Diagrams, Graphs, and Visual Imagination in Mathematics

What kind of experience does mathematics provide with? Is it purely logical or empirical? If logical, how could it give us nontrivial conclusions and be applied to other sciences? If empirical, how could it be absolutely certain? Where does this mathematical certainty come from?

This paper is part of a continuing discussion on the contributions Peirce's diagrammatic epistemology and logic have to a broad range of issues at the intersections of and philosophy of science, logic and cognition.

2008

In this paper, I discuss the role of diagrammatic thinking within the larger context of cognitive activity as framed by Peirce’s semiotic theory of and its underpinning realistic ontology. After a short overview of Kant’s scepticism in its historical context, I examine Peirce’s attempt to rescue perception as a way to reconceptualize the Kantian “manifold of senses”. I argue that Peirce’s redemption of perception led him to a series of problems that are as fundamental as those that Kant encountered. I contend that the understanding of the difficulties of Peirce’s epistemology allows us to better grasp the limits and possibilities of diagrammatic thinking.En este artículo se discute el papel que desempeña el concepto de pensamiento diagramático en el contexto de la actividad cognitiva, tal y como es concebida dentro del marco de la teoría semiótica de Peirce y su subyacente ontología realista. Luego de presentar una visión general del escepticismo kantiano en su contexto histórico, s...

This essay explores the Mathematics of Charles Sanders Peirce. We concentrate on his notational approaches to basic logic and his general ideas about Sign, Symbol and diagrammatic thought.

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.

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.

Published in The Transactions of the Charles S. Peirce Society, 61 (Summer 2005): 483-513. Also available in JStor

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education.

Cognitio Revista De Filosofia, 2005

Peirce regards mathematics as an informative science capable of really increasing our knowledge. This means that mathematics is not limited to conceptual analysis but possesses a real object of investigation. The core of Peirce's view of mathematics is that mathematical reasoning is not developed through general concepts alone but deals with an unavoidable element of individuality. The conclusion of a deductive inference can contain information that is not at all present in its premises and can only come into being through concrete work on the part of the mathematician. Peirce describes this work as observation and experimentation on individual diagrams. While the idea of an individual element in mathematics is already present in Kant (and can also be traced back to Aristotle), the different location Peirce assigns it attests to a marked difference in their conceptions, the basis of which lies in the difference between Kant and Peirce in categorial analysis. Peirce's semiotic approach to mathematics involves a shift from the plane of the object denoted to that of the sign itself. This holds true for geometric as well as algebraic inferences, which Peirce can equate in this respect. In both cases, the individual element of mathematics is thus to be found within the diagram itself with no reference to the object denoted. While the diagram is in any case a token, this cannot explain its essential individuality, to which end the indexical juxtaposition of its parts should be examined.