Stig Andur Pedersen

Visualization, Explanation and Reasoning Styles in Mathematics

This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.

Identification of Matrices in Science and Engineering

Engineering science is a scientific discipline that from the point of view of epistemology and the philosophy of science has been somewhat neglected. When engineering science was under philosophical scrutiny it often just involved the question of whether engineering is a spin-off of pure and applied science and their methods. We, however, hold that engineering is a science governed by its own epistemology, methodology and ontology. This point is systematically argued by comparing the different sciences with respect to a particular set of characterization criteria.

Proof Theory, History and Philosophical Significance

Preface. Contributing Authors. Introduction. Part 1: Review of Proof Theory. Highlights in Proof Theory S. Feferman. Part 2: The Background of Hilberts Proof Theory. The Empiricist Roots of Hilberts Axiomatic Approach L. Corry. The Calm Before the Storm: Hilberts Early Views on Foundations D. Rowe. Toward Finitist Proof Theory W. Sieg. Part 3: Brouwer and Weyl on Proof Theory and Philosophy of Mathematics. The Development of Brouwers Intuitionism D. van Dalen. Did Brouwers Intuitionistic Analysis Satisfy its own Epistemological Standards? M. Epple. The Significance of Weyls Das Kontinuum S. Feferman. Herman Weyl on the Concept of Continuum E. Scholz. Part 4: Modern Views and Results from Proof Theory. Relationships between Constructive, Predicative and Classical Systems of Analysis S. Feferman. Index.

Causality in medicine: towards a theory and terminology.

One of the cornerstones of modern medicine is the search for what causes diseases to develop. A conception of multifactorial disease causes has emerged over the years. Theories of disease causation, however, have not quite been developed in accordance with this view. It is the purpose of this paper to provide a fundamental explication of aspects of causation relevant for discussing causes of disease.The first part of the analysis will discuss discrimination between singular and general causality. Singular causality, as in the specific patient, is a relation between a concrete sequence of causally linked events. General causation, e.g. as in disease etiology, means various categories of causal relations between event types. The paper introduces the concept of a reference case serving as a source for causal inference, reaching beyond the concept of general causality.The second part of the analysis provides exemplification of a theory of causation suitable for discussing singular causation. The chain of events that induce a disease state can be identified as effective causal complexes, each complex composed of nonredundant components, which separately contribute to the effect of the complex, without the individual component being necessary or sufficient in itself to produce the effect. In the third part of the analysis the theory is elaborated further. Causes, defined as nonredundant components, can furthermore be differentiated according to their avoidability, according to theories about human error or by the potential of eradication.Multifactorial models of disease creates a need for systematic approaches to causal factors. The paper proposes a taxonomical terminology that serves this purpose.

The Tension Between Science and Engineering Design

Engineering design is an essential part of the process of constructing and maintaining modern complex systems as airplanes, power plants, and urban areas. As such engineering design must be based on scientific knowledge. But whereas it is the task of engineering design to assist in the realization of complex systems in their concrete real life context, it is the task of science and mathematics to find and justify new knowledge about the universal working of nature. In a science as physics mathematical structures and formalisms are developed and applied as means to identify and describe the form and nature of laws that govern the behavior of processes of very different scales in nature. This work requires comprehensive abstraction and idealization, and, as a consequence of that, advanced mathematical and physical theories are only valid in highly abstract and isolated systems. Consequently, these theories are far away from the concrete contexts that engineering design is about. In this paper we shall identify and discuss some of the epistemological problems that this tension between scientific idealization and engineering concretization may lead to.

Assessment and Discovery in the Limit of Scientific Inquiry

Acquisition of knowledge may come about in different ways. One step on the way to acquire knowledge would be to formulate a hypothesis and then evaluate the particular hypothesis in light of incoming evidence. Inductive logics, confirmation theory, and Popper’s deductivist epistemology all adopt this approach. Indeed, proponents of this “generate and test” epistemology have insisted that the core of scientific method is exhausted by the study of methods of hypothesis assessment. This lead Reichenbach to formulate the classical distinction between the context of justification and the context of discovery. Hempel later spoke of a logic of justification but only of a context of discovery just to emphasize the discrepancy. Whether a hypothesis is verified or refuted by the evidence is strictly a logical matter which can be settled “out of court” in a logical or approximately logical fashion. However, it seems to be the case of many, at least early, confirmation theorists or justificationists, like Hempel, that they did not insist on convergence to a correct hypothesis. For them, confirmation was to be an end in itself. In consequence, one could confirm forever heading nowhere near the correct answer.

Validation of simulation models

In philosophy of science, the interest for computational models and simulations has increased heavily during the past decades. Different positions regarding the validity of models have emerged but the views have not succeeded in capturing the diversity of validation methods. The wide variety of models with regards to their purpose, character, field of application and time dimension inherently calls for a similar diversity in validation approaches. A classification of models in terms of the mentioned elements is presented and used to shed light on possible types of validation leading to a categorisation of validation methods. Through different examples it is shown that the methods of validation depend on a number of things as the context of the model, the mathematical nature, the data available and the representational power of the model. In philosophy of science many discussions of validation of models has been somewhat narrow-minded reducing the notion of validation to establishment of truth. This article puts forward the diversity in applications of simulation models that demands a corresponding diversity in the notion of validation.

A note on innovation and justification

Within epistemology and the philosophy of science there is, in a number of cases, an a-symmetrical relation or even complementarity between innovation and justification. Innovations are not always justifiable, within the currently accepted body of scientific knowledge and readily justifiable innovations are seldom too interesting. This paper describes some such cases drawn from the history of science and attempts to classify different types of innovations.

Mathematics in Engineering and Science

Mathematics is an important aspect of natural science and engineering and many new mathematical concepts and theories have come about when researchers have been formulating and solving scientific or engineering problems. This interaction between mathematics and science/engineering took a new form towards the end of twentieth century in connection with the introduction of digital computers. In fact, some scientists believe that a new form of doing science has appeared: computational science. Norman J. Zabusky, e.g., argues that we are in the midst of a computational revolution that will change science and society as dramatically as the agricultural and industrial revolutions did (Zabusky, Phys Today 40(10), 1987). We shall discuss in what sense it is reasonable to talk about a new style of scientific reasoning and what this will mean for mathematical practice.

Ways of Worlds I–II Two Special Issues on Possible Worlds and Related Notions

Ways of Worlds II is the second of two special issues of Studia Logica on possible worlds and related notions. Historical and philosophical tracking of possible worlds was partly, but not exclusively, the business of the first issue which was launched as volume 82, no. 3, 2006. This volume featured papers by M. Cresswell, S.O. Hansson, D. Jacquette, A.-V. Pietarinen, A. Varzi and A. Zanardo. Ways of Worlds II focuses more on applications and alternative frameworks. In ‘First Order Classical Modal Logic’ by H. Arlo-Costa and E. Pacuit, the stage is initially set for following D. Scott and R. Montague’s alternative framework for modalities known as neighborhood semantics. Completeness results are proved for a variety of classical systems of first order logic. Then the neighborhood frames with constant domains are used to study some recently proposed epistemic modalities together with models for monadic operators of high probability both of which are either very difficult or down-right impossible to model in standard relational Kripke-semantics. The paper is concluded with the presentation of a general first order neighborhood semantics which is strong enough to provide characterization results for the full family of classical first order modal systems. It is convincingly argued that the neighborhood semantics program is the first platform for the semantic unification of classical first order modal systems. In A. Brandenburger’s and H. Jerome Keisler’s ‘An Impossibility Result Theorem on Beliefs in Games’ a doxastic self-referential paradox in games is identified which in turn gives rise to a game-theoretic impossibility result similar to Russell’s paradox. The paradox is formalized using belief models and a first order language and the impossibility theorem then says that ‘No belief model can be complete for a language that contains first-order logic’ ! This interesting and surprising result may be interpreted as saying that if