ࡱ>  |stuvwxyz{ bjbj .}}Ɯ+HH8tK&lm&:&'''':(0( mmomomomomomom$oxrHm<(''<(<(m''m111<(''mm1<(mm11TX'~(zU"Ymm0mUrh1RrDXXr%X4<(<(1<(<(<(<(<(mm1<(<(<(m<(<(<(<(r<(<(<(<(<(<(<(<(<(H Q: WHERE LIES THE REALITY OF MATHEMATICS FOR COMMON PEOPLE? Min Bahadur Shrestha  HYPERLINK "mailto:Shresthambs@gmail.com" Shresthambs@gmail.com Central Department of Education, Tribhuvan University, Nepal ABSTRACT It is common to say it is not mathematics that two plus two is always four, it is in politics and social activities that it may be otherwise". It is stated to insist that two plus two is always four in mathematics. It is a version common to common people and to mathematicians and philosophers since long ago. The Platonic philosophers defined such mathematical truths as true knowledge which are perfect, eternal and changeless, but the modern philosopher like Frege and Russell characterized them as absolute truth/exact truth on logical basis. Recently, question has been raised against it by saying that 2 + 2 = 4 is true not in absolute sense, rather it is true in accordance to our convenience and tradition. This is fallibilist position which can simply be understood to be denial of absolutism and there exist differences among fallibilists in the respect of status of mathematical knowledge. As I think, one main difference lies on whether or not we consider human relation to things/objects in the construction of mathematical objects and processes. Imre Lakatos' and Phillip Kitcher's contributions are taken as important contribution for the development of fallibilist/social constructivist philosophy of mathematics. Kitcher's viewpoint blends empiricism and evolutionism and give reason why 2 + 2 = 4. He says the basic core is experienced in physical acts of collecting, ordering, matching, and counting though it can be proved as a theorem in a formal axiomatic system. But the humanist/maverick philosopher Reuben Hersh (1997) says that such mathematical facts are true not in the object based sense but due to our convenience based on social construction. Hersh counters Quine's position that real numbers are really exist. But, to save mathematical knowledge from being mere fiction (created stories), Hersh Characterize mathematics as the study of the lawful, predictable, parts of the social-conceptual world. Although both Hersh and Ernest share many ideas as to the social constructivist views on mathematics, they seem to differ in some respect. As mentioned by Ernest (2006) in his article "Nominalism and Constructivism in Social Constructivism", the signs of mathematics are not just detached and empty symbols (signifiers) but always have meanings (signifieds) and that the numeral '3' connotes both the act of establishing 1-1 corresponding with prototypical triplet set (cardinality) and the act of enumerating a triplet set (ordinality). Social constructivism as the name literally indicates the construction of knowledge as social construction. In a sense, some may expect that it reflects the common people's conception of mathematical knowledge. The common people's concept of the mathematical reality (as evidenced by the truth of 2 + 2 = 4") seem to be based on its consistent valid utility in the physical acts of collecting, ordering, matching, and counting. Ernest (2014) acknowledge the counting process as proxy invariant, which means it is based on preconceptualized activities. So far as I think, the preconceptualized activity is driven by social necessity of physical acts, such as, collecting, ordering, matching and counting. Kitcher says mathematics is a lawful, comprehensible evolution from a basic core while Hersh says the study of mathematics as lawful, predictable, parts of the social-conceptual world apart from physical world. Hersh did so in order to shave mathematical knowledge from being mere fiction. As I think, ultimately, such characterization cannot shave mathematics from being mere fiction in the absence of human relation to thing/object. And, in the name of social construction of knowledge, it cannot represent what mathematics is for the common people. So, my inquiry is: Why not the mathematical objects, such as, the extension of counting numbers in such a sense can be thought of the study of the lawful, predictable parts of the physical world which consists of discrete, reasonably permanent, noninteracting objects. If it could be done so, mathematics would not be far from what common people find it. Keywords: absolute, fallible, infallible, humanist/maverick, nominalist, fictionalist, a priory, Ganita/Ganana(mathematics/counting). Introduction and Background In social and political discourses it is frequently heard people saying "It is not mathematics that two plus two is always four, it is in politics and social activities that it may be three or five, or otherwise. It is one of the common version heard in our Nepalese society and I think, it is common to all to some extent. It is commonly pronounced by common people as well as by politicians, social workers and learned scholars. The same version seems to be common among mathematics teachers and students. They seem to reflect similar versions in relation to mathematical truths. What is interesting to note is that it reflects people's conceptions on mathematical truths (Here, the term "People" simply indicates all those who can use at least basic arithmetic and measurements skills in their daily life activities or in their professions and vocations. It also includes other peoples, such as, skilled workers, office workers, intellectuals, teachers and students who have basic mathematical skills). It simply indicates that mathematics is exact. That is, mathematics is about exact relationships without any errors. What is intended in this writing is to account to and reflect on the nature of common people's version of mathematical truth. Since such mathematical truths might rest on many dimensions (such as, sociological, cultural, historical, practical, cognitive), the underlying reality needs to be examined with respect to such dimensions so as to examine the relative position of the fact. It is done in order to examine the validity of mathematical truth as is conceived by common people as well as by students and teachers of mathematics. Not only common people but also teachers and students seem to share the same opinion on the mathematical truths. It is notable why people are so much faithful to mathematical truths. Perhaps, basic mathematical truths stand so much obvious that people don't seem to have any questions regarding its validity and consistency. The truths of mathematics for common people seem to lie on its valid and reliable use in their everyday life: a common people may perceive the truth of arithmetical operations in selling and purchasing things in the market; a carpenter may perceive the truths of geometry and measurements in measuring the area, volume and capacity; and a tailor master may perceive the rules of mensuration in designing a shirt or coat. One can have different interpretation of their belief system, but the ratification of mathematical truths as obvious truth by common people need to be interpreted appropriately which would help to understand the subject at least from the reality they see. One should think that so much firm belief on the truth of mathematical facts rests on some firm belief system based on some grounds to assert them. There are different grounds for convincing oneself or to convince or persuades someone as to the truthfulness and validity of truths. There are at least six ways to convince oneself or to convince others as to accept a belief (Bell, 1977:291) and not only common people even professionals like technocrats and scientists have the basis of truths on their own grounds and they are different from the ways of asserting truth. Similar to common people, for technocrats, mathematics seems to be their valid and reliable aid/tool for them because it can be used conveniently and consistently in their endeavours. Rather than focusing on consistency of mathematical results on the basis of formal axiomatic structure as did by formalistic views of mathematics, they seem to view validity of mathematical truths in deriving their scientific or technical results which are compatible for scientific interpretation and usefulness. Referring to Ormell (1975), Ernest (1991: 114) mentions that scientists' and technologists' view that mathematics as a collection of tool to be utilized when needed as a 'black box' whose workings are not probed but consistently useful. What is commonly seem to be true of mathematics for common people lies on its applicability and usefulness of the mathematical results and their consistencies in the explanation and manipulation of our physical environments. To a large extent, such results are historical artifacts transmitted to succeeding generations through some form of instruction or demonstration. So, Knowingly or unknowingly, we are carrying out some traditions explicitly or implicitly which is ultimately based on some socio-historical foot-prints as well as philosophical underpinnings. This is why such consideration goes back to its ancestors so as to examine its true nature of the truth. The historical decedent of mathematical truths can be traced back to long ago in the version of popular saying by well-known scholars and philosophers, such as, Saint Augustine (354-430 B.C.), the Neo-Platonist philosopher whose view is remarkable to mention here and which reads as follows: "Seven and three are ten, not only now but always; nor was there ever a time when seven and three were not ten, nor will ever be a time when seven and three will not be ten. I say, therefore, that this incorruptible truth of number is common to me and to any person whatsoever" (As cited by Hersh, 1997: 103-104). The above version of Augustine indicates that the basic mathematical truths are a kind of universal truths. Such thinking goes back to ancient Greek philosophers Plato and Pythagoras, the result of which is that mathematics was regarded as the subject of absolute certainty at least up to the end of 19th century. Though mathematics seems to be considered as the subject of certain knowledge and mathematical truths are taken as the most exact results in many civilizations, the attribute of absolute certainty bestowed to mathematical truths (as an eternal truths) trace back to ancient Greek civilization. But, Hindus civilization which is one of the ancient civilization and which contributed significantly to mathematical development, does not hold the Greek concept of the absolute certainty for mathematical truths. Rather, it holds mathematical truths as supreme knowledge. For ancient Hindu religious philosophers, mathematical knowledge were supreme among the secular sastras (disciplines) as mentioned by B. B. Dutta and A. N. Singh (1939: 8) referring to Vedanga Jyotisha (supposed to be written in 1200 B.C.): As the crest on the heads of peacocks, as the gems on the hoods of snakes, so is Ganita (mathematics) at the top of sciences known as the Vedanga" What is interesting to note that unlike Greeks, Hindu mathematical tradition seems to have considered empirical basis too for the justification of mathematical knowledge. Such nature seems to be derived from the influence of the development of mathematics due to astronomer-mathematicians and astro-physical nature of mathematical development. Such interpretation is given importance so as to examine the basis of mathematical truths from different angles of views. It is also important to note that Hindus tradition of development of mathematics lend some similarity to quasi-empirical development of mathematical knowledge as perceived by Imre Lakatos (Lakatos, 1976). Lakatos's quasi-empirical development of knowledge describes how mathematical truths develop and get into such status-quo. It is not historical interpretation, rather it is developmental interpretation. Historical development of mathematical truths also provides us the clue for the nature of mathematical truths. As mentioned by C. B. Boyer (1968), it is clear that originally mathematics arose as the part of everyday life of man and at first the primitive notion of number, magnitude and form may have been related to contrasts rather than likeness (such as, the difference between one wolf and many, the inequality in size and the unlikeness in the shape of the objects) and then awareness of similarities in number and form which together must have given birth to science and mathematics (pp.1-2). What is notable is that there are two opposing theories of the beginning of mathematics: one representing the view of Herodotus and the other representing the view of Aristotle. The first, holding to an origin in practical necessity, the second, attributing in the priestly leisure and ritual for the beginning of mathematics. They have become sources to different philosophical underpinning on the basis of which mathematics have got different interpretations. Boyer writes, there is also common ground which support either theories: The rope- stretches by ancient Egyptian geometers were used both in laying out temples and in realigning the obliterated boundaries. The ancient Hindu priests are credited to develop geometry as the Sulvasutras in their attempts to construct sacrificial alters of definite shapes and sizes, which shows a kind of some similarity in the development of mathematics across the wider region of the globe. The term "Ganita" (mathematics) basically indicates the science of number as the term Ganita is derived from the term Ganana(counting) in Hindu civilization. Number is also given supreme position (Cooke, 1997:6) in western mathematical development. Such situation reflects some similarity in the development of mathematical thinking among very different cultures and which might provide some basis to throw light on the nature of the development of mathematical knowledge. Such consideration is made so as to throw some light in relation to the common people's views on mathematics. The main objective of this writing is to examine the ground on which reality of mathematics stands for common people. Although the main subject of this writing is common people's basis of belief on the reality of mathematical truths, preservice teachers'(M. Ed. students of mathematics/would be teachers') conceptions of reality of mathematical truths have also been considered as supplementary aids to clarify the situation more on the assumption that the preservice teachers' conception would represent the conception of would be teachers of mathematics as well as that of students of mathematics. Such consideration is partly based on my long experiences of teaching mathematics to in-service school mathematics teachers and teaching undergraduate and graduate students of mathematics education. My observations of common people's conception of mathematical truths is based on my experiences of observing public interactions/discussions informally and making interactions specially with shopkeepers, tailors, masons, carpenters, and others occasionally. It also includes my experience/observation of social transmission of mathematical knowledge from older people to younger people or from learnt people to learning pupils in the villages where most peoples were illiterate since I perceived the situation from the beginning of my informal and formal education. It is also impressed by how my own personal perception about the nature and truth of mathematics developed through distinct stages while I went through primary to university education. I remember that in my entire school life (in my mathematics classes), directly we never talk/ discuss about the nature of mathematical truth (we never talk about What is mathematics and what are mathematical truths?"). Sometimes, teachers used to say you should come to the correct answer to the problems in the form as given in the answers in the text and you should prove the theorem strictly in the format as it ought to be done". Further they used to say It is mathematics and it should be done exactly". It was the voice of mathematics teachers as well as seniors. I was much motivated by the logical consequences of deriving truths in proofs of the theorems in my high school Euclidean geometry, but we did never talk about the nature of basic truths, such as, axioms/postulates until the study of foundational nature of geometry at bachelor level in education. In a sense, mathematics was believed to be valid and absolutely true knowledge without any question on its structure until my first encounter to non-Euclidean geometry. The real exposure about the reality of mathematical knowledge first time came to me when I began to study The Philosophy of Mathematics" written by P. Ernest(1991) for the purpose of teaching one unit in the philosophy of mathematics(to M. Ed. students) one decade ago. The above consideration of my observation is taken so as to illustrate how silently and implicitly mathematical facts are transmitted as unquestionable knowledge by education and traditions. More or the less, similar phenomena of social-historical transmission seem to be the one main cause of descendent of mathematical truths even for common people. More than that, they believe in certainty of mathematical facts for common people as well as for teachers and students seem to lies on its consistent and valid usefulness of mathematics. On the whole, the main objective of this writing is to examine common peoples' including students'/teachers' thinking on the nature of mathematical truths as it appears commonly and then to examine them in the light of different views of thinking so as to examine its underpinning basis. For that purpose, the discussion is concentrated around the following aspects: Conception of mathematical reality among common people Conception of Mathematical Reality among students(Preservice Teachers) Different Philosophical positions(on mathematical reality) My reflection Conclusion Conception of Mathematical Reality among common people Mathematics is given supreme position among different disciplines as mentioned by Hindu mathematics. Mathematics is given great importance in western historical development of intellectual thinking and which is supposed to be the rediscovery of the ancient Greek intellectual thinking (Joseph, 2000, 2011; Pears, undated; Ernest, 2009 ). Historically, number is taken as supreme concept of mathematics. As the name Ganita (mathematics) seems to be derived from the word Ganana ""(counting) in the development of Hindu mathematics. Among mathematical objects, the number is given supreme position (Cooke 1997: 7). The supremacy of numbers and its operations gave rise to supremacy of arithmetic in the historical and philosophical development of mathematics. The greatest of the great mathematician Fredrick Gauss stated the importance of arithmetic by saying that mathematics is the queen of all the science and number theory is the queen of mathematics. The modern mathematician Kronecker attributed numbers as god's made. Many illustrations can be found which highlight the importance of arithmetic and numbers. As mathematical facts, arithmetic facts are most common representative not only among common people but also among students/teachers of mathematics. As a representative of the truths of arithmetic, the relation "2 + 2 = 4" is given as characteristics statement. It is not only a simple arithmetical fact that two plus two equals four, rather it is also a reflection of representation of certitude of mathematical truth. In social and political discourses it is frequently heard people saying It is not mathematics that two plus two is always four, it is in politics and social activities that may be otherwise. This statement means that two plus two is four exactly in mathematics. It is representative of the view that mathematics deals with exactness or mathematics is a science of exactitude. It is one of the common version heard in our Nepalese society and It is commonly pronounced by common people as well as by politicians, social workers and learned scholars. The same version seems to be common among teachers and students of mathematics. Taking this one as a representative case, let us examine more. As the basis of its reality, what is commonly observed that the reality is based on their experiences of manipulating of physical objects, such as, getting things together (in addition) and making them apart (separation in subtraction) in connection with physical manipulations of objects. This is commonly observed among common people in their discourses on numerical activities. Here are some typical observations (edited versions) observed by me in different instances occasionally: "Everybody knows that Two Plus Two is Four ! Why to tell times and again for such universal(obvious) truth? It is apparent as clear as the day sun light. You can verify such things in our everyday life activities" (Versions of common people, particularly, less educated physical labourers) When asked about its reality, they seem to take it as self-evident truth as evidenced by common practice of getting things together and making them apart. They seem to feel uncomfortable to explain more (possibly due to language problem). Further, their response shows that the facts (truths) are self-evident as shown by the practices and need not to be further verified. In building my home, I got opportunities to talk with masons, carpenters, plumbers and electric wire fitters about their thinking of arithmetical and mensurational calculations involved in their vocations (I was more involved with them because I took responsibility of contracting and involved in observing the working site almost regularly so as to build the home as per the specifications of the blue-print map recommended by the municipality). Here are some typical excerpts (edited version) obtained/observed during my consultations with them: After cost estimating for underground work, I said them to negotiate for the real cost. They said: Look Sir! (respecting me as the university teacher), You have stretched the tape with your hands with us in measuring every pieces of the task. we have correctly measured them and written here on the copy. Check our Hisab (calculation) one by one and find if there is any mistake. If not, there is no ground for bargaining on calculation. As a learnt teacher of mathematics, you know more how calculations are perfect/exact and no room for bargain." Consciously(by the way), to assess their thinking, I asked them: Are you so much sure in your calculation that you have made right estimation of the cost? Do you think that depending on the calculation would give justifiable cost?" They replied: "Do you see any other grounds other than measurement and calculation to find the real cost? we are sure, we have no doubt at all in such Hisab (calculations) and no one should have, because this is the fact and only way, and there is no way otherwise" Further, they said: "Look Sir, Hisab is Hisab (calculation is calculation) and is exactly correct. If you seek for some discount in the real cost, we will not disappoint you." It is one small section of discourse that came into being by the way. What is observed is that the numbers and mathematical calculations (based on fundamental operations) are granted as the correct and only tools to computations as are the scale and tape are used to measure length. It seems that the computational rules and measurements are the real and exact means to evaluate the physical objects for them. One other typical version can be taken from the conversation with masons building a wall structure perpendicular to a wall. To construct the wall perpendicularly to the given wall, the lead mason used the relation of Pythagorean triplets of lengths (3ft., 4ft., and 5ft.). To construct the wall perpendicularly to one end of just constructed part of the wall, he took 3ft. on that side of constructed part of the wall and he located a point perpendicular to the end of the wall at a distance of 4ft. from that point and 5ft. from the other end. He located the point meeting the condition of forming right triangle on the basis of side lengths. He used it as a technique to draw the wall perpendicular to the given wall at the given location. When asked about its accuracy and utility, he said: It is an exact practical way to construct the wall perpendicularly to the given position of the wall. Such techniques guide us to correctly perform our task. Those who cannot apply such techniques would not be an efficient mason. Its importance is so much that I cannot build the wall perpendicular to a wall without using it and it guides us to do most conveniently and accurate way." When I reiterated about it to assess their thinking more, the lead young mason said "Look Sir, really it is no matter of discussion in our circles of masons. You may have some doubt whether the wall would be straight right at each corners. But we are completely confirm and actually I don't need to measure times and again because we can build the wall straight right as shown by our tape and scale." Another lead mason who was more senior and who used to come infrequently to check the construction site, mentioned about his role and efficiency as the lead mason and said: Look sir, I would not have been an efficient mason and I would not have obtained so much opportunities to build such constructions if I were not a able to follow the blue print of the map of your home to be built into real construction. An efficient mason should read the map correctly and he should be able to perform the task of buildings correspondingly. With other many skills, an efficient mason should have mastery in estimating and constructing the two tasks: The first (comparatively rather simple) is to measure correctly and then to construct accurately a wall perpendicularly to the already constructed wall; and the second (more critical) is to determine the number of steps on the given length of the inclined plane(section) for the ladder and then to construct the ladder. All that firstly need correct Hisab (mathematical calculations) together with refined skills to do actually. In essence, Hisab is most fundamental which guides us toward the correct path. This is why only young and little more educated masons have been much successful in making modern construction." These mason had not completed school education and they mentioned that school education they got and most specially the mathematical calculation they learnt in the school formed the basis of their calculational skills required to perform their job efficiently. They mentioned that their school learnt mathematical skills lend them learn the techniques and skills through working under senior masons. They showed full confidence of mathematical truths as the real means of their workings. They consider them as the real means of doing their job efficiently and correctly without any question on its validity and consistency. The mason and the carpenter's perception on common arithmetical and mensurational skills show that the skills are as the basic skills to run their vocation efficiently and perfectly and correctly. Their views lend themselves to instrumental views on mathematics (Ernest, 1991:114) that mathematical techniques are set of rules to be followed exactly. Basic arithmetical skills seem to be representative of mathematical facts/skills for most common people for they are used in their daily life. For most common people, mathematical facts and skills are their most dependable aides for marketing articles. The fast computation/calculation made by the use of hand held calculators have changed the mode of calculation even with increased efficiency and accuracy, and thus making common use of mathematical skills available to larger mass than ever. Hand held calculators have become tool or instrument to accelerate the computations of numbers and hence arithmetic operations on numbers are instrumentalized. Such instrumentalization which is common among youngsters in school and in society seem to accept certainty of mathematical knowledge without any question on its truth. The commonly heard typical opinion in this respect can be cited as follows (edited version): Calculators have made calculations correct and very fast. It has been gift for us. Because, with its use, we have been able to do our job very fast and with most accuracy. Now a days, we can't think of calculation without calculators. Even, it is helpful in negotiating prices and managing business" (Young Shopkeepers' version) Even though students are asked to compute algorithms with numbers with paper and pencil, most senior students tend to use calculators at least to perform operations in course of solving problems. With the use of calculator, they can solve the problem fast"(Secondary school teachers teaching in urban area). Why not to use calculators in doing our homework ? Because it shaves time, it gives accurate results, and it is not harmful to use calculator to perform algorithms which takes much time in solving problems in mathematics. Most of us use calculator although school does not encourage us to use it" (Secondary school students). As seen in practice, hand held calculators(including the calculational functions in mobile phones) have been so much popular among young generation (including students) that common arithmetical calculations have been transformed into instrumental operations. The activities of instrumentalization in calculation seems to give rise cybernative characteristics to numerical computations. Rather than involving in cognitive skills of computing operations of arithmetic, the computation is based on instrumental skills of manipulation. The calculator has performed the role of input-output machine in which numbers are put to give results based on given operation. Such situation seems to lay a cybernative character to mathematical function rather than cognitive, which ultimately might lead to instrumentalist thinking in mathematics giving risen to blind support to certainty of mathematical facts studied in school curriculum. In the city area, such as, Kathmandu, where calculators and mobile phones have heavily been used, young people including school adolescents seem to develop such character in their style of working as expressed by school mathematics teachers. Such belief system on the reality of mathematical truths seem to dependent rather on utility both for the common people and students. Use of hand held calculators have further accelerated and added a new character of instrumentalist thinking in granting belief on the common truth of mathematics. Conception of Mathematical Reality among M.Ed. students (Pre-service Teachers of mathematics) The nature of mathematical facts (basis of mathematical truth) was taken into consideration more consciously in course of teaching the unit on the philosophy of mathematics to master degree students of mathematics education. On the basis of my experience of teaching in very large class(up to 150 students in each group) and on the basis of one- to one interview (as required by the requirement of practical part of the course), I got opportunities to inquire on students' thinking on the nature of mathematical truths. Here, are some typical responses expressed by students in connection to the nature of mathematical knowledge observed in classroom interaction as well as in interviews. Some typical responses (edited responses) have been mentioned below. When asked "What is mathematics?", the entire class could not make any response even saying repeatedly. And when asked What does mathematics deal about ?", with some hints, some said It deals with numbers and operations; it deals with geometrical objects, such as, point, line, plane and its relationships; and it also deals with measurements". Some of them mentioned that mathematics consists of numbers, operations, geometric shapes, rules, formulas and theorems. When asked "What are mathematical truths and why are they called so much true ? with follow up questions. In response to the question "What are mathematical truths?", many responded that numbers and their operational results, geometrical objects(such as, line, plane, triangle, quadrilateral, circle) and their theorems/measurements and many others as mathematical truths. In response to the question "why are they called so much true ?" many said we find them true because they are completely verified by their use of characterizing reality; that is, they have been correctly verified by their use in our daily life as well as in drawing correct results in science and technology". They further mentioned "mathematics gives us rules, techniques and formulas having correct results in many ways. It is due to such merits of mathematical knowledge, it deserves the status of exactitude and truth" "It is said that mathematical truths are absolutely correct and mathematics is said to be a subject of true knowledge. Do you know what does it mean to say mathematical knowledge as absolutely true knowledge ?". When interacted with such a question, at the outset, no one attempted to make interpretation in the beginning classes. Only after providing some hints with follow up questions(such as, "Does it mean true in all the situations and forever and everywhere ?"), some few students tried to explain. Their responses can be collected as It means it is quite true and true in all the situations/circumstances" When asked "Does it mean the universal truth ?", few notable students mentioned that they may be called so because they are true for ever and in all the circumstances". When asked the basis for it, their responses reflected that its accurateness rested on empirical grounds and it applicability in different spheres. On the whole, students' conception on the reality of mathematical truths seemed to based on its uses and applicability. They did not seem to have concept of a priory nature of mathematical truths because they equated axioms as self-evident truth as evidenced by physical interpretation(empirical grounds) rather than axiomatic. Partly, some of them reflected mathematics as the collection of correct and useful rules, techniques and formula used everywhere. Such situation indicate that the basis of mathematical reality seemed to be asserted on quasi-empirical and instrumental basis. The development of students' conceptions on the reality of mathematical truths were observed during the teaching of the lessons on the philosophy of mathematics. As the lessons advanced, students were seemed more motivated to the subject but with many queries on their faces. Since it was very large classroom teaching (before semester system introduced in 2013)and since some selected students were called on turn by turn to express their thinking, whatever is reported here should be taken as representative cases on probabilistic basis. Some typical students' responses(edited responses) have been stated below so as to assess their thinking on the reality of mathematical knowledge: When asked to the class Why Two plus Two is always Four ?", the class voiced "because, it is truth/facts of mathematics and all the truths of mathematics are always true". When asked "How do you know or believe that mathematical truths are always true ?", some said We know the truths of mathematics from its use in our daily life as well as their use in different fields and vocations". Some said Mathematical truths have been used to draw correct results not only in our practical life but also in different fields. The application of mathematics which provides correct results in everyday life as well as in vocational and scientific calculation is the basis for the certainty of mathematical truths". Some said "In fact, we have always considered them true without any question and no one asked such question to think more on it. In other words, we are taking it as granted truths having no doubt at all" When I reiterated as to the basis of the truths they think they are convinced with the truth of mathematics, they said we had no question about them because we found no contradiction/exception on their use/applications. They were taken as self-evident truths applicable everywhere ". One said actually, I do not remember any event in the class in which such matters were discussed" and other many students followed his opinion. Some of them requested me by saying Sir!, please, help us getting clear/confirm by giving more instruction because it is not the way we had tackled before". In course of teaching the nature of mathematical knowledge(a priori Vs. a posteriori) based on the philosophy of mathematics(Ernest, 1991), students were asked to read the nature of mathematical knowledge and then I said: We know already that the basic facts of addition consists of the addition of any two digits. That is, any two from the set containing 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which give hundred facts of addition. What does it mean to say basic facts of addition ?. Do they need to be proved or they should be taken without proof ? What is interesting is that one of them is stated: 1 + 1 = 2 as a theorem to be proved following the steps(Ernest 1991: 4-6)". Then students' were asked to read the steps of the proof followed with its justifications attentively and then present their ideas as to the development of the proof together with the nature of truth. On the next class, students views were assessed as follows. When asked Do you think that there is significance in proving such relation which is so much obvious to us ?", many said we do not really understand whether such proof is needed or not for such obvious fact which is common to primary school students ". Some said " We found the proof very tough for such simple facts of arithmetic and we could not understand clearly how the steps lead to the proof ". One student said I am most motivated by the development of proof for such basic fact of mathematics. I tried to follow the proof with enough attempts and I think, I understand the proof to some extent which I can explain now". He was called on to demonstrate and he demonstrated the steps of proof on the board. He was much impressed with the deductive style of proof adopted here and said that it was one example how proof could be developed to establish results in mathematics. It seems that there were few students in the class who could follow him. After the class, some of them followed me to discuss more about the nature of mathematical truths and the nature of proofs in mathematics. But when they were asked for the basis of reality of mathematical facts (such as, facts of addition and axioms of equality used in deriving the proofs), they frequently referred its reality in terms of its physical interpretation and its application in different fields. For example, many students said times and again: "Basic facts of arithmetic, such as addition, are the reflection/representation of putting things together while the facts of subtraction are the reflection/representation of making things apart (just reverse to addition). The physical balance used in weighing things yields the truths of axioms of equality and the importance of equation in mathematics. Actually, we had no doubt on basic mathematical truths either in arithmetic or in geometry which are clearly reflected by their use and by making drawings". Occasionally, some student said: Sir!, Is it not sufficient for us and for anybody to have firm believe on the truths of mathematics when the truths of mathematics have been found so much valid and consistent in our activities whether in markets/offices/working fields or in science and technology. Really, we had no question about it. First time, the question on the reality of geometric knowledge came in course of non-Euclidean geometry". From the discussions with students times and again in course of teaching, it was revealed that most students consider basic mathematical truths ultimately as the expression of the realities. On the whole, students seem to have the reality of mathematical facts based on utilitarian basis (empirical basis) similar to the common people. It is seen that the students (who have been school teacher for sometimes) seem to have opinions similar to that of common people on the nature of common mathematical truths. On the whole, what seems to me that the reality of common mathematical truths either for common people or for students/ teachers of mathematics ultimately rests on its utilitarian basis(empirical basis). With conversations with many school, college and university mathematics teachers, the same basis of its representation of correctness, certainty and truth seem to be common among them. A kind of commonality on the reality of mathematical knowledge among common people together with students and teachers of mathematics might indicate that the utility of mathematics has given it a common basis as that of physical realities yielded by physical science. Since knowledge is socially constructed and transmitted as cultural heritage(Vygotsky, 1978), the age old perception of the reality of mathematical knowledge among common people seem to be transmitted through generations. But one of source for the conviction on the certainty of mathematical facts seems to be dependent on its valid and consistent use. This seems to be one of the reason why common people using simple and basic mathematical skills have so much conviction on their own. Different Philosophical positions(on mathematical reality) As mentioned in the first section Conception of reality of mathematics among common people", attention has been paid on common people's conception on the reality of mathematics they think as such. In the second section, views of students (post-graduate students of mathematics education, the pre-service teachers of mathematics) have been included so as to represent the conception of reality of mathematics among students of mathematics as well as would be teachers of mathematics. This is done to examine the common people's conception of the reality of mathematics in the reference of thinking existed among the students of mathematics education. The views of common people(which encompasses wider range of people in different professions/vocations and fields using common mathematical facts and skills) are not isolated views rather they are dependent to social and intellectual development of the society in which teachers/students act as vital agent. This is why, in addition to common people, views of students of mathematics education on the reality of mathematical knowledge were taken into consideration. Here in this section, focus has been made on analyzing the basis of their thinking with respect to the main philosophical positions. As mentioned in the introductory part, the statement "The sum of 2 and 2 is always 4" is taken as an example of representative characterization of mathematical certainty. The single simple statement conveys very important massage for the nature of the truths of arithmetical facts in the field of mathematics. Common conversation saying It is not mathematics that 2 and 2 is always 4 " indicates that mathematics is different from other discourses and only mathematical discourses have exactness and certainty. In the entire field of knowledge, mathematical knowledge has been given separate and supreme position for its absolute certainty. What is very interesting is that the most common truth of arithmetic which is common in common public tongue is likely to be a fairy tell that came down to us through the long historical span. The Pythagoreans worshiped them; the Platonic thinkers took them as eternal but real objects; the Neo-Platonic great thinker, Augustine characterized as timeless/ tenseless truth(truth for ever); the modern great philosopher Kant characterized them as synthetic a priory; and the great modern logicists, Frege called them objects which are real but not abstract. But recently, social constructivist philosopher, such as, Paul Ernest, characterize them as social construction and the humanist/maverick philosopher, such as, Reuben Hersh, call them social construct like a dollar. There are also thinkers who say numbers are fiction. Numbers are so many things for so many thinkers. Question has also been raised Is 2 plus 2 equal to 4? by David Bloor (1994) in his article What can sociologist of knowledge say about 2 + 2 = 4 ? ". What is most common for common use has been most controversial. But, even now, there are philosophers like W.V. O. Quine who says that numbers are real. This is why this part of the article is focused to deal with different positions so as to clarify the issue to the concerned readers like my students who are facing problems and are seeking more discussions so as to through some light on the very nature of mathematical facts. So, in the following paragraph of this writing, attempts have been made to analyze the situation in the respect of different views. In doing so, I have also included some of my own reflection in course of teaching and learning of mathematics and its philosophy. Let me begin to think of the issue from the perspective of common people who utilize mathematical concepts and skills as part of their daily dealing. So far as I conceive, for them, mathematical skills and concepts they use in course of dealing with objects/articles and situations are techniques/skills leading them to certain and correct results in their activities. For them, mathematical skills and techniques are aids to accomplish their tasks efficiently and accurately. They seem to use common mathematical skills conveniently and boldly as a skilled worker use his/her tool to work with. Although recently, the use of calculators has deemphasized paper-pencil calculation of mathematical operations in public sphere, the use of calculations have dramatically been increased in course of urbanization of Nepalese societies and hence in effect, it has been more popular and common among increasing mass of people. The hand-held calculators/ electronic calculating aids have instrumentalized common people in using mathematical truths. What seems to be happening in effect is that they have been more instrumental in accepting the truths of mathematics and it seems that they have nothing to think about mathematical truths which works so surely in their tasks. In course of the classroom teaching on the lessons of philosophy of mathematics, frequently, I used to refer the common people's beliefs/practices on common mathematical truths that students observe in society. Once, one student said (and other followed) : Sir!, I dont understand clearly and we are getting query and even confuse as to the status of mathematical knowledge while we are studying more as to the fallibilists'/social constructivists' position of mathematical knowledge. He continued Sir!, why should we be so much worry as to the nature of mathematical truths which are so much faithful and applicable for us and for all common people". To reduce their worries and to motivate towards the lesson, I remained them times and again by quoting a version from the book "The Philosophy of Mathematics Education" (Ernest, 1991: 20) which mentions that the loss of certainty of mathematics does not loss the use of mathematical knowledge which is so much important for us. I repeated the same version among teachers and students of Central Department of Mathematics(Tribhuvan University, Nepal) recently few weeks ago in course of my presentation of an hour long talk on the reality of mathematical knowledge. What is noticeable was that some senior professors showed worries as I spoke about the fallibilist position of mathematical knowledge. One chairperson presiding the session spoke about it immediately after my presentation. We also discussed about how mathematics has always been helpful to guide us to interpret the phenomena and the physical world that is around us and too remote. What appears to us and What might be may be different. The difference seems to be more about mathematics(that is, about the philosophy of mathematics) rather than mathematics itself. So, there seems to be more differences among philosophers of mathematics rather than mathematicians themselves. How mathematics is viewed is primarily the concern of philosophy of mathematics which exerts much impression on the development of nature of mathematical knowledge. In this respect, Ernest (1991) mentions how mathematics is viewed is significant on many levels, but nowhere more so than in education and society (p. xii). He elaborate that depending on how mathematics is viewed(whether as absolute body of knowledge or as fallible social construction), mathematics has different roles to do. In relation to question What is mathematics ?", Imre Lakatos (1976) mentions that mathematics is what mathematicians do and have done with all the imperfection inherent in human activity or creation. Although, it is important interpretation as to the nature of mathematical knowledge, in does not directly address the question what is mathematical knowledge about?". Rather, it explains how mathematical knowledge is evolved. How mathematics is viewed depends largely on what mathematics is, and it is primarily the concern of philosophy of mathematics which exerts much impression on the development of nature of mathematical knowledge. So, the commonly held beliefs among common people on the reality of mathematical knowledge need to be examined in the light of philosophical underpinnings along with historical development. The commonly held opinions on the nature of mathematical knowledge seem to be largely based on the mathematics education of the western style as adopted in our school and college curricula. Hindu mathematical traditions which originated in south-east Asian reason, is said to superceded by western mathematical tradition (Eurocentric tradition) and it became less influencing among educated circles. This is why the western views of philosophical thinking seem to have dominant role in today's formation of people's thinking on mathematical truths. For this reason, common people's basis of thinking on the reality mathematics knowledge need to be examined more in the reference of prevailing western philosophical thinking although consideration of the impact of Hindu traditions is made so as to assess the impact. Broadly speaking, the western philosophy of mathematics can be grouped under the two major distinctions: Platonism and Social constructivism/Fallibilism. In the following paragraph, we will examine their role/place with respect to the issue of reality of mathematical knowledge as perceived commonly together with the impression of Hindu mathematical traditions which is even active in astronomical and ceremonial forms. And in discussion, their comparative positions would be considered. Platonic Position and its impression Unlike Platonism which views that the objects of mathematics have a real, objective existence in some ideal realm (Ernest, 1991: 29), common people thinking on the reality of mathematical truths seem to be based mainly on objective reality they perceive in dealing with the phenomena of physical and material worlds. What seems to be true for most common people who use mathematical techniques in their daily life (either as a common people or as some vocational or technical workers) is not about "what is it?", rather it is about "what it does ?". The importance of mathematics as conceived by the common people might be also helpful in deciding the true nature of mathematics as it began historically. The symbols of basic mathematical notions, such as, the notion of small counting numbers in different forms (such as, clay tablets, strokes, notches in bones ) in different civilizations but with similar correspondence might indicate its objective reality across the different cultures. As mentioned by Imre Lakatos, history lacking guidance of philosophy might lose its direction while philosophy without history will be empty. This indicates the determining role of history in philosophical development. Though the Platonic philosophy occupies very important place in the historical development of the philosophy of mathematics, it does not seem to reflect neither the historical development of mathematical conceptions nor the common people's conception of mathematics due to its eternal conception as to the reality of mathematical truth. Plato's view on mathematics seem to be guided by his aim to use mathematical realities to explore the realities of the universe. This is why Plato understood the physical universe to be organized in accordance with the mathematical ideas of number and geometry. For Plato, the mathematical objects, such as, circles, triangles and numbers are not merely formal quantitative structures imposed by the human mind on natural phenomena, nor are they only mechanically present in phenomena as a brute fact of their concrete being; rather they are eternal ideas which are invisible and apprehended by intelligence only, and which are formative causes and regulators of all empirically visible objects and process (R. Tarnas' comment as cited by Hersh, 1999). As mentioned in his literature, Plato was seeking some ideal truths that could be truly dependable. Along with some other attributes, mathematical attributes such as numbers and measurement were taken as most dependable in exploring the realities. In this respect, a paragraph is taken from Book X in Republic (edited by Adam, 2010: 35): The same objects appear straight when looked at out of the water, and crooked when in the water; and the concave becomes convex, owing to the illusion about colors to which the sight is liable. Thus every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of painting in light and shadow, the art of conjuring, and many other ingenious devices impose, having an effect upon us like a magic. In the above passage, Plato mentions that we cant depend on our perception because the same stick appears to be crooked in water and straight in air. For Plato, since our sense perception are not dependable, we should seek for attributes that are dependable. Since Plato was looking to use mathematical attributes as most dependable vehicle to explore reality, he conceived mathematical truths as having real and objective existence not in materialistic form but with ideal realm. Platonic thinking assumed the universal reality of mathematical knowledge as the external reality in form of an ideal realm independent of any inherent material world. Just like a virtue is common property not inherent in a person but bestowed upon a person, Platonic thinking posits that fundamental mathematical realities are already existed as are physical laws are already existed in the universe. Platonism came to neo-Platonism and its present form has been defined by leading philosophers, such as, Frege and Russell who have defined mathematical object, such as, numbers, as real and abstract but not physical nor mental. Rueben Hersh says it is new edition of Platonism. Hersh writes (1999:145): "What an astonishing kinship to Plato's Ideas! They were neither mental nor physical, but eternal and changeless. Frege is a Platonist as well as a Kantian. Frege's abstract objects include numbers" Hersh mentions that to define numbers as abstract but real objects, Frege denied physical and psychological basis of numbers. To do so, Hersh mentions that Frege tried to demolish all previous definitions given by empiricism, historicism and psychologist. He rejected empiricist view that numbers are things in the physical world and the position of psychologist that numbers are ideas in someone's head. He denied historicist view that numbers involve. The great Platonic philosopher Frege's intention to demolish all previous definitions of number was to show numbers free from its physical, social and historical development. In another sense, he wanted to cut off all the roots of numbers with physical and social phenomena. It seems that his views were guided by the Platonic views that numbers are real in eternal sense which cannot be seen and touched, but can be felt. This seems to be guided by the view that the existence of numbers including other mathematical objects in essence are external realities which are bestowed upon physical and social world. Although the Platonist philosopher Frege made great contribution to the definition of numbers in his historical work on foundation of arithmetic and he is given credit of being a first philosopher of mathematics who really defined numbers (Russell, 1920), he detached numbers from earthly phenomena; and thus detached from human and history on which it was developed. What seems to be interesting to me to mention here is the story of "Emperor's new cloth" where all the courtiers saw the emperor with magnificent dress, but a child said that the king was necked. In a similar manner, learnt people generally follow the intellectual norm which is governing as our intellectual and educational heritage. Unlike learnt people of mathematics(as that of courtiers in emperor's new cloth), common people's observation and experiences(as that of child observation of emperor's new cloth) on the nature and existence of mathematical truths might also yield some important information which could really say what is mathematics about. At least, it would really say what is mathematics for them rather than saying what is mathematics for philosophers and mathematicians. To some extent, mathematics may be different for different people for it has multiple faces and dimensions. In the above reference, it seems that Platonism does not speak about mathematics for which the subject was developed and on which basis it was developed. So, it could not speak even a bit the language and conception of common people. The question arise then How could then Platonism considered so much in the philosophy of mathematics ?". The mystic together with reasoning which involved Platonic thinking seems to provide philosophers/ thinkers a perennial/everlasting ground of thinking on which philosophy of mathematics dragged to have the three big schools of thinking. Reuben Hersh writes the attempts made by the three school in the first quarter of 20th century to establish mathematics on firm and sound ground is over and terminated by Gdel theorems(Hersh, 1999: 138). Hersh advocates for humanist/maverick philosophy of mathematics with mathematics of flesh and life not with dead eternality. Paul Ernest mentions about philosophical shift in mathematics and which is in the midst of a Kuhnian revolution (Ernest, 1991: xi). What seems to be is that the absolute status of mathematical knowledge is not going to be more issue today, rather the focus seems to be on social constructivism (Ernest; 1991, 97) as the philosophy of mathematics of mathematical knowledge. But, it does not mean that Platonic thinking which is so much perennial among mathematicians and philosophers is totally out from the consideration. In the history of mankind and mathematics, some views have been renewed, revised and improved so as to meet to changing situations. One of the veteran philosopher carrying out Platonism to its new edition is taken as W. V. O. Quine, the author of Quidities, a philosophical dictionary. In the last section, social constructivist/maverick/ fallibilist positions including to that of factionalist and Quinian positions have been considered so as to explain the underlying reality of mathematical knowledge as conceived by common people. For Quine, pure arithmetic is the pure science of numbers. Quine's position which seems to address quasi- empirical nature of mathematical objects(to some extent in some sense) seems to have some proximity with the common people's conception of common mathematical knowledge. Social Constructivism Vs. Quinian positions Social constructivism is a new conception in the philosophy of mathematics which views that mathematics is a social construction in which the objective reality of mathematics is social. Social constructivism as the philosophy of mathematics proposed by Paul Ernest (1991) is a novel philosophy of mathematics which largely draws upon conventionalism in accepting that the human language, rules and agreement play a key role both in establishing and justifying the truths of mathematics. It also takes into account quasi-empiricism and Lakatos's philosophical thesis which imply that mathematical knowledge is fallible and it grows through the process of conjectures and refutations (Ernest, 1991:42). Ernest's social constructivism as the philosophy of mathematics has been fully extended in his 1997 edition in which in addition to being almost three times the length in comparison to earlier one, there are a number of significant conceptual differences between them (Ernest, 1997: xiv). Reuben Hersh acknowledge it as the new philosophy of mathematics and says Ernest seems to be the first to speak of social constructivism in philosophy of mathematics" (Hersh, 1999: 228). Summarizing social constructivism, Hersh writes " In summary, the social constructivist thesis is that objective knowledge of mathematics exists in and through the social world of human, interactions and rules, supported by individuals' subjective knowledge of mathematics (and language and social life), which need constant re-creation. Hersh says subjective knowledge recreates objective knowledge, without the latter being reducible to the former. In a sense, social constructivist position accept the existence of objectivity in mathematics (which has been so common to common people), but the objectivity that seems to be so much universal among us is a social construction. In other word, the common people thinking on the objectivity of mathematical knowledge which seems to be an universal language on the globe is essentially a product of social phenomena across the globe. As to the universalization of mathematics, DAmbrosio (2006) writes The discipline known as mathematics is an ethnomathematics that originated and developed in Europe, having received some contributions from Indian and Islamic civilizations, and that arrived in its present form in the 16th and 17th centuries, from which point it began to be carried throughout and imposed upon the rest of the world. Today, this mathematics acquires a character of university" (pp. 56). Such interpretation gives us a historical glance of looking of how the universalization of mathematics was made. As mentioned by DAmbrosio, the universalization of mathematics was the first step of globalization. What is very interesting is that perhaps mathematics was the discipline that resembles more among different civilizations. It may have been one reason that mathematics became the first means of globalization as mentioned by DAmbrosio. Among many things that brought commonalities of mathematical notions among common people, the most common might have been emerged due to counting and measuring things as the history tells us. Similar basis might have been common to all individuals. Ernest (2006) writes: I understand signs in the semiotic sense of having or being composed of both a signifier, that which represents, often a material representation, and a signified, the meaning represented. Thus unlike in Hilberts formalism, the signs of mathematics are not just detached and empty symbols (signifiers) but always have meanings (signifieds), even if precisely specifying the characteristics and ontology of these entities is difficult and complex, and may involve ambiguity and multiplicity. Ernest as a nominalist hold such position in which he differs his position from Hilbert's position that mathematical entities, such as, numbers are just empty symbol without signifier. He mentions that mathematical symbols have always meaning (signified). Further clarifying his position, he writes: The objects of mathematics are signs, and furthermore the meanings of these signs are typically yet further signs. However, as indicated above, this is not to say that all that exists in mathematics is signs. There are in addition sign related activities: idealized human action on signs. Thus the numeral 3 connotes both the act of establishing a one-to-one correspondence with prototypical triplet set (cardinality), and the act of enumerating a triplet set (ordinality). Each of these connotations presupposes some elements of threeness, requiring the use of a representative triplet set." In the above paragraph, Ernest mention that in addition to sign related activities, there are idealized human activities on signs. He mentions that the numeral ' 3' connotes both the act of 1-1 correspondence with the cardinality triplet set and act of enumerating a triplet set. Here the query arise in relation to conservation and invariance of numbers: How does social constructivism consider invariance and conservation of numbers as considered by Piaget in his experiment ? Piaget's theory has been one of the basis of constructivist theory of learning, but the radical constructivism (of Glasersfeld) does not accept mathematical structures or structural problem descriptions as analytical tools apart from the constructed knowledge of a learner of problem solver (Goldin, 1995: 38). Ernest's interpretation of mathematical structures(such as, the well-known structure of counting numbers) as mentioned in his article "Certainty in Mathematics" (2014), has considered number as proxy invariant as follows: Contrary to our common sense that counting comes naturally, Ernest claims that the idea of counting material objects is not a naturally given one, as simple and obvious as it looks to the trained modern eye. He rather says that counting is based on a set of prior conceptualizations of the world that include the five assumptions as mentioned in his article. It is very important to be noted that Ernest addressed more to the causes of how historical development of mathematics produces a belief in the objectivity and certainty of mathematical knowledge. Ernest says such belief emerges because the origins of mathematics in counting and calculation require invariance, predictability and reliability in order to fulfil their social purposes. He distinguish between Platonic certainty of mathematics with his form of mathematical certainty consistent with social constructivism and other maverick philosophies of mathematics. From the point of view, it implies that the invariance and conservation of numbers through a long historical development (with its use in counting and calculation) has led to a belief as independently existing stable objects (with fixed and enduring properties), which is in essence is a function of socio-historical development. Ernest's interpretations seem to be much meaningful to explain the evolution of the structure and its invariance in mathematics. Such interpretation obviously implies that the common people belief that 2 + 2 = 4 is always correct everywhere, is the invariance which conserve the sum '4' not due to its inherent attribute but due to our social necessity. For any inquisitive mind, a question may arise here is What is that basis or motivating factor that gave mathematical attributes so much consistent explanation through the long socio-historical development across the different ancient civilizations? As mentioned by Renyi in his Socratic Dialogue in Mathematics (within 18 Unconventional Essays, edited by Hersh, 2009), a question arise: "why not two poets cannot write the same poem although different mathematicians can produce the same mathematics ?" These are some questions asked by the laymen (common people), students and teachers of mathematics as well as by mathematics educators. As far as I think personally, and as I share with my students (sometimes), mathematics (including science, such as, physical science) developed in the attempts to explore the world people need to live in and progress. The history of mathematics and mankind tells us that primitive peoples may have discovered numbers, such as, small counting numbers, for their necessity of counting objects. So, it seems that the cultural product of numbers should be seen in terms of its basis for which it was developed and on which it was developed. Therefore, the existence of numbers are the creation of mankind for the enumeration of objects. This is why any debate on the nature of numbers should not forget its physical basis. Any attempts to separate mathematics completely from physical interpretation might lose its difference from being mere fiction. Differentiating mathematical knowledge from being fiction, Hersh writes (1999:179): "Fictionalism rejects Platonism. In that sense, I'm a fictionalist. But Chihara, Field and I aren't in the same boat." Differentiating social constructivist position with that of fictionalist position as held by fictionalists as are Charles Chihara, Hartry Field, and Charles Castonguay who say real numbers do not exist, Hersh says in the above statement that he is a fictionalist to reject Platonism, the Quinian position that real number exist really, but he cannot sit together in the same boat with the fictionalist for they insist that mathematical knowledge are fictions. Clarifying his position, he writes: " The difficulty is failing to recognize different levels of existence. Numbers aren't physical objects. Yet they exist outside our individual consciousness. We encounter them as external entities. They're as real as homework grades or speeding tickets. They're real in the sense of social-cultural constructs. Their existence is as palpable as that of other social constructs that we must recognize or get our heads banged. That's why it's wrong to call numbers fiction, even though they possess neither physical nor transcendental reality." Hersh rejects both Platonism and fictionalism so as to save the position for his maverick thinking. He has spent much space to counter Quinian position for whom Hersh says he is Platonist of these days. He says, at present, W. V. O. Quine is unavoidable philosopher representing reality of mathematical truth who says real numbers really exist (Hersh, 1999:170). Quine argues that you are committed to bad faith if you say that real numbers are fictions. Hersh says Queen proved that the real number exists philosophically, not just mathematically. According to Quine, physics is inextricably interwoven with the real numbers, to such a pitch that it's impossible to make sense of physics without believing real numbers exist. Hersh quotes a paragraph from Quine's publication- The Scope and Language of Science: "Certain things we want to say in science may compel us to admit into the range of values of the variables of quantification not only physical objects but also classes and relations of them; also numbers, functions, and other objects of pure mathematics. For, mathematics not uninterpreted mathematics, but genuine set theory, logic, number theory, algebra of real and complex numbers, differential and integral calculus, and so on-is best looked upon as an integral part of science, on a par with the physics, economics, etc., in which mathematics is said to receive its applications." By regarding most part of the mathematics as an integral part of science, Quine has brought existence of mathematical entities to physical interpretation. For Quine, the existence of number lies on physical measurement. Quine does not agree with Russell in his view that in pure mathematics we never know what we are talking about nor whether what we are saying is true. He says, it is to Russell's credit that he soon dropped this idea; but many mathematicians have not. For Quine, pure mathematics is indeed the pure science of number (Quine, 1987:65): "The straight forward view of pure mathematics puts it more on 'a par with the rest of science. Pure arithmetic is indeed the pure science of pure number, but pure numbers participate in the subject matter of physics and economics on a par with bodies, electrons, and petroleum. " Quine's position has been countered by Hersh by saying that we use real numbers in physical theory out of convenience, tradition, and habit and not in the sense of Quine's ontological commitment. Hersh writes: "We use real numbers in physical theory out of convenience, tradition, and habit. For physical purposes we could start and end with finite, discrete models. Physical measurements are discrete, and finite in size and accuracy. To compute with them, we have discretized, finitized models physically indistinguishable from the real number model.Real numbers make calculus convenient. Mathematics is smoother and more pleasant in the garden of real numbers. But they aren't essential for theoretical physics, and they aren't used for real calculations." Hersh mentions that the existence of real numbers, such as, the concept of infinite and continuity are not needed in physics because physical measurement are discrete. His interpretation is that we use real numbers in physical theory due to our convenience, tradition and habit, but not due to intrinsic character of physical objects. Such interpretation lend to social construction of real numbers. Hersh has made extensive consideration as to social construction of mathematical knowledge in his epoch making book What is mathematics really ?" In relation to the common statement of mathematics, 2 + 2 = 4", he has made extensive consideration under the section Survey and Proposals" in which he examines the basis of so called universality of 2 + 2 = 4. Hersh writes that this statement has two meanings: So "two" and "four" have double meanings: as Counting Numbers or as pure numbers. The formula has a double meaning. It's about countingabout how discrete, reasonably permanent, noninteracting objects behave. And it's a theorem in pure arithmetic (Peano arithmetic if you like) (p.16). Hersh says, the pure numbers aroused out of counting number when Aristotle disconnected pure numbers from the real objects for number abstraction as the share concepts in the mind/brain. He insists that 2 + 2 = 4 is a fact based on shared concept. It shows that the shared concept historically descended as knowledge of social attribute through generations. It also shows that it is based on the concept of the collection of discrete, reasonably permanent noninteracting objects. In the article What can the sociologist of knowledge say about 2 + 2 = 4 ? " David Bloor (1994) mentions the truth as a cultural universal, that is, something everyone seems to believe. My Reflection Before drawing conclusion of this writing, let me express my own position/reflection on the reality of mathematical knowledge in connection with the common people's perception on the reality of mathematical knowledge they use. My position here means my reflection and my query to be clarified which I use to share even with my students during classroom discussions. Let me quote parts of some paragraph already mentioned. I am taking 2 + 2 = 4" as a representative case as done already. Let me starts by quoting again a paragraph from Ernest's article "Nominalism and Constructivism in Social Constructivism" in POME (2006), which is found to be more useful for me to consider the ontological and epistemological position of mathematical knowledge. Ernest(2006) writes: Thus unlike in Hilberts formalism, the signs of mathematics are not just detached and empty symbols (signifiers) but always have meanings (signifieds), There are in addition sign related activities: idealized human action on signs. Thus the numeral 3 connotes both the act of establishing a one-to-one correspondence with prototypical triplet set (cardinality), and the act of enumerating a triplet set (ordinality). The above statement mentions that the numeral '3' connotes both the act of establishing 1-1 corresponding with prototypical triplet set (cardinality) and the act of enumerating. The first line clearly mentions that the signs of mathematics are not empty symbols(the mere symbols only as in Hilbert's formalism where mathematical symbols are meaningless symbols in the created rules of a game), rather they are symbols (signifiers) together with signifieds, that is, idealized human actions on signs. The act of enumerating, ultimately, has to do much with the objects (physical or symbolic). As I feel, the proxy invariant nature of number is also the shadow/image of countability of physical object. But, these things seem to be silent to me in the paragraph mentioned above. As the human activity, mathematics cannot be viewed in isolation from its history and its application in the sciences and elsewhere (Ernest, 1991:35); and hence, as I think, numbers cannot be isolated from its practical basis of arranging/ordering and comparing objects. The common public believe on the process of counting seems to be based on such basis even if the idea of counting number is not naturally given one. Peoples who cannot read and write, and cannot count in sequence of counting numbers, can identify more chocolates from the less one. He/she may not have language to speak, but have ability to distinguish. According to Phillip Kitcher (whose publication is regarded to make significant contribution for the development of humanist/maverick philosophy of mathematics), all mathematical knowledge comes by a rationally explicable process of growth, starting from a basic core: the arithmetic of small numbers and the directly visual properties of simple plane figures (as cited by Hersh, 1999: 224). In relation to object and objective basis of mathematics, he mentions that the formula: 2 + 2 = 4 can be proved as a theorem in a formal axiomatic system, but it derives its force and conviction from its physical model of collecting coins or pebbles. I think, it is one of the objective basis (materialistic basis) for each children which give rise the learning of the concept of counting numbers and addition. David Bloor (1994), in his article "What can the sociologist of knowledge say about 2 + 2 =4?", mentions about the sociological interpretation for the fact 2 + 2 = 4. In the last phase of his writing, he raises the question Finally, what about the apparent universal use of 2 +2 = 4?", and then differentiate between the crucial partition between 'a thing' and 'the number one', and 'two things" and 'the number two'. He says "we could say things qua things belong to nature, while numbers belong to us, to society- though we must not forget that society is a part of nature". He, further says I do not deny that our innate ability to perceive in some sense puts us in contact with numerousness of things". He says like birds, we have innate abilities that can differentiate between two eggs from the three eggs which are importance for individual and collective lives (p.29). By referring to Wittgenstein view of philosophy, he seems to give the fact 2 +2 = 4 as the conventionalist interpretation in which the apparently universal use of 2 + 2 = 4 is the product of social convention. What is to be noted here is that the social convention has the root on the human's ability to perceive numerousness of things. That is, our innate ability that can differentiate numerousness of things which is the basis of number concept and which is ultimately materialistic objective basis for social construction. I think, this is what makes the difference between social basis of basic mathematical knowledge from mere social construction of knowledge. Ernest in his article "Certainty in Mathematics" (2014), has considered number as proxy invariant. His analysis shows that the very intuitively appealing process of counting (which gave rise to counting numbers) is by proxy invariant. It means that contrary to our common sense that counting comes naturally, Ernest claims that the idea of counting material objects is not a naturally given one, as simple and obvious as it looks to the trained modern eye. He rather says that counting is based on a set of prior conceptualizations of the world that include the five assumptions as mentioned in his article. But, what is notable is that the process of counting is based on prior conceptualizations of the world. Ernest says such belief emerges because the origins of mathematics in counting and calculation require invariance, predictability and reliability in order to fulfil their social purposes. The question arises What caused people to consider sequence in counting ?" Ernest mentions it is because counting and calculation require invariance, predictability and reliability. The further question is "Why then people needed invariance ?". I think, the necessity of ordering and comparing things gave way to the main basis of such invariance as it is reflected by the historical development of mathematical objects, such as, number and space. It is notable that the ancient Hindu priests developed geometric shapes of accurate measures for their purpose of sacrifices while the ancient Egyptian are said to develop geometric shapes and their measures in the attempts to measure land accurately that arose due to the flood of the river Nile. Though the two historical developments were guided by different purposes, they contributed to the task of the measurement of the land/space accurately. One may say that the measure of the land/space is a human action needed by social necessity and it could have taken in some other ways different from what we have now. But, what is to be noted that in any sense, the concept of the area (limited area) has to do much with the amount of the space. What is intended to convey here is that the development of mathematical concepts has human relation to things. In such a sense, social constructivist account of mathematical object also need to be dealt in relation to physical basis on which the human existence and civilizations began. It seems to me that social constructivist position as advocated by Reuben Hersh tends to cast the philosophy of mathematics completely on social phenomena independent of its physical basis. He seems completely in opposite direction to Quine's position who relies on physical interpretations. Quine may have hoped to give quasi-physical interpretation for the basis of mathematical knowledge so as to give Platonism a scientific basis instead of Platonic eternal basis. Hersh, on the other hand, seems to develop philosophy of mathematics solely on the basis of social phenomena. So far as I think, he intended to detach his maverick philosophy (as he prefer to say him maverick/humanist?") completely from materialistic interpretation so as to make the philosophy of mathematics fallible. This is why for Hersh, mathematical object like numbers are though not physical, they exist outside our individual consciousness and we encounter them as external entities as the school, dollar and homework grades as the social-cultural constructs (Hersh, 1997: 179). Hersh writings have been so mush impressive and instructive for me and it has opened much space to think. This is why I am putting my comment on his writing. What I feel here is that his characterization of number not as physical but as external/social reminded me to consider Frege characterization of number once again. So far as I think, Hersh needed to define it in such a way in order to shave mathematical knowledge from being fictions like a created story. If clear demarcation could not be made, it would be difficult to separate mathematical knowledge from being considered mere fictions(created stories). It is to be noted that Frege defined numbers as abstract objects which are real, but not physical and psychological. Frege's attributes of number as real and abstract, but not physical and psychological casted numbers into the domain of ideal ream. Such ream is not social or physical since it trace back to Platonic thinking which came as the blending of mathematics and theology from the ancient time of Pythagoras and Plato down to modern times of Descartes, Frege and Russell (Russell, 1957: 56). In essence, Frege attempts to define the number in such a way was ultimately guided by Platonic thinking that numbers are external reality, which are not inherited in physical and social world, rather the external reality is bestowed upon physical and social world (which is evidenced by Plato's ideas of separating knowledge and mere opinion in his study of forms).This is why Frege definition of number became so much obscure and mystic. The social constructivist/ maverick/ humanist philosophers have done great job to characterize the nature of mathematical knowledge as social construction. In a sense, they have liberated philosophy of mathematics from Platonic ideal thinking. In liberating the philosophy of mathematics, they based on social-cultural interpretation of mathematical thinking. So far as I think, in making such interpretations, physical/materialistic basis of human's social-cultural- historical development should not be ignored. It is because materialistic and quantitative phenomena has not only provided the basis of mathematical development, but also they have become underlying basis for many mathematical metaphors. But, Hersh characterization of mathematical object like numbers "which are not physical, which exist outside our individual consciousness and we encounter them as external entities as the school, dollar and homework grades as the social-cultural constructs", does not seem to involve physical characterization in his definition of number. Hersh's attempt shows that he want to give social constructivist interpretation of mathematics independent of physical basis. This is also evidenced by his insist that physical representation is the task of physics, not the mathematics (Hersh, 1997:19): of the lawful, predictable parts of the physical world has a name: "physics." of the lawful, predictable, parts of the social-conceptual world also has a name: "mathematics." This is why he seems to be a fictionalist although he has made demarcation with fictionalists in the respect of mathematical knowledge as mentioned already. He made this demarcation by stating of the lawful, predictable, parts of the social-conceptual world also has a name: mathematics.I think, any attempt to depart mathematics completely from physical/materialistic interpretation might ultimately lead mathematics more and more towards fictionalism and mere fictionalism which consider mathematical knowledge as mere created stories. To resolve the problem on the nature of mathematical knowledge, one could get significant hints from the historical development of non-European mathematics, such as, mathematical traditions developed in the south-Asian region. For example, Hindu mathematical traditions of the south-Asian reason, which shares common foundation of astronomy and mathematics in form of astronomical mathematical development may tell us about common and inseparable relation between science and mathematics at least in some aspects. Development of Hindu mathematics shows the intimate blending of science and mathematics in form of astro-mathematical development. This is why Hindu mathematical development (also referred as Indian mathematics) is considered as fundamentally different from Greek based western mathematical development due to its quasi-nature of mathematical development (Subramanium, Shrinivasa and Sriram, 2008). At last, let me quote Nietzsche- Foucault position as mentioned by Skovsmose in his article "Can Facts be Fabricated?published in POME (2010). Nietzsche-Foucault position is in extreme opposition to Descartes' assumption that a perfect harmony can be established between knowledge and the reality that knowledge is about. Skovsmose has cited Nietzsche's view on the status of mathematical knowledge: Let us introduce the refinement and rigor of mathematics into all sciences as far as this is at all possible, not in the faith that this will lead us to know things but in order to determine our human relation to things. Skovsmose in his article writes (2010): As mathematical constructs come to make part of our reality, this reality is changed. Through all forms of fabrication, it becomes a fabricated reality. It becomes a reality which includes many produced facts. Nietzsche's view implies that mathematics is about human relation to things. As mentioned by Skovsmose, when mathematical reality comes to make part of our reality, they are changed to become a fabricated reality. Such interpretation implies that mathematics represent something reality which is then fabricated in the process of developing towards some pre-identified goal, such as, towards the goal of establishing absolute certainty as did by Platonist. Among many indicators about the reality of mathematical knowledge, common people's thinking as to the truths of mathematics may be a basis. The common people's beliefs on the truth of mathematics, such as, 2 + 2 = 4, may have partly emerged due to the materialistic interpretation of the countability of objects(numerousness of things) as the physical attribute of the material world. In the above reference, my inquiry is: Why not the extension of counting numbers in such a sense can be thought of the study of the lawful, predictable parts of the physical world which consists of discrete, reasonably permanent, noninteracting objects. If it could be done so, mathematics would not be far from what common people find it. Conclusion The history of mathematics tells us that mathematical knowledge has been considered as a kind of certain/exact knowledge and it has been given unique place ever since the school of ancient Greek over 2000 years or in ancient Vedic civilizations of south-Asian reason. Although the conception of the nature of mathematical knowledge does not seem to be same in western and eastern intellectual development, it appears more or the less same for the common people who reap the benefit of mathematical facts. For most common people, the most common are the common facts on counting numbers together with fundamental operations of arithmetic. The most common representative telling the truths of mathematics has been the statement "The sum of 2 and 2 is always 4" as symbolized by "2+ 2 = 4 and nothing else ". The Platonic philosophers might have preconceived interest of establishing absolute certainty for their purpose of establishing eternal reality to mathematical truths, but for common people so much convinced with the basic facts of arithmetic, it is dependent on its consistent uses and applications in dealing/manipulating physical attributes directly or indirectly. My observations and interactions with common people indicates that they trust fully on mathematics as their most reliable tools and their basis of believe seem to based on its consistent and valid use and application. The use of hand held calculators among common people has made mathematical skills more common than ever; and in effect, the calculators have made the calculational skills more technical and instrumental. The students and teachers of mathematics seem to share more or the less the same confidence on the same basis. Being a product of human civilizations and its intellectual development, mathematical skills carry its foot print both from historical and philosophical development. As the philosophical carryover of Platonic view through more than two millennia, mathematical truths as timeless/tenseless certainty made the way by the contributions of many philosophers/ mathematicians. Among them, Augustine (4th century, AD) in noted one to speak of mathematical facts who said the sum of two numbers (such as 7 and 3 which is 10) as timeless and tenseless truth ever true. In the 20th century, Platonic view was carried out by Frege and Russell to the climax in the form of logicism. Hilbert and Brower took different roads to establish mathematics as absolute body of knowledge. Hersh says they all were checkmated by the Gdels Incompleteness theorems that mathematics cannot be shown to have absolute basis of certainty. The attempt is not completely checkmated and ended and it should not be ended for ever because every new interpretation might be a further check point of thinking. Hersh says, W. V. O. Quine is another living Platonist philosopher who is unavoidable and who says real numbers exist not only philosophically but also physically because physical measurement is intrinsically related to real numbers and we cannot make sense of physics without believing the real number exist. He says you are guilty of bad faith if you say that real numbers are fictions. Hersh has countered Quine's view that real numbers really exists. To reject Quine's position on the existence of real numbers, Hersh mentions Quine's leading statement that the reality of real physics implies the reality of mathematics. Ultimately, he wants to knock down Quine's assertion that physics is intrinsically interwoven with the real numbers. To rule out Quine's position on the existence of numbers (the basic ingredients of mathematics), Hersh used nominalist position (Hartry Field position) that there is alternative possibility that science does not require any part of mathematics that refers quantifiers over abstract entities(Hersh, 1999: 172). What is to be noted here is that the fictionalists like Hartry Field wish to formulate non- mathematical basis for the science. It seems notable that Hersh used nominalist position just to show that real numbers do not exists really and then he categorically counters Quine's position by considering infinite nature of real numbers and its infinite decimal representation. As Frege demolishes the claims of empiricism, psychologism and historicism categorically to establish his definition of numbers as real but not physical nor mental, Hersh, in similar way, demolishes the claims of Quine categorically one by one. For example, Hersh says physical measures are discrete, and finite in size and accuracy. He says the unaccountability of real numbers fascinates mathematicians and philosophers, but physically it is meaningless. In response to Quine's view that the real number system developed out of necessity, Hersh says Pi ( QUOTE  ) and  QUOTE   developed conceptually, but not but not physically and computationally. Hersh posited his position by saying that mathematics is smoother and more pleasant in the garden of real numbers, but they are not essential for theoretical physics (p.175). Hersh seems to do so in order to give social constructivist interpretation of mathematics independent of physical basis. Hersh consideration of mathematics as study of the lawful, predictable, parts of the social-conceptual world seems to be guided by his view which sits between the two extreme positions of constructionism and fictionalism. Reuben Hersh and Paul Ernest seem to share common view on social construction of mathematical knowledge. But Paul Ernest version that the numeral 3 connotes both the act of establishing a one-to-one correspondence with prototypical triplet set (cardinality) and the act of enumerating a triplet set (ordinality) (Ernest, 2006), and his identification of numbers as quasi-invariant has to do much with the objects (physical or symbolic). But, Hersh seems to completely depart mathematics from physical basis. This is why he intends to cast mathematics into the domain of the study of the lawful, predictable, parts of the social-conceptual world. He did so in order to shave mathematics completely from being fictions as created stories. I personally think, any attempt to depart mathematics completely from physical/materialistic interpretation might ultimately lead mathematics more and more towards fictionalism. This is because mathematics may lose its ground on which social construction could take place. To shave mathematics from being fiction, one should also consider it in relation to the phenomena for which it was created and on which basis it was extended. On the whole, there are different realities of mathematics from those of Platonic to constructivist and then to social constructivist, and up to fictionalist. For Platonist, mathematical objects, such as, numbers are ideal realities; for logicists Frege and Russell, numbers are real but abstract (Russell 1920); for formalist Hilbert, numbers are meaningless symbols; for constructivist, numbers are constructed reality (based on cognition of countable objects in Piaget's sense); for social constructivist, numbers are social construction; and for fictionalist, numbers are fictions like created stories. Not only for Phillip Kitcher, but also for Hilary Putnam, the 'objects' of pure mathematics are conditional upon material objects. For Kitcher, the basic core of mathematical knowledge is experienced in physical acts; and for Putnam, the objects of pure mathematics are conditional upon material objects though they are in a sense merely abstract possibilities. Except for Hilbert formalism and fictionalism, for most, numbers have meaning something in relation to objects. Hilbert did so intentionally to lay firm foundation for absolute certainty of mathematics because he might not have seen any other way to do so. Not only for empiricist (like Hum), but also for most other (except fictionalists and some social constructivist, such as Hersh), numbers seem to have somewhat object based sense in some form. For example, numbers have human relation to things in some fabricated form (as mentioned by Skovsmose in relation to Nietzsche- Foucault position) or as quasi-invariant as mentioned by Ernest (2014). If such consideration is made, the common people's concept on the truth of mathematics, such as, 2 + 2 = 4, can have some common ground with that of different views on mathematics at least in object based sense. Since common people's views on mathematical truths seems to be based on the matter what mathematics does rather than what mathematics is about for them, it seems that mathematical skills and techniques are most exact and useful tools to deal with quantitative/numerical aspects of their activities. For Imre Lakatos, mathematics is what mathematicians do and have done with all the imperfections inherent in any human activity or creations (Ernest, 1991:34). And for the common people, mathematics to the large extent is what it does for them to tackle numerical and quantitative aspect of their dealings. Mathematics may be different for different people depending on their concern. It is not even impossible that many people may live in the faith which may not capture the holistic truth (not even partially as for the six blind men of Hindustan) as shown by the story The Six Blind men and the Elephant" (as referenced by Hersh, 1999:141). If we follow the fictionalist line of thinking, there is nothing like numbers which exist really, or there exists multiple so called realities, or mathematical objects like numbers are not any thing about. If one thinks in such a way, the story of six blind men and the elephant cannot be relevant in the sense that there exists no unique pre-imaged object like the elephant which exists uniquely and on which the blinds make exploration. In such a case, Hersh would have no opportunity of pre- imaged structure of the real elephant. This happens so because the fictionalist (such as, Charles Chihara, Hartry Field, and Charles Castonguay) need to avoid mathematical objects from referring the objective world. This is why, Hersh has differentiated mathematical knowledge from being mere fictions by saying a social construct which is the study of the lawful, predictable, parts of the social-conceptual world. If this lawful and predictable part of the social construct is related to the materialistic and objective world (on which basis mathematics got unique realm), there would be something commonality among all including the common people (here considered) who experience about mathematics. Mathematics would then get wider ground than ever. If the common people's conception of common mathematical skills and knowledge could be represented and respected to some ground, mathematics would be considered as a part of their activities. Doing so could include not only common people's concepts on mathematics but also take into account of utilitarian stage of the historical mathematical development from which the ancient Greeks were credited to elevate the art and science of mathematics. 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#$()+,0189;<=>DEKLNO[\bcjkopwx~ݤݤݤݤݤݤݤݤݑݤݤݤݤݤݤݤݤݤݤݤݤݤݤ$hvqhY'CJOJQJaJmH sH $hvqhC!CJOJQJaJmH sH $hvqh CJOJQJaJmH sH $hvqh8:CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhDLcCJOJQJaJmH sH 9ƍǍ΍ύЍԍ֍׍ۍ܍#$&'./239:@AKLOPWX\]ghiݷݤݷݤݤݤݤݤݤݤݤݤݤݤݷݷݤݤݑݑ$hvqh'~ACJOJQJaJmH sH $hvqh" fCJOJQJaJmH sH $hvqhEXCJOJQJaJmH sH $hvqhCJCJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhC!CJOJQJaJmH sH 9imnwxʎЎюԎՎ"#%&ݷݤݤݤݑݑݑݑݑ$hvqhVCJOJQJaJmH sH $hvqh.)CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqhCJCJOJQJaJmH sH h<CJOJQJaJmH sH $hvqh'~ACJOJQJaJmH sH 9&239:>?GHNORS[\^_bcjkmnqrs|}ɏʏ̏͏ϏЏя؏ُ܏ݏݷݷʤݤݤݤݤݤʤݤݤݤݤ$hvqhCJCJOJQJaJmH sH $hvqhCJOJQJaJmH sH $hvqh.)CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhVCJOJQJaJmH sH @  !)*12:;ABLMOPTXYZ^_cdlmuvz{ززززززز؟؟؟،؟؟؟؟؟؟؟؟؟$hvqh CJOJQJaJmH sH $hvqh%'CJOJQJaJmH sH $hvqhU;CJOJQJaJmH sH $hvqh.)CJOJQJaJmH sH h<CJOJQJaJmH sH -hvqhCJB*CJOJQJaJmH php0sH 7Ɛǐ͐ΐАѐ֐אِڐ ./9:>?CDJKPQSTVWYZ`afgoprsvw~$hvqh CJOJQJaJmH sH $hvqh%'CJOJQJaJmH sH h<CJOJQJaJmH sH JđőǑȑΑϑӑԑ֑בؑۑܑʷݤݑݑݑݤ~~~~~~~ݤݤݤݤݤݤݤ$hvqh_nCJOJQJaJmH sH $hvqh}-CJOJQJaJmH sH 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$hvqhCJOJQJaJmH sH h<CJOJQJaJmH sH 9dghopvwxy{}~—×ėŗΗϗЗї֗חٗڗݗݷݤݤݤݤݤݤݤݤݤ݁ݤݤnnnn$hvqhP#CJOJQJaJmH sH $hvqhEXCJOJQJaJmH sH hvqCJOJQJaJmH sH $hvqh%\CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqh3CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhCJOJQJaJmH sH +ݗޗ"#&'-.01=>GHKLPQRSWXZ[]^bcjkopstyzݤ$hvqhg CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqh&CJOJQJaJmH sH $hvqhEXCJOJQJaJmH sH $hvqhP#CJOJQJaJmH sH h<CJOJQJaJmH sH 9ĘŘȘɘ͘ΘјҘ՘֘ ʷʷ$hvqhg CJOJQJaJmH sH $hvqh&CJOJQJaJmH sH $hvqh>CJOJQJaJmH sH $hvqhP#CJOJQJaJmH sH h<CJOJQJaJmH sH @'()-.2356:;<=>ABGHMNPQYZ\]^_`deijlmopvwyz~ʷ$hvqh*=CJOJQJaJmH sH $hvqhjCJOJQJaJmH sH $hvqh>CJOJQJaJmH sH $hvqhg CJOJQJaJmH sH h<CJOJQJaJmH sH @™ƙǙəʙЙљҙәԙՙ֙יؙٙڙۙܙݙޙ ݷ~'hvqh&5CJOJQJaJmH sH !h<5CJOJQJaJmH sH 'hvqhY.5CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqh&CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhj^CJOJQJaJmH sH ,Zәޙ!"'(עآstrs() $da$gd3c$d^a$gd3c$ & Fda$gd3c  "&'(0156;<>?@CDGHMNPQTUZ[cdhilmʷݤݑ~~~~~kkkkkkkkk$hvqh*=CJOJQJaJmH sH $hvqhjCJOJQJaJmH sH $hvqh CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqh&CJOJQJaJmH sH $hvqhj^CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhEXCJOJQJaJmH sH 'mz{|}šǚȚКњ՚֚ٚښ ѽ覽ѽ-hvqh8"B*CJOJQJaJmH phsH -hvqha~eB*CJOJQJaJmH phsH 'h<B*CJOJQJaJmH phsH -hvqhj^B*CJOJQJaJmH phsH -hvqh*=B*CJOJQJaJmH phsH 7 &'(,-2467:;@AEFKLNO[\^_fgkltuz{}~谜ĉĉĉĉĉĉĉĉĉĉĉĉvvv$hvqhe7CJOJQJaJmH sH $hvqhY.CJOJQJaJmH sH 'hvqh<5CJOJQJaJmH sH 'h3cB*CJOJQJaJmH phsH h<CJOJQJaJmH sH 'h<B*CJOJQJaJmH phsH -hvqh B*CJOJQJaJmH phsH )ƛǛ˛̛ћқԛ՛ۛܛޛߛ #$&'+,012356;<?@EFJKPQ$hvqhe7CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhj^CJOJQJaJmH sH PQTUYZ`afgijtuxy{|ǜȜ̜ΜϜМҜӜל؜ ݷݷ$hvqhs CJOJQJaJmH sH $hvqh^CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhe7CJOJQJaJmH sH H !"#$)*,-3478=>BCIJRSVWXY_`ablmpqvwyz~ÝĝƝǝӝԝםݷݷݷݷ$hvqhNCJOJQJaJmH sH $hvqhs CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqh^CJOJQJaJmH sH Hם؝ݝޝ #$&',-459:=>BCGHJKNORSZ[^_dejkmnwxz{$hvqhT)CJOJQJaJmH sH $hvqhEXCJOJQJaJmH sH $hvqhNCJOJQJaJmH sH h<CJOJQJaJmH sH HžÞŞƞɞʞϞОӞԞٞڞ !"'(+,2356BCIJʤ$hvqh6CJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqheYCJOJQJaJmH sH $hvqhT)CJOJQJaJmH sH h<CJOJQJaJmH sH @JMNQRXY[\bcefrsvw{|ğşʟ˟͟Ο֟ןڟ۟ݷݷݤݑݤݤݤݤݑݑ$hvqhHCJOJQJaJmH sH $hvqh78CJOJQJaJmH sH $hvqhj^CJOJQJaJmH sH $hvqhh2 CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqheYCJOJQJaJmH sH 9 $%()01349:<=@AIJXY\]`almopyzڷʤʤʤʤʤʤʤʤʤʤʤʤʤʤʑʑʑʑ$hvqh[CJOJQJaJmH sH $hvqh4iCJOJQJaJmH sH $hvqh~CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqh78CJOJQJaJmH sH $hvqhHCJOJQJaJmH sH 8 ĠŠΠРѠՠ֠ؠ٠ %&)*/02ݷݑݑݑݑݑݑ~$hvqh~CJOJQJaJmH sH $hvqhCJOJQJaJmH sH $hvqh`hCJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqhEUCJOJQJaJmH sH h<CJOJQJaJmH sH $hvqh`NCJOJQJaJmH sH 123>?BCFGQ`acdjkqrswxy}~ǡȡˡ̡ҡӡա֡ܡݡߡ$hvqhqCJOJQJaJmH sH $hvqh3cCJOJQJaJmH sH $hvqh~CJOJQJaJmH sH $hvqh`NCJOJQJaJmH sH h<CJOJQJaJmH sH @ !"$%()+,1245:;GHNOUVXYcdfgijrsxy|}$hvqh3cCJOJQJaJmH sH $hvqh~CJOJQJaJmH sH 'h<B*CJOJQJaJmH php0sH h<CJOJQJaJmH sH $hvqhqCJOJQJaJmH sH <¢âĢȢɢТѢբ֢آ٢ޢߢ  "#&'-.01ڷڷ~ڷڷڷڷڷڷڷڷڷڷڷڷڷ$hvqh3cCJOJQJaJmH sH $hvqh7-ACJOJQJaJmH sH $hvqh~CJOJQJaJmH sH h<CJOJQJaJmH sH $hvqhcCJOJQJaJmH sH $hvqhCJOJQJaJmH sH $hvqhCJOJQJaJmH sH 11<=ABEFLMOP[\`aefklnostyz}~ģţǣȣʣˣңӣ֣ף  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