Mathematical Creativity: A Peircean Abductive Proposal

The mainstay of Peirce's diagrammatic reasoning is deduction, for which he developed a comprehensive family of graphical logics termed existential graphs. He thought, however, that also the two other kinds of inferences, namely, abduction and induction, indirectly depend on diagrammatic reasoning, given that they depend for their justification on deductive reasoning. This chapter presents Peirce's theory of diagrams from the wider philosophical and scientific perspectives. In particular, the chapter addresses the issue of how abductive modes of reasoning too rest on the theory and philosophy of diagrams and how abduction contributes to the value of diagrammatic reasoning in science. The chapter highlights two fields of science that have enjoyed renewed potential arising from their analysis in terms of Peirce's theory of diagrammatic reasoning and abduction: cognitive science and exact mathematical and computational sciences.

T. Jappy (ed), The Bloomsbury Companion to Contemporary Peircean Semiotics, 2020

C.S. Peirce made relevant contributions in very different fields, but he was primarily interested in the logic of science, and more especially in what he called ‘abduction’ – as opposed to deduction and induction – which is the process whereby hypotheses are generated in order to explain surprising facts (Nubiola 2005). Although there are stirrings of it in Aristotle’s notion of apagoge, the modern idea of abduction comes from Peirce (Woods 2017: 137). In fact, Peirce considered abduction to be at the heart not only of scientific research, but also of all ordinary human activities, and in particular artistic creativity (Barrena 2015). In this chapter, first, the nature of abduction will be explained, providing several key textual sources from Peirce’s manuscripts. Second, scientific and artistic creativity will be described in some detail, highlighting that imagination, which plays a central role in abduction, is at the heart of reasoning. Finally, a number of contemporary applications of abduction from the philosophy of science, artificial intelligence and logic will be mentioned. We point out the richness of Peirce’s conception of abduction that far surpasses the contemporary accounts of creativity.

Diagrams, Graphs, and Visual Imagination in Mathematics, 2023

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. As we know from Kant, vision (as a part of human sensibility) and responsiveness to reasons (as supported by our overall conceptual capacities) are related with one another through the imagination. Mathematics is an expression of this relation based on our most fundamental intuitions about space and time. Peirce went a long way to develop Kant’s take on the nature of mathematics, and central to his interpretation of it was the idea of diagrammatic reasoning. According to Peirce, in practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. As thinking, in this case, is actually performed by means of manipulating images, seeing and understanding become one. Defining diagrammatic reasoning as a fusion of vision and thought helped Peirce find some intriguing answers to questions concerning the nature of mathematical knowledge, many of which could not even be as much as formulated by Kant. What is the role of observation in mathematics? How can we explain the fact that mathematical reasoning is deductive and, at the same time, capable of the discovery of new truths? How is mathematical necessity reconciled with the essential incompleteness and indeterminacy of our ordinary visual experience? What exactly is the relationship between the particularity of a mathematical diagram and the generality of the meaning it conveysand what is the difference (if any) between mathematics and natural languages in this respect? Etc. Peirce’s life-long, if unsystematic, work on the issues that are associated with the questions above created an intricate conceptual puzzle. The driving motivation of the research this book represents is to show that tackling this puzzle requires something more than sifting through the wealth of available historical and philosophical material. While the histories of science and philosophy do provide separate bits of the puzzle, Peirce’s theoretical interests, by his own admission, appear to be closely intertwined with certain facts of his personal history. In light of this, without considering relevant biographical data, in Peirce’s case, there is no way to understand how the pieces of the puzzle actually fit together. Due to the plurality of data impelled by the task, this book is addressed both to those specializing in philosophy, mathematics, and intellectual history, and to a wider audience that might be interested in what all those areas have in common in Peirce’s case. Last but not least, this book could not have been written without the support of family, friends, and colleagues. I am especially grateful to Eric Bredo, Marcel Danesi, Paul Forster, Nathan Houser, Henry Jackman, Steven Levine, Mark Migotti, and James O’Shea. Of great importance for the book were my conversations with Kathleen Hull and Thomas L. Short. As to the biographical part of the study, I am indebted to Joseph Brent, the author of the most comprehensive biography of Charles Peirce to date. Finally, I owe much more than I can tell to my constant companions and interlocutors, Zina Uzdenskaya and Gleb Kiryushchenko.

in Epistemology, Knowledge, and the Impact of Interaction, ed. by Juan Redmond, Angel Nepomuceno Fernàndez and Olga Pombo

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 again 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 and those diagrams 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.

2015

Creativity in mathematics is identified in many forms or we can say is made up of many components. One of these components is The Unexpected Links where one tries to solve a mathematical problem in a nontraditional manner that requires the formation of hidden bridges between distinct mathematical domains or even between seemingly far ideas within the same domain. In this article, we design problems that express unexpected links in mathematics and suit students of intermediate and secondary levels. We prove their feasibility through teachers’ testimonies and through introducing them in classrooms and collecting students’ attitudes with respect to understanding and interest. Results confirm that students can sense such component and that designed problems had caught teachers’ and students’ interest.

Mathematics (Education) in the Information Age, 2020

The Method of Scientific Discovery in Peirce’s Philosophy: Deduction, Induction, and Abduction, 2011

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, passing then to the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries.

Routledge Handbook on the Philosophy of Imagination, 2016

Abduction and the process of scientific discovery, 2007

When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce's original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics.

The Elements of Creativity and Giftedness in Mathematics, 2011

Creativity plays an essential role in mathematics. It helps to make plausible conjectures in developing mathematical theories. It is an essential factor when new ideas are formulated or when an idea is presented in a new way. Mathematical creativity is often considered as a mysterious phenomenon. Most mathematicians seem to be not interested in analyzing their own thinking processes and do not describe how they work or conceive their theories. Only a few (such as Poincaré and Hadamard) made effort to describe ideas related to mathematical creativity. Mathematical creativity can not occur in vacuum and needs a context in which the individual moves forward based on previous experiences. The purpose of this paper is to provide an overview of effective characteristics of mathematical creativity which briefly describe how creativity in mathematical developments could be occurred. In addition, it describes moving power of mathematical creativity.