CONJECTURES AND MANIPULATIONS
Action-Based Abduction in Science
Lorenzo Magnani 
lmagnani@cc.gatech.edu
Department of Philosophy and Computational Philosophy Laboratory,
University of Pavia, Piazza Botta 6
Pavia, 27100, Italy, 
and
Program in Philosophy, Science and Technology
Georgia Institute of Technology, 685 Cherry Street
Atlanta, GA, USA, 30332-0345

Introduction
An interesting and neglected point of contention about human reasoning is whether or not concrete manipulations of external objects influence the generation of hypotheses. I am focusing on the first features of what I call manipulative abduction showing how we can find in scientific and everyday reasoning methods of constructivity based on external models and action-based reasoning. I am analyzing the problem in the light of the so-called historical-cognitive method (Nersessian, 1998). While it tries to integrate findings from research on cognition and findings from historical-epistemological research into models of actual scientific practices, assessments of the fit between cognitive findings and historical-epistemological practices aid in elaborating richer and more realistic models of cognition. There are interesting parallels that can be exploited by cognitive scientists and AI researchers, the relevance of the concept of abduction can contribute to a central issue in cognitive science: hypothesis formation both in science and in everyday reasoning. 
Sentential Abduction
It is well known that many reasoning conclusions that do not proceed in a deductive manner are abductive. What I call theoretical (sentential and model-based) abduction (Magnani, 1999a and 2000) is, from a cognitive point of view, an internal process of reasoning. (What about the “external” ways of finding hypotheses? - cf. below the section on “manipulative 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 not certainly a deductive consequence of the glass being broken (a cat may well have been responsible for it). Hence, theoretical abduction 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. Moreover, we have to remember that although explanatory hypotheses can be elementary, there are also cases of composite, multipart hypotheses.
First, it is necessary to show the connections between abduction, induction, and deduction and to stress the significance of abduction to illustrate the problem-solving process. I think the example of diagnostic reasoning is an excellent way to introduce abduction. I have developed with others (Ramoni, et al., 1992) an epistemological model of medical reasoning, called the Select and Test Model (ST-MODEL) (Magnani, 1992) which can be described in terms of the classical notions of abduction, deduction and induction; it describes the different roles played by such basic inference types in developing various kinds of medical reasoning (diagnosis, therapy planning, monitoring) but can be extended and regarded also as an illustration of scientific theory change. The model is consistent with the Peircian view about the various stages of scientific inquiry in terms of “hypothesis” generation, deduction (prediction), and induction.
The type of inference called abduction was studied by Aristotelian syllogistics, as a form of ¹¿, and later on by mediaeval reworkers of syllogism. A hundred years ago, Peirce interpreted abduction essentially as an “inferential” creative process of generating a new hypothesis. Abduction and induction, viewed together as processes of production and generation of new hypotheses, are sometimes called reduction, that is ¹¿. 
There are two main epistemological meanings of the word abduction: 1) abduction that only generates “plausible” hypotheses (selective or creative) and 2) abduction considered as inference to the best explanation, which also evaluates hypotheses. 
To illustrate from the field of medical knowledge, the discovery of a new disease and the manifestations it causes can be considered as the result of a creative abductive inference. Therefore, creative abduction deals with the whole field of the growth of scientific knowledge. This is irrelevant in medical diagnosis where instead the task is to select from an encyclopedia of pre-stored diagnostic entities (Ramoni, et al., 1992). We can call both inferences ampliative, selective and creative, because in both cases the reasoning involved amplifies, or goes beyond, the information incorporated in the premises.
All we can expect of our “selective” abduction, is that it tends to produce hypotheses for further examination that have some chance of turning out to be the best explanation. Selective abduction will always produce hypotheses that give at least a partial explanation and therefore have a small amount of initial plausibility. In the syllogistic view (see below) concerning abduction as inference to the best explanation advocated by Peirce one might require that the final chosen explanation be the most “plausible”.
Since the time of John Stuart Mill (1843), the name given to all kinds of non deductive reasoning has been induction, considered as an aggregate of many methods for discovering causal relationships. Consequently induction in its widest sense is an ampliative process of the generalization of knowledge. Peirce (1955a) distinguished various types of induction: a common feature of all kinds of induction is the ability to compare individual statements: using induction it is possible to synthesize individual statements into general laws - inductive generalizations - in a defeasible way, but it is also possible to confirm or discount hypotheses. Following Peirce, I am clearly referring here to the latter type of induction, that in the ST-MODEL is used as the process of reducing the uncertainty of established hypotheses by comparing their consequences with observed facts. This perspective on hypothesis testing in terms of induction is also known in philosophy of science as the “hypothetico-deductive method” (Hempel, 1966) and is related to the idea of confirmation of scientific hypotheses, predominant in nepositivistic philosophy but also present in the falsificationist tradition (Popper, 1959).
Deduction is an inference that refers to a logical implication. Deduction may be distinguished from abduction and induction on the grounds that only in deduction is the truth of the conclusion of the inference guaranteed by the truth of the premises on which it is based. Deduction refers to the so-called non-defeasible arguments. It should be clear that, on the contrary, when we say that the premises of an argument provide partial support for the conclusion, we mean that if the premises were true, they would give us good reasons - but not conclusive reasons - to accept the conclusion. That is to say, although the premises, if true, provide some evidence to support the conclusion, the conclusion may still be false (arguments of this type are called inductive, or abductive, arguments).
All these distinctions need to be exemplified. To describe how the three inferences operate, it is useful to start with a very simple example dealing with diagnostic reasoning and illustrated (as Peirce initially did), in syllogistic terms: 

1.	If a patient is affected by a pneumonia, his/her level of white blood cells is increased. 
2.	John is affected by a pneumonia.
3.	John’s level of white blood cells is increased.
    (This syllogism is known as Barbara).

By deduction we can infer (3) from (1) and (2). Two other syllogisms can be obtained from Barbara if we exchange the conclusion (or Result, in Peircian terms) with either the major premise (the Rule) or the minor premise (the Case): by induction we can go from a finite set of facts, like (2) and (3), to a universally quantified generalization - also called categorical inductive generalization, like the piece of hematologic knowledge represented by (1). Starting from knowing – selecting – (1) and “observing” (3) we can infer (2) by performing a selective abduction. The abductive inference rule corresponds to the well-known fallacy called affirming the consequent (simplified to the propositional case)

It is useful to give another example, describing an inference very similar to the previous one:

1.	If a patient is affected by a beta-thalassemia, his/her level of hemoglobin A2 is increased. 
2.	John is affected by a beta-thalassemia.
3.	John’s level of hemoglobin A2 is increased.

Such an inference is valid, that is not affected by uncertainty, since the manifestation (3) is pathognomonic for beta-thalassemia (as expressed by the biconditional in   ). This is a special case, where there is not abduction because there is not “selection”, in general clinicians very often have to deal with manifestations which can be explained by different diagnostic hypotheses: in these case the inference rule corresponds to 

Thus, selective abduction is the making of a preliminary guess that introduces a set of plausible diagnostic hypotheses, followed by deduction to explore their consequences, and by induction to test them with available patient data, (1) to increase the likelihood of a hypothesis by noting evidence explained by that one, rather than by competing hypotheses, or (2) to refute all but one.
If during this first cycle new information emerges, hypotheses not previously considered can be suggested and a new cycle takes place. In this case the nonmonotonic character of abductive reasoning is clear and arises from the logical unsoundness of the inference rule: it draws defeasible conclusions from incomplete information. All recent logical accounts (“deductive”) concerning abduction have pointed out that it is a form of nonmonotonic reasoning. It is important to allow the guessing of explanations for a situation, in order to discount and abandon old hypotheses, so as to enable the tentative adoption of new ones, when new information about the situation makes them no longer the best. 
Uncertainty and imperfect information are fundamental characteristics of the knowledge relative to hypothetical reasoning. The nonmonotonic character of the ST-MODEL arises not only from the above mentioned nonmonotonic character of deductive inference type involved in it, but also from the logical unsoundness of the ascending part of the cycle guessing hypotheses to be tested. 
Many attempts have been made to model abduction by developing some formal tools in order to illustrate its computational properties and the relationships with the different forms of deductive reasoning. Some of the formal models of abductive reasoning are based on the theory of the epistemic state of an agent (Boutilier and Becher 1995), where the epistemic state of an individual is modeled as a consistent set of beliefs that can change by expansion and contraction (belief revision framework). This kind of sentential frameworks seems to exclusively deal with selective abduction (diagnostic reasoning) and relates to the idea of preserving consistency. If we want to provide a suitable framework for analyzing the most interesting cases of conceptual changes in science we do not have to limit ourselves to the sentential view of theoretical abduction but we have to consider a broader inferential one which encompasses both sentential and what I call model-based sides of creative abduction.
Hence, if we want to deal with the nomological and most interesting creative aspects of abduction we are first of all compelled to consider the whole field of the growth of scientific knowledge. Related to the high-level types of scientific conceptual change (Thagard 1992) are different varieties of model-based abductions (see, for examples, Magnani 1999b). 
Model-Based Abduction
What exactly is model-based abduction from a philosophical point of view? Peirce stated that all thinking is in signs, and signs can be icons, indices, or symbols. Moreover, all inference is a form of sign activity, where the word sign includes “feeling, image, conception, and other representation” (CP 5.283), and, in Kantian words, all synthetic forms of cognition. That is, a considerable part of the thinking activity is model-based. Of course model-based reasoning acquires its peculiar creative relevance when embedded in abductive processes. 
For Peirce a Kantian keyword is synthesis, where the intellect constitutes in its forms and in a harmonic way all the material delivered by the senses. Surely Kant did not consider synthesis as a form of inference but, notwithstanding the obvious differences, I think synthesis can be related to the Peircian concept of inference, and, consequently, of abduction. After all, when describing the ways the intellect follows to unify and constitute phenomena through imagination Kant itself makes use of the term rule (Kant 1929, A140, B179-180, 182), and also of the term procedure (Kant 1929, A140-B179-180, 182). We know that rules and procedures represent the central features of the modern concept of inference.
Most of these forms of constitution of phenomena are creative and, moreover, characterized in a model-based way. Let me show some examples of model-based inferences. It is well known the importance Peirce ascribed to diagrammatic thinking, as shown by his discovery of the powerful system of predicate logic based on diagrams or “existential graphs”. As we have already stressed, Peirce considers inferential any cognitive activity whatever, not only conscious abstract thought; he also includes perceptual knowledge and subconscious cognitive activity. For instance in subconscious mental activities visual representations play an immediate role.
We should remember, as Peirce noted, that abduction plays a role even in relatively simple visual phenomena. Visual abduction, a special form of non verbal abduction, occurs when hypotheses are instantly derived from a stored series of previous similar experiences. It covers a mental procedure that tapers into a non-inferential one, and falls into the category called “perception”. Philosophically, perception is viewed by Peirce as a fast and uncontrolled knowledge-production procedure. Perception, in fact, is a vehicle for the instantaneous retrieval of knowledge that was previously structured in our mind through inferential processes. Peirce says: “Abductive inference shades into perceptual judgment without any sharp line of demarcation between them” (Peirce, 1955b, p. 304). By perception, knowledge constructions are so instantly reorganized that they become habitual and diffuse and do not need any further testing: “[...] a fully accepted, simple, and interesting inference tends to obliterate all recognition of the uninteresting and complex premises from which it was derived” (CP 7.37). Many visual stimuli - that can be considered the “premises” of the involved abduction - are ambiguous, yet people are adept at imposing order on them: “We readily form such hypotheses as that an obscurely seen face belongs to a friend of ours, because we can thereby explain what has been observed” (Thagard, 1988, p. 53). This kind of image-based hypothesis formation can be considered as a form of visual (or iconic) abduction. Of course such subconscious visual abductions of everyday cognitive behavior are not of particular importance but we know that in science they may be very significant and lead to interesting new discoveries (Magnani, et al., 1994; Shelley, 1996). If perceptions are abductions they are withdrawable, just like the scientific hypotheses abductively found. They are “hypotheses” about data we can accept (sometimes this happens spontaneously) or carefully evaluate).
Peirce gives an interesting example of model-based abduction (Magnani, 1999a and 2001a) 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” (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 (cf. the following section on “action-based, manipulative abduction”). One more example is given by the fact that the perception of tone arises from the activity of the mind only after having noted the rapidity of the vibrations of the sound waves, but the possibility of individuating a tone happens only after having heard several of the sound impulses and after having judged their frequency. Consequently the sensation of pitch is made possible by previous experiences and cognitions stored in memory, so that one oscillation of the air would not produce a tone. 
To conclude, all knowing is inferring and inferring is not instantaneous, it happens in a process that needs an activity of comparisons involving many kinds of models in a more or less considerable lapse of time. All sensations or perceptions participate in the nature of a unifying hypothesis, that is, in abduction, in the case of emotions too: “Thus the various sounds made by the instruments of the orchestra strike upon the ear, and the result is a peculiar musical emotion, quite distinct from the sounds themselves. This emotion is essentially the same thing as a hypothetic inference, and every hypothetic inference involved the formation of such an emotion” (CP 2.643).
Following Nersessian (1999), I use the term “model-based reasoning” to indicate the construction and manipulation of various kinds of representations, not necessarily sentential and/or formal. She proposes the so-called cognitive history and philosophy of science approach, which affords a reframing of the problem of conceptual formation and change in science that not only provides philosophical insights but also pays attention to the practices employed by real human agents in constructing, communicating and replacing representation of a domain. Common examples of model-based reasoning are constructing and manipulating visual representations, thought experiment, analogical reasoning, but also the so-called “tunnel effect” (Cornuéjols, et al., 2000), occurring when models are built at the intersection of some operational interpretation domain – with its interpretation capabilities – and a new ill-known domain. 
We have to remember that visual and analogical reasoning are productive in scientific concept formation too, where the role they play in model-based abductive reasoning is very evident; scientific concepts do not pop out of heads, but are elaborated in a problem-solving process that involves the application of various procedures: this process is a reasoned process. Visual abduction, but also many kinds of abductions involving analogies, diagrams, thought experimenting, visual imagery, etc. in scientific discovery processes, can be just called model-based. Additional considerations about the intersections between abduction and model-based reasoning (especially in experiment and thought experiment) are illustrated by Gooding (1996): the ability to integrate information from various sources is crucial to scientific inference and typical of all kinds of model-based reasoning also when models and representations are “external”, like verbal accounts, drawings, various artifacts, narratives, etc. 
We know that scientific concept formation has been ignored because of the accepted view that no “logic of discovery” – either deductive, inductive, or abductive algorithms for generating scientific knowledge – is possible. The methods of discovery involve use of heuristic procedures (Peirce was talking of creative abduction as the capacity and the “method” of making good conjectures); cognitive psychology, artificial intelligence, and computational philosophy have established that heuristic procedures are reasoned (see the following section). Analogical reasoning is one such problem-solving procedure, and some reasoning from imagery is a form of analogical reasoning: Holyoak and Thagard (1995) elaborated an analysis of analogical reasoning that encompasses psychological, computational, and epistemological aspects. Hence the role of visual abduction and visual imagery in scientific discovery is very interesting. 
What happens when the abductive reasoning in science is strongly related to extra-theoretical actions and manipulations of “external” ojects? When abduction is “action-based” on external models? When thinking is “through doing” as illustrated in the simple case above of distinguishing the simple textures of cloth by feeling? To answer these questions I will delineate the first features of what I call manipulative abduction by showing how we can find in scientific and everyday reasoning methods of constructivity based on external models and actions.
Manipulative 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, like in the case given by Kant in the “Transcendental Doctrine of Method” (Magnani, 2001b). It refers to an extra-theoretical behavior that aims at creating communicable accounts of new experiences to integrate them into previously existing systems of experimental and linguistic (theoretical) practices. Gooding (1990) refers to this kind of concrete manipulative reasoning when he illustrates the role in science of the so-called “construals” that embody tacit inferences in procedures that are often apparatus and machine based. The embodiment is of course an expert manipulation of objects in a highly constrained experimental environment, and is directed by abductive movements that imply the strategic application of old and new templates of behavior mainly connected with extra-theoretical components, for instance emotional, esthetical, ethical, and economic.
The hypothetical character of construals is clear: they can be developed to examine further chances, or discarded, they are provisional creative organization of experience and some of them become in their turn hypothetical interpretations of experience, that is more theory-oriented, their reference is gradually stabilized in terms of established observational practices. Step by step the new interpretation - that at the beginning is completely “practice-laden” - relates to more “theoretical” modes of understanding (narrative, visual, diagrammatic, symbolic, conceptual, simulative), closer to the constructive effects of theoretical abduction. When the reference is stabilized the effects of incommensurability with other stabilized observations can become evident. But it is just the construal of certain phenomena that can be shared by the sustainers of rival theories. Gooding (1990) shows how Davy and Faraday could see the same attractive and repulsive actions at work in the phenomena they respectively produced; their discourse and practice as to the role of their construals of phenomena clearly demonstrate they did not inhabit different, incommensurable worlds in some cases. Moreover, the experience is constructed, reconstructed, and distributed across a social network of negotiations among the different scientists by means of construals. 
To illustrate this process - from manipulations, to narratives, to possible theoretical models (visual, diagrammatic, symbolic, mathematical) - in a previous work (Magnani, 2001a) I have considered some observational techniques and representations made by Faraday, Davy, and Biot concerning Oersted’s experiment about electromagnetism. They were able to create consensus because of their conjectural representations that enabled them to resolve phenomena into stable perceptual experiences. Some of these narratives are very interesting.
Epistemic Mediators
Recent research, taking an explicit ecological approach to the analysis and design of human-machine systems (Kirlik, 1998), has shown how expert performers use action in everyday life to create an external model of task dynamics that can be used in lieu of an internal model: “Knowing of not, a child shaking a birthday present to guess its contents is dithering, a common human response to perceptually impoverished conditions”. Not only a way for moving the world to desirable states, action performs an epistemic and not merely performatory role: people structure their worlds to simplify cognitive tasks but also in presence of incomplete information or of a diminished capacity to act upon the world when they surely have less opportunities to know. Epistemic action can also be described as resulting from the exploitation of latent constraint in the human-environment system. This additional constraint grants additional information: in the example of the child shaking a birthday present he “takes actions that will cause variables relating to the contents of the package to covary with perceptible acoustic and kinesthetic variables. Prior to shaking, there is no active constraint between these hidden and overt variables causing them of carry information about each other”. Similarly, “one must put a rental car ‘through its paces’ because the constraints active during normal, more reserved driving do not produce the perceptual information necessary to specify the dynamics of the automobile when driven under more forceful conditions ” (Kirlik, 1998). Moreover, a very interesting experiment is reported concerning short-order cooking at a restaurant grill in Atlanta: the example shows how cooks at various level of expertise use external models in the dynamic structure of the grill surface to get new information otherwise inaccessible.
We know that theoretical abduction certainly illustrates much of what is important in abductive reasoning, especially the objective of selecting and creating a set of hypotheses (diagnoses, causes, hypotheses) that are able to dispense good (preferred) explanations of data (observations), but fail to account for many cases of explanations occurring in science or in everyday reasoning when the exploitation of the environment is crucial. The concept of manipulative abduction is devoted to capture the role of action in many interesting situations: action provides otherwise unavailable information that enables the agent to solve problems by starting and performing a suitable abductive process of generation or selection of hypotheses.
Also natural phenomena can play the role of external artifactual models: the stars are not artifacts, but under Micronesian navigator’s manipulations of the images of them, the stars acquire a structure that “becomes one of the most important structured representational media of the Micronesian system” (Hutchins, 1995, p. 172). The external artifactual models are endowed with functional properties as components of a memory system crossing the boundary between person and environment (for example they are able to transform the tasks involved in allowing simple manipulations that promote further visual inferences at the level of model-based abduction). 
Not all epistemic and cognitive mediators are preserved, saved, and improved, as in the case of the ones created by Galileo at the beginning of modern science (see the following subsection). For example, in certain non epistemological everyday emergency situations some skillful mediators are elaborated to face possible dangers, but, because of the rarity of this kind of events, they are not saved and stabilized. Hutchins (1995, pp. 317-251) describes the interesting case of the failure of an electrical device, the gyrocompass, crucial for navigation, and the subsequent creation of substitutive contingent cognitive mediators. These cognitive mediators consist of additional computations, redistributions of cognitive roles, and finally, of the discovery of a new shared mediating artifact in terms of divisions of labor - the so-called modular sum that is able to face the situation.
Finally, we have to observe that many external things that usually are inert from the epistemological point of view can be transformed into epistemic or cognitive mediators. For example we can use our body: we can talk with ourselves, exploiting in this case the self-regulatory character of this action, we can use fingers and hands for counting; we can also use external “tools” like writing, narratives, others persons’ information, concrete models and diagrams, various kinds of pertinent artifacts. Hence, not all of the cognitive tools are inside the head, sometimes it is useful to use external objects and structures as epistemic devices. We indicated above that Micronesian navigator’s stars, that are natural objects, become very complicated epistemic artifacts, when inserted in the various cognitive manipulations (of seeing them) related to navigation.
Experiments and the “World of Paper”
Already in the Dialogues Concerning the Two Chief World Systems (1632), accentuating the role of observational manipulations Galileo presents an anatomist that, manipulating a cadaver, is able to get new, not speculative, information that goes beyond the “world of paper” of the Peripatetic philosophy. It is well known that recent philosophy of science has paid a great attention to the so-called theory-ladenness of scientific facts (Hanson, Popper, Kuhn). Nevertheless a lot of new information in science is reached by observations and experiments, and experiments are the fruit of various kinds of artifactual manipulations: the different strategies correspond to the expert manipulations of objects in a highly constrained experimental environment, directed by abductive movements that imply the application of old and new extra-theoretical templates of behavior. 
With Galileo’s achievements, we observe that human “scientific” thinking is related to the manipulation of a material and experimental environment that is no longer natural. Knowledge is finally seen as something cognitively distributed across scientists, their internal “minds”, and external artifacts and instruments. Experiments and instruments embody in their turn external crystallization of knowledge and practice. Modern science is made by this interplay of internal and external. An immediate consequence of Galileo’s ideas is the critique of the authority, that advocated the knowledge relevance of a “world of paper”, mainly internal from the cognitive point of view. Gooding observes: “It is ironical that while many philosophers admire science because it is empirical as well as rational, philosophical practice confines it to the literary view that Galileo rejected” (1990, xii). Galileo’s “book of nature” and his systematic use of the telescope are the revolutionary epistemic mediators that characterize the cognitive power of the new way of producing intelligibility. 
Manipulative Templates
We still know very little about what governs the action-based abduction. I plan to better delineate some of the manipulative templates of behavior that are active in creative abduction comparing scientific and everyday reasoning: 1. action elaborates a simplification of the reasoning task and a redistribution of effort across time when we “need to manipulate concrete things in order to understand structures which are otherwise too abstract” (Piaget 1974), or when we are in presence of redundant and unmanageable information; 2. action can be useful in 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 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; 4. 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: experimental manipulations exploit artificial apparatus to free new possible stable and repeatable sources of information about hidden knowledge and constraints; 5. in science experimental action shows a sensibility to the aspects of the phenomenon which can be regarded as curious or anomalous; manipulations have to be able to introduce potential inconsistencies in the received knowledge (Oersted’s report of his well-known experiment about electromagnetism is devoted to describe some anomalous aspects that did not depend on any particular theory of the nature of electricity and magnetism; 6. action exhibits a preliminary sensibility to the dynamical character of the phenomenon, and not to entities and their properties, common aim of manipulations is to practically reorder the dynamic sequence of events in a static spatial one that should promote a subsequent bird’s-eye view (narrative or visual-diagrammatic). 
For example, it is clear that in geometrical construction many of the requirements above are fulfilled. Geometrical constructions present situations that are curious and “at the limit”. They 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 the features typical of manipulative reasoning illustrated above, such as the simplification of the task and the capacity to get visual information otherwise unavailable.
Geometrical construction is a kind of              manipulative abduction
Let’s quote an interesting Peirce’s passage about constructions. Peirce says that mathematical and geometrical reasoning “consists in constructing a diagram according to a general precept, 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” (CP, 1.54). Not dissimilarly Kant says that in geometrical construction “[…] I must not restrict my attention to what I am actually thinking in my concept of a triangle (this is nothing more than the mere definition); I must pass beyond it to properties which are not contained in this concept, but yet belong to it” (Kant, 1929, A718-B746, p. 580).
We have seen that manipulative abduction is a kind of abduction, usually model-based, that exploits external models endowed with delegated (and often implicit) cognitive roles and attributes.
1. The model (diagram) is external and the strategy that organizes the manipulations is unknown a priori.
2. The result achieved is new (if we, for instance, refer to the constructions of the first creators of geometry), and adds properties not contained before in the concept (the Kantian to “pass beyond” or “advance beyond” the given concept, Kant, 1929, A154-B193/194, p. 192).
Mechanizing Manipulative Abduction
Humans and other animals make a great use of perceptual reasoning and kinesthetic abilities. We can catch a thrown ball, cross a busy street, read a musical score, go through a passage by imaging if we can contort out bodies to the way required, evaluate shape by touch, recognize that an obscurely seen face belongs to a friend of ours, etc. Usually the “computations” required to achieve these tasks are not accessible to a conscious description. Mathematical reasoning uses language explanations, but also non-linguistic notational devices and models. Geometrical constructions represent an example of this kind of extra-linguistic machinery we know as characterized in a model-based and manipulative - abductive -way. 
Automatic geometrical constructions as epistemic mediators
A very interesting artificial intelligence computer program has been built, ARCHIMEDES (Lindsay, 1994, 1998), that represents geometrical diagrams (points, line segments, polygons, and circles) both as pixels arrays and as propositional statements. For example a triangle will be represented from a propositional description as a set of marked pixels in an array, together with a set of data naming the given triangle and storing facts about it (for instance that it is right) and constraints upon it (perhaps that it remains right throughout this use of the diagram). Hence, a computational equivalent of a physical diagram is represented, plus some human propositional knowledge about it. 
The program is able to manipulate and modify its own representations of diagrams, that is it is able to make geometrical constructions (called “simulation constructions”): adding parts or elements, moving components about, translating and rotating by preserving metric properties, of course subordinated to the given specific constraints and to the whole structure of the two-dimensional space. Some knowledge of algebra is added, and of the taxonomic hierarchy of geometric figures (all square are rectangles, etc.); moreover, it is also added additional knowledge like side-angle-side congruency theorem and the sum of the interior angles of a triangle, knowledge of problem solving strategies and heuristics, knowledge of logic (for example: a universal statement can be disproved by a single counterexample) (Lindsay, 2000b).
When the program manipulates the specific diagram, it records the new information that comes out, then it can for example detect sets of area equivalences, and so on: for example, it is able to verify that a demonstration of the Pythagorean Theorem is correct, mirroring its truth in terms of constructions and manipulations. To account for the universality of geometrical theorems and propositions many different methods for learning and “generalizing” the specific instance of the constructed diagram are exploited (Lindsay, 1988, pp. 260-264). 
These methods come from a kind of predicative knowledge of course “exogenous” to the mere diagrammatic representation: generalization is not a possible product of the pure diagrammatic understanding. For instance, one suggestion is to break the problem into cases, to individuate a “representative” instance for each case, and to demonstrate that this conclusion holds for each of these instances. Another one is to exploit the simulative aspects of constructions by “running experiments” that show how some parts of a diagram co-vary with changes in others: the observation of the interaction of the diagrams parts as one property is varied allows us to grasp and understand the “universal” value of some geometric relations and results (for example congruency theorem, asymptotic behaviors, periodic relations, and some symmetric relations). It is interesting to note that one of the construction manipulations proposed by the program to verify the Pythagorean Theorem intends to show that it is not true of (some examples of) non-right triangles.
An extension of the program (described in Lindsay, 2000a), aims to autonomously build constructions that can demonstrate a given proposition, rather than simply verify them. It accomplishes the further complex task of discovering conjectures that can lead to the constructions of demonstrations, illustrating the possible role of diagrammatic reasoning in creativity. 
Automatic “thinking through doing”
Geometric constructions are certainly epistemic mediators that exploit the semantics of two dimensional diagrams (rather than the syntax of formal propositions) to perform various manipulative abductive tasks (discover a new property or new proposition/hypothesis, selecting suitable sequences of constructions as able to convincingly verifying theorems, etc). 
Hence, geometrical construction, one of the most ancient exploitations of two-dimensional diagrams for both practical and mathematical problem solving, is “embodied” in a computational program, that is, finally, in a machine. From the epistemological point of view it is important to note that the program shows how it is possible to delineate the rules and the procedures that underlie the diagrams as models of propositions about space, that is able to capture the structure of space. The kind of reasoning described is very rich and takes advantage of almost all the resources of two-dimensional space (going beyond the simple use of topological properties like in the case of Euler/Venn diagrams). Moreover, the physical diagram necessarily (that is for the objective reasons of its materiality) preserves topological and geometric properties of two-dimensional space. The following passage is very clear: 
Geometric diagrams […] are intended to reflect all structural properties of two dimensional space, although this requires that the observer ignore line width and so forth and deal with the intended idealization. Since they are recorded in space (on a piece of paper for example) the mapping is iconic. The intuitive structure of two-dimensional space is partially captured by Euclid’s definitions, postulates and axioms. Although it is now known that this is not the veridical description of actual space, for objects of human scale it is adequate both for reasoning and for successful intercourse with the environment, just as Newtonian mechanics is adequate for modeling physical processes that occur at a human scale though it is inaccurate at very large and very small scale […] (Lindsay, 2000b).
We know that the structure of intuitive space is also embraced by analytic geometry. Lindsay observes that, in general, it could be better to use an analytic representation because conventional digital computers are “a natural match for numerical representation”. If we consider the various ways of representing geometrical diagrams and their behavior - the analytic representation is an example - it is important to point out their real cognitive nature. I think we have to agree with the following position: “This does not mean, however, that diagrams represented numerically are not really diagrams. What makes them diagrams is not bits or voltages or axioms or CCD signals. What makes them diagrams is that they capture the structure of space. This is another way of saying that they enforce constraints on the behavior of the representations that reflect restrictions on the behavior of objects in space” (ibid.).
The computational embodiment generates a kind of “squared” epistemic mediator: geometrical constructions, as epistemic mediators, are further mediated. It is not simply reproduced a mental performance like in more traditional AI systems, but just a way of “thinking through doing”, as illustrated by manipulative abduction. Humans can think using geometrical constructions also without “doing”, for instance in the case of “thinking through drawing” at the level of imagination. Kant (and Proclus) were perfectly clear when they referred to the role of imagination as the condition of possibility of the empirical drawing itself (Magnani, 2001b). 
It has been established that imagination is able to perform some tasks at intermediate levels of complexity. Humans, when working from an external physical display, are aided by actual pencil shading or tracing. When working from mental images humans have greater difficulty keeping track of what has been inspected. Moreover, in the case of imaginary geometrical representations (or in data structures in a computer) we can use whatever rules we want (for instance the rules that reflect the motion of rigid bodies but also the ones that violate them and do not happen - or are not yet seen to happen - in the physical world). This of course does not mean that the mind can imagine anything whatsoever, or everything with equal facility: “[…] human mind is indeed predisposed to handle certain types of imagery and simulations better than others, presumably those kinds for which evolution has best prepared us, especially the motion of rigid objects” (Lindsay, 1988, p. 267).
In the past some researchers emphasized the role of visualizations in geometrical reasoning. For example in 1847 Byrne prepared an amazing and attractive edition of the first six books (which range from the most elementary plane geometry to the theory of proportions) of the Elements of Euclid “in which coloured diagrams and symbols are used instead of letters for the greater ease of learners”. The aim was to try to use as little text - and in particular, labels - as possible. Byrne considered the use of colors in geometrical constructions a suitable tool to this aim. Interesting exploitations of the visual devices of the Internet are some Java applications/manipulations where it is possible to “construct by clicking”.
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