Perini ( 2012) concludes from such considerations that, strictly speaking and contrary to first appearances, diagrams as physical entities do not serve a modelling function. If true, then modelling with diagrams is quite unlike modelling with scale models, where the physical object is manipulated. In fact, Perini ( 2012) maintains that chemical formulas are mostly modified in the chemists’ minds, not on paper. But manipulating diagrams on paper also involves mental manipulations. Insofar as they are physical entities, chemical formulas seem to function much like two-dimensional versions of scale models (Goodwin 2008, 2009). What functions as the model: physical marks or mental images? The external, physical format of Berzelian formulas made them easy to modify and turned them into efficient research tools (Klein 2003). This guarantees that manipulations to diagrams are non-arbitrary and yield meaningful results. when they are sets of meaningful symbols, which are modified according to syntactic rules that respect the symbols’ meanings. In this paper, I will argue that analogous constraints arise when diagrams have the properties of ‘interpreted formal systems’ (Haugeland 1985), i.e. The challenge, which received little attention so far, is to identify constraints that are functionally analogous to the physical constraints of scale models. In the absence of constraints on permissible modifications, it is therefore hard to see how diagrams could yield valuable inferences about their targets. But diagrams like chemical formulas offer no physical resistance to removing a letter here, a numeral there, or adding a baroque squiggle for good measure. In the case of scale models, part of the story is the ease/difficulty with which it is possible to contort them into certain shapes (Goodwin 2008 Toon 2011). Understanding what justifies the inferences is therefore important. What justifies diagram- to- target inferences? Scientists employ models because they allow inferences about what they represent. I will argue that a strict division of labour can underpin the modelling role of diagrams. Adjudicating between these options requires a closer look at the respective roles of syntax and semantics. But another possibility is a division of labour, whereby (1) manipulating diagrams is a matter of following syntactic rules and (2) drawing inferences is a matter of attributing meanings to syntactic marks. It is possible, for instance, that both syntax and semantics guide the manipulation of diagrams, as Klein ( 2001) maintains for Berzelian formulas. But the specific contributions of these properties are unclear. What does a diagram’s syntax and semantics contribute to its modelling role? Diagrams have syntactic and semantic properties (Goodman 1976 Perini 2005b) and both are probably relevant when diagrams are employed as models (Klein 2001, 2003). This paper aims to address this gap by focusing on four central questions: While such studies have demonstrated that diagrams can function as representational models (or ‘models’ in what follows), much remains open about how they do so. Another example are structural formulas of cyclical compounds, where the compounds’ stability was investigated by measuring on paper geometric features of the formulas (Goodwin 2008). Berzelian formulas like ‘H 2O’ depicted chemical substances and allowed that marks on paper, like ‘H’ and ‘2’, be re-arranged in order to learn about the chemical reactions in which the substances were involved (Klein 2001, 2003). Chemical formulas have received particular attention in this respect. As representational models, diagrams are manipulated in lieu of their representational targets (Giere 2002 Downes 2012). as systems that aim to represent the world and, in addition, enable scientific investigations to be carried out on the model, rather than on reality itself. Some diagrams can also be employed as “representational models” (Frigg and Hartmann 2012), i.e. Many scientific diagrams purport to depict features of the world.
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