Visualization and Understanding in Mathematics
Department of Mathematics and Computer Science,
University of Southern Denmark, Odense,
We address understanding in mathematics in relation to visualization. We will use a description of understanding suggested by Ajdukiewicz (as is also explored in Sierpinska,1994), where understanding is described in the following way: “a person P understands an expression if on hearing it he directs his thoughts to an object other than the word in question” (Ajdukiewicz, quote from Sierpinska 1994, pp. 28-29). We illustrate this description by examples where we take what is understood to be some expression (e.g., a definition or a proof) and the object that the thoughts are directed to consists of (possibly mental) diagrams or pictures. The major example comes from a part of analysis, more precisely from free probability theory, that was introduced by Voiculescu in the 1980’s. The example is intended to illustrate that mathematicians also think in terms of pictures and diagrams, but that these often are removed in the final representation of the results. If we grant that the description of understanding given above is accurate, then I suggest that understanding for research mathematicians is not necessarily different from understanding for students. Finally, I will relate the description of understanding to previous work on the nature of mathematical objects.
The topics of visualization and explanation in mathematics are currently discussed in the philosophy of mathematics. Although the topic of understanding can be seen as connected both to visualization and explanation, not much has been written about it. This paper considers the topic of understanding in relation to visualization.
I will explore a description of understanding formulated by Ajdukiewicz. This description is also explored in Sierpinska, (1994). Understanding is described in the following way: “a person P understands an expression if on hearing it he directs his thoughts to an object other than the word in question” (Ajdukiewicz, quote from Sierpinska 1994, pp. 28-29).
Although there may be many ways to understand this statement, we shall here only be interested in when the object that the thoughts are directed to is a (possibly mental) diagram or picture. I will present a number of examples illustrating how I understand this statement. The major example is taken from actual research mathematics and is intended to show that mathematicians also think in terms of pictures. Because of readability, this example is omitted in the present paper. A full version can be found in (Carter 2008). One consequence is that understanding for a mathematician can be described in the same way as understanding for a student.
In the last part of the paper, I will connect the description of understanding to a philosophical description of mathematical objects presented in (Carter, 2004).
Visualization, explanation and understanding
Visualization, in terms of using pictures and diagrams in proofs, has played an immense role in mathematics. When talking about proofs relying on diagrams one can not but mention Euclid’s Elements. However, during the 19th century, the use of pictures in proofs became discredited. Mathematicians of the time worked on the increase of rigour in mathematical reasoning which meant abandoning the use of intuition and geometric reasoning. One example of this thought at play is Cauchy’s Cours d’Analyse from1821. Mancosu (2005) explains how the use of diagrams has again won acceptance because of developments in, for example mathematics, mathematics education and computer science.
With respect to understanding proofs, pictures seem to be very useful. There are even examples of proofs using only pictures. One example is the proof of the equality 1+2+…n = n(n+1)/2 by drawing a picture consisting of dots representing the numbers 1, 2, 3,… . The sum can be obtained from the area of this figure:
Explanation is, of course, connected to understanding, as explanations are given in order to provide understanding. When discussing explanation in mathematics, it is often referred to a classic paper by Steiner (1978). Steiner also discusses proofs and draws a distinction between proofs that demonstrate and proofs that explain. Steiner’s aim is to describe what is meant by an explanatory proof. One of the descriptions that Steiner discusses is that a proof is explanatory if it can be visualized. But he finds that this criterion is too subjective. Instead he settles for the following description: A proof is explanatory if it makes “reference to a characterizing property of an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on this property” (p. 143). Note that it may also be possible to visualize in a picture or diagram how a result depends on properties of certain objects. Consider the diagram of the dots representing the sum 1+2+…n = n(n+1)/2 above. This diagram shows that the numbers can be arranged so that they represent part of a square, and this gives the desired result.
There are also other descriptions of explanation in mathematics. Sometimes when mathematicians talk about explanations, they refer to facts that are more general than what is to be explained. As an example of this could be the fact that complex analysis explains the behavior of certain things in real analysis. I doubt, however, that this description is very different from Steiner’s characterization. By this characterization, a proof of a fact, for example in real analysis, would have to refer to certain properties of the objects embedded in complex analysis.
Turning to the question, whether there is a difference between understanding for mathematicians and students, referring to more general or underlying facts could hardly make things easier to understand for students. Hersh (1993) claims that proofs play different roles for mathematicians and for students. He writes that the role of proofs for a mathematician is convincing, whereas for students it is understanding. This claim seems to be questioned by other mathematicians: “There is no doubt about it, many results are, in the first place, proved by sheer brute force. … Subsequently other people, impressed with the result, look at it, try to understand it and finally dress it up in a manner which makes it look appealing, makes it look elegant. …If you want other people to understand the essential ingredient of an argument, it ought in principle to be simple and elegant” (Atiyah (1974), p. 234).
I wish to insist that understanding can be described in the same way for students and for mathematicians. We now investigate the description of understanding referred to above. We will exclusively consider this description in the case where what is understood is some expression (e.g., a definition or a proof) and the object that the thoughts are directed to consists of (possibly mental) diagrams or pictures. In the next section, we shall give a number of examples illustrating this description.
Note that several authors have claimed that there are different levels of understanding. Connes and Changeux (1995) distinguish 3 levels. The first level consists of mechanical operations, operations that follows a certain procedure and could also be performed by a computer. The second level of understanding involves being able to adapt and criticize a method to a particular problem. This kind of activity results in an understanding of both the aim and the mechanism of the procedure (Connes and Changeux, p. 87). Finally the third level of understanding refers to the creation of new methods or results. I do not find it appropriate to characterize the first level as understanding. For a person to understand something, it requires that the person understands the underlying mechanisms.
Understanding using pictures
A simple example illustrating the act of understanding an expression by directing ones thoughts towards a picture consists of an understanding of the fact that 7 x 8 = 56. A picture that could illustrate this is a rectangle, where 7 and 8 corresponds to the (length of the) sides and 56 is the area. It could also be pictured as 7 rows of 8 dots. Such a picture could also help understand (visualize) other properties of the product, for example, that 8 x 7 = 7 x 8 . Note that the pictures may provide more information than what is contained in the expression. In the case of the product, we can visualize that multiplication is commutative.
Understanding and the nature of mathematical objects
Finally, I wish to show how the description of understanding presented here connects to my previous work on characterizing mathematical objects.
In `Ontology and Mathematical Practice’ (2004), I present a position on the nature of mathematical objects based on a case study in modern mathematics. According to this position mathematical objects are introduced or constructed by mathematicians. Claiming that mathematical objects are human constructions raises the question whether there is any difference between mathematical objects and other objects that are invented by human beings, such as fictional characters. I claim that there is a difference. One difference lies in the fact that for mathematical objects it is possible to distinguish between two acts of presenting a given object. Take as an example, Mount Blanc. It is possible to describe this mountain by the expression `the highest mountain in Europe’. It is also possible to travel to this mountain and point to it. Thus there is a difference between describing the object and exhibiting the object. This is not possible for fictional characters.
For some mathematical objects, it is possible to exhibit the object by a picture. Consider, for example, a circle. It can be defined as a curve fulfilling the equation x2 +y2 = 1, but it can also be drawn. Obviously, it is not always the case that a mathematical object can be visualized in such a way. However, I argue in the paper that it is still possible to distinguish between two different acts of presenting an object.
In the case study, we have discussed above, there are obvious exhibitions of the permutations and equivalence classes. Thus, one may state that in the case where there is a visual exhibition of a mathematical object, then one can understand the definition of this object by directing ones thoughts to the exhibition. In this way, it is possible to link the description of what it is to be a mathematical object to what it takes to understand a definition of the object.
Carter, J. (2004): Ontology and Mathematical Practice. Philosophia Mathematica 12, 244-267.
Carter, J. (2008): Visualization and understanding in mathematics. In: Sriraman, Michelsen, Beckman and Freiman: Proceedings of the 2nd International Symposium on Mathematics and its Connections to the Arts and Sciences, Odense. Centre for Science and Mathematics Education, University of Southern Denmark. Vol 4.
Changeux, J.-P. and Connes, A. (1995): Conversations on Mind, Matter, and Mathematics, Princeton: Princeton University Press.
Hersh, R. (1993): Proving is convincing and explaining. Educational Studies in Mathematics 24, 389-399.
Mancosu, P. (2005): Mathematical reasoning and visualization (Visualization in logic and mathematics. In: Mancosu, Jørgensen and Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 13-30). Dordrecht: Springer.
Panza, M (2003): Mathematical proofs. Synthese 134, 119-158.
Sierpinska, A. (1994): Understanding in Mathematics. London: Falmer Press.
Steiner, M. (1978): Mathematical explanation. Philosophical Studies 34, 135-151.
Tappenden, J. (2005): Proof Style and understanding in mathematics I: Visualization, unification and axiom of choice. In: Mancosu, Jørgensen and Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 147-207). Dordrecht: Springer.
 A few authors have addressed these topics, for example, J. Tappenden (2005) and J. Avigad (Understanding proofs. To appear in The Philosophy of Mathematical Practice forthcoming for Oxford University Press). Avigad explains understanding in terms of certain abilities, i.e. describing what you should be able to do in order to claim that you have understood a proof. Tappenden’s approach is more in line with this paper, as he makes a case that visualization plays a role in explaining fruitfulness of certain frameworks.
 Interestingely, Connes also places much of what goes on in high-school mathematics, “tracing graphs of curves and doing kinematic calculations” in this level.
 This distinction is taken from Panza (2003).