Math_ed

Nature of Mathematics and the Relevance of Sociocultural Context in the Teaching and Learning of Mathematics Amber Habib Mathematical Sciences Foundation New Delhi 1

Abstract. This article is a response to the thesis that the nature of mathematical knowledge is such that it is essentially divorced from any specific socio-cultural context – hence any social inequities associated with mathematics education must be entirely due to socio-cultural factors and independent of any feature or aspect of mathematics itself. This argument is based on the claim that the essence of mathematics is that it is based on logical deduction from axioms, and this procedure is entirely objective in that all human minds would carry it out in the same way. I argue that this view of mathematics is partial and that, in any case, the question of concern is not so much the nature of mathematical knowledge as the nature of mathematical activity. The dispute is not purely philosophical. The view that the essence of mathematics is logical deduction from axioms, when it enters the domain of mathematics education, has many consequences – and not all are obviously beneficial.

What is Mathematics? It is evident that school mathematics education, as it stands today, is fraught with many difficulties and failures. A question of particular interest is the extent to which these difficulties and failures are correlated with socio-cultural factors and, furthermore, whether the nature of mathematics itself has a role to play in such correlations. One response is to deny outright even the possibility of an affirmative answer to the last question. For example, (Dhankar, 2009) argues that the nature of mathematical knowledge is such that it is divorced from any specific context – hence any social inequities associated with mathematics education must be entirely due to socio-cultural factors and independent of any feature or aspect of mathematics itself. This argument is based on the claim that the essence of mathematics is that it is based on logical deduction from axioms, and this procedure is entirely objective in that all human minds would carry it out in the same way. I argue that this view of mathematics is partial and that, in any case, the question of concern is not so much the nature of mathematical knowledge as the nature of mathematical activity.

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This is an expanded version of some comments made at the July 2009 Workshop on Middle School Mathematics conducted by the Eklavya Foundation and the Centre for Science Education and Communication, University of Delhi. The original comments were a response to (Dhankar, 2009).

Let us begin by considering the question “What is mathematics?” This is also the title of one of the classic texts on mathematics, authored by Richard Courant2 and Herbert Robbins3. (Courant & Robbins, 1941) The preface to “What is Mathematics?” has a lively response to the question, and I shall reproduce many parts of it here. We begin with the well-accepted idea that the origins of mathematics lie in human needs: “All mathematical development has its psychological roots in more or less practical requirements. But once started under the pressure of necessary applications, it inevitably gains momentum in itself and transcends the confines of immediate utility. This trend… appears in ancient history as well as in many contributions to modern mathematics by engineers and physicists.” Let us look at some episodes from the history of mathematics, starting with the amassing of geometric information in ancient Mesopotamia, followed by its organization in Greece into structured knowledge. Courant & Robbins note that in the latter stage the Greeks became aware of many “difficulties inherent in the mathematical concepts of continuity, motion, and infinity, and in the problem of measuring arbitrary quantities by given units.” Out of their attempts to grapple with these difficulties arose, first, Eudoxus’ theory of the geometrical continuum and then Euclid’s crystallization of the deductivepostulational trend in his Elements. Euclid’s Elements are unanimously hailed as a great triumph of the human intellect, yet Courant & Robbins point out that this universal acclaim had its ill effects: “For almost two thousand years, the weight of Greek geometrical tradition retarded the evolution of the number concept and of algebraic manipulation, which later formed the basis of modern science.” On the one hand, axiomatization creates knowledge out of information. On the other, it impedes creativity. The most famous example is Calculus, whose development in the 17th and 18th centuries was possible only by deliberate disregard of the principle of deduction from axioms. “Logically precise reasoning, starting from clear definitions and noncontradictory, “evident” axioms, seemed immaterial to the new pioneers of mathematical science. In a veritable orgy of intuitive guesswork, of cogent reasoning interwoven with nonsensical mysticism, with a blind confidence in

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Richard Courant – a student of David Hilbert – was one of the leading mathematicians of his time. He founded the Courant Institute of Mathematical Sciences in New York to carry forward his vision of mathematics as an organic whole encompassing both its pure and applied manifestations. Of the 5 winners of the Abel Prize – Mathematics’ equivalent of the Nobel Prize – 3 have been from the Courant Institute. 3 Herbert Robbins worked initially in Topology and then in Mathematical Statistics, at Columbia University. The “Robbins problem” in algebra stumped mathematicians for over 60 years till it was solved by a computer in 1996.

the superhuman power of formal procedure, they conquered a mathematical world of immense riches.” Another striking example, not taken up by Courant & Robbins, is that of Indian mathematics. Highly regarded by contemporaries in other parts of the world, mathematics in ancient and medieval India emphasized result over foundation. Its underlying philosophy has been termed “computational positivism” (Narasimha, 2003) and holds that true knowledge is achieved when computation agrees with observation (as opposed to being correctly derived from assumptions). Nilakantha, one of the leaders of the Kerala school of mathematics (15th Century), stated that “logical reasoning is of little substance, and often indecisive.”4 While this approach is most easily comprehended in the context of applications to areas such as astronomy, it extends to pure mathematics as well. Its epitome in more recent times was Srinivasa Ramanujan. Indeed, the notion of rigour, of the criteria which will be used to determine whether an argument is logical, has itself varied. For example, Euclidean geometry long stood as the ideal example of rigour. Around the end of the 19th century, David Hilbert undertook the task of verifying that all of Euclid’s conclusions actually follow from his declared axioms. He found that he had to expand the number of axioms from 5 to 20! Now, wouldn’t it be odd to say that Euclidean geometry was not mathematics till Hilbert published his work in 1899? This discussion should have already established that the “axiomatist” view of mathematics ignores much of value. If it were adopted, large swathes of the most significant mathematical activity would have to be declared non-mathematical. It must be conceded, at the very least, that the question of the nature of mathematics has received quite different answers at different times and places in our history. Yet if the answer is allowed to vary with time and geography, why should it not also vary with culture or social grouping? An a priori rejection of the possibility is certainly misguided. It is now time to address another aspect of the question “What is mathematics?” Namely, is this question to be answered by isolating mathematics from any context and then looking at only its internal structure? Consider Euclidean geometry. In its structure there are certain fundamental objects (such as points and lines) and relations between them (For example, “Two distinct lines cannot meet at more than one point.”). The fundamental objects are described by definitions while their relations are captured by axioms. Any proposition has to be a consequence of these axioms and definitions in order to be accepted as proved in this system. Now look at Euclid’s definition of a point:

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“Logical reasoning” is interpreted here as deduction from axioms, not merely the use of logical steps while reasoning. (Narasimha, 2003)

A point is that which has no part. A “part” is a more complicated notion than a point. As a definition this is not very satisfying. Indeed, one of the remarkable things about the Elements is that the definitions of the fundamental objects are not used in the proofs – only the axioms are. The consequences of this fact went unnoticed for almost two thousand years. What it tells us is that an object is understood by its relations with other things. Or, as Courant & Robbins put it, “For all purposes of scientific observation an object exhausts itself in the totality of possible relations to the perceiving subject or instrument.” “To renounce the goal of comprehending the “thing in itself”, of knowing the “ultimate truth”… may be a psychological hardship for naïve enthusiasts, but in fact it was one of the most fruitful turns in modern thinking.” If we apply this viewpoint to mathematics itself, we see the pointlessness of trying to understand it purely in isolation, bereft of all contexts. It is not surprising that this endeavour leads to a sterile rejection of even attempting to investigate possible connections between mathematics and society. “Creative minds forget dogmatic philosophical beliefs whenever adherence to them would impede constructive achievement… It is not philosophy but active experience in mathematics itself that can answer the question: What is mathematics?”

Mathematics and Logic When Courant & Robbins published their book, the pendulum had swung towards an extreme belief in the virtues of axiomatisation, following success in fitting real numbers and calculus into its framework, so that they felt compelled to issue a warning: “There seems to be a great danger in the prevailing over-emphasis on the deductive-postulational character of mathematics… If the crystallized deductive form is the goal, intuition and construction are at least the driving forces.” “Although the axiomatic form is an ideal, it is a dangerous fallacy to believe that axiomatics constitutes the essence of mathematics. The constructive intuition of the mathematician brings to mathematics a non-deductive and irrational element which makes it comparable to music and art.” One might say that the belief in axiomatism as the essence of mathematics creates a mechanical, sterile and static image of mathematics, devoid of beauty and adventure. Is

mathematics only about logic? If so, how is it different from logic? In the axiomatist view, it becomes at best a part of logic. Yet the truth is that the working mathematician is not finally bound by either logic or philosophy. For example, consider the question of the nature of mathematical objects and results. Do they have an existence of their own outside human thought? Do they belong to the individual mind, or are they a social construct? These are attractive questions, but a mathematician will not long persist in exploring their subtleties. At some point, there will be a shrug of the shoulder and “I don’t care what they are, I just want to know how they act.” In the pursuit of this particular kind of curiosity the mathematician, as already illustrated, will also dispense with pure reason and resort to leaps of imagination. Even in current times, some of the most famous mathematicians are known not only for the proofs they have provided but for the conjectures they have been brave enough to make. One of them is William Thurston, winner of the Fields Medal, whose Geometrization Conjecture reshaped mathematicians’ vision of geometry and topology.5 In (Thurston, 2006) he has presented a view of mathematical activity which amounts to a refutation of the axiomatist belief: 1. A mathematician’s pursuit is the acquisition and communication of human understanding of mathematical knowledge. Proofs are an essential part of this process, but the notion of “proof” depends on social context and is not reducible to formal symbolic proof. For example, a proof may be accepted by the community of mathematicians merely on the basis of experts’ opinions. A fully formal version of the proof will most likely never come into existence. 2. In any case, the formal symbolic proof represents a loss of human understanding of mathematical knowledge, as it is divested of the “mental models” which led to the proof. The correctness of a proof is not by itself enough to make it valuable to the community. 3. The foundations of mathematics are not strong in the formal sense: “On the most fundamental level the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions.” What gives mathematics stability, in spite of the shaky foundations, are human processes: “The reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.”

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For example, it motivated Perelman’s recent proof of the Poincare Conjecture.

One thing that separates mathematics from logic is the special role of examples (or, as Thurston puts it, “mental models”) in the former. A good mathematician is distinguished not only by her mastery of abstract techniques of proof, but by her library of examples that give a base from which imagination can be launched. It is examples that also keep us from straying into mistaken proofs and results. And when we think “example” are we not edging over to “context” and even “socio-cultural context”?

Axiomatism and Education So far we have argued that it is a mistake to believe that formal proofs constitute the essence of mathematics. Our real interest is in mathematics education – What are the consequences of an attitude to education that starts with this (mistaken) belief? Let us start with what we can take to be positive effects of an emphasis on axiomatic deduction. One is that in this framework, Mathematics acquires fixed and clear rules of judgement. Perhaps it is the only subject where a student can be completely sure of his ground, and can even defend it against his teacher’s attacks. Thus, it can lead to a sense of empowerment. Another is that once the rules are sufficiently well-understood they can form the basis of creative explorations. By efficiently structuring knowledge, they can free the student from a clutter of information and allow him to focus on the essential principles. Of course, these are the very reasons that make axiomatic systems attractive to the professional mathematician. The failing is not in the axiomatic systems, as in the belief that they are all there is to mathematics. As we have shown, the axiomatist view of mathematics is partial. It follows that an education based on this view also gives a partial view of the discipline of mathematics. It removes the human element and promotes the celebration of the ability to follow given rules wherever they may take you. It is impossible to refrain from quoting Courant & Robbins yet again: “A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person.” Admirers of mathematics sometimes compare it with poetry (The other natural sciences presumably play the role of prose). G H Hardy (Hardy, 1940) wrote that while poetry involves the patterns formed by words, mathematics is concerned with the patterns

formed by ideas. Apart from similarities in their nature, the two subjects share a similar fate when it comes to school education.

Is Mathematics Oppressive?

Bibliography Courant, R., & Robbins, H. (1941). What is Mathematics? Second Edition, revised by Ian Stewart, Oxford University Press, 1996. Dhankar, R. (2009). Nature of Mathematics: and some other issues. Notes for lecture delivered at Workshop on Middle School Mathematics, Centre for Science Education and Communication, University of Delhi, July 2009. Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. Horgan, J. (1993, October). The Death of Proof. Scientific American , 93-103. Narasimha, R. (2003, August 30). Axiomatism and Computational Positivism. Economic and Political Weekly , 3650-56. Rowlands, S., Graham, T., & Berry, J. (2001). An Objectivist Critique of Relativism in Mathematics Education. Science & Education , 10, 215-241. Thurston, W. P. (2006). On Proof and Progress in Mathematics. In R. Hersh, 18 Unconventional Essays on the Nature of Mathematics (pp. 37-55). New York: Springer.