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Part I : Conceptual and Theorical Issues

# The Role of Representations in Mathematical Reasoning1

Jessica Carter
p. 55-70

## Résumés

Cet article discute le rôle des représentations dans les preuves mathématiques. Il est suggéré ici que les représentations nous permettent de diviser une preuve en plusieurs parties plus faciles à traiter. Nous illustrerons cela avec un exemple de la pratique mathématique actuelle qui consiste à trouver la valeur d'une expression en la divisant graduellement en parties plus simples. Par ailleurs, j'explique le rôle que jouent les icônes et les indices dans cette procédure. Les icônes assurent la similarité entre l'expression et les résultats obtenus. En revanche, les indices agissent comme des indicateurs qui permettent de réinsérer les résultats aux emplacements appropriés.

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# Introduction

• 1 I wish to thank Frederik Christiansen as well as the anonymous reviewers for insightful comments.

1The use of representations in mathematical reasoning is the focus of a number of papers and books in both the philosophy of mathematics and in mathematics education. Ken Manders has long stressed the use of artefacts in mathematics, see for example his [Manders 2008]. More recently the topic has been discussed by Grosholz in her book Representation and Productive Ambiguity in Mathematics and the Sciences [Grosholz 2007], where it is claimed that ambiguous representation is a prerequisite for the fruitfulness of mathematics as well as natural sciences. In [Duval 2006] it is argued that the ability to shift between different representations is fundamental for mathematical understanding. This paper seeks to clarify the role of representations in mathematical proofs. In [Carter 2010] a proof is described as using an interplay between different kinds of representations. By representations, I mean diagrams, symbols, definitions and expressions that can be written down. One role that representations play is to enable us to break down proofs into manageable parts and thus to focus on certain details of a proof, by removing irrelevant information. [Carter 2010, 3]

2Here I wish to show just how representations enable us to break a proof down, and explain in more detail the role that icons and indices play in this process.

• 2 Ken Manders [Manders 1999] deals more extensively with a similar point. He introduces the notions o (...)

3My claim is simply that one role of iconic representations, i.e., representations that resemble what they represent, is to ensure that there is a link between the obtained result and the original expression. Use of indices makes it possible to reinsert the result obtained, by dealing with the manageable parts, in the appropriate place. A different aspect that will be addressed concerns what could be denoted as 'compound definitions'. By those, I mean definitions consisting of more than one component. In the case considered, we are dealing with matrices with certain entries. So one component of the definition tells us we are dealing with matrices, the second component concerns the entries of the matrices. The point here is that at a particular step of the proof one need only to focus on one component—thus removing irrelevant information.2

4This process will be illustrated by an example from current mathematical practice, more precisely a result from Free Probability Theory. What is found is the value of an expression. By doing this my approach differs from previous work in this field in two respects. First, most examples in the literature are historical examples. Second, as many authors are concerned with visualisation, many examples concern geometry—and geometric diagrams. The example presented here concerns an algebraic expression.

• 3 For more interpretations of Peirce's philosophy of mathematics I recommend the recent book edited b (...)

5Before turning to the case study, I will give a brief introduction to what will be needed from Peirce's theory of signs. Note my point is not to present— or argue for—a reading that Peirce would necessarily have endorsed. I present an interpretation of his theory that I believe is useful when addressing certain aspects of mathematical reasoning.3

# 1 Semiotics

6The fundamental concept is that of a sign. A sign—or a representation—is something that stands for something else. According to Peirce, for something to act as a sign more is required; a sign is not a sign unless it is interpreted as a sign. A sign, therefore, is a triad (as everything else is understood in terms of three):

• 4 References to Peirce are to his Collected Papers, 1965. 2.228 means paragraph 228 in the second boo (...)

A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen. [Peirce 1965, 2.228, italics in original]4

7What is notable in this description is that, in addition to the sign and the object that it represents, there is a third part, the interpretant. Thus a sign can be pictured as follows:

• 5 According to Peirce's Phaneroscopy, sign, interpretant and object can take part in firstness (possi (...)

8Given this triad one may ask, for example, about the nature of the sign in question and about relations between sign and object.5 What is relevant for our purpose is the relation between the sign and the object; one may ask: In what capacity does the sign represent a given entity? According to Peirce's categories there are three possibilities, named icon, index and symbol. For a first description, icons represent by virtue of likeness. A picture is an example. An index represents by being actually connected to what it represents. The movement of the leaves and branches of the tree outside my window is a sign to me that it will be windy later when I cycle home. Finally a symbol represents by the assignment of a rule that determines its interpretant. Words are symbols, as well as, for example, π.

9Peirce makes further subdivisions. Icons may in turn represent in three ways, depending on whether the sign shares certain qualities with the represented; or it shows some actual relations; or by virtue of "parallelism in something else" [Peirce 1965, 2.227]. These different kinds are denoted images, diagrams and metaphors.

10Indices represent by being connected to the represented, either by a real physical—causal—connection, or by a purposeful act of connecting the signs. As an example of an index in the first sense, Peirce refers to a weather cock that is taken as a sign of the direction of the wind. For the second sense, Peirce mentions the geometers' use of letters, in the sense of signposts, in diagrams as indices.

• 6 This point was made by one of the reviewers.

11Symbols obviously play a major role in mathematics as most expressions are built up from symbols. This is also the case with the expression that we will consider later. Here we will not be much concerned with the role that these play. I mention one curiosity:6 usually the symbol π is taken to denote the real number that denotes the relation between the circumference and the diameter of a circle. In the case study we shall deal with, π denotes a permutation. The rationale for this is that Greek letters are often used to denote mathematical entities and it is 'p' for permutation. The letters that we shall consider are the outcome of a "purposeful act of connecting a sign to its object", and they will therefore act as indices:

Geometricians mark letters against the different parts of their diagrams and then use these letters to indicate those parts. [Peirce 1965, 2.285]

12In this sense, Peirce comments that "indices are absolutely indispensable in mathematics" [Peirce 1965, 3.305].

13In addition Peirce explicitly highlighted the role of icons—"likenesses"—in mathematics:

The reasoning of mathematicians will be found to turn chiefly upon the use of likenesses, which are the very hinges of the gates of their science. The utility of likenesses to mathematicians consists in their suggesting in a very precise way, new aspects of supposed states of things... [Peirce 1965, 2.281]

14In the case study we shall deal with, there will be two senses of iconicity at play.

15Central to Peirce's conception of reasoning in mathematics is that all such reasoning is diagrammatic—and as such iconic. A mathematical theorem will contain certain hypotheses. By fixing reference with certain indices, it is possible to produce a diagram that displays the relations of these referents. In statements concerning basic geometry, the diagram could be a geometric diagram as we know them from Euclid. In other parts of mathematics diagrams take a different form. One of Peirce's favourite examples from set theory concerns the theorem that—in modern terms—the cardinality of a set is strictly less than the cardinality of its power set. Denoting the set by C, it is possible to express the relations in a diagram as follows: There is no 1-1 correspondence between the members of C and P (C ). This is a statement about the relation between a set, C and the set of subsets of C, i.e., a diagram. It is thus clear that Peirce employs the term 'diagram' in a much wider sense than usual. Even spoken language can be diagrammatic:

the wordy and loose deductions of the philosophers may make use rather of auditory diagrams, if I may be allowed the expression, than [of] visual ones. [Peirce 1976, 164]

16Generally a proof proceeds by—either seeing the conclusion of the theorem directly in the diagram—or by performing experiments on the diagram which will in the end lead to the conclusion. In the above example, the crucial step in the experiment consists in first assuming that the theorem is false, saying that there is a 1-1 correspondence f : C P (C ) and then forming the set M={sC : f (s) C}.

17In our case study we see examples of diagrammatic reasoning in this sense. In addition, there will also be examples of reasoning using iconicity in the sense of a parallelism in something else, i.e. as a metaphor:

Particularly deserving of notice are icons in which the likeness is aided by conventional rules. Thus, an algebraic formula is an icon, rendered such by the rules of commutation, association, and distribution of the symbols. [Peirce 1965, 2.279]

18To illustrate this consider the following identity: (a + b)2=a2+2ab+b2. To see that this holds, we use (conventional) rules that hold for multiplication and sum of numbers, and what it means for something to be "squared", i.e., to be multiplied by itself. We also need to know that ab = ba. Most notably, we need to know, or be acquainted with, what the letters a and b are taken to represent, so they function as indices. If they represent numbers, as in this case, the commutative rule holds, if something else, this rule need not hold. Since—with the aid of these conventional rules—we obtain new insight, Peirce stresses that iconicity is at stake:

This capacity of revealing unexpected truth is precisely that wherein the utility of algebraical formulae consists, so that the iconic character is the prevailing one. [Peirce 1965, 2.279]

19Before we turn to the main example, I shall illustrate some of the main ideas with a simpler example. The point is to simplify the expression sin(tan-1(x)). The first step is to represent tan1(x) geometrically, see the picture below. To simplify, we assume x>0.

20Denote by θ the angle tan-1(x). In this diagram, we can also identify sin(tan-1(x)) = sin(θ). To simplify, we denote this line segment a. From the diagram we see that we have two similar triangles, BAC and BED. Using the fact that the relation between corresponding sides of such triangles are identical, we can write the following equation:

21Summing up: In the first step we represent part of the expression in a geometric diagram, i.e., we display relations of the theorem in a diagram. It is also possible to identify the sought for expression, sin(tan-1(x)), in this diagram as the length of a side in a certain triangle. By lettering these sides appropriately—using indices—we can express a relation between known x and unknown quantities a, b, and thus produce a new diagram. This diagram is an equation involving real numbers. Using rules for how to operate on these, we arrive at the final expression for a. Since a was the index representing the desired expression, we can reinsert this and obtain the result:

22Note the use of iconicity, both in the sense of producing certain diagrams, and as using conventional rules. In this example we also see that each step has a certain focus. When dealing with the first diagram, the focus is first on similar triangles. Next we focus on the right angled triangle. In the final step(s), we solve an equation involving real numbers. This is one aspect of what I refer to as "using representations in order to break a proof down into manageable parts". Finally note the extensive use of indices. I explicitly mentioned this with respect to the 'a' that is used to denote the expression.

# 2 Case from Free Probability Theory

23We will consider U. Haagerup's and S. Thorbjørnsen's paper [Haagerup & Thorbjørnsen 1999]. In this paper the following expression is studied

24A Gaussian random matrix is a matrix whose entries are complex valued random variables of the form fij + i gij. The Bi's are thus of the following form:

25The entries, fij ,gij, form families of independently but equally distributed Gaussian random variables.

26This expression is first reduced to

27where

• B1, …Bp are independent matrices of the same type as B.

• The π appearing in the subindices is a permutation on the set {1, 2, …p}.

28It is the following expression that will be the topic of our investigation:

29In this particular case we take the mean of the entries of the matrices to be 0 and their variance 1, i.e., fij,gij ~ N(0,1). It is shown that the expression is equal to (π̂ is a permutation obtained from π, k(π̂) and l(π̂) denote certain natural numbers, that will be explained below):

30The proof is done in a number of steps that will be presented here, demonstrating how it is broken down into manageable parts. The focus is on the overall steps, keeping the mathematical details to a minimum. The proof can be construed as consisting of two major parts (as illustrated in picture 7). In one, the permutation π is reworked into a different permutation, π̂. In the second, the focus is on the expression (1). We start with this expression.

• 7 We also need the fact that the expectation of the sum is the sum of expectations.
 1 In the first step the trace of the product of the matrices is found. In order to obtain this, the letters, B, are taken to represent ordinary matrices. We then use rules for multiplication of matrices and take the trace of the resulting matrix.7 Letting 6(u, v, k) designate the u, v'th entry of the matrix Bk, fixing reference by indices, the following expression (3) is obtained:

31Note the indices of ui are divided into even and uneven parts, and i runs from 1 to 2p. Odd indices keep track of columns and even ones keep track of the rows of the matrices. Recall that the entries 6(u,v,k) are (complex valued) Gaussian distributed random variables. They are distributed so that the value of these products—if not zero—is 1. This means that we simply can count how many of these terms in the sum are not zero. We may make an initial guess about the value of the expression simply by looking at the number of possible choices of the ui's. If i is odd, ui can take all values between 1 and n. If i is even, ui can take any value between 1 and m. Adding up, since there are p each of them, there are npmp different products. To find the value of the expression, we need to work out how many of these products are not zero.

• 8 This follows since the expression E ({f + ig}m{f - ig}n ) is zero unless m = n.
 2 Next the entries b(u,v,k) are taken to designate complex valued random variables. The condition that the expectation of such a product is different from zero is that the product must consist of pairs of entries with their conjugate.8 This gives the following condition:

b(u2i,u2i+1,π(i)) = b(u2π(i),u2π(i)-1,π(i)), i {1,2,…,p}

or

u2i = u2n(i) and u2i+1 = u2n(i)-1

 3 By calculations on these two conditions and use of the definition of n one obtains uj = uπ̂ (j)+1. This equality tells us that when a product is non-zero, then this equality holds. The question is then how many groups of different uj for j ∈ {1,2,…, 2p} there are. Mathematically this corresponds to determining the number of equivalence classes under the equivalence relation j ~ π̂ π̂ (j)+ 1 on the set {1,2,…, 2p}.

32Note that there is also a notational shift where j { 1 , 2,…, 2p} replaces 2i+1 and 2i.

33It is seen that, through these steps, there is a gradual reduction of the original expression (1). The first expression concerns the expectation of a product of Gaussian Random variables. In the reduced expression - uj = u π̂(j)+1 - uj as well as the indices j represent natural numbers. More precisely they represent variables that take as values certain natural numbers. (If j is odd uj can take values from 1 to n, if even uj can take values from 1 to m. j can take values from 1 to 2p.) π in the indices represents a permutation. The task of finding the value of the expression is finally reduced to considering an identity concerning natural numbers. This is one of the simpler details referred to in the above description of the role of representations in proofs. Note also that each step has a certain focus. In step one, the focus is on properties of matrices, forgetting that their entries are random variables. In the second step, the focus shifts to the entries, the complex valued random variables, and as already noticed, in the third step, we deal mainly with natural numbers.

34Turning to the role of icons, we see that, at first, part of the theorem (2) is displayed in the expression (1). In step 1 above we use iconicity in the sense of a parallelism in something else. We take the indices, Bi, to denote matrices and use rules that tell us how to multiply these in order to obtain the product of the random variables.

35At step 2 iconicity is used in the sense of producing a (new) diagram that depicts certain relations that will tell us something about the solution of the problem. Looking at expression (3), we see that it is a sum. The setup is such that each summand—if not zero—is one. So we need to count the number of terms that are not zero. The desired relations are obtained by asking for the condition—what are relations between random variables— so that product is not zero. The formulation of this condition becomes the new diagram. The rest of the proof works out these conditions using an interplay between these two roles of icons, until the basic step concerning natural numbers is arrived at.

36The second part of the proof concerns the permutation π : {1,2,…,p} — {1,2,…,p} which is transformed into a new permutation π̂ on {1,2,…, 2p}. This also reflects the change of sub-indices in the expression above. One way to arrive at the expression for n is to write the product of the matrices without the permutation:

C1* C2C2*p-1C2p

37But we still wish to keep track of which matrices are identical, so we compare the above expression with the original one:

B1*Bπ(1)Bp*Bπ (p)

• 9 π = (12)(34) means that it is the map π : {1, 2, 3,4} {1, 2, 3,4}, where the number 1 is mapped to (...)

38Before illustrating the general idea we consider an example, letting p = 4 and considering the permutation π = (12)(34)9:

C1*C2C3*C4C5*C6C7*C8,

is compared to

B1*B2B2*B1B3*B4B4*B3

39We see that the first and fourth B have the same index 1, which means they are identical, similarly the matrices in the second and third place are identical, continuing until we have:

 C1 = C4 C2 = C4 C5 = C8 C6 = C7

40This particular example is shown in picture 3 below.

41In general comparing the two expressions one obtains the following identities:

C2i-1 = Bi = Bπ (π-1(i)) = C2π-1(i)

and

C2i = B π (i) = C2 π (i)-1

42Calculations on these equations give the permutation π̂:

43Note that π̂ takes odd numbers into even numbers and vice versa. These permutations can be pictured by diagrams. If p = 4, and π is the permutation

44(12)(34):

Picture 3. The diagram on the left represents π = (12)(34) and the diagram on the right represents the corresponding π̂.

45For another example take the permutation (13)(24):

Picture 4. The diagram on the left represents π = (13)(24) and the diagram on the right represents the corresponding π̂.

46Recall the equivalence relation obtained above. The number of equivalence classes can also be illustrated by diagrams. The relation is j~π̂ π̂ (j)+1 and we use it on the two permutations above. The first gives the following equivalence classes:

Picture 5. The diagram on the left represents π̂ from picture 3 and the diagram on the right represents equivalence classes. Numbers connected by a line are in the same equivalence class. These are calculated as follows: 1~π̂ π̂ (1) + 1 = 5, so 1 is seen to be related to 5. Similarly 5~π̂ π̂ (5)+ 1 = 8 + 1 = 1, calculating mod 8. Thus 1 and 5 form an equivalence class.

47In this case, there are 5 equivalence classes, of which 2 contain even numbers and 3 contain odd numbers. From the definition of the equivalence relation, it can be seen that equivalence classes will always contain exclusively even numbers or odd numbers. Thus we may define k(π̂) to be the number of equivalence classes containing even numbers, and l(π̂) to be the number of equivalence classes containing odd numbers. Then if the value (2) of the expression is recalled, it is seen that if p is 4, and the permutation is as just discussed, the value of the expression is m2n3.

48The second example gives:

Picture 6. The diagram on the left represents π̂ from picture 4 and the diagram on the right represent equivalence classes. Numbers connected by a line are in the same equivalence class. These are calculated as in the former example.

49Here are only 3 equivalence classes, which give the value m2 n to the expression. Since in the two cases there is a different number of equivalence classes, one may ask whether there is a way to determine how many there are for a particular permutation. This question is examined by the authors. It turns out that if the lines do not cross, as in the first example, there will always be 5 equivalence classes when p = 4. In general there will be p + 1 equivalence classes. This is another detail of the proof that can be studied partly using diagrams as addressed elsewhere [Carter 2010]. Here I will mention a single point concerning iconicity of the diagrams. The property of being a crossing or non crossing permutation has bearings on the result of the expression. This property is at first visible in the diagrams. It is possible, though, to formulate an algebraic analogue of being a crossing permutation. In this sense, since the diagram displays this relation, it is an iconic representation.

50Moving on to explain the role of icons and indices, consider first the diagram below representing the process described above:

Picture 7

51We see again that the expression is finally, via a reformulation of the permutation, reworked to the equality uj = u π̂ (j)+1. The answer is then found from the equivalence relation j~π̂ π̂(j)+ 1, calculating the number of equivalence classes. We have thus seen how the original expression is gradually broken down to study specific details.

1. The claims about the role of icons and indices were:

2. Icons, i.e., representations that resemble what they represent, enable the use of an obtained result in the original expression.

52The role of indices is to make it possible to reinsert the result in the appropriate place.

53The major letters used are the following:

• B, in the first step representing matrices, in the second step representing Gaussian random matrices.

• b, representing random variables.

• π, representing a permutation.

• uj, representing natural numbers.

54In the first step of the proof, we take the Bi's to represent matrices and it is our acquaintance with these that give us rules on how to multiply, and find the trace of, matrices. These rules are used to obtain the second expression, the sum of expectations of the product of the entries. Observe that to obtain the first equality, we need both acquaintance with what the letters, Bi, are taken to represent together with relations that hold for these kind of entities. This was the description given above for iconicity in terms of a mediated third. The same picture holds in step three, where calculations are performed on the conditions

u2i = u2π(i) and u2i+1 = u2π(i)-1,

55leading to the expression uj = u π̂(j)+1. Here we take uj to represent numbers, and use relations on these to obtain the desired equality. (This is simplified somewhat in order not to drown the main point in mathematical details.)

56The second step is different. Here we note that the b(u,v,k)'s represent random variables, and that the expression involves products of complex random variables with their conjugates. From this expression, we extract certain relations that must hold in order for the products to be different from zero. These relations allow us to produce a new diagram.

57Even though there is partial likeness along the way, there is a puzzle that needs to be assembled in the end. That this is possible depends on the indexical use of the representations, i.e., the role of representations as "sign-posts", which is illustrated in the diagram.

• π̂ points to π in the original expression.

• uj = u π̂ (j)+1 points—if the change of subindices is recalled—to the entries b(u2i,u2i-1,k).

58In conclusion, my claim was that the value of a complex expression could be found—using representations—in a series of steps to break it down to manageable parts. In each step it is possible to select a particular focus, and thus "remove irrelevant information". Use of iconicity along the way, ensures that the final expressions or results tell us something about the original expression, whereas the use of symbols as indices lets us recall where to reinsert the obtained results.

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## Bibliographie

Carter, Jessica
2010 Diagrams and proofs in analysis, International Studies in the Philosophy of Science, 24(1), 1-14.

Duval, Raymond
2006 A cognitive analysis of problems of comprehension in a learning of mathematics, Educational Studies, 61, 103-131.

Grosholz, Emily R.
2007 Representation and Productive Ambiguity in Mathematics and the Sciences, New York: Oxford University Press.

Haagerup, Uffe & Thorbjornsen, Steen
1999 Random matrices and K-theory for exact C*-algebras, Documenta Mathematica, 4, 341-450.

Manders, Kenneth
1999 Euclid or Descartes: Representation and responsiveness, unpublished.
2008 The Euclidean diagram, in
The Philosophy of Mathematical Practice, edited by Mancosu, Paolo, Oxford: Oxford University Press.

Moore, Matthew E.
2010
New Essays on Peirce's Mathematical Philosophy, Chicago and La Salle: Open Court Publishing Company.

Peirce, Charles Sanders
1965 Collected Papers, Cambridge: The Belknap Press of Harvard University Press.
1976
The New Elements of Mathematics, vol. IV, The Hague: Mouton.

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## Notes

1 I wish to thank Frederik Christiansen as well as the anonymous reviewers for insightful comments.

2 Ken Manders [Manders 1999] deals more extensively with a similar point. He introduces the notions of responsiveness and indifference in order to address the topic of progress in mathematics. Human beings have limited resources for paying attention. As a consequence, one measure for the progress of a method is in terms of the number of types of response rendered superfluous by it.

3 For more interpretations of Peirce's philosophy of mathematics I recommend the recent book edited by Matthew Moore [Moore 2010].

4 References to Peirce are to his Collected Papers, 1965. 2.228 means paragraph 228 in the second book.

5 According to Peirce's Phaneroscopy, sign, interpretant and object can take part in firstness (possibility), secondness (existence), and thirdness (law), which—at first— gives rise to 10 different categories of signs.

6 This point was made by one of the reviewers.

7 We also need the fact that the expectation of the sum is the sum of expectations.

8 This follows since the expression E ({f + ig}m{f - ig}n ) is zero unless m = n.

9 π = (12)(34) means that it is the map π : {1, 2, 3,4} {1, 2, 3,4}, where the number 1 is mapped to 2, 2 to 1, and 3 is mapped to 4, and 4 to 3.

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Légende Picture 3. The diagram on the left represents π = (12)(34) and the diagram on the right represents the corresponding π̂.
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Légende Picture 4. The diagram on the left represents π = (13)(24) and the diagram on the right represents the corresponding π̂.
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Légende Picture 5. The diagram on the left represents π̂ from picture 3 and the diagram on the right represents equivalence classes. Numbers connected by a line are in the same equivalence class. These are calculated as follows: 1~π̂ π̂ (1) + 1 = 5, so 1 is seen to be related to 5. Similarly 5~π̂ π̂ (5)+ 1 = 8 + 1 = 1, calculating mod 8. Thus 1 and 5 form an equivalence class.
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Légende Picture 6. The diagram on the left represents π̂ from picture 4 and the diagram on the right represent equivalence classes. Numbers connected by a line are in the same equivalence class. These are calculated as in the former example.
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## Pour citer cet article

### Référence papier

Jessica Carter, « The Role of Representations in Mathematical Reasoning »Philosophia Scientiæ, 16-1 | 2012, 55-70.

### Référence électronique

Jessica Carter, « The Role of Representations in Mathematical Reasoning »Philosophia Scientiæ [En ligne], 16-1 | 2012, mis en ligne le 01 avril 2015, consulté le 26 novembre 2020. URL : http://journals.openedition.org/philosophiascientiae/716; DOI: https://doi.org/10.4000/philosophiascientiae.716

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## Auteur

### Jessica Carter

University of Southern Denmark, Odense (Denmark)

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