Online research in philosophy

This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.

Scientists use a variety of modes of representation in their work, but philosophers have studied mainly sentences expressing propositions. I ask whether diagrams are mere conveniences in expressing propositions or whether they are a distinct, ineliminable mode of representation in scientific texts. The case of path analysis, a statistical method for quantitatively assessing the relative degree of causal determination of variation as expressed in a causal path diagram, is discussed. Path analysis presents a worst case for arguments against eliminability since path diagrams are usually presumed to be mathematically or logically “equivalent” in an important sense to sets of linear path equations. I argue that path diagrams are strongly generative, i.e., that they add analytical power to path analysis beyond what is supplied by linear equations, and therefore that they are ineliminable in a strong scientific sense.

.  In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

Molecular biologists and biochemists often use diagrams to present hypotheses. Analysis of diagrams shows that their content can be expressed with linguistic representations. Why do biologists use visual representations instead? One reason is simple comprehensibility: some diagrams present information which is readily understood from the diagram format, but which would not be comprehensible if the same information was expressed linguistically. But often diagrams are used even when concise, comprehensible linguistic alternatives are available. I explain this phenomenon by showing why diagrammatic representation is especially well suited for a particular kind of explanation common in molecular biology and biochemistry: namely, functional analysis, in which a capacity of the system is explained in terms of capacities of its component parts.

One effect of information technology is the increasing need to present information visually. The trend raises intriguing questions. What is the logical status of reasoning that employs visualization? What are the cognitive advantages and pitfalls of this reasoning? What kinds of tools can be developed to aid in the use of visual representation? This newest volume on the Studies in Logic and Computation series addresses the logical aspects of the visualization of information. The authors of these specially commissioned papers explore the properties of diagrams, charts, and maps, and their use in problem solving and teaching basic reasoning skills. As computers make visual representations more commonplace, it is important for professionals, researchers and students in computer science, philosophy, and logic to develop an understanding of these tools; this book can clarify the relationship between visuals and information.

There are two distinct interpretations of the role that Feynman diagrams play in physics: (i) they are calculational devices, a type of notation designed to keep track of complicated mathematical expressions; and (ii) they are representational devices, a type of picture. I argue that Feynman diagrams not only have a calculational function but also represent: they are in some sense pictures. I defend my view through addressing two objections and in so doing I offer an account of representation that explains why Feynman diagrams represent. The account that I advocate is a version of that defended by Kendall Walton, which provides us with a basic characterization of the way that representations in general work and is particularly useful for understanding distinctively pictorial representations - in Walton's terms, depictions. The question of the epistemic function of Feynman diagrams as pictorial representations is left for another time.

Diagrams make it possible to present scientific facts in more abstract and generalized form. While some detail is lost, simplified and accessible knowledge is gained. E. B. Wilson's work in cytology provides a case study of changing uses of diagrams and accompanying abstraction. In his early work, Wilson presented his data in photographs, which he saw as coming closest to “fact.” As he gained confidence in his interpretations, and as he sought to provide a generalized textbook account of cell development, he relied on increasingly abstract diagrams. In addition, he came to see that highly abstract and even schematic drawings could provide more than pictures directly from life.

This article puts forward the notion of “evolving diagram” as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter “diagrams*”) in the context of “sketch theory,” a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.

A logical system is studied whose well-formed representations consist of diagrams rather than formulas. The system, due to Shin [2, 3], is shown to be complete by an argument concerning maximally consistent sets of diagrams. The argument is complicated by the lack of a straight forward counterpart of atomic formulas for diagrams, and by the lack of a counterpart of negation for most diagrams.

In this paper, I aim to identify Peirce?s great contribution to logical diagrams and its limit.Peirce is the first person who believed that the same logical status can be given to diagrams as to symbolic systems.Even though this belief led him to invent his own graphical system, Existential Graphs, the success or failure of this system does not determine the value of Peirce?s general insights about logical diagrams.In order to make this point clear, I will show that Peirce?s revolutionary ideas about diagrams not only overcame some important defects of Venn diagrams but opened a new horizon for logical diagrams.Finally, I will point out where Peirce?s new horizon for logical diagrams stopped and will claim that this limit is mainly responsible for the discrepancy between Peirce?s and others? estimates of his contribution to logical diagrams.