Nature and issues related to mathematics knowledge

1.  “Nature and issues related to mathematics knowledge”:

1.1.    Nature of mathematics Knowledge:-

                    About the nature of mathematics, there are many views like mathematics is formal, logical, intuitionist, symbolic, fallibility, absolutist, inductive, deductive, generality, applied, abstract etc. The nature of mathematics can be expressed in many ways argues that different conceptions of mathematics influence in which society views mathematics. They affects the way grow to view mathematics and its role in their world. In other words, nature of mathematics knowledge.

Mathematics has long been taken as the source of the most certain knowledge known to humankind. Before inquiring into the nature of mathematical knowledge, it is first necessary to consider the nature of knowledge in general.

 Thus we begin by asking, what is knowledge? How to analyze mathematics? What is mathematics really?  How and what is the structure of mathematics? How is it formed? What are its main elements? The new conception form from the answer of these above mentioned questions gives a clear view regarding the nature of mathematics. Reason includes deductive logic and definitions which are used, in conjunction with an assumed set of mathematical axioms or postulates, as a basis from which to infer mathematical knowledge. Thus the foundation of mathematical knowledge, that is the grounds for asserting the truth of mathematical propositions, consists of deductive proof.

 The different views on nature of mathematics knowledge are listed below.

     I.            Absolutist’s  Nature of Mathematics Knowledge:

        

                      The absolutist’s nature on mathematical knowledge is that it consists of certain and unchangeable truths. According to this view, mathematical knowledge is made up of absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms, such as ‘All bachelors are unmarried’.

This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths. I shall subsequently argue that each of these assumptions weakens the claim of certainty for mathematical knowledge.

The nature of mathematics explores Logic and mathematics are considered to be the domain of certainty, tautology, analytical thinking and fiat.

                      II.            Logistic Nature of Mathematics Knowledge:

                  

                            Laicism believes that the nature of mathematics is open and logical. It’s close to views on the nature of mathematics. It disagrees with the rigidity of mathematical principal and knowledge. In mathematical practice, axiom are not the starting point, they are not the key of the knowledge process, but unclear concepts and sometimes, are hypothesis or conjectures.

At the hands of Bertrand Russell the claims of logicism received the clearest and most explicit formulation. There are two claims:

   ü All the concepts of mathematics can ultimately be reduced to logical

concepts, provided that these are taken to include the concepts of set theory

or some system of similar power, such as Russell’s Theory of Types.

  ü All mathematical truths can be proved from the axioms and rules of inference

of logic alone.

The purpose of these claims is clear. If all of mathematics can be expressed in purely

logical terms and proved from logical principles alone, then the certainty of mathematical knowledge can be reduced to that of logic. Logic was considered to provide a certain foundation for truth, apart from over-ambitious attempts to extend logic, such as Frege’s Fifth Law.

According to this nature, mathematics is not only a canon developed in an organized ways but also it is a form of discipline emerged from inductive to deductive method through concrete and abstract process.

         III.            Formalist Nature of Mathematics Knowledge:

                     formalism is the view that mathematics is a meaningless formal game played with marks on paper. Formalist programme aimed to translate mathematics into uninterpreted formal systems. By means of a restricted but meaningful meta-mathematics the formal systems were to be shown to be adequate for mathematics,

by deriving formal counterparts of all mathematical truths, and to be safe for  mathematics, through consistency proofs.The formalist thesis comprises two claims.

       ·       Pure mathematics can be expressed as uninterpreted formal systems, in which the truths of mathematics are represented by formal theorems.

       ·       The safety of these formal systems can be demonstrated in terms of their freedom from inconsistency, by means of meta-mathematics.

 For formal proof, based in consistent formal mathematical systems, would have provided a touchstone for mathematical truth. However, it can be seen that both the claims of formalism have been refuted. Not all the truths of mathematics can be represented as theorems in formal systems, and furthermore, the systems themselves cannot be guaranteed safe.

         IV.            Platonist Nature of Mathematics Knowledge:

        The Platonist view portrays mathematics as static body of knowledge, “bound together by filaments of logic and meaning” .Platonism is the view that the objects of mathematics have a realm and objective existence in some ideas real. Platonist maintains that the objects and structures of mathematics have a real existence independent of humanity, and that doing mathematics is the process of discovering their pre-existing relationships.

According to Platonists, mathematical knowledge consists of description of these object and relationships and structures connecting them. Platonism evidently provides a solution to the problem of the objectivity of mathematics. It accounts both for its truths and the existence of its objects, as well as for the apparent autonomy of mathematics, which obeys its own inner law and logic.

Mathematical Platonism assumes mathematical entities that they are abstract and they are independent of all our rational activities. There is separate reality of mathematics. Theorem and axiom are discovered not invented. Platonism is nature as one of the branch of absolutist theories. Above all Platonism emphasis on the extensive and abstract use of mathematics and it is complex and beyond imagination of human beings.

                  V.            Fallibilistic’s Nature of Mathematics Knowledge:

                   The fallibilistic’s nature of mathematics knowledge cannot go beyond the human study, but it doesn’s exists in the study of human beings. Mathematics is a product of social process. Mathematics is made by mean and has all the fallibility and uncertainty. It does not exist outside the human mind, and it take its qualities from the mind of men who created it. The nature of mathematics is dynamic. It’s functions actively on the basis of human activities and their study. It subjects matters are changeable according to time, place and context.

This is the view that mathematical truth is fallible and corrigible, and can never be regarded as beyond revision and correction. The fallibilist thesis thus has two equivalent forms, one positive and one negative. The negative form concerns the rejection of absolutism: mathematical knowledge is not absolute truth, and does not have absolute validity. The positive form is that mathematical knowledge is corrigible and perpetually open to revision.

Mathematics studies as the outcome of social process, so, it’s nature is changeable. This nature is associated with constructivist and post-modernist through in education. Mathematics is related to human activities and his values.

       VI.            Intuitionist Nature of Mathematics Knowledge:

                 Mathematics is conceived as an intellectual activity in which mathematical concepts are seen as mental construction regulated by natural law. These constructions are regulated as abstract objects that do not necessary dependent on proof. It explains logic of what is mathematics? How is it nature?. It also disagrees with the concept that mathematics is unchangeable and far from having any drawback and out of the human approach.

Philosophy of mathematics should be defined as a mental activity and not as asset of theorems. Mathematics is not only the collection of theorems but also it is the activities of human mind. Thus, it can be conclude that mathematics can not go aways from human activities. It is based on the logic of an individual. Mathematics rejects the ideas that any things can be the ultimate truth. It emphasizes on human creations. Nonetheless , it can be both inductive and deductive but changeable.

VII.            Social Constructivist Nature of mathematics:

                    Social constructivist nature of mathematics giving an alternative way to absolutist. The nature of mathematics is not absolute or unchangeable but it has fallible and historically shifting character. In the social constructivism, social phenomena are taken as the main source of mathematical knowledge. Cultural, public and collective knowledge and not as personal,  private or individual beliefs nor as external and absolute.

A novel central features of social constructivist is that it adopts conversation as the basic understanding representational form for its epistemology. Thus his position views mathematics as basically linguistic, textual and semiotic, but embedded in the social world of human interaction.

VIII.            A Prior and A Posterior Nature of mathematics:

                   The term”a prior” and “a posterior” are used primarily to denote the foundation upon which a proposition is known. A given proposition is knowable a prior if it can be known independent of any experience other than the experience of learning the language in which the proposition is expressed, where as a proposition that is knowable a posterior is known on the basis of experience. For example, the proposition that all bachelors are unmarried is a prior, and the proposition that is raining outside now is a posterior.

The prior nature claims there are alternative methods of gaining knowledge by serving the tie to reality; it allows any idea to be accepted. And posterior knowledge includes things like the orbits of planets, the contents of an atom or the placement of an object in your home.    

1.2.    Issues Related To Nature of Mathematics Knowledge:

                           There are two classes of states of consciousness that differ from each other in origin and nature, and in the end toward which they aim. One class merely expresses our organisms and the object to which they are most directly related. Strictly individual, the states of consciousness of this class connect us only with ourselves, and we can no more detach them from us than we can detach ourselves from our bodies. The states of consciousness of the other class, on the contrary, come to us from society; they transfer society into us and connect us with something that surpasses us.

Being collective, they are impersonal; they turn us toward ends that we hold in common with other men; it is through them and them alone that we can communicate with others. In brief, this duality corresponds to the double existence that we lead concurrently: the one purely individual and rooted in our organism, the other social and nothing but an extension of society".

Concepts have their own life, said Durkheim. "When once born they obey laws all their own. They attract each other, repel each other, unite, divide themselves and multiply"

                                                             i.            Subjective vs. objective:

                                               The social constructivist philosophy of mathematics, the relationships between subjective and objective knowledge of mathematics is central, and these realms are mutually dependent, and serve to recreate each other.

Objective mathematics knowledge is reconstructed as objectives knowledge by individual through interaction with teacher and other person. Subjective mathematical knowledge has an impact on objective knowledge. The creation not only at the edge of mathematical knowledge, but also throughout the body of mathematical knowledge.

The social constructivist thesis is that objective knowledge of mathematics exists on and through the social world of human actions, interaction and rules, supported by individual subjective knowledge of mathematics, which needs constant re-creation.

Subjective knowledge re-creates objective knowledge without the latter being reducible to the former.

                                                           ii.            Ethno-mathematics vs. Global mathematics:

                                                  Ethno-mathematics is the study of the relationship between mathematics and culture. It examines a diverse range of ideas including mathematical models, numeric practices, quantifiers, measurements, calculations, and patterns found in culture, as well as education policies and pedagogy regarding mathematics education. “The goal of ethno-mathematics is to contribute both to the understanding of culture and the understanding of mathematics, but mainly to appreciating the connections between the two”.

The global mathematics is worldwide perspective. They are algebra, statics, geometry, complex analysis, arithmetic and topology. “Although many people assume that the U.S. will always be a world leader in science and technology, this may not continue to be the case inasmuch as great minds and ideas exist throughout the world.  Math appears to be the subject in which accomplishment in secondary school is particularly significant for both an individual’s and a country’s economic well-being.

                                                       iii.            Invention vs. Discovery:

                                    What has gone unnoticed in this debate is that there is a parallel and equally fundamental dispute over whether mathematics is discovered or invented. The absolutist view of mathematics sees it as universal, objective and certain, with mathematical truths being discovered through the intuition of the mathematician and then being established by proof.

Many modern writers on mathematics share this view, including Roger Penrose in The Emperor’s New Mind, and John Barrow in Pi in the Sky, as indeed do most mathematicians. The absolutists support a ‘discovery’ view and argue that mathematical ‘objects’ and knowledge are necessary, perfect and eternal, and remark on the ‘unreasonable effectiveness’ of mathematics in providing the conceptual framework for science.

They claim that mathematics must be woven into the very fabric of the world, for since it is a pure endeavour removed from everyday experience how else could it describe so perfectly the patterns found in nature?

                                                        iv.            Mathematics vs. Reality:

                                  The nature of mathematics and of its relationship with reality, I would like to quote you two extracts from interviews I carried out for Libération, 1 one of a mathematician you are familiar with Alain Connes and the other of a Belgian theoretical physicist and epistemologist Dominique Lambert. Here is what Alain Connes said:

             “There are two opposing extreme viewpoints about mathematical activity. The first one, which I am entirely in agreement with, follows the Platonists: it states that there is a raw, primitive mathematical reality which predates its discovery. A world whose exploration requires the creation of tools, in the same way as ships had to be invented to cross oceans. Mathematicians will, therefore, invent, create theories whose purpose is to shed some light on this preexisting reality. The second viewpoint is that of formalists; they deny mathematics any preexistence, considering that it is a formal game, founded on axioms and logical deductions, hence a purely human creation. This viewpoint seems more natural to non-mathematicians, who are reluctant to assume an unknown world which they do not perceive. People understand that mathematics is a language, but not that it is an external reality outside the human mind. Yet, the great discoveries of the 20th century, in particular Godel’s work, have shown that the formalist viewpoint is unsustainable. Whatever be the means of exploration, the formal system used, there will always be mathematical truths beyond it, and mathematical reality cannot be reduced to the logical consequences of a formal system”.

A mathematician seemingly invents with imagination as his only guide and mathematical rules as the only rules. Thirty years later, his invention helps to describe a particle or space-time. Why? Mathematics are efficient. Very much so. To the point of causing turmoil among physicists, the biggest “clients” of mathematics.

REFRENCE:-

           -         Earnest Paul (1993). The philosophy of mathematics education. USA: the falmer press.

            -         Acharya Bed Raj (2013). Studies in mathematics education. Dikshant prakashan.