**by Axel Arturo Barceló Aspeitia**

Many philosophers of mathematics have been puzzled by the fact that mathematicians’ notebooks and blackboards are full of diagrams, but they rarely appear on published research materials. Until recently, the traditional explanation was that diagrams were good enough for heuristic purposes, that is, to informally understand, explain, explore and teach phenomena, but not fit for rigorous mathematical work. In more recent years, however, a competing account has emerged, where diagrams have been shown to be as rigorous as formulas and where their exclusion from the finished results of mathematicians is presented as based on a prejudice against non-formal methods in mathematics.

My take on the phenomenon, however, is different. I agree with the traditional account that diagrams are better fit for heuristic work than for the needs of published work; however, I also agree with the more recent re-assessments of diagrams in their assessment that they are as fit for the development of rigorous mathematical proofs and theories as formulas. This is because I take it that presenting proofs and theories is a task of a fundamentally different sort that understanding, explaining or teaching them. Thus the requirements for one are substantially different from the other, and the difference is so large that it does not boil down to one being more rigorous than the other. In particular, presenting proofs and theories is a communicative task and as such requires our representations to be easily understood by many, while exploring theories and finding proofs in them is the kind of work that is done either by ourselves and in close proximity with others, in other words, they are tasks that take place in heavily contextualized situations. Consequently, the representations we use in these situations can fruitfully exploit the information available in such contexts and need not be meaningful outside them. In other words, it does not matter if the diagram on the board is not understandable by anyone outside the discussion it was drawn for. However, diagrams in printed media, require being more explicit, not so much in themselves, but in their written context. In other words, in order to properly interpret a diagram in a written context, the contextual information necessary for their interpretation has to be mostly explicitly given in the text. This makes other synthetic means of putting that same information across, like formulas, a more efficacious tool. Thus, the difference is not one of rigorous vs non-rigorous, but between widely and narrow audiences, between poor and rich contexts.

When writing about diagrammatic reasoning, it is not rare to make a difference between what I elsewhere (2016) called the epistemic and ergonomic aspects of visual representations, which roughly corresponds also to the distinction Larkin and Simon (1987) make between their information content and its computational character, i.e., how is such information extracted from them, and in particular, how quickly and easy it is (the same distinction appears in Zhang 1997, Kulvcvki 2013, and Bechtel 2017). Thus, the reasons why diagrams are better suit for the mathematical notebook and blackboard that for the pages of the research journal are not epistemic, but ergonomic: in the context and for the goals of exploration and analysis, diagrams are easier to use than formulas; in the context son and for the goals of a published research paper, it is easier to use formulas. Diagrams could also be used, but they would be too cumbersome.

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