We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared.

Argumentamos en contra de la afirmación de que el uso de diagramas en la geometría euclidiana da lugar a vacíos o lagunas en las pruebas. En primer lugar, mostramos que es un error evaluar sus méritos a través de las lentes de la reconstrucción formal de Hilbert. En segundo lugar, esclarecemos las habilidades empleadas en las inferencias basadas en los diagramas en los Elementos, y mostramos que los diagramas son herramientas matemáticas respetables. Finalmente, complementamos nuestro análisis con una revisión de resultados experimentales recientes que pretenden mostrar que la práctica diagramática euclidiana no solo está estrictamente regimentada, sino que también está enraizada en ciertas habilidades cognitivas universalmente compartidas. complex and varied argumentative structures is presented as an alternative to the idea of «inference to the best explanation».

A widely held view in the philosophy of mathematics attributes to diagrams a merely heuristic or illustrative—and thus dispensable—role in mathematical demonstrations. This tradition conceives of demonstrations as syntactic objects made up of finite and inspectable arrangements of sentences (Tennant 1986). Thus, demonstrations that make essential use of diagrams—i.e. that make use of information provided by drawings—came to be viewed as non-rigorous and of no interest to philosophy (Borwein 2008); such was the case with the demonstrations found in Euclid’s

This view can be found in the work of many nineteenth-century authors, such as Pasch, who affirms that “the theorem is only truly demonstrated if the proof is completely independent of the figure” (Pasch 1882/1926, 43

Legris (2012) remarks that the rejection of the epistemic relevance of diagrams has its roots in the development of symbolic logic and the discussions about the foundations of mathematics that took place in the late nineteenth and early twentieth centuries.

Given this definition of proof—pervasive in logic textbooks of the time, such as Carnap (1939), and Church (1956)—the standard views on mathematical rigor and knowledge were tied to the possibility of a formal reconstruction of demonstrations. As Ferreirós (2016) summarizes: “Clarifying what proof and rigor meant was the task of logical analysis, which led to the twentieth-century identification of ‘rigor’ with respect for the rules of a logical calculus, and ‘proof’ with certain sequences of strings of symbols in formal systems” (24).

However, as the author points out, defining those notions in those terms brings about multiple tensions when we confront actual mathematical practice. Firstly, mathematicians from the past did not share that conception of rigor (indeed, Euclid’s

These are some reasons why Ferreirós (2016) suggests that understanding how we attain mathematical knowledge crucially depends on an analysis of

In this paper, we start with Ferreirós’ definition of mathematical practice and argue that, from that perspective, Euclidean geometry is self-sufficient and rigorous, i.e. that assumptions based in diagrams do not give rise to argument gaps as has been claimed. With that in mind, we first present, following Lassalle Casanave (

We then go on to explain why, within the Euclidean practice, diagram use is a controlled and reliable mathematical procedure. In order to do so, we turn our attention to the work of Manders (2008a, 2008b) about the systematic use of specific aspects of diagrams as a basis of inferences in that practice. We complement Manders’ analysis with a tentative list of abilities and competences involved in the skilled use of diagrams within the Euclidean tradition. The result of our discussion is that the Euclidean practice is much more regimented and constrained than its critics have supposed.

The previous discussion might make it seem as if the abilities required for diagram-based reasoning in the

Ferreirós (2016, chap. 5) claims, against commentators (especially mathematicians and philosophers from the late nineteenth and early twentieth centuries), that the subject matter of Euclidean geometry is distinct from that of Hilbert’s reconstructions. According to Ferreirós, the appearance of complete continuity between those two frameworks obscures the fact that they showcase distinct ways of doing mathematics. It is an anachronism to assess the merits of Euclidean geometry through the particular objectives and practices of modern axiomatic theories.

In this section, we intend to argue that interpreting the formal system presented in Hilbert’s (1980)

When we look at the first six books of the

Building on Lassalle Casanave’s (

1.In Euclid, the theoretical language (e.g. of the definitions) is sharply distinguished from the practical one (e.g. of the postulates), as well as theorems from problems. In Hilbert, practical propositions are reduced to theoretical ones by means of axioms and theorems.

2.In Euclid, the theory provides its objects by means of postulates or constructions. The construction postulates and, generally speaking, the

3.The Euclidean distinction between solving a problem by means of a construction and demonstrating the soundness of that solution disappears in Hilbert’s reconstruction, where the only activity is the demonstration of existential theorems.

4.Hilbert’s axioms replace diagrams as sources of justification. The figures used by Hilbert are only pedagogical tools. All proofs must be based on pre-established axioms.

In his later writings,

Regardless of that, it is often not even clear in which sense the word ‘formal’ is being employed in these contexts. Lassalle Casanave (

Considerations like these point to the fact that Euclid’s

These observations count heavily against the misguided idea that Hilbert’s formal reconstruction is simply an amelioration of Euclid’s

When we take a closer look at distinct practices of mathematical demonstration, both Euclidean plane geometry and its axiomatic counterparts come out as self-sufficient mathematical theories on their own. They are distinct theories which exploit, in different ways, the capacities of mathematical agents, each with their own methods and merits (Ferreirós 2016, 119). The hypothesis is that the Euclidean diagram-based practice itself shows that there are no gaps in its demonstrations. Instead, it presents us with a fruitful and perspicuous way of proving results by using diagrams. As Manders says:

Euclidean diagram use forces us to confront mathematical demonstrative practice, in a much richer form than is implicit in the notions of mathematical theory and formal proof on which so much recent work in philosophy of mathematics is based; and to confront rigorous demonstrative use of non-propositional representation. The philosophical opportunities are extraordinary. (2008a, 68)

The possibility that non-propositional resources be used in legitimate and rigorous proofs affords us a richer understanding of the notion of demonstration and mathematical knowledge. From that perspective, if one is concerned with Euclidean geometry, the right question to ask is: inside Euclid’s framework, how should one proceed in order to justify inferences on the basis of a diagram? In our view, the ability and competence in dealing with diagrammatical reasoning is a key factor for answering that question. With this in mind, in the next section we examine what kind of diagrammatic elements are considered in Euclidean demonstrations and point out some abilities involved in the diagrammatic reasoning present in this practice.

As one can see already in the first proof in the

Manders (2008a, 2008b) goes in that direction in his enlightening study of the use of diagrams in Euclidean practice. As he emphasizes, his concern is not assessing whether Euclidean geometry is a good theory, since that is something he takes for granted given its unquestionable and long-lasting success. His concern is explaining how diagrams can be legitimate tools in Euclidean proofs, contradicting the prevailing view by claiming that such use is fundamentally controlled and safe.

Central to Mander’s analysis is his influential distinction between

A key observation by Manders is that Euclid’s diagrams are used only as a source of co-exact information. Euclid never infers exact information from the drawing:

Typical alleged ‘fallacies of diagram use’ rest on taking it for granted that an—to the eye apparently realized but false—exact condition would be read off from a diagram; but the practice never allows an exact condition to be read off from the diagram. Typical ‘gaps in Euclid’ involve reading off some explicit co-exact feature from a diagram; and this is permissible (..). (2008b, 91)

In this section, we argue that the use of diagrams in the

In order to investigate the cognitive basis of diagram use in the Euclidean practice, we explore recent studies in cross-cultural geometrical cognition. Our emphasis will be on two particular experiments that provide empirical evidence for the existence of basic intuitions of visuospatial relations corresponding to one of the core competences involved in Euclidean geometry enumerated in the previous section: the recognition of co-exact relations in a drawing. Specifically, our interest is in experiments comparing the abilities to recognize geometric information of populations with formal education (and acquainted with tools such as compass and ruler) with populations without these characteristics. The purpose of these experiments is to verify whether some of these abilities are present in human beings independently of their culture, education and environment.

One notable experiment was performed by Dehaene

Van der Ham

In summary, both experiments strongly suggest that the accuracy of co-exact information judgments is significantly higher than that of exact ones (Van der Ham

In this paper we have opposed the traditional view of diagram use in proofs and have argued that they can be mathematically rigorous tools of geometric reasoning. We focused mainly in the particular case of Euclidean geometry, where diagrams are commonly used as source of information in proof steps. Our case in favor of the Euclidean practice is two-fold. First, we showed that the use of diagrams within that practice is strictly regimented so as not to give rise to errors. Second, we complemented our analysis of diagram use by reviewing recent experimental results purporting to show that some important abilities over which the Euclidean diagram-based practice is based are universal traits of human spatial cognition.

We started by showing that the traditional view culminates in a conception of proof as formal proof, one in which diagrams cannot play any significant role. But, as we intend to have argued, this view does not stand by itself when we conceive of mathematical knowledge as a product of different historically situated mathematical practices. We argued that, from that perspective, Euclidean geometry is self-sufficient and rigorous, that the use of diagrams is controlled and, consequently, legitimate as a source of inferences. With that in mind, we structured this paper in three sections which we can summarize as follows:

1.In the first section, we contrasted Hilbert’s geometry (exhibited in the

2.In the second section, we explored the abilities and competences required for engaging in the diagram-based reasoning presented in the

3.In the third section, we complemented our analysis of Euclidean diagram use by taking into account recent experimental results according to which the judgments of co-exact relations is much more reliable than that of exact ones, independently of one’s level of formal education or training in geometry. This fits very well with the fact that Euclid exclusively relied on these aspects in his diagram-based proofs. As we have emphasized, this does not allow us to make the stronger claim that Euclidean geometry emerges spontaneously from human beings’ cognitive architecture. Instead, our hypothesis is that classical geometry is grounded on basic perceptual abilities which are then carefully developed and regimented inside of a shared framework. We closed the paper with some suggestions for future investigation.

This research was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, grant 2016/20480-5 and 2014/23191-9). We wish to thank José Ferreirós, María de Paz, Matheus Valente, Rogerio Passos Severo, Abel Lassalle Casanave, Emiliano Boccardi and Marco Ruffino for insightful discussions and suggestions on earlier drafts of this paper, as well as the referees for their helpful comments.

See Hilbert’s 1894 lecture in Hallett & Majer (2004).

See Giaquinto (2007) for concrete examples of mathematical cases where diagrams seemed to lead to fallacious conclusions and an explanation of why these cases do not constitute a good induction base for the claim that diagrams can often lead to fallacious conclusions.

About the changes in the notion of mathematical rigor throughout the history, see Kleiner (1991). In that paper, the author claims that the patterns of mathematical rigor can vary and that it is not always the case that the changes consist in an increase in rigor. An illustrative case of rigor decrease is the use of infinitesimal calculus by Euler.

One example is Gauss’ presentation of four distinct proofs for the Fundamental Algebra Theorem and the six distinct proofs for quadratic and higher reciprocity laws in number theory (examples taken from Ferreirós 2016, 10). A more detailed study about these issues can be found in Lemmermeyer (2000) and Goldstein

On the appearance and motivations of the philosophy of mathematical practice, as well as a good source of important references within that tradition, see Mancosu (2008) and Giardino (2017).

As was observed by an anonymous referee, some properties of Euclidean objects are tacitly given and never made explicit by the text. However, we believe it is a part of the abilities and competences of a geometer being able to recognize the diagrammatically relevant aspects of a drawing and being able to use them in the argument. This will become clearer when we introduce Manders’ distinction between the diagrammatic and the textual aspects of an Euclidean proof. In any case, see Giaquinto (2011) for a discussion of the thesis that some properties of the Euclidean space are simply assumed.

This conception of formal theories was given by Hilbert in the context of the Foundations but it also applies to his posterior works where theories are formal in the purely syntactic sense. The open interpretation of the primitive symbols (point, line, plane) paves the way for the appearance of strange objects, e.g. ones that do not obey the Euclidean relations. For example, if the axiom corresponding to the side-angle-side congruence criterion is substituted for a weaker one, we get a geometry where isosceles triangles do not need to have equal base angles (Euclid’s proposition I.4) (Appendix II of the

On the second section of chapter 4, Lassalle Casanave (

For a discussion of the functions of symbols in Hilbert’s formalism, see Lassalle Casanave (2015).

In the area of logic, see, for example, the logico-syntactical formalization of Euclidean geometry presented by Tarski & Givant (1999). In the area of mathematics, one example of formalism that gives a great deal of attention to the presentation of the formal proofs can be found in the works of Landau in analysis (see the collected works of Edmund Landau (Heath-Brown 1989)). We would like to thank an anonymous referee for illuminating us on this point by means of very useful suggestions.

About Hilbert’s works in the 1920’s and Hilbert’s Program, see Sieg (2014).

Tarski had paid great attention to logico-syntactical formalizations of mathematics. Even though this is outside the scope of the present paper, we would like to call attention to the role of increasing logical sophistication inside this story, a topic which deserves further historical investigation. About this point, besides Tarski’s own works, we suggest Pieri’s work on the foundations of geometry (see Pieri’s collected works, 1980, under the title

About the notion of symbolic and theoretical frameworks, see Ferreirós (2016, chap. 3).

For more on this idea, see Lassalle Casanave and Panza (2015)

Among the examples offered by critics as evidence that diagram use is not reliable, we find the famous fallacy such that “every triangle is isosceles”. This example is discussed in recent analysis of authors such as Norman (2003) and Manders (2008b), where the authors try to show why it does not amount to a criticism of Euclidean proofs. The point is that every time that the fallacious proof is exhibited, only one of the possible diagram configurations is provided. Norman (2003) shows that, in one of the possible configurations of the diagram, one information extracted from the figure in the fallacious proof does not appear, i.e. it is not co-exact information in Manders’ terminology. Heath (1921) calls attention to a Euclid’s work entitled Pseudry or Pseudographemata which we know only from secondary sources such as Proclus (who called it the Book of Fallacies). That book might have been intended to make the geometer more attuned to the possibility of avoiding fallacies such as this (Heath 1921, 430-431).

We would like to thank an anonymous referee for calling attention to these relevant points and references.

The Mundurukus’ performance was lower but still higher-than-chance for slides dealing with paradigmatically exact properties, such as symmetries and metric properties. The two domains where the Mundurukus’ performance was poor were in “a series of slides assessing geometrical transformations, for instance, by depicting two triangles in a mirror-symmetry relation; and another two slides in which the intruder shape was a randomly oriented mirror image of the other shapes” (Dehaene

Van der Ham et al’s experiment was in many senses a realization of the suggestions put forward by Hamami & Mumma (2013).

Nevertheless, the Dutch participants showed higher accuracy across the tests than the other participants. The authors contend that this difference cannot be explained by the two groups’ distinct average levels of formal education, since the Senegalese population without formal education had a highly similar performance to the Senegalese population that had received basic mathematical training relevant to the tasks at hand. Instead, the authors argue that the difference in performance between the two nationalities must be due to “general cultural differences in how people use spatial information in their daily activities” (Van der Ham

About Euclid’s concerns in minimizing disagreement and improving the reliability of his proofs, see Proclus (1970).

According to authors like Spelke

Overmann (2013), for example, is a cognitive archaeologist who saw, with ethnographic support—quoted the sources in her paper-, that populations with low level of material complexity (size population, social classes, religion, and so on) did not (and typically do not) develop a number system to count beyond five. We do not know of any study on the relations between material complexity and geometry, but we think it is fair to believe that the results would be comparable to these of Overmann’s.

A view that is analogous to ours has been defended, with the support of empirical results, for the related case of numerical cognition (Núñez 2011; 2017). In these works, the author distinguishes between (what he calls) quantical cognition from (properly) numerical cognition. Both are cognitive abilities having to do with the recognition of quantities, however, the former differs from the latter in that it is a mere “inexact and non-symbolic” (Núñez 2017, 404) capacity for discriminating quantities, being detectable even in primates and pre-linguistic infants, while the latter is essentially symbolic, allowing for complex processes such as general exact quantification. Núñez’s thesis is that the former type of cognition is a biologically evolved capacity that is innate at least in the sense that it is not learned and not contingent on a particular culture. The latter, on the other hand, “demands crucial ingredients—cultural traits: specific cultural concerns and practices, and the use of symbolic reference” (

The recent literature already shows signs of heading in that direction, especially in cognitive archaeology studies on the development of mathematics (de Cruz, 2012; Overmann, 2013; 2016). Similarly, Keller (2004) distinguishes three periods from geometry’s arcaheo-ancestral origins until the appearance of Euclidean geometry. In that sense, it is important to take into account some recent studies in the area of Cultural Evolution (Laland 2017), and, more particularly, in evolutionary and cognitive archaeology (Bruner y Lozano 2014 a, 2014 b; Laland & O’Brien 2010, and Coolidge and Wynn, 2016) that could be useful to understand the archaeo-cognitive genesis of proto-mathematics, and its relation with later mathematical developments.

See Giaquinto (2007).