__Jessica
Carter__

Department
of Mathematics and Computer Science,

Denmark.

*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.[1] 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,
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

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.[2]
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.

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.

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.[3] Take as an example,

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 x^{2} +y^{2} = 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.

**References:**

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 2 ^{nd}
International Symposium on Mathematics and its Connections to the Arts and
Sciences, *.
Centre for Science and Mathematics
Education,

Changeux,
J.-P. and Connes, A. (1995): *Conversations
on Mind, Matter, and Mathematics*, Princeton:

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).

Panza,
M (2003): Mathematical proofs. *Synthese
134*, 119-158.

Sierpinska,
A. (1994): *Understanding in Mathematics*.

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).

[1] 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.

[2] Interestingely, Connes also
places much of what goes on in high-school mathematics, ``tracing graphs of
curves and doing kinematic calculations'' in this level.

[3] This distinction is taken from Panza (2003).