Abstract
The themes introduced in this chapter illustrate some important aspects of geometrical construction critical to correctly posing the problem of the relationships between geometry and cognition (cf. the following chapter). In the history of philosophy there are at least three main ways for designing the role of geometrical construction in hypothesis generation, always considered in the perspective of problem solving performances. All aim at demonstrating that the activity of geometrical construction is paradoxical, either illusory or obscure, implicit, and not analyzable.
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In the following section and in section 4, these steps of problematical analysis will be characterized as abductive.
On the difference between knowledge and mere true opinion, and other methodological and epistemological considerations on a modern interpretation of Plato’s Meno, cf. Glymour (1992).
“SOC. Do you see what a captious argument you are introducing — that, forsooth, a man cannot inquire either about what he knows or about what he does not know? For he cannot inquire about what he knows, because he knows it, and in that case is in no need of inquiry; nor again can he inquire about what he does not know, since he does not know about what he is to inquire” (Plato, 1977, 80e, p. 301).
That is a kind of definition that prescribes “what you are to do in order to gain perceptual acquaintance with the object of the world” (CP, 2.330).
This kind of reasoning is also called by Peirce “theorematic” and it is a kind of “deduction” necessary to derive significant theorems: “is one which, having represented the conditions of the conclusion in a diagram, performs an ingenious experiment upon the diagram, and by observation of the diagram, so modified, ascertains the truth of the conclusion”. The “corollarial” reasoning, mechanical and not creative, “is one which represents the condition of the conclusion in a diagram and finds from the observation of this diagram, as it is, the truth of the conclusion” (Peirce, CP, 2.267, cf. also Hoffmann, 1999).
This event constitutes in its turn an anomaly that needs to be solved/explained.
On the special meaning of the adjective “inferential” used by Peirce see section 2.2 below.
It is clear that the two meanings are related to the distinction between hypothesis generation and hypothesis evaluation, so abduction is the process of generating explanatory hypotheses, and induction matches the hypothetico-deductive method of hypothesis testing (1st meaning). However, we have to remember (as we have already stressed) that sometimes in the literature (and also in Peirce’s texts) the word abduction is also referred to the whole cycle, that is as an inference to the best explanation (2nd meaning).
To follow Peirce’s terminology, in both cases the reasoning preformed is “theorematic”, but in the second case is not creative at all, because it is a proof of an already given theorem.
A logical system is monotonic if the function Theo that relates every set of wffs to the set of their theorems holds the following property: for every set of premises S and for every set of premises S’, S ⊆ S’ implies Teo(S) (S’). Traditional deductive logics are always monotonic: intuitively, adding new premises (axioms) will never invalidate old conclusions. In a nonmonotonic system, when axioms, or premises, increase, their theorems do not (cf. Ginsberg, 1987; Ramoni, Magnani, and Stefanelli, 1989; Lukaszewicz, 1990; Magnani and Gennari, 1997). Following this deductive nonmonotonic view of abduction, we can stress the fact that in actual abductive medical reasoning, when we increase symptoms and patients’ data [premises], we are compelled to abandon previously derived plausible diagnostic hypotheses [theorems], as already — epistemologically — illustrated by the ST-MODEL.
Many approaches have been proposed. For example, the so-called theory of explanatory coherence (Thagard, 1989, 1992) introduces seven principles which encompass the type of plausibility that occurs upon accepting new hypotheses and theories in science. Josephson (2000) has stressed that evaluation in abductive reasoning has to be referred to the following criteria: 1. How a hypothesis surpasses the alternatives. 2. How the hypothesis is good in itself. 3. Its confidence in the accuracy of the data. 4. How thorough was the search for alternative explanations.
Some cognitive limitations of the sentential models of theoretical abduction and a more complete illustration of abductive reasoning are given in Magnani, 2001.
The so-called constructive and generic modeling in scientific discovery are illustrated in Nersessian, et al. (1997).
Cf. for more details Magnani, 2001, chapter 5.
Circles denote concepts (mentally represented) that can be communicated, squares denote things in the material world (bits of apparatus, observable phenomena) that can be manipulated — lines denote actions.
Cf. Minski, 1985 and Thagard, 1997.
Further aspects of experiment design and its relationship with the problem of communication in science during the transition from the personal to the public domain are given in Gooding and Addis (1999): only a small subset of many observations and measurements performed by individuals of research teams acquire the status of real and public phenomena. Moreover, additional properties of the agent in a scientific experimental setting are described: 1. ability to discriminate between observed results, 2. ability to make judgments about the likelihood of the occurrence of a result, 3. flexibility of the agent’s change in perception of the world and his consequent capacity to respond to new information, 4. degrees of competence to build an experiment and observe the results, from novices to experts.
I derive this expression from the cognitive anthropologist Hutchins (1995), who coined the expression “mediating structure” to refer to various external tools that can be built to cog-nitively help the activity of navigating in modern but also in “primitive” settings. Any written procedure is a simple example of a cognitive “mediating structure” with possible cognitive aims: “Language, cultural knowledge, mental models, arithmetic procedures, and rules of logic are all mediating structures too. So are traffic lights, supermarkets layouts, and the contexts we arrange for one another’s behavior. Mediating structures can be embodied in artifacts, in ideas, in systems of social interactions […]” (pp. 290–291).
It is difficult to preserve precise spatial relationships using mental imagery, especially when one set of them has to be moved relative to another.
It is Hutchins (1995, p. 114) that uses the expression “cognitive ecology” when explaining the role of internal and external cognitive navigation tools. More suggestions on manipulative abduction can be derived by the contributions collected in the recent Morgan and Morrison (1999), dealing with the mediating role of scientific models between theory and the “real world”.
Another example is given by the gestures that are also activated in talking, sometimes sequentially, sometimes in an overlapping fashion.
Together with the exploitation of mathematical models.
In the following chapter I will illustrate the role played in the “origins of geometry” by the interactions with the external world and by the manipulations of external objects by means of instruments and artifacts (sections 1 and 2).
Of course in the case we are using diagrams to demonstrate already known theorems (for instance in didactic settings), the strategy of manipulations is not necessary unknown and the result is not new.
Also Aliseda (1997) provides interesting use of the semantic tableaux as a constructive representation of theories, where abductive expansions and revisions, derived from the belief revision framework, operate over them. The tableaux are so viewed as a kind of reasoning where the effect of “deduction” is performed by means of abductive strategies.
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Magnani, L. (2001). Geometry, Problem Solving, Abduction. In: Philosophy and Geometry. The Western Ontario Series in Philosophy of Science, vol 66. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9622-5_6
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