Abduction and diagrammatic reasoning in a theory of scientific discovery
This paper sketches a theory of scientific discoveries that is mainly based on two concepts that Charles Peirce developed: abduction and diagrammatic reasoning. Both are problematic. While “abduction” describes the process of creating a new idea, it does not, on the one hand, explain how this process is possible and, on the other, is not precisely enough defined to distinguish different forms of creating new ideas. “Diagrammatic reasoning,” the process of constructing relational representations of knowledge areas, experimenting with them, and observing the results, can be interpreted, on the one hand, as a methodology to describe the possibility of discoveries, but its focus is limited to mathematics. The theory sketched here develops an extended version of diagrammatic reasoning as a general theory of scientific discoveries in which eight different forms of abduction play a central role.
Charles Peirce’s concept of “abduction” is wellknown in philosophy of science as a contribution to the question how scientific discoveries are possible (e.g., Hanson 1972 <1958>; Simon 1979; Nickles 1980b; Nickles 1980a; Grmek et al. 1981; Jason 1988; Kleiner 1993; Haaparanta 1993; Meheus & Nickles 1999; Magnani et al. 1999; Magnani 2001; Magnani & Nersessian 2002). However, as I will show in the first part of this paper, many things Peirce himself says about abduction are more confusing than helpful when it comes to explain how “the process of forming an explanatory hypothesis,” as he defines “abduction” (Peirce CP 5.171), might be possible.
Therefore, the second part of this paper focuses on another concept Peirce developed for the first time: “diagrammatic reasoning.” The central idea of this kind of reasoning is that we have to represent what we know about something in order to see problems, what again stimulates creativity. Those creative possibilities, on the other side, can be specified by a distinction of different forms of abduction.
The distinctions I am suggesting here are based on two further concepts Peirce developed: “hypostatic abstraction” and the “theoric transformation” of a problem. “Hypostatic abstraction” can be defined as creating a new sign for a new object by transforming a concrete predicate into an abstract noun. While in “prescissive abstraction” we abstract from features like color and width to define a geometrical line, in “hypostatic abstraction” we turn what can be a predicate of many things—honey is sweet, strawberries are sweet, sugar is sweet—into “a subject of thought” (CP 5.534): “sweetness” (CP 4.235). Since “hypostatization”—from the Greek hypostasis—is the same as “reification” in Latin, I will use here this latter concept which is better established today. Whether “hypostatization” or “reification,” both concepts mean creating a new thing out of what is not a thing. Since all concepts of our languages are outcomes of reification performed at some time in the history of our cultures, reification would be one of the most fundamental concepts to describe the genesis of knowledge.
The second Peircean term I will discuss here as interesting to describe scientific discoveries is “theoric transformation.” For Peirce, “theoric” refers to the Greek “theoria” (original meaning: “vision”). He translates this term as “the power of looking at facts from a novel point of view” (Peirce MS 318: CSP 50 = ISP 42). “Theoric” reasoning consists “in the transformation of the problem,—or its statement,—due to viewing it from another point of view” (ibid., CSP 68 = ISP 225). Thus, a “theoric transformation,” or a “theoric step” in a deductive argument, means changing the perspective. We are looking at the same data, or the same representation, but in a way that opens up completely new horizons of interpretation. Peirce hints at the wellknown fact that especially developing the idea of a proof in mathematics often depends on a “theoric” shift (CP 4.612). But we can use this concept also beyond the limits of mathematics. For example, when Aldo Leopold saw for the first time that ecological relations are not simply causal relations—remove the wolves to enlarge the deer population—but that he has “to think like a mountain” in order to being able to manage an ecosystem as a multidimensional configuration (Norton 2005, p. 213ff.), he performed a “theoric transformation,” a “perceptual shift” (219).
The problems we are facing with Peirce’s concept of abduction can be illustrated when we simply take a look at one his best known definitions of this term:
Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis. (Peirce CP 5.171)
The second half of this quote is not part of the definition, but an explanation for it. However, it adds something to this definition because it says implicitly that there are only three logical operations for Peirce, a claim that he confirms in another remark where he says that “there are but three elementary kinds of reasoning”: abduction, deduction, and induction (CP 8.209). This means, however, that any form of “reasoning,” or “logical operation,” that is neither deduction nor induction has to be abduction. The problem that arises here is that the concept of abduction becomes very broad. We do not only find abduction in science as the process of “examining a mass of facts and in allowing these facts to suggest a theory” (CP 8.209), but also in any perception “when I so much as express in a sentence anything I see” (Peirce LOS, p. 899f.; cf. CP 5.182ff., 8.64). Even when “a chicken first emerges from the shell” and “does not try fifty random ways of appeasing its hunger, but within five minutes is picking up food, choosing as it picks, and picking what it aims to pick,” this is “just like abductive inference” (Peirce LOS, p. 899f.).
A second problem concerns the claim that abduction is supposed to be a “logical operation.” It has been discussed elsewhere in detail so that there is no need to go again into this (Hoffmann 1999). And a third problem concerns the question what exactly abduction is supposed to do. In the quote above, Peirce hints, on the one hand, at the “the process of forming an explanatory hypothesis” and, on the other, at introducing a “new idea.” This, however, allows different interpretations. First, it is possible that the two concepts refer to two different operations, because we can form an “explanatory hypothesis” without generating a “new idea.” If any perception is a case of abduction, we would generate a lot of “explanatory hypotheses” regarding sensory inputs though hardly any “new idea.” For example, when reading a word, the word we read is a hypothesis that “explains” a sequence of letters. In this case, we form an explanatory hypothesis without introducing a new idea since the idea we associate with a certain sequence of letters exists already in our mind.
Second, it is not clear whether the “idea” we introduce is only new for us as individuals or new for our civilization. And third, an “idea” can either be what we discussed above as the result of a “reification,” that is something that can be represented by a singular concept, or by a symbol, or it could be a new perspective on the same data as produced by a “theoric transformation.” In sum, Peirce’s innocent and wellknown definition can describe six very different forms of abduction (cf. Table 1).

If “idea” means something that can be represented by a singular symbol (based on reification) 
If
“idea” means the (theoric transformation) 
If an explanation is possible by referring to an idea that exists already in our mind 
identifying abduction (as in reading) 
Gestalt shift 
If we create an idea that is new for us, but that exists already as a part of our culture’s knowledge 
analogical reifying abduction 
analogical theoric abduction 
If we create an idea that is entirely new 
creational reifying abduction 
creational theoric abduction 
Table 1: Six forms of abduction that are possible based on CP 5.171 alone
The most important problem with Peirce’s concept of abduction, however, concerns the question how our ability to infer abductively can be explained. Peirce hints at many places at an “instinctive” power of “guessing rightly” (cf. Rescher 1995; Fann 1970, pp. 35–38), and also at “the uncontrolled part of the mind” (CP 5.194; cf. Semetsky, 2005). But since none of these approaches is sufficiently elaborated, the possibility of abduction remains at the end unexplained. In this situation, we should focus on methods that might, at least, facilitate abduction. Just this is the role “diagrammatic reasoning” can play in a general theory of scientific discoveries.
Peirce defines “diagrammatic reasoning” as a threestep process: (a) constructing a representation, (b) experimenting with it, and (c) observing the results (Peirce NEM IV 47f.; cf. Stjernfelt 2000; Hoffmann 2003; 2004). However, the essential feature of diagrammatic reasoning that makes this method so interesting for the description of scientific discoveries is not this threestep process, but rather that the whole process has to be performed by means—and within the limits—of a given “system of representation” (CP 4.418). When Peirce defined a “diagram” as “a representamen which is predominantly an icon of relations” that “should be carried out upon a perfectly consistent system of representation” (CP 4.418), he worked on a series of definitions to formulate the foundations of his socalled “Existential Graphs,” a graphical notation of logic intended to replace algebraic notations (cf. Roberts 1973; Ketner 1996 <1990>; Shin 2002). This system of representation is characterized by a set of conventions to represent propositions and logical relations between those propositions, and a set of rules for the transformation of graphs.
There is no question that one needs “a perfectly consistent system of representation” for representing logical implications and validity, and the same is true for representation systems in mathematics. If we want to prove in Euclidean geometry, for example, that the triangle’s inner angles sum up to 180°, we can use a parallel to the triangle’s base as auxiliary line to perform the proof. The possibility of such a parallel, however, is provided only in Euclidean geometry, not in nonEuclidean ones. That is, the specific means available in this system of representation determine the outcome of any transformation of a Euclidean figure we perform in accordance with the rules of this system. At the same time, however, those operations are also constrained by the means available. If we change the system of representation—as has been done in the development of nonEuclidean geometries—we open up entirely new possibilities (cf. Hoffmann 2004). In short, a proof in mathematics is only as perfect and consistent as the representation system in which it is performed.
The consistency of representation systems is also essential when we use the concept of diagrammatic reasoning—beyond the limits of logic and mathematics—for a general theory of scientific discoveries. Not only axiomatic systems in mathematics have to be consistent, but also, for instance, the description of styles in art, the grammar we construct to understand the syntax of our everyday languages, and theories in science.
Besides consistency, there are two further features that are common to all those systems of representation. On the one hand, we need them to design a particular representation. We need an axiomatic system to construct a proof in mathematics; we need a scientific theory to formulate a hypothesis or observational statement; we need the grammar of our everyday language to formulate a normal sentence; and we need a style to draw a painting in art—although in both the latter cases knowledge about the system of representation might be mostly implicit. On the other hand, all these systems of representation play a normative role: in logic and mathematics we can check the validity of an inference, or a proof, by means of the rules and conventions defined by a certain system of representation; in science the plausibility of a particular statement depends on its theoretical background; in our everyday languages the correctness of expressions is determined by grammar; and in art the definition of styles is a means of classification, for example.
The central role of the chosen representation system for diagrammatic reasoning becomes visible when we consider its function for each of the three steps by which diagrammatic reasoning can be defined (cf. Figure 1). As this “map definition” of diagrammatic reasoning shows, the outcome of each of the three steps depends—in different ways—on the system of representation we choose in a certain situation to represent a problem, or the knowledge area we are focusing on.
The consistency and normativity of the chosen system of representation are decisive when it comes to the possibility of discoveries. Peirce once highlighted as a core idea of this pragmatism that all reasonings—and especially mathematical reasonings
turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a “practical consideration.” (CP 5.9)
Such an “inevitableness” depends obviously on the normative character of the representational system in which such reasoning is performed, as is evident from mathematics: That 2 plus 2 equals 4 results from the rules and conventions of arithmetic as the chosen system of representation.
This “inevitable experience” is the most important precondition of discovering something new by diagrammatic reasoning. As has been showed elsewhere with regard to mathematics, we can distinguish two possibilities: on the one hand, the process of unfolding new implications of a diagram within a given system of representation and, on the other, the process of developing representational systems themselves that can open up new horizons and possibilities—as it has been the case in the development of nonEuclidean geometries (Hoffmann 2004).
However, if we try to generalize what is plausible with regard to mathematics, we are facing a serious problem. While we can hardly overemphasize the role of representations in any science, it is harder to determine what exactly the “systems of representation” in science are that we need to explain the possibility of scientific discoveries by means of diagrammatic reasoning. While in mathematics and logic we can mostly formulate a clear distinction between a “consistent system of representation” (e.g., an axiomatic system, or a notation) and a “diagram” constructed by means of this system, this is not so easy in the sciences. Of course, we could say that any science uses different—more or less formal—languages so that its representations depend on the “grammar” of those languages. But just this is part of the problem. There are quite a lot of those “languages,” and they are more or less relevant for what happens in scientific activity. We would assume, for example, that it does not matter whether we formulate Einstein’s theory of relativity in English or in Chinese. Sometimes, however, it matters, for example when we formulate a sociological theory that is based on ideas, or concepts, for which there is no equivalent in another culture.
The problem is, thus, that there are very different systems of representation that we often use at the same time when representing scientific knowledge, and that not all of them determine and constrain the possibilities of scientific activities in the same way as the Euclidean language determines proof possibilities in geometry. It seems to be impossible to propose a classification of all systems of representation we are using in different disciplines, and to determine in general which is relevant in which way.
The only way I can see to cope with this problem is not to start with a “top down” classification of representation systems, but to proceed “bottom up.” It should be sufficient to define the term “system of representation” only formally as that consistent set of conditions we have to presuppose to understand a representation, and to look at diagrammatic representations themselves guided by the question ‘What kind of representation system do we need to make sense out of this expression?’. When we read a scientific text in which Sun, Moon, and Mars are called “planets,” for example, a consistent representation system that could produce such an utterance would be the Ptolemaic astronomy, but not the Copernican system of representation with its classification of celestial bodies.
From a general point of view, the term “system of representation” refers first of all to those conditions of understanding. Therefore, those systems can only be determined based on an analysis of concrete representations, they are relative to what they are supposed to determine. It depends on the concrete diagram what we have to identify as a relevant system of representation. Relevant is what we need to understand the diagram.
In order to show how, based on these considerations, the possibility of scientific discoveries can be explained, I suggest extending the definition of diagrammatic reasoning proposed in Figure 1 above. We have to continue the series of steps that are described in this definition. A complete description of what follows after the third step described in Figure 1 is presented in Figure 2. Remember, the first three steps that are already mentioned by Peirce are: constructing a diagram, experimenting with it, and observing—and analyzing—the outcome of those experiments. The continuation of the process depends, obviously, on the result of the third step. As elaborated in Figure 2, we can distinguish three possible results, each combined with a certain implication that will then again lead—at least in two cases—to further steps.
The basic ideas of this enlarged model of diagrammatic reasoning can be summarized as follows:
1. There is only one situation in which the process of diagrammatic reasoning comes—temporarily—to a halt: when we get the first possible result of observing and analyzing the outcome of an experiment, that is “the outcome does not contradict our expectations” (see Figure 2). In this situation we see that what we observe in the experiment is a necessary implication of the original diagram, and we can be happy that no contradiction occurs. There is no reason to continue the process. We learned something new, although nothing that could surprise us.
2. Everything else besides this first possibility leads over a series of further steps finally back again to the first step, the construction of a diagram. This means, the whole process is with regard to the activities described in Figures 1 and 2 an endless loop that stops only when the situation mentioned in (1.) occurs.
3. However, regarding the representational means available, each of these loops continues with the first step on a more advanced level: we are either able to construct a new diagram by the means of the representation system that we used already from the very beginning, or we are able to construct a new diagram based on a new system of representation.
4. The possibility of such an advancement is the result of scientific discoveries that are differentiated in this model as eight forms of abduction (1. to 8. in the second half of Figure 2). The terminology according to which these forms are classified goes back to the concepts “reification” and “theoric transformation” as defined at the end of the introduction, and to the terms I coined based on that with regard to abduction in Table 1. Each of these forms of abduction enlarge the set of representational—and therefore also: cognitive (cf. Hoffmann & Roth forthcoming)—means available for diagrammatic reasoning.
5. The distinction of all the possibilities differentiated in Figure 2 is based on an analytical interest to separate different forms of scientific discoveries—or elements of those discoveries—as clear as possible. In practice, however, these forms will usually be intertwined, I guess.
The essential point is, however, that each of these eight possible forms of abduction leads to an enlargement of the representational and cognitive means that are then available for further processes of diagrammatic reasoning. And this again is what scientific progress is all about.
Figure
SEQ "Figure" \*Arabic 1: A definition of diagrammatic reasoning. All maps are
created with
IHMC Cmap tools: http://cmap.ihmc.us/
Figure SEQ
"Figure" \*Arabic 2: The process of diagrammatic reasoning, extended version,
continuing the series of three steps described in Figure 1.
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