Abstract
Einstein famously said, “Imagination is more important than knowledge”. But how to study imagination and how to represent and communicate what the content of imagination may be in the context of scientific discovery? In 1908 Peirce stated that deduction consists of “two sub-stages”, logical analysis and mathematical reasoning. Mathematical reasoning is further divisible into “corollarial and theorematic reasoning”, the latter concerning an invention of a new icon, or “imaginary object diagram”, while the former results from “previous logical analyses and mathematically reasoned conclusions”. The iconic moment is clearly stated here, as well as the imaginative character of theorematic reasoning. But translating propositions into a suitable diagrammatic language is also needed: A diagram is for Peirce “a concrete but possibly changing mental image of such a thing as it represents”. “A model”, he held, “may be employed to aid the imagination; but the essential thing to be performed is the act of imagining” (MS 616, 1906). Peirce had observed that the importance of imagination in scientific investigation is in supplying an inquirer, not with any fiction but, in quite stark contrast to what fiction is, with “an inkling of truth”. Since Peirce’s limit notion of truth precludes gaining any direct insight into the truth, in rational inquiry the question of what the truth may be or what it could be needs to be tackled by imagination. This imaginative faculty is aided by diagrams which are iconic in nature. The inquirers who imagine the truth “dream of explanations and laws”. Imagination becomes a crucial part of the method for attaining truth, that is, of the logic of science and scientific inquiry, so much so that Peirce took it that “next after the passion to learn there is no quality so indispensable to the successful prosecution of science as imagination”. In this paper we investigate aspects of scientific reasoning and discovery that seem irreplaceably dependent on a Peircean understanding of imagination, abductive reasoning and diagrammatic representations.
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Notes
- 1.
https://www.youtube.com/watch?v=RXABcv9djQ0 (see also Feynman 1964, 156).
- 2.
Peirce told his students to possess a “secret” about necessary consequences, which is “a very useful thing to know, although most logicians are entirely ignorant of it. It is that not even the simplest necessary consequence can be drawn except by the aid of Observation, namely, the observation of some feature of something of the nature of a diagram, whether on paper or in the imagination. I draw a distinction between Corollarial consequences and Theorematic consequences. A corollarial consequence is one the truth of which will become evident simply upon attentive observation of a diagram constructed so as to represent the conditions stated in the conclusion. A theorematic consequence is one which only becomes evident after some experiment has been performed upon the diagram, such as the addition to it of parts not necessarily referred to in the statement of the conclusion” (MS 455–6, Lowell Lecture II, 1903).
- 3.
For example Nersessian (2002, 137) simply dismisses abduction as that mode of reasoning worthy of further consideration in model-based accounts in scientific discovery, on the grounds that it has not been specified what the nature of the underlying processes of abductive inference really is. That is, the charge is that abductive inference does not seem to follow rules. But abduction does have rules, although they are rules of different kind from rules of inference in deductive or inductive arguments, or even from those of reasoning by analogical modes (see Pietarinen and Bellucci 2014 on what Peirce’s notion of retroduction/abduction in relation to the other two stages of reasoning seems to amount to).
- 4.
By hypostatic abstraction we convert a term that connotes (a predicate) into one that denotes (a subject). We transfer matter from the signification to the denotation or, as Peirce sometimes says, from the interpretant to the object. Hypostatic abstraction is not just the principle engine of mathematical reasoning. Hypostatic abstraction is a very primordial ingredient of every form of thinking whatsoever.
- 5.
Hookway (2010) has argued that Peirce’s insistence on the idea of a ‘form of relation’ suggests a position akin to structuralism in the philosophy of mathematics. Pietarinen (2010a) argues against that view on a number of counts, including structuralism’s neglect of (i) experimentation and observation on the diagrammatic forms of relations, (ii) the kinds of reality of objects in structuralism which are all-important in Peirce’s theory of signs and philosophy of mathematics, (iii) the continuity of forms, as well as (iv) the essentially hypothetical and modal notions that characterise mathematical assertions. Pietarinen (2014) suggest a closer alliance of Peirce’s pragmatist philosophy of mathematics with that of modal-structuralism, although that, too, suffers from nominalism with respect to the semantics of its key objects, namely those of possible worlds (see also Pietarinen 2005, 2011).
- 6.
We leave for further occasion a discussion on the relevance of metaphors in the contexts of discovery, abductive and the model-based reasoning in science, noting only that metaphors are also icons for Peirce: they are the third category of “hypoicons”, in addition to the first of images and the second of diagrams (Pietarinen 2008).
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Acknowledgments
Research supported by Estonian Research Council Project PUT267 and the Academy of Finland project no. 12786, Diagrammatic Mind: Logical and Communicative Aspects of Iconicity, Principal Investigator Ahti-Veikko Pietarinen. The second author is the main author of Sect. 21.4, the first the main author of other sections. The paper was completed thanks to the 2014–2015 Foreign Experts Program of China at Xiamen University. Early version of the paper, bearing the title “On the Possibility of the Logic of Real Discovery”, was presented as a keynote to the International Symposium of Epistemology Logic and Language, organized by the Centre for Philosophy of Science at the University of Lisbon, Portugal, in October 2012. We thank the editors, organizers and the audience for comments and discussion.
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Pietarinen, AV., Bellucci, F. (2016). The Iconic Moment. Towards a Peircean Theory of Diagrammatic Imagination. In: Redmond, J., Pombo Martins, O., Nepomuceno Fernández, Á. (eds) Epistemology, Knowledge and the Impact of Interaction. Logic, Epistemology, and the Unity of Science, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-26506-3_21
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