For more than two millennia, Euclid's Elements set the standard for rigorous mathematical reasoning. The reasoning practice the text embodied is essentially diagrammatic, and this aspect of it has been captured formally in a logical system termed Eu [2, 3]. In this paper, we review empirical and theoretical works in mathematical cognition and the psychology of reasoning in the light of Eu. We argue that cognitive intuitions of Euclidean geometry might play a role in the interpretation of diagrams, and we show that neither the mental rules nor the mental models approaches to reasoning constitutes by itself a good candidate for investigating geometrical reasoning. We conclude that a cognitive framework for investigating geometrical reasoning empirically will have to account for both the interpretation of diagrams and the reasoning with diagrammatic information. The framework developed by Stenning and van Lambalgen [1] is a good candidate for this purpose.
A distinctive feature of elementary Euclidean geometry is the natural and intuitive character of its proofs. The prominent place of the subject within the history and education of mathematics attests to this. It was taken to be the foundational starting point for mathematics from the time of its birth in ancient Greece up until the 19th century. And it remains within mathematics education as a subject that serves to initiate students in the method of deductive proof. No other species of mathematical reasoning seem as basic and transparent as that which concerns the properties of gures in Euclidean space.
One may not expect a formal analysis of the reasoning to shed much light
on this distinctive feature of it, as the formal and the intuitive are typically
thought to oppose one another. Recently, however, a formal analysis, termed
Eu, has been developed which challenges this opposition [
In this paper, we explore the potential Eu has in this respect by confronting it with empirical and theoretical works in the elds of mathematical cognition and the psychology of reasoning. Our investigation is organized around the two following issues: 1. What are the interpretative processes on diagrams involved in the reasoning practice of Euclidean geometry and what are their possible cognitive roots? 2. What would be an appropriate cognitive framework to represent and investigate the constructive and deductive aspects of the reasoning practice of Euclidean geometry? By providing a formal model of the reasoning practice of Euclidean geometry, Eu provides us with a tool to address these two issues. We proceed as follows. To address the rst issue, we rst state the interpretative capacities that according to the norms xed by Eu are necessary to extract, from diagrams, information for geometrical reasoning. We then present empirical works on the cognitive bases of Euclidean geometry, and suggest that cognitive intuitions might play a role in the interpretative aspects of diagrams in geometrical reasoning. To address the second issue, we compare the construction and inference rules of Eu with two major frameworks in the psychology of reasoning|the mental rules and the mental models theories. We argue that both have strengths and weaknesses as a cognitive account of geometrical reasoning as analyzed by Eu, but that one will need to go beyond them to provide a framework for investigating geometrical reasoning empirically.
The two main issues are of course intimately related. In a last section, we
argue that the framework developed by Stenning and van Lambalgen in [
Eu is based on the seminal paper [
According to Manders, Euclid is not relying on geometric intuition illicitly in
his proofs; he is rather employing a systematic method of diagrammatic proof.
His analysis reveals that diagrams serve a principled, theoretical role in Euclid's
mathematics. Only a restricted range of a diagram's spatial properties are
permitted to justify inferences for Euclid, and these self-imposed restrictions can be
explained as serving the purpose of mathematical control. Eu [
Eu has two symbol types: diagrammatic symbols and sentential symbols A. The sentential symbols A are de ned as they are in rst-order logic. They express relations between geometric magnitudes in a con guration. The diagrammatic symbols are de ned as n n arrays for any n. Rules for a well-formed diagram specify how points, lines and circles can be distinguished within such arrays. The points, lines and circles of Euclid's diagrams thus have formal analogues in Eu diagrams. The positions the elements of Euclid's diagrams can have to one another are modeled directly by the position their formal analogues can have within a Eu diagram.
The content of a diagram within a derivation is xed via a relation of diagram equivalence. Roughly, two Eu diagrams 1 and 2 are equivalent if there is a bijection between its elements which preserves their non-metric positional relations.3 The equivalence relation is intended to capture what Manders terms the co-exact properties of a Euclidean diagram. A close examination of the Elements shows that Euclid refrains from making any inferences that depend on a diagram's metric properties. At the same time, Euclid does rely on diagrams for the non-metric positional relations they exhibit|or in Manders' terms, the co-exact relations they exhibit. Diagrams, it turns out, can serve as adequate representations of such relations in proofs.
Eu exhibits this by depicting geometric proof as running on two tracks: a diagrammatic one, and a sentential one. The role of the sentential one is to record metric information about the gure and provide a means for inferring this kind of information. The role of the diagrammatic track is to record non-metric positional information of the gure, and to provide a means for inferring this kind of information about it. Rules for building and transforming diagrams in derivations are sensitive only to properties invariant under diagram equivalence. It is in this way that the relation of diagram equivalence xes the content of diagrams in derivations.
What is derived in Eu are expressions of the form 1; A1 ! 2; A2 where 1 and 2 are diagrams, and A1 and A2 are sentences. The geometric claim this is stipulated to express is the following: ?Given a con guration satisfying the non-metric positional relations depicted in 1 and the metric relations expressed in A1, then one can obtain a con guration satisfying the positional relations depicted by 2 and metric relations speci ed by A2. 3 For a more detailed discussion of Eu diagrams and diagram equivalence, we refer the reader respectively to sections A and B of the appendix.
Diagrams in Euclid's Elements are mere pictures on a piece of paper. How can a visual experience triggered by looking at a picture lead to a cognitive representation that can play a role in geometrical reasoning? From a cognitive perspective, taking seriously the use of diagrams in reasoning requires an account of the way diagrams are interpreted in order to play a role in geometrical reasoning. The formal system Eu provides a theoretical answer to this question. Here we compare this theoretical account with recent works in mathematical cognition probing the existence of cognitive intuitions of Euclidean geometry. We will argue that the Eu analysis of geometrical reasoning suggests a possible role for these cognitive intuitions in the interpretation of diagrams. 3.1
As described in section 2, the key insight behind the Eu analysis of the diagrammatic reasoning practice of Euclid's Elements is the observation that appeals to diagrams in proofs are highly controlled: proofs in Euclid's Elements only make use of the co-exact properties of diagrams. From a cognitive perspective, the practice of extracting the co-exact properties from a visual diagram is far from a trivial one. According to Eu, the use of diagrams presupposes the two following cognitive abilities: 1. The capacity to categorize elements of the diagram using normative concepts: points, linear elements and circles. 2. The capacity to abstract away irrelevant information from the visual-perceptual experience of the diagram.
The formal syntax of Eu, combined with the equivalence relation between Eu diagrams, can be interpreted as specifying these capacities precisely. More speci cally: { The rst capacity amounts to the ability to see in the diagram the elements p, l, c of a particular Eu diagram hn; p; l; ci, which respectively denote the sets of points, lines and circles. { The second capacity amounts to the ability to see in the particular diagram positional relations that are invariant under diagram equivalence.
Thus, the interpretation of a visual diagram according to Eu results in a formal object which consists in an equivalence class of Eu diagrams. It is precisely on these formal objects, the interpreted diagrams, that inference rules operate in the Eu formalization of reasoning in elementary geometry. 3.2
Recently, several empirical studies in mathematical cognition [5{7] have directly
addressed the issue of the cognitive roots of Euclidean geometry. These studies
might be classi ed in two categories: one approach consists in providing empirical
evidence for the existence of abstract geometric concepts [
The two studies [
The second approach for detecting the presence of intuitions of Euclidean
geometry is based on the framework of transformational geometry [
The two approaches reported here to investigate cognitive intuitions of Euclidean geometry bring out cognitive abilities which seem related to the abilities for the interpretation of diagrams postulated by Eu. This observation suggests a possible role for the cognitive intuitions of Euclidean geometry in the reasoning practice of elementary geometry, as we will now see. 3.3
How might the cognitive intuitions of Euclidean geometry relate to the deductive
aspects of Euclid's theory of geometry? In a correspondence in Science [
The axioms of geometry introduced by Euclid [. . . ] de ne concepts with spatial content, such that any theorem or demonstration of Euclidean geometry can be realized in the construction of a gure. Just as intuitions about numerosity may have inspired the early mathematicians to develop mathematical theories of number and arithmetic, Euclid may have appealed to universal intuitions of space when constructing his theory of geometry. [10, p. 320]
Interestingly, stating precisely the postulated cognitive abilities involved in the interpretation of diagrams according to Eu might suggest a role for the cognitive intuitions of Euclidean geometry in the reasoning practice of Euclid's Elements.
Firstly, we have noticed that one of the key abilities in the interpretation of diagrams, according to Eu, is the capacity to categorize or type the di erent gures of the visual diagram using the normative concepts of geometry. This ability seems to be universal, according to the empirical data that we reported in the previous section, as the Mundurucu people, without previous formal education in Euclidean geometry, seem to possess an abstract conceptual systems which contains the key normative concepts of elementary geometry, and are able to use it to categorize elements of visual diagrams.
Secondly, interpretation of diagrams in Eu requires an important capacity of abstraction which is formally represented by an equivalence relation between diagrams. This equivalence relation aims to capture the idea that some features of the diagram are not relevant for reasoning: for instance, the same diagram rotated, translated or widened, would play exactly the same role in a geometrical proof of the Elements. This capacity of abstraction seems to connect with the cognitive ability of perceiving abstract geometric features, such as angle and length proportions, while abstracting away irrelevant information from the point of view of Euclidean geometry.
We now turn to the deductive and constructive aspects of geometrical reasoning. In section 5, we argue that a plausible cognitive framework for an empirical investigation of geometrical reasoning will have to bring together the interpretative, deductive and constructive aspects of geometrical reasoning to be faithful to the mathematical practice of Euclidean geometry. 4
In this section, we begin by presenting the Eu formalization of constructive and deductive steps in Elementary geometry, and then discuss the capacity of the mental rules and the mental models theories to represent these reasoning steps. We conclude that an adequate framework for an empirical investigation of geometrical reasoning will have to go beyond these two theories.
The Eu proof rules specify how to derive 1; A1 ! 2; A2 expressions. They specify, in particular, the operations that can be performed on the pair 1; A1 to produce a new pair 2; A2. Given the intended interpretation of 1; A1 ! 2; A2 given by ?, the fundamental restriction on these rules is that they be geometrically sound. In other words, if 2; A2 is derivable from ;A1, then with any con guration satisfying the geometric conditions represented by 1A1 one must either have a con guration satisfying the conditions represented by 2; A2 (if 1 = 2), or be able to construct a con guration satisfying the conditions represented by 2; A2 (if 2 contains objects not in 1).
The formal process whereby 2; A2 is derived has two stages: a construction and demonstration. Hence Eu has two kinds of proof rules: construction rules and inference rules. The construction stage models the application of a proof's construction on a given diagram. The demonstration stage models the inferences made from the assumed metric relations and the particular diagram produced by the construction. The soundness restriction thus applies only to the inference rules. The construction rules together codify a method for producing a representation that can serve as a vehicle of inference. The demonstration rules codify the inferences that can be made from such a vehicle.4 4.2
The mental rules theory [
The formalization of geometrical reasoning provided by Eu shares one
important feature with the mental rules theory: Eu represents geometrical reasoning
in terms of syntactic rules of inference. This is made possible by considering
diagrams as a kind of syntax, and then by stipulating rules that control inferences
that are drawn from diagrams. Thus, Eu diagrams might be seen as representing
something like the logical form of concrete visual diagrams. One could perhaps
say that Eu diagrams represent their geometric form, and that Eu inference
rules operate on these forms. From this point of view, the formalization of
geometrical reasoning provided by Eu appears to be in direct line with the the
mental rules theory of reasoning.
4 For the formal details, see sections 1.4.1 and 1.4.2 of [
Nevertheless, the mental rules theory seems to run into troubles when we consider the construction operations on diagrams, which are central to the reasoning practice of Euclidean geometry, and which are formalized by Eu's construction rules. One might still argue that Eu construction rules are syntactic operations on Eu diagrams, since diagrams are included its syntax, and so Eu construction rules t the framework of mental rules theory. However, this does not seem correct as the mental rules are considered to be deduction rules; the soundness restriction applies to them. They are thereby of a very di erent nature than Eu's construction rules. Consequently, even though the mental rules theory could suitably represent inferential steps in geometrical reasoning, it seems that the theory lacks the necessary resources to account for the construction operations on diagrams, an aspect fundamental to the reasoning practice in Euclidean geometry. 4.3
The mental models theory [
Contrary to the mental rules theory, the mental models theory seems suited to provide an account of the representation of the di erent construction operations on diagrams: visual diagrams used in geometrical proofs can be conceived as mental models entertained by the reasoner. Mental models associated to diagrams would then be of a visual-spatial nature, re ecting the spatial relations between the di erent elements of the diagram. This just seems to be a description of Eu diagrams, which encode the information that can be legitimately used in geometrical reasoning.
Accordingly, if Eu diagrams are conceived as mental models, the construction operations on diagrams would be explained in terms of the ways mental models are constructed. Nevertheless, one might worry about the capacity of the mental models theory to account for the speci c use of diagrams in geometrical reasoning. One of the main points of Eu is to exhibit precisely that diagrammatic information enters legitimate mathematical inferences in a very controlled way. In order to account for the use of diagrams in geometrical reasoning, the mental models theory would have to be supplemented with a regimentation of the information that can be legitimately extracted from diagrams conceived as mental models. The formal system Eu shows that this can be done using syntactic rules that operate on diagrams represented as syntactic objects. 4.4
Geometrical reasoning, as practiced in Euclid's Elements, constitutes an
interesting test for current theories in the psychology of reasoning. Our previous
comparison predicts that the mental rules and mental models theories present
both advantages and disadvantages as an account of the reasoning practice of
geometrical reasoning with diagrams. Indeed, it seems that these two theories
are actually complementary in their capacity to account for geometrical
reasoning as described by Eu: the mental rules theory seems adequate to represent
inferences in geometrical reasoning, while the mental models theory seems
adequate to represent the diagrammatic constructions that support such inferences.
Thus, it seems that to provide a framework for an empirical investigation of
geometrical reasoning with diagrams one will need to go beyond these two
theories. Such a move is also suggested, although for di erent reasons, in [
According to the Eu analysis of geometrical reasoning, interpretation and
reasoning with diagrams are intimately related. This observation, originating in the
study of the reasoning practice in Euclid's Elements [
We [. . . ] view reasoning as consisting of two stages: rst one has to establish the domain about which one reasons and its formal properties (what we will call `reasoning to an interpretation') and only after this initial step has been taken can ones reasoning be guided by formal laws (what we will call `reasoning from an interpretation'). [1, p. 28]
Geometrical reasoning with diagrams, as formalized by Eu, precisely ts within this framework: reasoning to an interpretation corresponds to the process of interpreting a visual diagram along with an associated metric assertion, resulting in Eu into a pair ; A; reasoning from an interpretation is then represented as the application of the formal rules of Eu to prove geometric claims.
This perspective uni es the two main issues addressed in this paper. Our main
conclusions can then be restated as follows: (i) intuitions of Euclidean geometry
as studied by mathematical cognition are likely to play a role in reasoning to an
interpretation of a diagrams, (ii) the mental rules and mental models theories
of reasoning are inadequate for representing reasoning from an interpretation,
as none of them is able to account for both deductive and constructive aspects
of geometrical reasoning. In the perspective of Stenning and van Lambalgen [
The empirical investigation of the cognitive bases of Euclidean geometry is a multi-disciplinary enterprise involving both mathematical cognition and the psychology of reasoning, and which shall bene t from works in formal logic and the nature of mathematical practice. In this paper, we used the formal system Eu to review existing empirical and theoretical works in cognitive science with respect to this enterprise. Our investigation shows: (i) the necessity of dealing jointly with interpretation and reasoning, (ii) the relevance of works on the cognitive bases of Euclidean geometry for the interpretation of diagrams and (iii) the necessity to go beyond the mental rules vs mental models distinction for accounting for both constructive and deductive aspects of geometrical reasoning.
The syntactic structure of diagrams in Eu has no natural analogue in standard logic. Their underlying form is a square array of dots of arbitrary nite dimension.
The arrays provide the planar background for an Eu diagram. Within them geometric elements|points, linear elements, and circles| are distinguished. A point is simply a single array entry. An example of a diagram with a single point in it is Linear elements are subsets of array entries de ned by linear equations expressed in terms of the array entries. (The equation can be bounded. If it is bounded on one side, the geometric element is a ray. If it is bounded on two sides, the geometric element is a segment.) An example of a diagram with a point and linear element is Finally, a circle is a convex polygon within the array, along with a point inside it distinguished as its center. An example of a diagram with a point, linear element and a circle is
The size of a diagram's underlying array and the geometric elements distinguished within it, comprise a diagram's identity. Accordingly, a diagram in Eu is a tuple
hn; p; l; ci where n, a natural number, is the length of the underlying array's sides, and p; l and c are the sets of points, linear elements and circles of the diagram, respectively.
Like the relation symbols which comprise metric assertions, the diagrams have slots for variables. A diagram in which the slots are lled is termed a labeled diagram. The slots a diagram has depends on the geometric elements constituting it. In particular, there is a place for a variable beside a point, beside the end of a linear element (which can be an endpoint or endarrow), and beside a circle. One possible labeling for the above diagram is thus
A
E
D B
C Having labeled diagrams within Eu is essential, for otherwise it would be impossible for diagrams and metric assertions to interact in the course of a proof. We can notate any labeled diagram as
hn; p; l; ci[A; R]
where A denotes a sequence of variables and R a rule matching each variable to
each slot in the diagram.
The de nition of diagram equivalence is based on the notions of a diagram's
completion and that of co-exact map. The completion 0 of a diagram is simply
the diagram obtained by adding to intersection points to the intersections
present in it. (See page 38-40 of [
The key idea behind the de nition of diagram equivalence, then, is that of of a co-exact map. A necessary condition for their to be such a map between diagrams 1 and 2 is that both diagrams contain the same number of points, linear elements and circles, and these objects are labeled by the same variables and variable pairs.
Suppose 1 and 2 are two such diagrams, and x is the bijection that maps an element in 1 to the element in 2 with the same label. In virtue of their position with respect to the underlying array of 1, the elements of 1 satisfy various geometric relations. The bijection x is a co-exact map if and only if it preserves a certain sub-set of these relations. Precisely x is a co-exact map if and only if it satis es the following nine conditions, where P and Q are points of 1, N and M are linear elements of 1 (i.e. segments, rays, or lines) and C1 and C2 are circles of 1.