APEC-TSUKUBA International Conference III
Tokyo and Kanazawa, December 9-14, 2007.

Why focusing in representation.
From a semiotic to a purely mathematical approach.
Arturo Mena-Lorca

Abstract. Focusing on representation might be considered something thathas to be done to favor learning, but that, at the end, might be external tomathematical activity itself. In this paper I call to mind that, on thecontrary, representation is not only something that mathematicians also do,but that is in fact at the heart of the mathematical activity. For that, I startfrom what we regard as some compelling although broad suggestions ofhistorical nature, then briefly look at some psychological aspects ofrepresentations as stated in Raymond Duval’s semiotic representation
theory, and finally present how intimately related are the pedagogicalissues of the matter to the abstract contemporary mathematical activity.

Historical approachingThinking about the importance of representation to Mathematics, perhaps thevery first example that comes to mind is that of Decimal system.
Of course, there is many a reason for that, and it is an interesting naïve exercise
to try to imagine what the development of Arithmetic and Mathematics at large would
have become without Decimal system: It might be argued that one can make morecalculations than what is usually expected, say, with Roman numerals; but it would beeasier to assert that those computations would be harder to perform to everybody andhence, from a pedagogical perspective, less convenient. Anyhow, mankind has chosen
Decimal system.
An interesting additional example of the importance of representation comes out
by considering the struggling of mathematicians in the 16th century to turn the cosa
(their rendering for Latin res, coming in turn from the al-jabr’s name ˇsay’ for the
unknown magnitude) in something more manageable: Regiomanto’s census and demptis(1464), abbreviated by Luca Pacioli (1494) and also Stevin (1585) and Klau (1593);
abstracted by Viète (1593) and finally turned in Descartes’ “x” (1637). There is morethan a point here: one, of course, is the accessibility of calculations, another one is theconnection of the process barely suggested above and the mathematical constructions
that are behind it (Complex numbers, e.g.).
In the same vein one might want to consider the comparatively burdensome
verbose Aristote’s and Megaric-Stoic Logic, take in account the switch made by
Leibniz’s target “Calculemus” (1666) – as opposite to merely arguing – and then turnto the more accessible notations built in the second half of the 19th century, and up tocommon usage Russell-Whitehead’s version (1910). Again, the point here would not- Arturo Mena-Lorca 1 -only be accessibility of the matter and proficiency in the calculations but also theoreticaldevelopment.
More directly related to the first case, perhaps, and more interesting for our topic
is the following: up to the beginning of the 20th century, mathematicians would write(and think) a function from A to B as f (A) ⊂ B. Now, Saunders Mac Lane claimed thatthe first person who wrote f (A) →B, instead, was probably Witold Hurewicz around
1940, and that the arrow expressed well a central interest of topology, and thus
prompted the idea of a functor, “Thus, a notation, (the arrow) led to a concept(category)". (Mac Lane1, 1971, p. 9). Category theory is, so to say, an all embracing
theory in mathematics, and involves a far reaching idea of representation, as we will see.

The pedagogical facts
It is natural to expect that history (even from a rather broad view, as in some ofthe examples above) would illuminate a subject’s importance. On the other hand, it isalso natural to assume that, if mankind has had to struggle with some particular aspectsof its scientific endeavor, there is a chance that every individual should experiencecertain pains when learning them – although perhaps using more paved routes, and withmore abstract constrains.
Anyhow, the beauty of the whole undertakings shown by historical approaches
somehow shadows when confronted with a teaching situation. As for representation inmathematics, for example, it is so appealing to consider the possibility of a second
degree equation having real roots and how many of these should be, by means ofdrawing the Cartesian graph of the quadratic function involved, but it is far from easy to
get kids to grasp this matter.

Duval’s approach.
The well known Raymond Duval’s semiotic theory is a quite useful tool to look
into the importance of focusing into representation. For the sake of the discussion, I will
try to remind some essentials of it and of the evidences given. In doing so, I’ll attempt
not to go into the details of definitions and the like, and, at the same time, not tooversimplify. Thus, I will surely fail; but, on the other hand, I hope that I may be able toshow that his centering in representation is very close to the heart of mathematics2.
Duval’s approach is of psychological foundation:
He takes the notion of representation register developed firstly for little kids
beliefs and explanations on natural phenomena in the 20’s, followed by Piaget’s
evocation of absent objects (30’s to 50’s), to the ideas developed by works onacquisition of mathematical knowledge and over learning systems (80’s), and turns it
into a specific tool for teaching and learning – especially, in Mathematics.
1

Mac Lane was one of the creators of the theory and also one of the leading mathematicians of the 20th
century.
2
In any case, this does not imply that he has taken his ideas from the mathematics that I’ll talk about later,
rather, that going deeply into mathematic thinking he found by himself paths that are at the core ofcontemporary mathematical issues.

- Arturo Mena-Lorca 2 -He is concerned with the ‘fundamental cognitive actions’: conceptualization,
reasoning, problem resolution, text comprehension; Mathematics is a privileged field
for studying such activities.
The question posed is in which way information coming from the exterior can
get into the internal system, i.e., what description made up with the help of symbols
susceptible of being used by the system allows to capture the information coming fromoutside.
For that, he starts by distinguishing between mental and semiotic
representations: A mental representation is all set of images and concepts that anindividual may have about an object or a situation, and what is related to them. A
semiotic representation is a production made by the use of signs belonging to a systemof representations which has its own constraints of meaning and functioning.
So one would ask if these kinds of representations fulfill the same functions in
intellectual development and cognitive activities; it turns out that semiotic
representations do comply with the functions that mental representations satisfy, butalso with other functions that are not satisfied by these. This is illustrated by the factthat the main semiotic organization is natural language, but some fundamental cognitive
activities require of systems of expression and representation that are different from it.
In fact, a semiotic representation is not just a means for an individual toexteriorize his/her inner representations – to make them visible or accesible to others –but it is essential to the way under which information is discovered and taken into
account in a representation system. Knowledge progress is always related to creationand develpoment or new specific semiotic systems that coexist with natural languages(which is evident in formation of scientific knowledge in general).
A distinction is made between semiosis, that is, aprehension and production of asemiotic representation, and noesis the cognitive act of the conceptual aprehension of an
object. The distinction been made, the fundamental quest would be what is therelationship between them, if any – they might be independent. It is better for us totranslate this question to the mathematical realm.

Duval on Mathematics.
Certainly, Mathematics – or, more precisely, a mathematical theory – is in itself
a semiotic system; thus, in Mathematics, semiosis is not only necessary forcommunication, but it is also indispensable for the activity itself. Also, Duval claims,
mathematical objects are not accessible by perception: they need semiotic registers,semiotic systems.
Duval’s main result is of particular importance for the mathematical activity, andhence for its learning and teaching: “There is no noesis without semiosis, it is semiosis
that determines the possibility of exercising noesis” 3. (Duval, 1995, p. 4)

3“Il n'y a pas de noésis sans sémiosis, c'est la sémiosis qui détermine les conditions de possibilitéd'exercice de la noesis”.

- Arturo Mena-Lorca 3 -But, of course, a representation in Mathematics does not play the same role thanrepresentations do in every other realm. In fact, as we now, mathematical activity is
characterized for always being proceeding from one step the following, and symbolicaland other properly mathematical representation systems allow to do this progressing inthem – they are not just immobile pictures.
With a similar view, Duval turns his attention to two other cognitive activities
inherent to any representation – other than being a perceptible marc or set of marcs
identifiable as depiction of something in a determined system: these are to transform therepresentations according with the rules proper to the system (so that otherrepresentations may be obtained in the system), and to convert the representations
produced in a system into another (which may help making explicit other meanings ofwhat is represented).
Already from an immediate pedagogical perspective, the aforementioned
activities are, of course, quite important: for a big percentage of students,
comprehension of contents is limited to the representation used. More importantly,
Duval goes on to saying that there is no comprehension in Mathematics if distinction is
not made between the object and its representation.
Now, common teaching of Mathematics shows the persistence of representations
that do not come from a unique semiotic system. Duval claims that, in fact, inMathematics, at least two semiotic systems are required to produce the representation ofan object, a situation or a process. In general terms, a plurality of semiotic systems
allows diversifying the representations of one object, which augments cognitive
capacities of the person and hence his/her mental representations; but, as it is wellknown in Mathematics, confusing the representations will always imply loss ofcomprehension.
The passing of one system to another requires simultaneous movilization fromone system to another, that is often far from evident to students 4 . That passage isspontaneous when the systems are congruent, that is: when there is semantic
correspondence between the significant units that constitute them, conversion of asignficant unit in the source representation in a unique significant unit in the target
representation, and equal possible order of aprehension of the significant units in both
representations.
Anyhow, there is many a time when there is no congruence between different
representations of the same object. Thus, for improving learning, there is a need ofdoing specific work in a diversity of representation systems, of making use of theproperties of each one and also of making comparisons by puting the system incorrespondance and examine their mutual ‘translation’. If the teacher just centers incontents, coordination of systems is hardly achieved; on the other hand, when learningin the diversity described, students show changes in initiative and performance, andthere is a raise of interest on the task.
If we go deeper, Duval claims, there is no noesis without a plurality – albeit
potential – of semiotic systems and their coordination by the individual.
4

Linear Algebra is but a quite illuminating example of this, and of the pedagogical difficulties that thisconveys to students, especially if the instructor is not aware of them.

- Arturo Mena-Lorca 4 -From history to abstract mathematics
It is well know that former mathematicians were also philosophers (at least since
Thales, say). They would ponder about the nature (essence, substance, reality) ofmathematical objects.
We don’t claim that this is not the case today: realism, e. g, is well alive in themathematical community. Anyhow, perspective has changed, and from a quite specificmathematical point of view, that we briefly describe in the following.
As algebraists and other mathematicians have shown, even if you construct it,
you don’t really work with an object: you always end up with an (in a general way)equivalent class of isomorphic objects, each one of which is replaceable by anyone elseof the class. Hence, you always work with a ‘presentation’ of ‘the’ object. This showsthat the problem of representation is at the core of Mathematics itself. Let’s take a closer
look at this:Suppose that you work with Integers number. You may start (following
Brahmagupta, 628) considering that you have a business, and that at the end of a day
you want to know how much came in and how much came out: the day ended up in
fortune (his word) of debt. This in turn will ‘naturally’ take you to consider pairs ofnatural numbers: (3,5) is a way of representing that that day you sold 3 and spent 5 inthe process. Now, since selling 3 and spending 5 is, as a result, equivalent to selling 6
and spending 8, you go on to consider equivalence classes of pairs: (3,5) and (6,8) arerepresentative of a class of results of a business day, one that may be represented by adistinguished member, (0,2), that allows to have a better picture of the day. Also, if you
want to recapitulate what happened in a lapse of some days, you will have to add up thefortunes and debts of each one of them: (3,5) + ( 14,6) = (17,11), equivalent to (6,0)
shows that you overcame your initial bad luck.
As it happens, because both of practical and intellectual requirements, you willdevelop that system of yours, and look for properties: commutativity and the like.
Later on, your drive will take you to consider the properties themselves, from a
broader perspective: The system is a ring – there are others. You have uniquefactorization – there are other such structures. You have long division – still, there areothers.
The purely mathematical question is, then, what is the fundamental feature of theInteger numbers, what are the properties that characterize their system, so that no othersimilar ring would satisfy them.
Now, if you come about an answer to that question, you will not be able to avoid
the following: if you construct another object with the very same properties, you will
have to accept that it will be ‘the same’ one: as in the case of pairs, the whole systemthat you are working with is but a representative of an equivalence class – perhaps you
chose a convenient one. This reference to equivalent classes may not seem entirely
natural, but, in the end, when you are working, you just look for the properties, not for
the nature of the elements.

- Arturo Mena-Lorca 5 -The aforementioned is somehow easier to grasp in the case of Real numbers: you
may construct them5 in several ways: taking into account just the customary ways ofbuilding them from Natural numbers, you would end up with 2x1x1x5 = 10 equivalent
objects6. Each one of them is characterized for being a complete totally ordered field7.
Summing up: (in a quite specific way) one always works not really with objects
but with equivalent classes of objects; it is the properties that count. (Thinking of adding
the real numbers 2 and 3, say, to go back to the elements themselves8) would be really
cumbersome and useless.

Mathematical representation for Mathematics
It is clear that what we were discussing has a lot to do with Duval’s congruenceof semiotic systems. But let’s go on searching for analogous of his transformation andconversion in the mathematical realm.
In any theory, other than the objects themselves, you always have specific‘arrows’ between them: if you are working with sets, there is the functions; in Linear
Algebra, you have linear transformations between vector spaces; then, continuousfunctions, differentiable functions, group homomorphisms, etc. These arrows, in theirown way, ‘respect’ the object’s structure: some specific procedures that you can do withelements in the source object can be carried out with the ‘images’ of these elements inthe target object; this is the general notion of a (homo)morphism.
There are this kind of arrows from the object to itself, and also from the object toother objects. There are arrows of these that are reversible (as in bijective functions),
let’s call them isomorphisms. Thus, Duval’s psychological concepts are genuinely close
to the mathematical way of thinking.
Moreover, sometimes you want to know about a property of some object andfind it easier to get by taking advantage of that ‘respecting’ of specific arrows. This may
be explicitely stated or not: for instance, when one thinks about an n-dimensional vector
space, one just thinks of the n-tuples (of real numbers, say); the cyclic group on nelements we have in mind is always the same 9 . In both cases we have used an
isomorphism – for instance, between any n-dimensional space and the corresponding ntuples. The heart of the matter is that you not only ‘represent’, one by one, the elementsof an object in another object, but that you can work in the second one as if it were thefirst: in psychological terms, a pair of congruent registers.

5

A pedagogical matter, Hilbert would say.
30 for Complex numbers
7
A very precise expression: notice that the Complex numbers system is a field, is totally ordered, and iscomplete.
8
For instance, to take the corresponding Cauchy sequences of Rational numbers, or the corresponding
Dedekind cuts.
9
In considering the elementary cyclic structure of, say, the days of the week, or of a watch, such groups
are present, although in an implicit way.
6

- Arturo Mena-Lorca 6 -Interestingly enough, when mathematicians study, for example, groups ingeneral – abstract ones, say – they try to represent (their word) them as permutationgroups or as matrix groups (or linear groups), which are better known10.
In fact, one may see that it is arrows that receive the more attention: say,continuous functions, or differentiable functions, or linear transformations. This switchfrom objects to arrows is very deep, and, of course, it starts as a tool for studying theobjects.
Now, sometimes such a resource takes you farther than expected, and you end
up in another kind of structure.
One of the firsts of such situation occurred to topologists some time ago: They
knew that they could continuously deform, say, a bowl into a plate; but they didn’tknow whether that could be done between a tire and a ball. This was a difficult topic,
one that could not be solved inside the theory of topological spaces. Instead, they had tobuild aspecific and ‘fat’ arrow (in fact, a functor) from topological spaces to groups, anarrow that converted topological spaces into groups, and continuous maps into grouphomomorphisms. The properties of this functor implied that you could not continuously
deform a tire into a ball, because that would imply certain conditions in the target
objects that couldn’t occur.
This in turn opened the door to think, in a quite broad way, the posibility ofrepresenting objects of one class in objects of another sort – arrows between themincluded.
The higly abstract theory that deals with this, the Theory of Categories, allows totake the idea of representation much farther, as suggested before: it is not only that theelements of an object (say, a vector space) may be in an appropriate correspondencewith elements of another of the same kind (so that to be able to look into the target foranswers for the source), but also that all the objects of some kind (say, all topologicalspaces) may be put in a convenient correspondence with the objects of another kind (allof the groups, e.g.) so that making possible to get information about the sources bylooking at the targets.
The problem has turned into a purely mathematical one, especially, perhaps,
since the 1960’s, when some mathematicians11 succeeded in representing categories as
categories of familiar concrete structures and their arrows12 – a problem that is a remoteanalogous of thinking, for instance, groups as matrix groups.
As it may be suspected, these last pursuits not only aim at purely descriptive
results, but also at, so to say, uncovering unknown facts about these objects – and about
the objects inside them.

10

The general idea being older, this way of studying started formally around 1890, with G. F. Frobenius.
Notably, Czech algebraists.
12
More precisely, the question is to embed arbitrary small categories in a faithful and full way – that is, ina reversible way – in more familiar categories;
11- Arturo Mena-Lorca 7 -Conclusion
Given the broadness of the subject and the space constraint, I use the word
conclusion just as a means to imply that this paper is ending. Nevertheless, something
may be added, even if only as a faint clue for further reflection.
A far shot from direct pedagogical concerns, the last part of our commentary stillshows several important points in connection with the idea of representing mathematicalobjects:
Firstly, representing is a natural mathematical idea, one that one may encounter,
for example, in any higher level class of Mathematics: even if the instructor has nopedagogical intentions, the theory that he/she and his/her students deal with has, in its
own way, a natural idea of representation, one that allows better calculations,comprehension, make inquires.
Secondly, if you try to look deeper in the nature of mathematical objects, youwill find that, as it were, they will avoid your direct grasping of them, and you will
surely have to accept that you don’t really work with elements: mathematics nature ismore abstract, as we now, and we have to admit that we always work with accessible
representations of elusive objects.
The bottom line is that representing is intrinsic to mathematical activity, and so,whenever we try to represent an object, we are not taking some extra-mathematicalaction, but, on the contrary, even if that representation does not belong to amathematical theory, the endeavor of representing objects and working with therepresentations has the right to be considered as of mathematical nature.
At any level, it is not possible to learn Mathematics without some kind ofrepresentation, and working with more than one of these and making connections
between them improve the outcomes. After all, this is the way in which possibly every
person who have dealt with Mathematics has been doing, all over History.

References

Duval, Raymond (1995), Sémiosis et pensée humaine. Registres sémiotiques etapprentissages intellectuels, Berne, Peter Lang.
Mc Lane, Saunders (1971), Categories for the working mathematician, Springer-Verlag,
New York.

- Arturo Mena-Lorca 8 -