More than a hundred years ago, the great American philosopher Charles Sanders Peirce coined the term ``abduction'' to refer to inference that involves the generation and evaluation of explanatory hypotheses. Peirce says that mathematical and geometrical reasoning ``consists in constructing a diagram according to a general precept1, in observing certain relations between parts of that diagram not explicitly required by the precept, showing that these relations will hold for all such diagrams, and in formulating this conclusion in general terms. All valid necessary reasoning is in fact thus diagrammatic'' (Peirce, 1958, CP, 1.54). We contend that a considerable part of scientific reasoning is a kind of abductive reasoning.
What is abduction? Many reasoning conclusions that do not proceed in a deductive manner are abductive. For instance, if we see a broken horizontal glass on the floor we might explain this fact by postulating the effect of wind shortly before: this is certainly not a deductive consequence of the glass being broken (a cat may well have been responsible for it). Hence, abduction (Magnani, 2001) is the process of inferring certain facts and/or laws and hypotheses that render some sentences plausible, that explain or discover some (eventually new) phenomenon or observation; it is the process of reasoning in which explanatory hypotheses are formed and evaluated.
Following Nersessian (1995a, 1995b), we use the term ``model-based reasoning'' to indicate the construction and manipulation of various kinds of representations, not mainly sentential and/or formal, but mental and/or related to external models. Obvious examples of model-based reasoning are constructing and manipulating visual representations, thought experiment, analogical reasoning, occurring when models are built at the intersection of some operational interpretation domain - with its interpretation capabilities - and a new ill-known domain, for example, in mathematical reasoning.
Peirce gives an interesting example of a simple model-based abduction related to sense activity: ``A man can distinguish different textures of cloth by feeling: but not immediately, for he requires to move fingers over the cloth, which shows that he is obliged to compare sensations of one instant with those of another'' (Peirce, 1958, CP, 5.221). This idea surely suggests that abductive movements also have interesting extra-theoretical characteristics and that there is a role in abductive reasoning for various kinds of manipulations of external objects. When manipulative aspects of external models prevail, like in the case of manipulating diagrams in the blackboard, we face what we call manipulative abduction (or action-based abduction).
Manipulative abduction happens when we are thinking through doing and not only, in a pragmatic sense, about doing. For instance, when we are creating geometry constructing and manipulating a triangle. In the case of natural sciences the idea of manipulative abduction goes beyond the well-known role of experiments as capable of forming new scientific laws by means of the results (nature's answers to the investigator's question) they present, or of merely playing a predictive role (in confirmation and in falsification).
It is indeed interesting to note that in mathematics model-based and manipulative abductions are present. For example, geometrical constructions present situations that are curious and ``at the limit''. These are constitutively dynamic, artificial, and offer various contingent ways of epistemic acting, like looking from different perspectives, comparing subsequent appearances, discarding, choosing, re-ordering, and evaluating. Moreover, they present some of the features indicated below, typical of the so-called abductive epistemic mediators (Magnani, 2001): simplification of the task and the capacity to get visual information otherwise unavailable.
Epistemic mediators exhibit very interesting features (for example, we can find the first three in geometrical constructions): 1. action elaborates a simplification of the reasoning task and a redistribution of effort across time (Hutchins, 1995), when we need to manipulate concrete things in order to understand structures which are otherwise too abstract, or when we are in the presence of redundant and unmanageable information; 2. action can be useful in the presence of incomplete or inconsistent information - not only from the ``perceptual'' point of view - or of a diminished capacity to act upon the world: it is used to get more data to restore coherence and to improve deficient knowledge; 3. action enables us to build external artifactual models of task mechanisms instead of the corresponding internal ones, that are adequate to adapt the environment to agent's needs. 4. action as a control of sense data illustrates how we can change the position of our body (and/or of the external objects) and how to exploit various kinds of prostheses (Galileo's telescope, technological instruments and interfaces) to get various new kinds of stimulation: action provides some tactile and visual information (e.g. in surgery), otherwise unavailable.
Diagrams serve an important role in abduction because they can be manipulated. In mathematics diagrams play various roles in a typical abductive way. Two of them are central: