Math: Pure and Applied


The Objective Indefiniteness Interpretation of Quantum Mechanics. The common-sense view of reality is expressed logically in Boolean subset logic (each element is either definitely in or not in a subset, i.e., either definitely has or does not have a property). But quantum mechanics does not agree with this "properties all the way down" picture of micro-reality. Are there other coherent alternative views of reality? A logic of partitions, dual to the Boolean logic of subsets (partitions are dual to subsets), was recently developed along with a logical version of information theory. In view of the subset-partition duality, partition logic is the alternative to Boolean subset logic and thus it abstractly describes the alternative dual view of micro-reality. Perhaps QM is compatible with this dual view? Indeed, when the mathematics of partitions using sets is "lifted" from sets to vector spaces, then it yields the mathematics and relations of quantum mechanics. Thus the vision of micro-reality abstractly characterized by partition logic matches that described by quantum mechanics. The key concept explicated by partition logic is the old idea of "objective indefiniteness" (emphasized by Shimony). Thus partition logic, logical information theory, and the lifting program provide the back story so that the old idea then yields the objective indefiniteness interpretation of quantum mechanics. Here is a set of slides for the talk based on the paper.

A Common Fallacy in Quantum Mechanics: Why Delayed Choice Experiments do NOT imply Retrocausality. There is a common fallacy, here called the separation fallacy, that is involved in the interpretation of quantum experiments involving a certain type of separation such as the: double-slit experiments, which-way interferometer experiments, polarization analyzer experiments, Stern-Gerlach experiments, and quantum eraser experiments. It is the separation fallacy that leads not only to flawed textbook accounts of these experiments but to flawed inferences about retrocausality in the context of "delayed choice" versions of separation experiments.

The Logic of Partitions: Introduction to the Dual of the Logic of Subsets. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms. Modern categorical logic as well as the Kripke models of intuitionistic logic suggest that the interpretation of classical "propositional" logic should be the logic of subsets of a given universe set. The propositional interpretation is isomorphic to the special case where the truth and falsity of propositions behave like the subsets of a one-element set. If classical "propositional" logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical "propositional" logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system for partition logic. (Reprint from: Review of Symbolic Logic, Vol. 3, No. 2 June, 287-350).

Counting Distinctions: On the Conceptual Foundations of Shannon's Information Theory. Ordinary "propositional" logic can be interpreted as the logic of subsets. The concept of a partition on a universe set U is dual to the concept of a subset of the universe set in the category-theoretic sense of duality between epimorphisms and monomorpisms. The dual to the notion of an element being in a subset is that of a distinction being made by a partition (i.e., a pair of elements being in distinct blocks of the partition). The conceptual beginnings of probability theory was to move beyond the logic of subsets on a universe by assigning a "probability" to each subset of a finite universe which was the number of elements in the subset normalized by the size of the universe set (using the Laplacian assumption of each element having equal probability). This paper works out the corresponding conceptual development starting with the logic of partitions. Each partition on U is assigned a "logical entropy" which is the number of distinctions made by the partition normalized by the number of ordered pairs in UxU. This notion of logical entropy is then precisely related to Shannon's notion of entropy showing that information theory is conceptually based on the logical notion of distinctions. (Reprint from: Synthese)

Adjoints and Emergence: Applications of a new theory of adjoint functors. Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens to focus on what is important in mathematics. The central feature of an adjunction is what might be called "determination through universals" based on universal mapping properties. Given the importance of adjoint funtors in mathematics, it seems appropriate to look for empirical applications. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language). [Reprint from: Axiomathes, March 2007] See slides for a talk on adjoint functors.

Adjoint Functors and Heteromorphisms. Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important in mathematics. Our focus here is to present a theory of adjoint functors. The basis for the theory is laid by first showing that the object-to-object "heteromorphisms" between the objects of different categories (e.g., insertion of generators as a set to group map) can be rigorously treated within category theory. The heteromorphic theory shows that all adjunctions arise from the birepresentations of the heteromorphisms between the objects of different categories.

A Theory of Adjoint Functors ---with some Thoughts about their Philosophical Significance. This new paper approaches the question "What is category theory" by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of "chimera" morphisms or "heteromorphisms" between objects in different categories. This theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical concepts and some central themes in the life sciences and in social philosophy. This paper is from: What is Category Theory? Edited by Giandomenico Sica, Polimetrica, Milan, 2006.

Concrete Universals in Category Theory This old 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.  While category theory can be quite forbidding to the non-specialist, simple examples can be used based on inclusion between sets.  Given two sets A and B, consider the property of being a set X that is contained in A and is contained in B.  In other words, the property is the property of being both a subset of A and a subset of B.  In this example, the "participation" relation is the subset relation.  There is a set, namely the intersection, meet, or overlap of A and B, denoted A?B, that has the property (so it is a "concrete" instance of the property), and it is universal in the sense that any other set has the property if and only if it participates in the universal example:

concreteness:       A?B is a subset of both A and B, and

universality:          X is a subset of A?B if and only if X is contained in both A and B.

Thus the intersection A?B is the concrete universal for the property of being a subset of A and a subset of B.  I argue that category theory is relevant to foundations as the theory of concrete universals.  Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal.

Series-Parallel Duality and Financial Mathematics.  Series-parallel duality has been previously studied in electrical circuit theory and combinatorial theory.  The parallel sum arose naturally when resistors are connected in parallel instead of series.  Given two resistors with the positive real resistances of a and b, their combined resistance is a+b when connected in series and (1/a + 1/b)? when connected in parallel.  The full colon (:) notation will be used for the parallel sum, a:b = (1/a + 1/b)?.  Any equation on the positive reals concerning the two sums and multiplication, can be "dualized" by applying the "take-reciprocals" map to obtain another equation.  Each number is replaced by its reciprocal and the two additions are interchanged.  The principal application considered in the essay is series-parallel duality in financial arithmetic.  The basic result is that the parallel sum of the one-shot "balloon" payments at different times that would pay off a given loan is the equal amortization payment that would pay off the loan if paid at each of those times.  By carrying through this interpretation, we see that each equation in financial arithmetic can be paired with a dual equation to reveal the structure of series-parallel duality within financial arithmetic.  The duality in economics is convex duality, and it is shown that series-parallel duality is the "derivative" of convex duality. (Click on title to open memo or paper.)

Series-Parallel Duality in the Mathematics of Appraisal. It has long been noticed that the basic functions in the mathematics of real estate appraisal and in financial arithmetic come in pairs where one function is the reciprocal of the other. For instance, the "payments to amortize a principal of one" is the reciprocal of the "principal amortized by payments of one." This paper shows that this phenomenon is an example of the series-parallel duality ordinarily studied in electrical circuit theory and combinatorial mathematics. Reprint from: FSR Forum (Financial Studies association Rotterdam) (February 2008): 13-18.

Double-Entry Accounting: The Mathematical Formulation and Generalization .  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 was not known, at least not in the field of accounting.  The mathematical basis behind DEB (algebraic operations on ordered pairs of numbers) was developed in the nineteenth century by Sir William Rowan Hamilton as an abstract mathematical construction to deal with complex numbers and fractions.  The particular example of the ordered pairs construction that is relevant to DEB ("group of differences" in technical terms) is the one used in undergraduate algebra courses to construct a number system with subtraction by using operations on ordered pairs of non-negative numbers.  All that is required to see the connection with DEB is to identify these ordered pairs with the two-sided T-accounts of DEB (debits on the left side and credits on the right side).  Yet with the exception of a paragraph in a semipopular book by D.E. Littlewood, the author has not been able to find a single mathematics book, elementary or advanced, popular or esoteric, which notes that the group of differences construction has been used in the business world for about five centuries.  And the mathematical basis for DEB is totally unknown in the separate world of accounting.  The mathematical formulation of DEB also allows it to be generalized to vectors of incommensurate physical quantities so the DEB can finally be "priceless." (Click on title to open paper.)

Generalized Double-Entry Accounting: Showing What is "Double" in the Double-Entry Method.  This paper develops the mathematical formulation of double-entry accounting in its additive form for scalars and vectors and in a multiplicative form where the two-sided "T-accounts" are fractions.  The paper shows that the "double" in the double-entry method is the two-sidedness of the ordered pairs ("T-accounts") used in additive, multiplicative, and more general forms of the double-entry method.

The Mathematics of Double Entry Bookkeeping. This is a scan of the first publication in a mathematics journal of the mathematical treatment of double-entry bookkeeping using the group of differences construction. This treatment is restricted to scalars (value accounting) and does not cover the easy generalization to vectors.[Reprint from: Ellerman, David 1985. The Mathematics of Double Entry Bookkeeping. Mathematics Magazine. 58 (Sept. 1985): 226-233.]

Double Entry Multidimensional Accounting. This is a scan of the first journal publication of the mathematical formulation and generalization to vectors of double-entry bookkeeping. [Reprint from: Ellerman, David 1986. Double Entry Multidimensional Accounting. Omega. 14 (1 (January 1986)): 13-22.] The complete book-length treatment is in my 1982 book: Ellerman, David 1982. Economics, Accounting, and Property Theory. Lexington MA: D.C. Heath, which can be downloaded here.

Towards an Arbitrage Interpretation of Optimization Theory.  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?  We argue that the solution to the "find the markets in the math" problem is to reconceptualize equilibrium as the absence of profitable arbitrage instead of the equating of supply and demand.  With each proposed solution to a classical constrained optimization problem, there is an associated market.  The maximand is one commodity, and each constraint provides another commodity on this market.  Given a marginal variation in one commodity, one can define the marginal change is any other given commodity so the market has a set of exchange rates between the commodities.  The usual necessary conditions for the proposed solution to solve the maximization problem are the same as the conditions for this mathematically defined "market" to be arbitrage-free.  The prices that emerge from the arbitrage-free system of exchange rates (normalized with the maximand as numeraire) are precisely the Lagrange multipliers.  We also show that the cofactors of a matrix describing the marginal variations can be taken as the prices (before being normalized) so the Lagrange multipliers can always be presented as ratios of cofactors. (Click on title to open memo or paper.)

Math on Spreadsheets.  A surprising number of mathematical algorithms can be programmed on spreadsheets such as Excel.  Since spreadsheets are widely available, the algorithms can then be used by researchers, teachers, and students without using specialized software.  Since any algorithm of any sophistication requires circular references and iterations, the trick is to use the fixed order of recalculating cells during iteration (usually left to right across columns, top to bottom down rows) and to construct an iteration counter within the spreadsheet.  With an iteration counter, the formulas can distinguish between the initial iteration when the variables are first given seeded values and the later iterations when the variables use the results of earlier iterations. (Click on "Math on Spreadsheets" title for downloadable spreadsheets).

Some Downloadable Books and Papers by Others:

Introduction to Probability and Random Processes. This is a scanned copy of Gian-Carlo Rota's and Kenneth Baclawski's Introduction to Probability and Random Processes manuscript in its 1979 version. Through the years, Gian-Carlo Rota strove to 'perfect' the manuscript until his untimely death in 1999. At that time, only an Italian version had been published. The book is a product of Rota's seminal work in combinatorial theory which resulted in his unique approach to probability theory. In addition, the remarkable word problems about applications of probabilistic concepts are comparable to William Feller's classic book. With the approval of Prof. Baclawski (Northeastern University), I scanned in the 1979 mimeo version of the manuscript and posted it here so that this unique text would be more widely available. It is a 3 meg PDF file. (Right) click here or on the title above to download the book.

A General Theory of Fibre Spaces with Structure Sheaf. (Grothendieck's "Kansas Paper") This is a scan of one of Alexandre Grothendieck few publications in English, a technical report  published (without copyright as an NSF report) by the Dept. of Mathematics, University of Kansas, 1958.