Mathematics

## A Fundamental Duality in the Mathematical and Natural Sciences

This is an essay in what might be called “mathematical metaphysics.” There is a fundamental duality that runs through mathematics and the natural sciences, from logic to biology.

## Keiretsu, Proportional Representation, and Input-Output Theory

This is Chapter 9 in my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

This essay grew out of an attempt to model mathematically the possible cross-ownership arrangements that might arise between privatizing firms in the former Yugoslavia [see Ellerman 1991].  The cross-ownership arrangements resemble the groups of Japanese companies called keiretsu.  There is cross ownership between the companies in the group as well as some ownership outside the group that is traded on the stock market.  In spite of the partial outside ownership, the keiretsu often behave as “self-owning” groups.  If firm A owns shares in B, then the management in A usually signs over its proxy on shares in B to the management in firm B.  And the management in B does likewise with respect to the managers in A.  Thus within certain constraints, each firm can act like a “self-owning” firm, not totally unlike the self-managing firms of the former Yugoslavia.

## Parallel Addition, Series-Parallel Duality, and Financial Mathematics

This is Chapter 12 in my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

## Valuation rings: A better algebraic treatment of Boolean algebras

This is Chapter 11 in my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

## Finding the Markets in the Math: Arbitrage and Optimization Theory

This is Chapter 10 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

One of the fundamental insights of mainstream neoclassical economics is the connection between competitive market prices and the Lagrange multipliers of optimization theory in mathematics.  Yet this insight has not been well developed.  In the standard theory of markets, competitive prices result from the equilibrium of supply and demand schedules.  But in a constrained optimization problem, there seems to be no mathematical version of supply and demand functions so that the Lagrange multipliers would be seen as equilibrium prices.  How can one “find the markets in the math” so that Lagrange multipliers will emerge as equilibrium market prices?

## Category Theory as the Theory of Concrete Universals

This is Chapter 8 of my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

This essay deals with a connection between a relatively recent (1940s and 1950s) field of mathematics, category theory, and a hitherto vague notion of philosophical logic usually associated with Plato, the self-predicative universal or concrete universal.  Consider the following example of “bad Platonic metaphysics.”

Given all the entities that have a certain property, there is one entity among them that exemplifies the property in an absolutely perfect and universal way.  It is called the “concrete universal.”  There is a relationship of “participation” or “resemblance” so that all the other entities that have the property “participate in” or “resemble” that perfect example, the concrete universal.

All of this and much more “bad metaphysics” turns out to be precisely modeled in category theory.

## The Semantics Differentiation of Minds and Machines

This is Chapter 7 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

The watershed event in the philosophy of mind (particularly as it relates to artificial intelligence or AI) during the last decade was John Searle’s 1980 article “Minds, Brains and Programs.”  This chapter was written about the same time and independently of Searle’s but it was updated in 1985 to take Searle’s work into account.  Searle’s exposition was based on his now-famous “Chinese Room Argument”—an intuition pump that boils down to a nontechnical explanation of the difference between syntax (formal symbol manipulation) and semantics (using symbols based on their intended interpretation).  Searle argues, in opposition to “hard AI,” that computers can at best only simulate but never duplicate minds because computers are inherently syntactical (symbol manipulators) while the mind is a semantic device.

The syntax-semantics distinction is hardly new; it was hammered out in philosophical logic during the first part of this century and it is fundamental in computer science itself.  The purpose of our paper is to analyze the minds-machines question using simple arguments based on the syntax-semantics distinction from logic and computer science (sans “Chinese Room”).  I arrive at essentially the same results as Searle—with some simplification and sharpening of the argument for readers with some knowledge of logic or computer science.

## Double-Entry Bookkeeping: Mathematical Formulation and Generalization

This is Chapter 6 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

The essay on double-entry bookkeeping (DEB) is intellectually interesting for several reasons in spite of the well-known soporific aspects of bookkeeping.  Several of the essays in the volume explicitly employ the analogy between additive and multiplicative operations (i.e., the common group-theoretic properties of additive groups of numbers and multiplicative groups of nonzero numbers).  For instance, given the system of multiplying whole numbers or integers, there is no operation inverse to multiplication (i.e., there is no division).  But there is a standard method of enlarging the system to allow division.  Consider pairs of whole numbers a/b (with b ¹ 0) and define multiplication in the obvious way: (a/b)(c/d) = (ac)/(bd).  These ordered pairs of integers are the “fractions” and they allow the operation of division (“multiply by the reciprocal”).

Now substitute addition for multiplication.  We start with the additive system of positive numbers along with zero (i.e., the non-negative numbers) where is no inverse operation to addition (i.e., there is no subtraction).  To enlarge the domain of non-negative numbers to include subtraction, consider ordered pairs [a // b] and define addition in the analogous way: [a // b] + [c // d] = [a+c // b+d].  This enlarged system of additive operations on ordered pairs of non-negative numbers allows subtraction (“add on the reversed pair”).  The origin of the intellectual trespassing into DEB was the observation that these ordered pair were simply the T-accounts of DEB.

Aside from illustrating the interplay of additive-multiplicative themes, the essay illustrates one of the most astonishing examples of intellectual insulation between disciplines, in this case, between accounting and mathematics.  Double-entry bookkeeping was developed during the fifteenth century and was first recorded as a system by the Italian mathematician Luca Pacioli in 1494.  Double-entry bookkeeping has been used as the accounting system in market-based enterprises of any size throughout the world for several centuries.  Incredibly, however, the mathematical basis for DEB is not known, at least not in the field of accounting.

## Are Marginal Products Created ex Nihilo?

This is Chapter 5 from my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

When an orthodox economist considers the principle of people getting the fruits of labor, he or she will invariably interpret it in terms of marginal productivity.  The orthodox claim is that under the conditions of competitive equilibrium, each unit of labor “gets what it produces.”  Well-meaning capitalist liberals emphasize that actual capitalism may be neither competitive nor in equilibrium, and in any case, there are enormous difficulties in measuring the “marginal product of each factor of production.”  In other words, they accept that interpretation of marginal productivity theory in principle but fuss about its applicability in practice.