This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

Some have suggested that images can be arguments. Images can certainly bolster the acceptability of individual premises. We worry, though, that the static nature of images prevents them from ever playing a genuinely argumentative role. To show this, we call attention to a dilemma. The conclusion of a visual argument will either be explicit or implicit. If a visual argument includes its conclusion, then that conclusion must be demarcated from the premise or otherwise the argument will beg the question. If (...) a visual argument does not include its conclusion, then the premises on display must license that specific conclusion and not its opposite, in accordance with some demonstrable rationale. We show how major examples from the literature fail to escape this dilemma. Drawing inspiration from the graphical logic of C. S. Peirce, we suggest instead that images can be manipulated in a way that overcomes the dilemma. Diagrammatic reasoning can take one stepwise from an initial visual layout to a conclusion—thereby providing a principled rationale that bars opposite conclusions—and the visual inscription of this correct conclusion can come afterward in time—thereby distinguishing the conclusion from the premises. Even though this practical application of Peirce’s logical ideas to informal contexts requires that one make adjustments, we believe it points to a dynamic conception of visual argumentation that will prove more fertile in the long run. (shrink)

In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. I (...) consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity. (shrink)

The question, “What norms are appropriate for the evaluation of the probative merits of visual arguments?” underlies the investigation of this paper. The notions of argument and of multimodal visual argument employed in the study are explained. Then four multimodal visual arguments are analyzed and their probative merits assessed. It turns out to be possible to judge these qualities using the same criteria that apply to verbally expressed arguments. Since the sample is small and not claimed to be representative, this (...) finding can at best be regarded as suggestive for the probative assessment of multimodal visual arguments in general. (shrink)