Ole Immanuel Franksen

Group Representations of Finite Polyvalent Logic a Case Study Using APL Notation

Abstract Aiming at a unification of the theory of finite polyvalent logic the conventional axiomatic approach is substituted by an alternative group- theoretical formulation. Among the systems treated, besides Boolean or divalent logic, are those of Kleene, Bochvar, Post, and Lukasiewicz. APL is adopted as the mathematical notation throughout.

Mathematical programming in economics by physical analogies Part I: The analogy between engineering and economics

This article is the first of a series of three in which we establish and solve a physical analogy of the economic model underlying mathematical programming. Basically, the first article is a reexamination of the fundamental assumptions underlying quasi-static models in economics and engineering, with a view to the estab lishment of a conceptual framework common to both disciplines. At the microscopic level it is demonstrated that the assumptions of perfect competition can be cast in a form analogous to the one used in statistical physics. At the macroscopic level it is shown that measurements in economics and physics can be classified in identical manners with the result that derived relationships, like Ohms law and demand curves or electric power and total revenue, can be made analogous concepts. On this basis it is then asserted that underlying economics we find two basic laws which, apart from a single change in sign, are completely analogous to the well-known first and second laws of thermodynamics.

The nature of data—from measurements to systems

In spite of the fact that the concept of data is at the foundation of the sciences, we do not present it in our computer education in the context of a scientific theory, based on laws or immutable principles. Rather, by drilling computer programming, we teach it as dependent upon current technology, destined to be obsolete in 10–15 years. The aim of this paper is to reverse this trend. In a historical perspective and across a broad spectrum of disciplines, we trace the development of data from the basic notion of measurement to the description of theoretical systems and models. From this survey we deduce a common framework of ideas which, developed by the great scientists of the past, suggests how we may establish a modern theory of data, conceived as a distinct geometry characterized by data laws or invariances.

On fuzzy sets, subjective measurements and utility

Fuzzy reasoning is founded on subjective measurements specified as grades of membership of property categories called fuzzy sets. These membership gradings, it is assumed, may be expressed numerically by functions or corresponding discrete representations the values of which submit to the conventional arithmetic operations. This paper raises the question as to the empirical justification of these assumptions. That is, what empirical support can be established for this approach considering the properties of subjective measurements in psychophysics and those of utility in modern microeconomics or management science. Based on a presentation of the evidence demonstrated in these disciplines a power function seems to be the tentative form of the membership gradings of fuzzy sets representing a large variety of psychophysical continua and the corporate utility under risk. However, practically no empirical evidence was found to support the submission of such power function representations to arithmetic operations. Hence, there is an urgent need to establish more empirical facts on the assignments of subjective membership gradings and, in particular, the combinations of such gradings.

Colligation or the logical inference of interconnection

The concept of interconnection is fundamental to the modelling of discrete, physical systems. On the basis of centuries of scientific experience, everyone will agree that the concept is part of a logically consistent approach, permitting us to draw conclusions, verifiable by observation, from basic laws or assumptions. Yet interconnection as an abstract concept seems to be without scientific underpinning in pure logic. Adopting a historical viewpoint, our aim is to show that the reasoning of interconnection may be identified with a neglected kind of logical inference, called ‘colligation’ by Charles Sanders Peirce.

Invariance under Nesting - An Aspect of Array-based Logic with Relation to Grassmann and Peirce.

Hermann G. Grassmann’s Die lineale Ausdehnungslehre (1844) made no impact on logic, except perhaps for the indirect influence it exerted upon Ernst Schroder and a few others through the logical publications of Hermann’s brother, Robert Grassmann (Grattan-Guinness 1996; Peckhaus 1996). For although Grassmann saw great generality in his remarkable algebraic-geometrical formulations, logic and its key question of inference were not his concern. Yet, in old age about 1909, Charles Sanders Peirce would remark that “in my memoir of 1870 ... I made no reference to Grassmann ... which I am all but absolutely sure that I should have done had I been acquainted with either of Grassmann’s volumes. So I infer that the too exclusive admiration of Hamilton in our household prevented my acquaintance with that great system” (Hartshorne et al. 1932, IV, §669, p. 566).

Mathematical programming in economics by physical analogies: Part III: System equilibrium and mathematical programming

This article is the third of a series of three in which we establish and solve a physical analogy of the economic model underlying mathematical programming. In the third article the concept of equilibrium is de rived from the basic notion of a static mechanical system at rest, and it is shown how, by refining this concept, it can be extended to dynamic physical systems. In this process it is also explained why an extremum value of a state-function determines an equilibrium state of a system. From an economic viewpoint the physical concept of equilibrium is identified with that of a perfectly competi tive market, and it is demonstrated how this concept dif fers from the equilibrium concept of a monopolist. For comparison, industries with one and two outputs are considered. From a mathematical programming viewpoint both linear and quadratic problems are discussed in terms of methods of classical mechanics. In particular it is shown that the simplex method can be derived from Fouriers inequality for equilibrium on a boundary and that the Kuhn-Tucker conditions are statements analogous to Kirchhoffs mesh law for an electrical network.

Mathematical programming in economics by physical analogies Part II: The economic network concept

REVIEW OF PART I AND PREVIEW OF PART II This article is the second of a series of three in which we establish and solve a physical analogy of the economic model underlying mathematical programming. Basically, the first article (published in SIMULATION last month) was a reexamination of the fundamental assumptions underlying quasi-static models in economics and engineering, with a view to the establishment of a conceptual framework common to both disciplines. At the microscopic level it was demonstrated that the assumptions of perfect competition can be cast in a form analogous to the one used in statistical physics. At the macroscopic level it was shown that measure ments in economics and physics can be classified in iden tical manners with the result that derived relationships, like Ohms law and demand curves or electric power and total revenue, can be made analogous concepts. On this basis it was then asserted that underlying economics we find two basic laws which, apart from a single change in sign, are completely analogous to the well-known First and Second Laws of thermodynamics. The present article is primarily a reformulation of the Walrasian economic model, which underlies mathemati cal programming, into an analogue electrical network. In accordance with the tradition of physics, the Wal rasian system of equations is derived from the postulates of the First and Second Laws of economics. The advan tage of this approach, which differs from conventional economic expositions, is that it permits a nearly auto matic establishment, term for term, of the corresponding electrical network model. The constraints and state-functions of the electrical analogue are formulated by modern network techniques in order to separate, in the economic formulation of the Walrasian system, the analytical aspects from those more general aspects which are involved in the design of an economic production system.

Mr. Babbage, the difference engine, and the problem of notation: An account of the origin of recursiveness and conditionals in computer programming

Abstract Contrary to the opinion of Lady Lovelace and others the origin of programmed conditionals and recursiveness may be traced back to the Difference Engine of Charles Babbage. To prove this is the aim of the present inquiry which is supported by contemporary quotations, based on a rather extensive historical research, and by computational experiments performed on an interactive simulation model in APL of the Difference Engine. It is attempted in the article to document the use of APL in such a manner that the few figures of particular interest for readers knowlegeable of APL, may be skipped without any general loss of meaning by the common reader.

Structural aspects of controllability and Sobservability—II. digraph decomposition

Abstract The graph theoretic aspects of controllability and observability are examined and related to the tensorial formulation of Part I of the paper. Particular emphasis is given to the significance of the system digraph decomposition and the relevance of this to certain system algebraic properties of interest in control theory.