thehelp of this axiom the theory of natural numbers is moredeeply founded. This is an artifcial device to maintainRussells theory that numbers are constructed out of theactual classes in the world.As it has been already mentioned Frege-Russelldefnition of number enables one to defne zero frst andother higher numbers subsequently in the following way:Property of not being identical with itself may be called theproperty of empty collection and the class of classes.similar to the empty collection is the number zero. Next we(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISmay defne one as the class of classes similar to thecollection whose only number is zero and similarly highernumbers may also be defned in terms of the lowernumbers.4.Remarks on Frege-Russells defnition of Number : We have seen that Peanos defnition ofnumber is not satisfactory. We fnd that Frege-Russelldefnition too is far from being satisfactory .Firstly, we fndthat like Peano they too have based their defnition on theconcept of similarity of sets. But it is not always easy toestablish similarity between two collections. If we have twocollections, say a collection of cups and another collection ofsaucers, we can verify the similarity between thesecollections by placing each cup on a saucer and fnding thatno saucer is left without a cup. Here one-onecorrespondence is established. But if the cups are closed in(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISone box and the saucers in another box, no suchcorrespondence is visualized unless the boxes are openedand the cups are placed on saucers. The question ariseswhether similarity existed between the two collections beforethe cups were actually placed on the saucers. Obviously wefeel that such similarity exists from before. So we see that itis possible to exhibit the correspondence between two setsonly when similarity exists between them from before.Hence we conclude that correspondence simply verifessimilarity. Let us now consider the collection of planets andthat of the muses. It is not at all possible to establish acorrespondence between these collections. Hence theproposed defnition of similarity gives only a sufcient butnot necessary condition of similarity and restricts themeaning of similarity too narrowly. (80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS Secondly, in order to determine similarity betweentwo classes, we have to know the proper meaning of theword class. It has two meanings viz. list and concept. If itmeans list, i.e. a collection, then correspondence can beestablished between the members of two collections only ifthey have equal number of members. But if class meansconcept, then two concepts can be put in one-onecorrespondence even if one set is an extension of the otheri.e., even if two classes are not equinumbered.According to Frege two sets must either be similaror not similar from a purely logical basis. But if some one isasked to tell the number of stars twinkling in the sky indark night at particular moment, he has no method todetermine the number; for, while establishingcorrespondence with any other collection, say, a collection ofmustard seeds, some stars will disappear and some new(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISones will appear without his notice. At best he can say thatthere are many stars. Hence we fnd that sometimes wehave to indicate the number of a collection by numeralsmany, very many and few. But Frege seems to pay noattention towards this aspect. From the second part of Freges defnition andthe manner of description for the number zero, one, two etc.it appears that number states something about the conceptand not about the counted things themselves. But do wealways remember this fact in our language or statements?In any command, say, 3 books the command does not saythat the class of books to be asked for is an element of theclass three. Our command or language is unaware of thisinterpretation. It cannot be expressed in the subjectpredicate form. Frege-Russell defnition unnecessarilyrestricts the concept of number to the subject predicate(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISform of our proposition [51]5.Number and NumeralWe have seen that it is not easy to handle the conceptof correspondence. Some may try to escape this difcultyby identifying numbers with numerals, so that number onemay be identifed with the number I, number two with thenumeral II and so on. But this way of representation sufersfrom various defects.Firstly, we see that though very small numerals may bedemonstrated in this way, very large numerals cannot be sodemonstrated. Hence there is a distinction between visualnumbers and counted numbers. Secondly, we know thatnormally the use of numerals is associated with countingby which process children learn the numerals. Further wesee that the numeral I may be of diferent forms anddiferent colours; it may be handwritten or typewritten or(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISprinted; it may be in Davangari or Roman or Arabic script.But the number one does not have these properties. Againwe see that a number may be odd or even, prime orcomposite, squared or nonsquared but the numerals cannothave these properties. Hence it is clear that number andnumeral cannot be identical.We can say that the relation between a numberand a numeral is similar to that between a proposition anda sentence [18] A sentence is the physical representation ofa proposition but is not identical with it. Various sentencesmay represent the same proposition. Similarly, several formsof numerals may represent the same number. But theconcept of proposition is equally difcult to becharacterized. It is sometimes held that a proposition issomething like the mental image of a sentence whichbelongs to the external world. This opinion confuses a(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISproposition with a sentence; for whatever there is in themind it must be some representation of the proposition andnot of sentence, since the frst difers from the written orthe spoken word because it is not a communication. In thesame way, if we say that number is something not to bedefned but to be known by intuition as some image of anumeral, then again confusion arises between number andnumeral.6.Conclusion :As we go deep into the analysis of the number concept,the problem becomes more and more complicated. Just likethe defnition of time, it is not possible to give the defnitionof number in a formula or a sentence. But its nature canbe studied and judged by describing the uses of the wordsnumber and numeral. The concept ofnumber may becompared to that of a point which has diferent meanings(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISin diferent branches of science but the plain and simpleword point loses its precise meaning and assumes a vagueand indistinct description. Similarly the concept of numberalso become vague and indistinct if it is not allowed to bedefned by a certain calculus. The individual numberconcepts form a family, whose terms have a familysimilarity. The word number does not designate a concept(in the sense of logic) but a family of concepts [51]. Thismeans that the individual number types are related to eachother in many ways even though they may not have aproperty or trait in common. As Frege holds, numbers arealready there somehow so that its discovery is like thediscovery of a continent that exists even before the actualdiscovery.Thus we see that in spite of the valuable attemptsmade by ancient as well as modern thinkers a clear but(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISdefnition of number has not been possible.CHAPTER - VICRITICAL STUDY OF PEANOS AXIOMS ON NATURALNUMBERSIn the present chapter we propose to studycritically the axiomatic system of Peano and to examine if theseare suitable characterizations for the natural numbers.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS1.Classical Interpretation of the Peano System :In any attempt to give an axiomatic characterization of amathematical system one would like to ensure that the system isnonempty; and besides the logical constants one would like touse minimum number of individuals as well as predicates. Onewould also like to compelte his tasks, if possible, with monadicpredicates only. But no infnite feld can be characterized byaxioms which containpredicates of one variable only [34].Hence in order to characterize an infnite system he has to useat least one predicate of two variables and he would be in ahappy position if his task is accomplished with only one suchpredicate.With this idea in mind Peao postulated his famous axiomsfor the natural numbers in Arithmetices principia. These axiomsmay be expressed in the following logical form:(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISAn alternative formulation of A5 may be given by These axioms are based on Dedekinds postulatesand by means of these axioms Peano claimed to reduce all puremathematics to the theory of natural numbers. In the classical interpretation of the Peano system itproposes to characterize the natural number system, which isan infnite system of denumerable cardinality and may betermed as the standrard model of the Peano system. In theseaxioms Peano has used only three undefned terms viz. O, Nand S. N stands for the natural number system, O, standsfor the natural number zero and Sx stands for the immediatesuccessor of number x viz. x+1. o and N are the only primitiveindividuals used in the system; S is used as function of one(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISvariable, which may also be treated as a predicate constantoftwo variables according to which the alternative formulation ofA3 may beThe last axiom A5 is usually known as the InductionAxiom and the monadic predicate P used in this axiom is apredicate variable. We do not know any particulars of P exceptthat if P belongs to O and whenever it belongs to any number x,it also belongs to Sx, then P belongs to all numbers, or, in thealternative formulation we have that if M be any subset of Nwhich contains O and if xiscontained in M, Sx is alsocontained in M then M contains all elements of N. In order togive a clear picture of this aspect of N one may point out to acollection of blocks standing in juxtaposition so that a forwardpush on the block representing the element x of N forces it toknock over the next block representingSx. When the blockrepresenting O is pushed forward it knocks the block(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISrepresenting SO, which again knocks the block representingSSO and so on this, process continues without any halt. In thisunending process all the blocks are knocked over as a result ofthe pushing over of the frst block, so that if the frst block falls,all the blocks fall.[52].2.Progressions as Models for the Peano SystemBesides the classical interpretation of the Peano systemthere are other interpretations also. The primitive terms O, Nand S occur in the system as variables in the sense that theseterms ar capable of various interpretations and so the systemdoes not prossess a unique model. In fact any progressionnot necessarily of numbers, may be seen to verify theseaxioms [5]and in this case N represents the progresson, orepresents xo, the frst element of the progression and Sxpstands for xp+1 , the element appearing in the progression justnext to xp.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISSome authors have found fault with the system in that itdoes not contain anything in itself to distinguish between itsdiferent interpretations [40]. But it may be seen that anyprogression which is a model for the system is isomorphic withits standard model viz. the natural number system. Hence allsuch models may be considered abstractly as identical modelsof the Peano system and it is no fault of the system for notcontaining anything in itself to distinguish one model fromanother (isomorphic) model.3.Inadequacy of the Peano System for Characterization of NAt the frst sight the Peano system appears to be elegant;for, in order to characterize the infnite system N, Peano usesonly two individuals,viz. O and N, only one monadicpredicateP and the only dyadic predicate S and he also gurantees thenonemptiness of N by means ofthe frst axiom. Categoricity ofthis system may also be deduced from that of the Dedekindspostulates [3] which form basis of the Peano system. Mututal(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISindependence of the axioms of the system has also beenestablished by Peano and Padoa[40]. Moreover, the capacity ofthe system to have various models confrms the potentiality andrichness of the system also.But on close examination of the Induction axiomand the arguments involved in the proof o the categoricity of thesystem we fnd that much of its elegance is lost. Firstly, we fndthat the categoricity of the system can be established only if ourlanguage is very very strong so as to admit all sorts of sets andpredicates as individuals and it is not possible to establish thecategoricity in the ordinary language. Secondly, we have seenthat the predicate Pused in the Induction axiom is not welldefned and that it occurs as a predicate variable. But predicatevariables can occur only in the second order language of logic.From this we infer that Peano proposes to characterize N in thesecond order language. Also we know that there corresponds asubset M on N for every predicate P and vice-versa and hence(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISthe use of P as universally quantifed predicate variable in theInduction axiom determines the set of all subsets of N, which isa nonenunerable set. This suggests that Peano proposes tocharacterize the denumerable system N with the help ofnondenumerable system of axioms, and this can never beconsidered a suitable characterization of N. Thus the systemfails to give a satisfactory characterization N.4.Categoricity of the Peano System : When we examine the reasons for failure of thePeanosystem, naturally we would blame the strong andunrestriected language used in the axioms. And if we want to getout of the trouble our frst impulse would be to restrict ourselvesto some weaker languagewhich admits only such predicates ashold under certain restrictions. The question is what sort oflanguage we shall use and what would be the nature of theadmissible predicates.Let us arbitrarily formulate a frst order structure M,(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnonisomorphic with N, which admits only those predicateswhich hold either for a fnite set of objects of M or for thecomplement of a fnite set of objects of M. Suppose that Mconsists of all natural numbers and all fractions of the form for allso thatand let P be an admissible predicate. We proceeded to show thatthe system M is model for the Peano system.We see that(i)(ii)(iii)(iv)Which means that the system M verifes the frst four(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISaxioms of the Peano system. We now proceed to prove that Mverifes the Induction axiom also. Supposing P(0) holds andP(u+1) holds whenever P(u) holds, we get that P holds for allelements of N. Now let P(u) hold for some u . Then P (u)most hold for all succeeding elements in M-N. If not, let ussuppose that P(u) does not hold for u= (2x+1)/2 for a certainN m. Then P (u) does not hold for u = (2m-1)/2 also; for , if itholds for the latter, it also holds for the former, which is thesuccessor of the latter and similarly proceeding we can showthat P(u) does not hold for anyelement of M-N before and upto(2m+1)/2. This means that P does not hold for an inftie subsetof M and at the same time it holds also for an infnite subset ofM, which contradictsthe admissibility of P. Hence P(u) musthold for all u and since it holds for all natural numbers, itmust hold for all u . Thus we have proved that M verifes theInduction axiom of Peano also. Hence the Peano systempossesses a model M which is not isomorphic with its standardmodel N.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISThus in an attempt to get out of one trouble we are againthrown into another trouble where the very categority of thesystem is destroyed.5.Completeness of Axiomatic Systems :We have seen that the categoricity of the Peano systemcannot be established in the ordinary language of logic. We havealso seen that none of the above system is a perfectcharacterization of the natural number system. TheodolfSkolem(1915) has established that any formalization of arithmetic failsto characterize the number system completely and admits asvalues of the number variables a class of entities of which thenatural numbers from only the initial segment[34] According toLowenheim and Skolem no axiomaticsystem is completebecause if any infnite system is described by an axiomaticsystem A, a countable model andalso a model of highercardinality are possible models of A. Hence we conclude thatevery axiomatic system of Arithemetic is incomplete as well as(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESISnoncategorical.Uptill now we have been engaged in dealing with naturalnumbers only. We now propose to introduce a theory of Real andComplex numbers with the knowledge of natural numbers in thenext two chapters.(80)PATNAUNIVERSITYPATNAUNIVERSITY Ph.D. THESIS