=Paper=
{{Paper
|id=None
|storemode=property
|title=Computing with Numbers, Cognition and the Epistemic Value of Iconicity
|pdfUrl=https://ceur-ws.org/Vol-954/paper9.pdf
|volume=Vol-954
}}
==Computing with Numbers, Cognition and the Epistemic Value of Iconicity==
https://ceur-ws.org/Vol-954/paper9.pdf
Computing with Numbers, Cognition and the
Epistemic Value of Iconicity
José Gustavo Morales
National University of Córdoba & National Research Council (CONICET)
Abstract. In my paper I will rely on research by Grosholz (2007) con-
sidering, in particular, her thesis of the irreducibility of iconic represen-
tation in mathematics. Against this background, my aim is to discuss
the epistemic value of iconicity in the case of different representations
of numbers in elementary arithmetic. In order to make my point, I will
bring in a case study selected from Leibniz’s work with notations and the
lessons Leibniz draws in the context of his number-theoretic considera-
tions.
Keywords: Iconic Representation, Notations, Visual Reasoning, Num-
bers, Epistemic Value
1 Introduction
According to the standard view in twentieth century philosophy of logic and mathe-
matics, reasoning in the formal sciences is best characterized as purely syntactic so that
1) the body of mathematical knowledge can be seen as built up systematically and 2) a
purely syntactic presentation guarantees formal rigor and transparency by making ex-
plicit all relevant epistemic contents in proving mathematical results. Full-explicitness
goes hand in hand with the elimination of any reference to the context of work.1 From
such assumptions follows that figures and, more generally, iconic ingredients are to be
eliminated from a fully systematic presentation.
This view of the role of representation in the formal sciences has been called into
question by a number of recent investigations. According to a critical line of research
developed by Grosholz (2007), the research mathematician engages in formal reason-
ing that is broadly conceived as sets of problem-solving activities which make use of
an irreducible variety of modes of representation or working tools such as notations,
tables, figures, etc. but also discursive reasoning in natural language. Such modes of
representation depend on the specific context of work as well as acquired cognitive
abilities of the agent which include background knowledge that remains largely im-
plicit. As the critics point out “syntactic reconstructions” of formal reasoning may be
complemented by formal semantics but the price to pay is the elimination of discursive
language and forms of know-how relevant to the successful use of notations and other
working tools, as well as rich dialogical aspects of mathematical practice leading to in-
novation. In particular, the syntactic view requires that figures be eliminated in favor
of a one dimensional formal reconstruction which is purely symbolic.
1
For a discussion of the standard view see [6], [5], and [4].
R.K. Rendsvig and S. Katrenko (Eds.): ESSLLI 2012 Student Session Proceedings, CEUR Work-
shop Proceedings vol.: see http://ceur-ws.org/, 2012, pp. 84–92.
The Epistemic Value of Iconicity 85
2 Aim of My Paper
In my paper I will rely on research by Grosholz (2007) considering, in particular,
her thesis of the irreducibility of iconic representation in mathematics. Against this
background, my aim is to discuss the epistemic value of iconicity in the case of different
representations of numbers in elementary arithmetic. In order to make my point, I will
bring in a case study selected from Leibniz’s work with notations and the lessons
Leibniz draws in the context of his number-theoretic considerations. The reason for
my selection of this particular case study is twofold. Firstly, throughout his work as a
mathematician, Leibniz emphasizes the importance of visual reasoning while relying on
a variety of tools which display rich iconic aspects in the implementation of problem-
solving activities. Secondly, Leibniz discusses the peculiar iconicity of representations
of numbers; in particular, he illuminates the issue of the epistemic value of different
numerical systems by discussing the cognitive benefits of the binary system vis-à-
vis the system of Arabic numerals. For Leibniz some notations are more fruitful than
others, moreover, simplicity and economy is amongst the epistemic virtues of notational
systems. In the case under consideration – my case study – Leibniz argues for the view
that the iconic aspects present in binary notation reveal structural relations of natural
numbers that are concealed in other numerical modes of representation such as the
system of Arabic numerals.
3 The Idea of Iconic Representation
Let me start by focusing on the idea of iconic representation. Representations may be
iconic, symbolic and indexical depending upon their role in reasoning with signs in spe-
cific contexts of work.2 According to the received view representations are iconic when
they resemble the things they represent. In the case of arithmetic 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 canon-
ical 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 artic-
ulate likeness by visual or spatial means [6, p. 262].
In order to determine whether a representation is iconic or symbolic, the discursive
context of research 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 me focus on the idea that representations “articulate likeness by visual or
spatial means” in the case of arithmetic. Grosholz suggests that even highly abstract
symbolic reasoning goes hand in hand with certain forms of visualizations. To many
this sounds polemical at best. Granted to the critic that “shape is irreducible” in
2
This tripartite distinction goes back to Charles Peirce’s theory of signs. For a
discussion of the distinction, see [6, p. 25].
86 J. G. Morales
geometry as it is the case with geometrical figures. But what is the role of iconicity in
the representation of numbers, and more generally, what is involved in visualizing in
arithmetic?
Giardino (2010) offers a useful characterization of the cognitive activity of “visual-
izing” in the formal sciences. In visualizing, she explains, we are decoding articulated
information which is embedded in a representation, such articulation is a 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 [3, p. 37].
According to this perspective, the ability to read off what is referred to in a rep-
resentation 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 representa-
tion and the relevant background knowledge which often remains implicit. The starting
poing of this process is brought about by representations that are iconic in a rudimen-
tary way, namely, they have spatial isolation and organize information by spatial and
visual means; and they are indivisible things. Borrowing Goodman’s terms we may say
that representations have ‘graphic suggestiveness’.
4 The Role of Iconicity in the Representation of
Numbers
Against this background, I will bring in my case study in order to consider the role
that iconicity plays in the representation of natural numbers both by Arabic numerals
(0, 1, . . . , 9) and by binary numerals (0, 1). In a number of fragments, Leibniz discusses
both notational systems. He compares them with regard to usefulness for computation
and heuristic value leading to discovery of novelties. I begin by asking about the inter-
est in choosing a particular representation of numbers in the context of arithmetical
problem-solving activities. Once more, I will rely here on the Leibnizian view as dis-
cussed by Grosholz. According to this view the use of different modes of representation
in the formal sciences allows us to explore “productive and explanatory conditions of
intelligibility for the things we are thinking about” [7, p. 333].3 The objects of study
of mathematics are abstract (“intelligible”), but they are not transparent but prob-
lematic, and they are inexhaustible requiring mathematical analysis which will clarify
and further develop conceptual structures aided by the appropriate working tools.4 In
the case of number-theoretic research different modes of representation may reveal dif-
ferent aspects of things leading to discovery of new properties and the design of more
elaborated tools.
3
The roots of this perpective are, as Grosholz recognizes, in Leinbiz’s theory of
expression developed around 1676-1684.
4
See [6, p. 47 and p. 130].
The Epistemic Value of Iconicity 87
5 Representation of Numbers
From this perspective, let us now consider the case of the representation of natural
numbers. A natural number is, Grosholz writes, “either the unit or a multiplicity of
units in one number” [6, p. 262]. Following this idea a very iconic representation of six
could look like this:
//////
On the one hand, this “picture” contains – or shows – the multiplicity of units
contained in the number six. On the other hand, the unity of the number six is expressed
by some strategy of differentiation from the rest of objects surrounding it on the page
(or surface). In this particular case, the six strokes are spatially organized to achieve
this aim. But this works only with small numbers and if we want to represent, for
instance, the number twenty-four, the iconicity of the strokes collapses partly because
the reader cannot easily visualize the number that was meant to be thereby represented.
In opposition to this rudimentary representation of numbers by means of strokes, take
the Arabic numeral “6” which does not reveal the multiplicity of units contained in
the natural number six but exhibits instead the unity of the number itself. Arabic
numerals exhibit each number as a unity and just for this reason they are iconic too.
If what is at stake are big numbers, representation by strokes becomes useless and the
weaker iconicity of Arabic numerals appears as more productive in problem solving
activities such as basic computing with numbers. In other words, iconicity is a matter
of degree depending on each context of work as well as background knowledge relevant
to the problem-solving activity. When it comes to arithmetical operations the “graphic
suggestiveness” of strokes may not be the last word while the “maneuverability” of
Arabic numerals seems more decisive as it allows us to operate with precision and
easiness. In Arabic notation each individual mark is “dense” in the sense that in each
character there is a lot of information codified in highly compressed way.
6 Leibniz and His Preference for the Binary Sys-
tem
In this section, I will look at Leibniz’s discussion of the binary notation broadly exposed
in “Explication de l’arithmetique binaire” (1703). I consider this case study of great
interest because it brings to light specific aspects which are central to the topic of my
paper, in particular, the epistemic value of iconicity in the representation of number
systems in specific problem-solving contexts of work. Leibniz emphasizes what he sees
as the cognitive virtues of his favorite notational system in arithmetic, the binary
notation. This notation displays some properties of numbers by means of only two
characters, namely, 0 and 1 and the following four rules: 1+1 = 10, 1+0 = 1 (addition)
and 1 · 1 = 1, 1 · 0 = 0 (multiplication). In binary notation when we reach two, we must
start again; thus, in this notation two is expressed by 10, four by 100, eight by 1000
and so on.
Time and again Leibniz points to the values of this binary notation, economy and
simplicity of the system. All of arithmetic can be expressed by only two characters and
a few rules for the manipulation of them. Having presented the law for the construction
of the system, Leibniz explains the benefits of it comparing it with the more familiar
decimal system:
88 J. G. Morales
Mais au lieu de la progression de dix en dix, j’ai employé depuis plusieurs
années la progression la plus simple de toutes, qui va de deux en deux; ayant
trouvé qu’elle sert à la perfection de la science des Nombres [10, Vol. VII, p.
223].
For Leibniz, the economy and simplicity of his binary system seems to run contrary
to the decimal system of Arabic numeration. Simplicity and easiness go hand in hand,
as in every operation with binary notation the elements of the system are made fully
explicit, while in Arabic notation we must always appeal to memory:
Et toutes ces opérations sont si aisées (. . . ) [o]n n’a point besoin non plus
de rien apprendre par coeur ici, comme il faut faire dans le calcul ordinaire,
ou il faut scavoir, par exemple, que 6 et 7 pris ensemble font 13; et que 5
multiplié par 3 donne 15 (. . . ) Mais ici tout cela se trouve et se prouve de
source... [10, Vol. VII, p. 225].
Consider the case of three multiplied by three. In order to solve this case by means
of the decimal system, Leibniz argues, we need to appeal to memory; we must recall
the multiplication table for 3 which gives us the correct outcome, and the same goes
for any numeral of the Arabic system from 0 to 9. In contrast, the same operation
made within the binary system always makes explicit all applications of the rules for
any operation we perform. In this case we do not need to rely upon memory but only
on the combination of characters fully deployed on the page. Thus, in decimal we know
by memory that 3 · 3 = 9 while in binary notation we “see” it (Fig. 1), where “11”
expresses the natural number three and “1001” stands for the number nine.
Fig. 1. Leibniz, Mathematische Schriften, VII, ed. Gerhardt, p. 225.
Moreover, Leibniz insists, the binary system reveals structural relations among
characters, and in the same text “Explication de l’arithmetique binaire”, Leibniz goes
on to note that in virtue of the economy and simplicity of the binary system we are
The Epistemic Value of Iconicity 89
able to easily visualize structural relations between numbers thus uncovering novelties.
The example he presents here to illustrate his point is that of geometric progression of
ratio two:
On voit ici d’un coup d’oeil la raison d’une propriété célébre de la progression
Géométrique double en Nombres entiers. . . [10, Vol. VII, p. 224, italics included
in quoted edition].
Let us take the following geometric progression “deux-en-deux” of natural numbers
7, 14, 28. Next, we express those numbers as sums of powers of two. Accordingly, within
the decimal system of Arabic numerals we then have the following progressions:
a)
4+2+1=7
b)
8 + 4 + 2 = 14
c)
16 + 8 + 4 = 28
Finally, we proceed to decompose a), b) and c) into powers of two:
a’)
20 + 2 1 + 2 2 = 2 0 · 7
b’)
21 + 2 2 + 2 3 = 2 1 · 7
c’)
22 + 2 3 + 2 4 = 2 2 · 7
In the first case, the three lines do not provide any information about the pattern
that lead to a), b), c). Instead, those lines require familiarity with all elements of the
system as well as familiarity with the operation in question (addition).
Similarly in the second case, the three lines do not provide any information about
the pattern underlying the progression. In each of the three lines, the right side of a’),
b’), c’) does not indicate the outcome. We must find out how to express the outcome
as power of two. In all three cases we find that the outcome cannot be expressed as
power of two and that it is therefore necessary to introduce a new element: the factor
7. Of course, we must know the multiplication table for seven as well.
Let us now go back to the binary notation and consider how the spatial distribution
of a), b), c) is expressed by binaries (see Fig. 2).
In opposition to the decimal system of Arabic numerals, within the binary system
it is unnecessary to analyze the case in two separate steps. This is so because the
characters and the order they exhibit on the page make visible the pattern underlying
the progression. We only need to know the rules for addition and the system characters
(“0” and “1”).
Finally, Leibniz points to another feature of binaries in relation with the construc-
tion of the system. It is the simplicity and economy of the binary that according to
him brings forth a remarkable periodicity and order. In making this point the author
again emphasizes visual aspects and spatial configuration of characters:
(. . . ) les nombres étant réduits aux plus simples principes, comme 0 & 1, il
paroı̂t partout un ordre merveilleux. Par exemple, dans la Table même des
Nombres, on voit en chaque colonne régner des périodes qui recommencent
toujours [10, Vol. VII, p. 226, italics included in quoted edition].
90 J. G. Morales
Fig. 2. Geometric progression of ratio two as expressed by binaries.
Leibniz groups together numbers that fall under 21 , 22 , 23 , etc. I include below a
segment of a larger table used by Leibniz to show three groups of numbers (surrounded
by vertical and horizontals lines), namely, 0, 1; 2, 3 and 4 – 7. Each group is a cycle
which is iterated in the next cycle, and so on ad infinitum as we can easily see.5
7 Conclusion
For Leibniz mathematical research starts with the search for suitable signs (“char-
acters”) and the design of good notations (or “characteristics”) by means of which
structural relations of intelligible objects of study could be explored; a good “char-
acteristic” should allow us to uncover different aspects of things by means of a sort
of reasoning with signs. When Leibniz remarks this in a brief text of 1683-1684 his
example of a “more perfect characteristic” is the binary notation vis-à-vis the decimal
system.6 In order to understand Leibniz preference for the binary system, we recall
here, it is useful to focus on the importance of visual reasoning in problem-solving
contexts of work. According to Leibniz, problem-solving activities and the discovery
of new properties is the goal of mathematical analysis in the case of number theory,
one of the areas of research he was most interested in pursuing. Such goal strongly
motivates the design of notational systems with a view to obtain fruitful results. As
already pointed out, for Leibniz the binary system is characterized by its simplicity
and economy so that in each operation every element (“0”, “1”) is displayed for the
eye to see without any need to rely on memory as in the case of operating with Arabic
numerals. Leibniz observed that in certain contexts of work the spatial distribution of
such binary elements reveals patterns which are relevant to the resolution of the prob-
5
[10, Vol. VII, p. 224].
6
Exempli gratia perfectior est caracterı́stica numerorum bimalis quam decimales
vel alia quaecunque, quia in bimali – ex characteribus – omnia demonstrari possunt
quae de numeris asseruntur, in decimali vero non item. [2, p. 284].
The Epistemic Value of Iconicity 91
Fig. 3. Leibniz, Mathematische Schriften, VII, ed. Gerhardt, p. 225.
lem under consideration, or to the discovery of new properties that would otherwise
remain hidden. Such is the case of the geometrical progression “deux-en-deux”, which
as we have just noted, can be easily seen only when expressed by means of the binary
system. No doubt that for practical considerations of everyday life the decimal system
of Arabic numerals may be easier to calculate with, nonetheless Leibniz was fascinated
by the binary as facilitating algorithmic structures which like the calculus engaged the
issue of infinite series.
To conclude, one of the things we can learn from my case study is that in com-
puting with numbers – a way of “reasoning with signs” – we always require systems
of signs or characters but some of them are more fruitful than others, some are easier
to calculate with but beyond the specific epistemic virtues they may have, all of them
include important iconic features that are most relevant to cognition. This conclusion,
in particular, calls into question the old idea that working with algorithmic structures
– computing with numbers – is a purely mechanical affaire which excludes iconicity.
Acknowledgments.
I should like to thank Norma Goethe for extended comments and suggestions. Many
thanks go also to two anonymous referees for helpful observations.
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