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Mathematics and Visuality
Sundar Sarukkai
National Institute of Advanced Studies
Indian Institute of Science Campus
Bangalore 560012, India
sarukkai1@yahoo.com
Abstract: It is well-known that disciplines such as philosophy and science depend significantly on the use of visual metaphors. Such a process can be seen at work even in the creation of mathematical discourse. There are many concepts, including abstract definitions in a field like topology, which depend on the way we visually make sense of the world. The visual impact on mathematics can also be seen in the way the mathematical discourse is written. The writing strategies of mathematics, those which influence the way students are taught to write mathematics, are also indebted to specific human visual capabilities. There are two important cognitive processes involved in these actions: one related to the capacity to learn language in general and the other the capacity for artistic cognition, thereby suggesting the need for new methods, drawing on teaching language and art, in the teaching of mathematics.
What exactly is the relationship between the nature of mathematics and strategies of teaching and learning the subject? Obviously, how some content is taught will depend on the nature of the content. But in the case of mathematics, what is the nature of mathematics that needs to be understood as far as pedagogy is concerned? Analogously, consider the nature of science debate in teaching science. On the one hand, it might be felt that teaching science is not just to teach the content of science but also to include elements of the nature of science. One can in principle apply the same argument to mathematics teaching and argue that teaching mathematics is to teach not only the content but also elements of the nature of mathematics. While I do not want to go into the merits or demerits of the nature of science argument and whether it is indeed relevant for science teaching, I believe that in the case of mathematics teaching, the issue of the nature of mathematics is indeed of profound importance. In this brief report, I will not discuss the different views on the nature of mathematical reality such as Platonism. I will instead focus on how the mathematics text is read and written.
Mathematics is a special language in that it can capture relations in more precise ways than natural language can. The uniqueness of mathematical objects, the way it deals with relations and the rules that constitute mathematics are very well exemplified in mathematical writing. In what follows, I will argue that one can learn mathematics effectively by understanding its writing and discursive strategies. Such a mode of learning makes mathematical learning somewhat similar to the activity of learning languages and therefore tools from that discipline can be used in mathematical learning and vice versa. I will also suggest that pictorial cognition plays a very important role in mathematical learning and this is also illustrated by considering some aspects of mathematical writing.
The Writing of Mathematics
Let me begin with a simple question: how does one read mathematics? When students are taught about numbers the first thing that probably strikes them is the different nature of the notations different from what they are used to in their own languages. These symbols like 1, 2, 3 and so on are written as if they are alphabets but do not read like alphabets. This is because the sound we make when we read them is not the same as we do when we read alphabets. For example, when we see a or b, the sound we make when we read them is exactly that of the symbol. That is, we do not look at a and read it as apple. When we look at a word we read it as a word, as a combination of sounds which are in some sense reducible to the sounds of the alphabet. So we might read apple as a set of sounds starting with aa, then pp and so on.
The numeral 2 is written as if it is an alphabet. But what is the sound corresponding to this written numeral? It is not a single sound like that of an alphabet. We read it like we read a word and the word in this case is two. We speak 2 through its word-like association two. We do not speak a through other word-like associations. Thus, although 2 is presented to us as a single mark on the paper, the speaking of that word betrays the fact that it is not really readable like alphabets of verbal language.
2 is not an alphabet in another sense. It stands for an object. That is the first lesson we are taught about numbers. We are told that 2 is a number. It is an object belonging to the class of numbers. And as such, it names these objects. Thus, 2 is a name for some object called numbers. Herein begins the first confusion about numbers. 2 looks like an alphabet but it is not. We are told to imagine that it stands for an object but we dont commonly use alphabets to name objects. Naming, in natural language, is in principle associated with words.
Let me illustrate this difference by considering translation. How does one translate alphabets and symbols? We are aware of the complexity inherent in any act of translation. Translating even a single word is a complex process. When one is translating a word in a given language (source language) we need to find equivalent word/words in the target language. There are different theories of translation which engages with this problem. One specific problem is related to the transmission of meaning of a word in one language into similar meanings of the translated word.
Often, it is believed that numerals do not have to be translated. So, in science and mathematics texts which are translated from one language to another, it is often the practice that the natural language terms are translated whereas the numerals and other symbolic notations are retained. While we take this move for granted, it should also allow us to ask: What kinds of units are not usually translated? Proper names are often not translated. So when a novel in Italian is translated into Kannada the language will be changed but the names of the characters will often remain the same. Proper name, in this context, functions exactly like the numeral. However, it is the reverse of the numeral 2 hides a word within it whereas a proper name hides a symbol within it.
Alphabets in general are not translatable. We also do not translate sounds when we translate a word. We do not search for the alphabetic equivalent of one language when we translate. That is, if I want to translate fruit into Kannada, I do not look for translation of f (that is, the alphabetic equivalent of f in Kannada), then r, then u and so on. We do not translate alphabets because there is nothing to translate. Meaning is necessary for translation and symbols, by definition, are creatures which are in principle meaningless.
What else is not translatable? Pictures. We do not translate paintings. Comic books, when translated, have the pictures intact but have the words translated into another language. Why is there the belief that pictures do not have to be translated? Is it that they are immediately accessible in the way words are not? Is it that they communicate meaning not through convention as in language but by mere perception? If so, then the fact that symbolic writing in mathematics is not translated when the natural language component is translated does this imply that mathematical symbols function also like pictures?
Mathematics is essentially a system of symbols. As symbols, they arise in a semantic vacuum. This is the first problem that arises in teaching mathematics. After teaching language, which is fundamentally based on the idea of meaning, here we are in the situation of teaching something which looks like language but yet does not encode meaning in the way it is written. This is a discursive strategy of mathematics and has relevance to the nature of mathematics. That is, doing mathematics and learning mathematics is as much about its objects and operators as it is about how these are written and communicated.
Therefore, we can now conceive of a way to teach mathematics that focuses attention on the way mathematics is written instead of beginning from the abstract and conceptual world of mathematics. Not that these are independent there is a good reason why mathematics writes its discourse the way it does. One of which is the belief that symbols and alphabetic representation stand for mathematical objects.
The first use of symbolic writing is through the replacement of a symbol for a word. Thus, we often find in a mathematical text expressions like, Let n be a number, Let A be a matrix and so on. We can immediately note that mathematical objects like numbers, sets, matrices, groups are all reduced to symbols. So herein there seems to be a strategy of naming or writing the name for objects as symbols instead of words.
Doing mathematics includes the process of creating new mathematical entities. If all these entities are to be given the status of a symbol, then it is clear that we will run out of symbols very soon. After finishing with English alphabets, mathematicians could perhaps consider another set of alphabets, say, the Greek one. But the problem of natural languages like English and Greek is that all of them have a fixed set of symbols, that is, a fixed number of alphabets. Either mathematics can keep on borrowing alphabets of all the languages in the world or it can come up with a writing strategy to create new alphabets.
In fact, this is the distinctive fact of mathematical writing: the attempt to keep on creating new alphabets. This is the reason why mathematical writing is not mere symbolic, it is an art of creating new symbols. This is what makes mathematical texts look so different.
Unlike natural language like English where the alphabets correspond to some sound, the mathematical alphabets correspond to larger linguistic expressions. We read an alphabet like f(x) as fx, a combination of two alphabetic sounds. Strictly speaking, we should be reading it as f, open bracket, x, close bracket but the brackets are silent. The brackets are silent only in so far as I am reading to myself from seeing the symbol. If I want to communicate to somebody who is not seeing the text that I am reading the symbol f(x) I will have to read as function of x or at least f of x or f open bracket x close bracket.
In the context of teaching mathematics, here is an immediate source of ambiguity for the student. One does not read the writing of mathematics like reading the natural language text. What is written is not what is read. The reading has to be much more than the symbol one sees. The reading of mathematical symbols is more closer to the reading of pictures than of alphabetical writing. But it is not independent of alphabetic writing since to read the symbol properly it is necessary to use an expanded linguistic expression like in the case of reading f(x).
Reading a picture of a matrix is more complex. In this case it is best read only as a picture. Consider a 2 X 2 matrix with elements a, b, c and d. Let us say that a and b are on the first row, and c and d on the second row. This way of ordering also means that a and c are on the first column and b and d are on the second column. The matrix is an ordering of these four elements and these elements are enclosed by a bracket. One cannot read the matrix as it is written. But we immediately understand the nature of the matrix by seeing this pictorial way of writing it.
Consider another simple example. A vector can be symbolised by the alphabet v. But the alphabet has to undergo some modification if it has to symbolize the velocity. The fact that the velocity is a vector is encoded by putting an arrow on top of v or using a bold face in print to indicate this nature. This is what I have referred to as geometrization of words. One way to understand this process is in the following manner. Science begins by picturing a phenomenon for example, the diagram corresponding to the oscillation of a pendulum. Creating pictures like this is an integral part of doing science and the use of mathematics in the sciences begins from such pictorial representations. Now imagine the possibility of picturing words and concepts. The act of creative symbolization is precisely this activity of attempting to picture the word or concept just like we picture the physical world. In such an act, meaning gets encoded as in a picture and not as in a word. Such a move has profound consequences for understanding mathematics.
Mathematical discourse also has a discourse of operators and the rules by which one operates on these symbols. It would not be possible for me to elaborate on the mode of talking about operators in a mathematical text in this brief report but it must be noted that the rules of mathematics are also made visible through the manipulations on these alphabetic symbols.
Visual Encoding in Mathematics
The relationship between mathematics and the visual is illustrated in this process of writing. First of all, mathematics whose objects of discourse are apparently about a Platonic, non-physical world and independent of human senses is surprisingly filled with visual metaphors. The pictorial cognition that is inherent in reading mathematical writing should alert us to the importance of the idea of the visual in mathematics. One might argue that writing is only representational and that mathematical writing is really about mathematical concepts. But this position will be insensitive to the ways by which the character of writing influences how we make meaning of the objects including the meanings that arise through operations. In other words, what happens to the object under an operation can often be discerned in the way the symbol is written.
Moreover, it is the case that mathematical concepts (even if seen as outside their written representation) are also visually encoded. Sets, as collection, are a simple example. Sets are denoted usually by the use of brackets. Brackets as visual marks enclose, collect and bring together elements. The relationship with the visual is also well captured in calculus: the picture of a graph, the drawing of a tangent, the picturing of an angle, the representation of a coordinate system and so on. We learn and make sense of mathematical concepts and operation through these many visual representations. A good example from higher mathematics comes from diagrams that pictorially represent commutativity or non-commutativity. For example, group theory is filled with such pictorial descriptions. To dismiss the visuality in mathematics is to miss the point about the essential role such descriptions play in learning and creating new mathematics.
Topology is another interesting example of a discipline which is filled with visual metaphors. The concepts that define this field such as neighbourhood, continuity, dense sets, open sets, real line and so on are based upon various encoded meaning related to vision. Thus, mathematics is intrinsically based on the use of visual ideas both in the creation of its concepts as well as in its representational strategies.
The fact that visual themes influence mathematics has significant impact on mathematics teaching and learning. First of all, the student is in a situation where she has to learn to read alphabet-like symbols as word-like sounds. Secondly, she has to grasp the universality of these symbols in the sense that the written text of natural language can change but the symbols dont. Students who learn natural language learn the process of writing as stringing together words. The basic idea of calculation is very similar to the act of writing. One can teach calculation in mathematics in terms of how one teaches writing and composition to children. Calculation has a set of objects, there are rules of combination and calculation creates a narrative. But there is something unique in the form of this writing. Writing in natural language puts words together. The act of combining words is an operation in itself. Different ways of combining are possible under different rules. In mathematical calculation seen as writing we need to note that there are no words strung together. The very process of writing in mathematics is to create new alphabets and new strings of alphabets. The process of writing is not just adding known words together but to modify the word itself.
These processes indicate the need to teach mathematics in a different manner: by first presenting it as another form of language with unique strategies of creating its alphabets and writing, by showing how pictorial cognition plays an important role in understanding the nature of the mathematical symbols, by emphasising its relation to natural language (therefore using techniques used to teach natural language) and finally to describe the processes of calculation as one based on rule-following, where rule-following is not about logical principles alone but also one where the mathematical is made visible in the written characters on the page. Thus, the cognitive elements necessary for mathematics teaching are not only those associated with natural language teaching but also involve elements of artistic cognition.
References
Gentzler, E. 1993. Contemporary Translation Theories. New York: Routledge.
Sarukkai, S. 2001. Mathematics, Language and Translation. META, 46 (4), pp. 664-674.
Sarukkai, S. 2002. Translating the World: Science and Language. Lanham: University Press of America.
Sarukkai, S. 2008. Philosophy of Science and its Implications for Science and Mathematics Education, in The epiSTEME Reviews, Eds. B. Choksi et al, Macmillan India, pp. 35 61.
See Sarukkai (2001) for a discussion on the relation between mathematics and translation. Sarukkai (2002) has a more comprehensive discussion of the relation between science and language, and argues that translation is an essential mode of understanding scientific/mathematical texts.
For example, see Gentzler (1993).
Although the use of symbols is necessary for mathematics in particular because of the disengagement with the question of meaning in such symbols the symbols begin to get meaning associated with them over use.
See Sarukkai (2002).
See Sarukkai (2002) for more details.
It is well-known that disciplines such as science and philosophy depend significantly on visual metaphors. The claim in this report is that mathematics is also similarly indebted to the ideas of human vision.
See Sarukkai (2008) for a more detailed analysis.
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