=Paper=
{{Paper
|id=None
|storemode=property
|title=The Role of Shape in Problem-Solving Activities in Mathematics
|pdfUrl=https://ceur-ws.org/Vol-1007/paper9.pdf
|volume=Vol-1007
|dblpUrl=https://dblp.org/rec/conf/shapes/MoralesS13
}}
==The Role of Shape in Problem-Solving Activities in Mathematics==
https://ceur-ws.org/Vol-1007/paper9.pdf
The Role of Shape in Problem-Solving
Activities in Mathematics
José Gustavo MORALESa,1 and Matías SARACHO b2
a
National University of Cordoba / CONICET
b
National University of Cordoba
Abstract. In our paper we will rely on research by Grosholz (2007) considering
her thesis of the irreducibility of iconic representation in mathematics. Against this
background, our aim will be to discuss the epistemic value of “shape” or iconicity
in diagrammatic representations in geometry. We show that iconic aspects of
diagrams reveal structural relations underlying the method to solve quadrature
problems developed by Leibniz (1675/76). As a concluding remark, we shall argue
that in retrieving the information embedded in a diagram the reader must establish
a meaningful relationship between the information supplied by the diagram and the
relevant background knowledge which often remains implicit.
Keywords. Iconic Representation, Diagrams, Visualization, Leibniz, General
Method, Background Knowledge.
Introduction
In our paper we rely on research by Grosholz (2007) considering, in particular, her
thesis of the irreducibility of iconic representation in mathematics. Against this
background, our aim is to discuss the epistemic value of “shape” or iconicity in the
representations of diagrams in the case of geometry. In order to illustrate our point, we
bring in a case-study selected from Leibniz´s work with diagrams in problem-solving
activities in connection with a “master problem, the Squaring of the Circle – or the
precise determination of the area of the circle”, a problem which remains insoluble by
ruler and compass construction within Euclidean geometry. 3 Our main reason to focus
on Leibniz is as follows. On the one hand, throughout his work as a mathematician,
Leibniz relies on a variety of tools which display rich iconic aspects in the
implementation of problem-solving activities. On the other hand, it is precisely in the
case of geometry where Leibniz makes important contributions. Reasoning with
diagrams plays a central role in this particular case. In order to solve certain
geometrical problems which could not be solved within the framework of Euclidean
geometry, Leibniz devises a method that proceeds by transforming a certain
mathematically intractable curve into a more tractable curve which is amenable to
1
José Gustavo Morales, National University of Cordoba, School of Philosophy, Cordoba (5000),
Argentina; E-mail: gust.914@gmail.com.
2
Matías Saracho, National University of Cordoba, School of Philosophy, Cordoba (5000), Argentina;
E-mail: matias.m00@gmail.com
3
See [3, p. 36].
117
calculation. This method is sometimes called the method of “transmutation” as it is
based upon the transformation of one curvilinear figure into another.
For Leibniz depending upon the context of research some methodological tools are
more fruitful than others, moreover, simplicity and economy is also amongst the
epistemic virtues guiding the design of methods for problem-solving activities. In our
case-study, we show how Leibniz devises a method which allows him to re-conceive a
given curve by “transforming” it into a more tractable curve as part of his strategy to
calculate the area of curves that may contain irrational numbers (the real number π in
the case of the circle). In particular, we aim to show that iconic aspects of diagrams
reveal structural relations underlying the process of “transformation” developed by
Leibniz in Quadrature arithmetique du circle, de la ellipse et de l’ hyperbole
(1675/76).4
1. The Idea of “Shape” As Iconic Representation
Let us start by focusing on the idea of “shape” in the sense of “iconic representation”.
Representations may be iconic, symbolic and indexical depending upon their role in
reasoning with signs in specific contexts of work.5 According to the traditional view
representations are iconic when they resemble the things they represent. In many cases
this characterization appears as doubtful because of its appeal to a vague idea of
similarity which would seem untenable when representations of numbers are involved.
But Grosholz argues that in mathematics iconicity is often an irreducible ingredient, as
she writes,
In many cases, the iconic representation is indispensable. This is often, though
not always, because shape is irreducible; in many important cases, the
canonical representation of a mathematical entity is or involves a basic
geometrical figure. At the same time, representations that are ‘faithful to’ the
things they represent may often be quite symbolic, and the likenesses they
manifest may not be inherently visual or spatial, though the representations
are, and articulate likeness by visual or spatial means [3, p. 262].
In order to determine whether a representation is iconic or symbolic, the context of
research with its fundamental background knowledge needs to be taken into account in
each particular case, in other words, iconicity cannot simply be read off the
representation in isolation of the context of use. We find here a more subtle
understanding of “iconicity” than the traditional view. Let us focus on the idea that
representations “articulate likeness by visual or spatial means”. Grosholz suggests that
even highly abstract symbolic reasoning goes hand in hand with certain forms of
visualizations.
Giardino (2010) offers a useful characterization of the cognitive activity of
“visualizing” in the formal sciences. In visualizing, she explains, we are decoding
articulated information which is embedded in a representation, such articulation is a
4
In this paper, we shall be refereeing to the French translation (Parmentier 2004) of Leibniz original
text De Quadratura Arithmetica (1675/76).
5
The distinction goes back to Charles Peirce´s theory of signs. For a brief discussion of this distinction,
see [3, p. 25].
118
specific kind of spatial organization that lends unicity to a representation turning it
intelligible. In other words, spatial organization is not just a matter of physical display
on the surface (paper or table) but “intelligible spatiality” which may require
substantial background knowledge:
(...) to give a visualization is to give a contribution to the organization of the
available information (...) in visualizing, we are referring also to background
knowledge with the aim of getting to a global and synoptic representation of
the problem [1, p. 37].
According to this perspective, the ability to read off what is referred to in a
representation depends on some background knowledge and expertise of the reader.
Such cognitive act is successful only if the user is able to decode the encrypted
information of a representation while establishing a meaningful relationship between
the representation and the relevant background knowledge which often remains implicit.
The starting point of this process is brought about by representations that are iconic in a
rudimentary way, namely, they have spatial isolation and organize information by
spatial and visual means; and they are indivisible things. In the next section we turn to
our case study taken from Leibniz’s work in geometry which we hope will help to
illustrate some of the above considerations.
2. Our Case Study - Leibniz’s De Quadratura Arithmetica (1675/76)
In Quadrature arithmétique du cercle, de l'ellipse et de l'hyperbole [8] Leibniz
provides a general method whereby “quadrature” problems for curvilinear figures can
be solved. The first seven propositions of this work form a unity and as Leibniz himself
emphasizes, Proposition 7 is the "fruit" of all that has gone before [8, p. 35]. In this
context of work, Leibniz presents the reader a diagram (Fig. 1).
119
Figure 1. Leibniz Quadrature arithmetique du circle, de l’ellipse et de l’huperbole 1675/76, [8, p. 65].
While for the untrained eye this diagram appears as a set of highly entangled
shapes, for Leibniz, the diagram should offer the reader an overall assessment of the
way his proposed method works. In order to show the most salient aspects of Leibniz’s
method we shall try to make more explicit some of the features displayed in Figure 1.
We proceed to put the original diagram “under the microscope” dissecting it into four
diagrams (Figures 2-5). This will allow us to see some of the most relevant steps
involved in the resolution of the problem under consideration. These visualizations
together with the indications as to how to “read” Figures 2-5 may then be seen as
offering a brief outline of Leibniz´s method.
Figure 2 Figure 3
Figure 4 Figure 5
Leibniz aims to show that the area of a curvilinear figure C – which cannot be
calculated - may be determined by constructing a second figure D, whose area can be
calculated. A crucial step in Leibniz reasoning relies upon certain geometrical results
known since Euclid which allow us to assume that the ratio between C and D is known
to us. This step in the reasoning is represented in the diagram by two different shapes
that we have highlighted in Figure 2. On the one hand, we see an enclosed area
120
delimited by segments A1C, A3C and the arc 1C2C3C – which represents the area C,
unknown to us. On the other hand, we see another enclosed area delimited by segments
1B1D, 1B3B, 3B3D and the curve 1D2D3D which represents the area of the second figure
D. Finally, we can also see some specific lines that represent geometrical relations
between both figures according to Euclidean geometry. 6
With a view to determine the area of curvilinear figure C we first need to find the
area of figure D. Leibniz proceeds to decompose D into a finite number of elemental
parts - the rectangles 1N1B2B1S and 2N2B3B2S - which are then added up. We have
highlighted this procedure in Figure 3. As we can also see the sum of rectangles makes
up a new shape or figure which Leibniz calls “espace gradiforme”.7 At this stage of the
reasoning, the construction of such “space” is crucial for Leibniz’s problem-solving
strategy. Instead of an exact calculation of the area of D, Leibniz approximates the area
of D by calculating the area of such “espace gradiforme”, so that the difference
between both figures will be less than any assignable number.
Next, the newly constructed “espace gradiforme” is transposed upon figure C (See
Figures 4 and 5). This procedure can be described in two steps.
The first step consists in decomposing the curvilinear figure C into “triangles”
which we highlighted in Figure 4. Note that the number of triangles will be greater than
any arbitrarily assignable number as it is possible to decompose the figure into
arbitrarily many triangles where the whole set of triangles has the single vertex A. Here
Leibniz takes distance from other techniques used at the time. While Cavalieri, for
instance, often decomposed curvilinear figures into parallelograms, Leibniz proceeds to
resolve the problem by decomposing curvilinear figures into triangles (for an
illustration of this difference see Figure 6). Accordingly, instead of rectangles or
parallelograms, the elemental parts in this case will be triangles, as Leibniz points out
in Scholium 1 of the treatise:
(...) on peut en effet également décomposer en triangles des figures
curvilignes qu’à l’exemple d’autres grands savants Cavalieri ne décomposait
souvent qu’en parallélogrammes, sans utiliser, à ma connaissance, une
résolution générale en triangles [8, p. 39].
6
Leibniz relies upon a generalization of Euclid’s Elements (Proposition 1, Book I) to justify his
reasoning when assuming that the “triangle” A2C3C equals one half of the “rectangle” 2B2N. 2B3B (A2C3C =
1/2 2B2N. 2B3B). See [9].
7
The expression Leibniz used in the original Latin is "spatium gradiforme" [8, p. 69].
121
Figure 6. Leibniz’s method as opposed to Cavalieri’s method.
The second step consists in the construction of the “espace gradiforme” upon C
(See Figure 5). To this end, Leibniz uses the rectangle with sides 2B2N and 2N2S which
can be constructed from a given triangle A2C3C relying on certain well-established
geometrical relations which hold so that the ratio between the areas of figures C and D
is ½. It is precisely in this context where Leibniz relies upon results already established
by Euclid.8
Let us now return to Leibniz’s original diagram corresponding to Proposition 7
(See Figure 1). With Euclid’s results concerning structural relations between two types
of shapes - triangles and rectangles – in mind, we are justified to establish a correlation
between triangles A1C2C, A2C3C,… and corresponding rectangles 1B1N2B1S,
2B2N3B2S,.... For instance, the triangle A2C3C corresponds to the rectangle 2D2B3B2S.
Next, we recall that the area of figure D can be approximated by the sum of the (finite
number of) elemental parts – rectangles – the original figure D was decomposed into.
Finally, the area of the curvilinear figure C can be calculated by applying the ratio
of ½ upon the area of figure D. According to Leibniz, the calculation obtained by this
method is not exact but one may consider it is a precise determination of the area of the
curvilinear figure C. To sum up, it is by recognizing certain geometrical relations
holding between triangles and rectangles that one can see that the precise determination
of the area of the curvilinear shape will depend upon the value of the approximation of
the area of D. The latter, in turn, can be calculated on the basis of the “espace
gradiforme”, the new shape designed by Leibniz which is required to approximate the
value of D.
3. Concluding Remarks
In this section we finally consider some of the requirements which are imposed upon
the reader in order to be able to perform the relevant “cognitive act” of successfully
decoding a visualization that includes “shapes” in the context of problem solving
activities in mathematics. Again, we shall focus on the diagram of our case-study
(Figure 1).
Diagrams are shapes that represent by spatial and visual means. Their intelligibility
partly depends on their integrity and shape, features that make a diagram intrinsically
iconic. But diagrams also often combine, as Grosholz argues, iconic aspects with
8
See footnote 6 (above).
122
symbolic ingredients. If diagrams were just iconic, they would be but a copy – a more
or less faithful picture - of what we intent to refer to. However, diagrams are inherently
general, a drawing of, say, curvilinear shape without being just a drawing of a
particular curve on this particular page of a text. On the one hand, diagrams resemble a
particular shape, on the other hand, they represent a whole set of (instances of) a certain
shape and are in this sense general. To clarify this feature of diagrams we distinguish
following M. Giaquinto between “discrete” and “indiscrete” representations,
(...) diagrams very frequently do represent their objects as having properties
that, though not ruled out by the specification, are not demanded by it. In fact
this is often unavoidable. Verbal descriptions can be discrete, in that they
supply no more information than is needed. But visual representations are
typically indiscrete, because for many properties or kinds F, a visual
representation cannot represent something as being F without representing it
as being F in a particular way [1, p. 28].
“Indiscrete representations” as opposed to “discrete representations” are
representations that represent by spatial and visual means including the combination of
iconic aspects as well as symbolic ingredients. As a consequence of this important
feature of diagrams, it follows that both particular instances and generality go hand in
hand. Returning to our case-study and Leibniz’s diagram, we may offer the following
three observations in this regard:
The diagram that goes with proposition 7 (Figure 1) exhibits a circular shape.
We may consider that Leibniz’s method to calculate the area for this
curvilinear shape works only for this particular curve. But Leibniz intends to
use his method as a general method so as to include any curvilinear shape, as
he writes in the Schollium to proposition XI:
La proposition 7 m’a fourni le moyen de construire une infinité de figures
de longueur infinie égales au double d’un segment ou d’un secteur (…)
d’une courbe donnée quelconque, et ceci d’une infinité de manières
(Leibniz 1676, p. 97).9
In the diagram (Figure 1) the curvilinear shape C is actually divided into only
four points, namely, 1C2C3C4C. However, it is possible to divide the arc C into
as many points as we want.
If the number of points is large enough, the diagram will be less faithful to the
particular instance that it pretends to represent and when the magnitude of
segment A1C is less than any assignable number, we have the limit-case. At
this point, the space 1CA3C2C (called “triligne” by Leibniz) can be assumed as
a space composed by curve 1C2C3C and the straight line A3C (called “secteur”
by Leibniz)10.
9
Leibniz specifies the class of curves which fall under the domain of application of his method in
Proposition 6 of Quadrature arithmetique.
10
See [8, p. 97].
123
Note that in our case-study, the reader has to select only part of the information
furnished by the diagram; he/she has to be able to discern the relevant information
contained in the diagram in the light of the problem under consideration. In particular,
it is necessary to distinguish in the diagram between iconic ingredients and symbolic
ingredients. What exactly is required of the reader to be able to decode the relevant
information encrypted in the diagram? To answer this question we return here to
Giardino’s observation that “to give a visualization is to give a contribution to the
organization of the available information”. First the reader needs to consider the
context in which the diagram is inserted. As already noted, part of the context is made
explicit by remarks written in natural language as it is the case in the written text
accompanying the diagram [8, pp. 65, 67]. In the written text, Leibniz explains how to
construct the diagram he shows together with Proposition 7 (our Figure 1). But such
description is hardly enough, as the reader still needs to rely on substantial information
– background knowledge concerning relevant chapters of the history of geometry – in
order to get “a global and synoptic representation of the problem”. However the
relevant background knowledge cannot be made fully explicit, at least not “all at once”.
The expertise of the community of mathematicians which includes different traditions
of research and, in a broad sense, the history of mathematics, provides different tools
and techniques which need to be acquired by teaching and learning. For instance, in our
case-study Leibniz’s diagram relies heavily on procedures and techniques whose origin
goes back to Euclid and Archimedes but also recalls the work of some of his
contemporaries such as Cavalieri's "theorem of indivisibles" and Pascal's
"characteristic triangle", which is used by Leibniz in order to “transform” triangles into
rectangles. Finally, as we may divide the arc C into "as many points as we want",
sometimes the diagram is meant to be read as an infinitesimal configuration and at this
point the symbolic dimension of the diagram comes into play so that in each case the
trained eye of the reader will be required to be able to recognize the roles of these
different dimensions.
References
[1] Giaquinto, M. (2008) Visualizing in Mathematics, in P. Mancosu, The Philosophy of Mathematical
Practice, Oxford University Press, 2008
[2] Giardino, V. (2010), Intuition and Visualization in Mathematical Problem Solving, Dordrecht, Springer,
Topoi (2010) 29:29–39.
[3] Grosholz E. (2007) Representation and productive ambiguity in mathematics and the sciences. Oxford
New York, 2007.
[4] Grosholz, E. (2010) Reference and Analysis: The Representation of Time in Galileo, Newton, and
Leibniz, Arthur O. Lovejoy Lecture, Journal of the History of Ideas, Volume 72, Number 3 (July 2011),
pp. 333-350.
[5] Ippoliti, E.(2008) Inferenze ampliative. Visualizzazione, analogia e rappresentazioni multiple,
Morrisville, North Carolina (USA), 2008.
[6] Lakoff, G., Núñez (2000), Where Mathematics Comes From. How the Embodied Mind Brings
Mathematics into Being. Basic Books, The Perseus Books Group, 2000.
[7] Leibniz, W. G., (1849 - 1863), Mathematische Schriften, vol. I – VII, ed. Gerhardt Hildesheim: Olms,
1962.
[8] Leibniz, G.W. (1675/1676), Quadrature arithmétique du cercle, de l’ellipse et de l’hyperbole, Paris,
Librairie Philosophique J. VRIN, French Translation by Marc Parmentier, 2004.
[9] Parmentier, M. (2001), Démonstrations et infiniment petits dans la Quadratura arithmetica de Leibniz,
Rev. Hist. Sci. 54/3. 275-289.
124