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.

We advance a theory of the representational role of Euclidean diagrams according to which they are samples of co-exact features. We contrast our theory with two other conceptions, the instantial conception and Macbeth's iconic view, with respect to how well they accommodate three fundamental constraints on theories of the Euclidean diagrammatic practice-(i) that Euclidean diagrams are used in proofs whose results are wholly general, (ii) that Euclidean diagrams indicate the co-exact features that the geometer is allowed to infer from them and (iii) that Euclidean diagrams play the same role in both direct proofs and indirect proofs by reductio-and argue that our view is the one best suited to account for them. We conclude by illustrating the virtues of our conception of Euclidean diagrams as samples by means of an analysis of Saccheri's quadrilateral. KeywordsEuclidean diagrams • Reductio ad absurdum • Co-exact information • Iconicity • Samples B Tamires Dal Magro

Abstract. Kuhn continued producing very high quality philosophy after The structure of scientific revolutions (1962) until his death in 1996. This paper describes the changes his thought underwent regarding three theses: (1) that scientific development is not cumulative, (2) that paradigms are incommensurable, and (3) that the choice of new paradigms in periods of revolution is not based solely on observations and logical reasoning. These three theses were the subject of controversies in the initial reception of Kuhn's work, motivating strong criticisms of the image of science allegedly defended by author. In response to those criticisms Kuhn emphasized the realistic aspect of his philosophy and reformulated in more linguistic terms theses (1) and (2), but he left unchanged thesis (3). We argue that the changes introduced in (1) and (2) rendered Kuhn's views more accurate and less comprehensive, and that thesis (3) was remained inadequately developed.Keywords: Kuhn; scientific revolutions; incommensurability; scientific rationality. IntroduçãoA estrutura das revoluções científicas (Kuhn 1962; doravante: Estrutura) é amplamente considerada um divisor de águas na história da filosofia da ciência. Além de romper com alguns padrões de análise que predominaram na filosofia da ciência da primeira metade do século vinte, que tendiam a privilegiar discussões e abordagens abstratas e metodológicas, mostrou, talvez definitivamente, que qualquer concepção adequada da ciência tem de levar em conta também a sua história.1 Na sua caracterização da atividade científica, Kuhn defendeu diversas teses que foram particularmente influentes na literatura posterior. Destacaremos aqui três delas: (1) a tese de que o progresso da ciência não é cumulativo, mas contém rupturas fundamentais, chamadas de 'revoluções científicas'; (2) a tese de que teorias de diferentes paradigmas científicos são incomensuráveis; e (3) a tese de que a avaliação e escolha de hipóteses científicas não é guiada somente por critérios lógicos e observacionais -especialmente nos períodos revolucionários. No que diz respeito a essa terceira tese, Kuhn afirma que as decisões científicas podem sofrer influência de elementos subjetivos como a capacidade de persuasão dos defensores de uma hipótese ou a sua aceitabilidade prévia no interior de uma comunidade científica. Elementos sociais e psicológicos internos à comunidade científica podem justapor-se a critérios lógicos e observacionais de escolha. Ilustrações disso podem ser encontradas em diversos textos de Kuhn. 2 Em um livro publicado alguns anos antes da Estrutura, narrando Principia 17(1): 183-216 (2013).

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We review Danielle Macbeth's book Realizing Reason, published by Oxford University Press in 2014. This extensive book is composed by nine chapters in which Macbeth critically presents the development of mathematical practices in the Western world -from its founding in Ancient Greece's diagrammatic practices to the apogee of mathematical logic in the nineteenth and twentieth-centuries -while offering a revaluation of its present stage by means of a reconsideration of Gottlob Frege's philosophical contributions. In this review, we present a summary of each chapter's contents and make general considerations about them. ___________________________________________________________________________________________________ Danielle Macbeth's Realizing Reason is a tour de force about the history of mathematical knowledge from ancient Euclidean geometry to the late 19 th century and early 20 th century developments on mathematical logic. It is an