Jordi Cat

In the aftermath of the Spanish War, the Catalan philosopher Eugeni d'Ors and the Spanish philosopher José Ortega y Gasset offered a reading of Einstein’s theory of relativity in which discussions of unity and plurality connected their respective synthetic philosophies and nationalist projects of political and cultural analysis and reform. Besides their philosophical views, references to Einstein tracked their territorial concerns and personal circumstances in the relations between Catalonia, Spain and Europe. Einstein’s theory symbolized the saving connection between the classical and the modern, Europe and the Mediterranean, and science and philosophy. In this paper I focus on the case of Ors in relation to Catalan nationalism and the project he called Noucentismeand the case of Ortega in relation to Spanish nationalism and his political philosophy in España Invertebrada.

The premise and focus of this book is the basic role of categorization as a practice in linguistic and pictorial representation –also acknowledging other roles derived from or alternative to representation. Symbolic use cuts across the role of pictures and language: analogical pictures and diagrams, include elements their contribute their content symbolically and the interpretation of pictures often depends on the auxiliary role of linguistic symbols. To articulate how the pictorial case challenges the unity of application of fuzziness criteria, including fuzzy set theory, I have appealed to three distinctions: image/picture, intrinsic/extrinsic and truth/fit.

In this chapter I use the notions introduced above to lay out conditions of vagueness in pictures, the diversity of roles and scenarios and differences from vague linguistic predicates and representation.

Most quantities or measures of interest modeling some form of uncertainty correspond to the intrinsic kind. Fuzzy image analysis is based on the premise that the properties of edge, boundary region or tonal relations in images are not generally represented in sharp terms. In general, the selection of relevant features and their standards is constrained by the aims and standards of the technical practice of analysis or processing and its specific applications, e.g., as formulated by performance indices.

Application of Mathematics in the Representation of Images: From Geometry to Set Theory

In philosophical debate, how to best understand the vagueness of predicates remains a controversial matter. A renewed defense of epistemic interpretations, in terms of deficient states of knowledge with insufficient information or a semantic relation of indeterminacy of interpretation (whether as content or predicate), has been met with a critical reaction and prompted a defense of vagueness’ objective character. On objectivist views such as the one Smith has recently defended, vagueness is not a semantic relation, but an ontic feature of the things denoted by the subject of predication. The feature is then expressed by a semantic relation of partial truth to the predicative proposition, in accordance with fuzzy set theory.

I have pointed to the rather obvious fact that pictures, unlike linguistic symbols, have a specifically and distinctively visual character that is central to its uses. That is, pictures, like words, play many roles, as does categorization, not just representation, and for the performance of their functions pictures rely directly or indirectly on their perceptual properties. In this case, we can locate the conditions for vagueness of representation in the categorization of perceptual entities and properties.

A putative form of vagueness in perceptual presentation/representation is blur. From the standpoint of accuracy—in representationalism or intentionalism—blurred vision is like hallucinations, an instance of inaccuracy, of misrepresentation. Things are not the way they are perceived or represented as being. Alternatively, it has been argued that along with error, blur has its own distinctive phenomenological conditions of presentation, in a broader sense, and recognition. For authors such as Crane, Pace and Smith, blur challenges the transparency of visual representation and its content. An alternative view to misrepresentation or underrepresentation is that the phenomenological property of blur is a form of overrepresentation. I want to suggest a different approach.

Vagueness of appearance and depiction is a property of the categorization of images. Through categorization, whatever pictures do, they may do approximately and vaguely. And what images can do, that is, what they do for us and we can do with them, depends on what we think their roles are. In general, pictures play a role in ordinary and scientific argument and in cognition more broadly. They are key to identifying, documenting, tracking and exploring visible properties of empirical systems and phenomena, also and to visualizing and communicating empirical and theoretical information; they can be emotionally compelling, aesthetically powerful, and exhibit and enforce values and biases. This is no less relevant in the study of systems, individual or generic, whose relevant properties are spatial, chromatic or structural. Relevant examples differ widely in medium, mode of production and use; they include photographs, drawings, data charts, diagrams, animations, film recordings, computer generated images, etc. Pictures in many such cases are meant to support inferences, recognition processes and carry heuristic and evidence value. We think with them and through them.

In this chapter I pick up on previous results and extend the discussion in chapter 5 to tackle these questions: How do pictures and their fuzziness enter these activities? How can fuzzy set theory accommodate fuzzy visual thinking? Performing cognitive tasks involves a variety images, some of them are fuzzy perceptions, others are fuzzy pictures. As I have done in my broad discussion of representation and thinking, also here I want to include visual thinking in scientific contexts. Set theory can be used in the formalization of reasoning from images or with images associated with the representational and inferential use of symbolic diagrams, also in the formalization of thinking in a broader set of tasks that include computation and problem-solving using both fuzzy diagrams and fuzzy analogical pictures.