The logical entropy formula 
Given a partition 
History of the Logical Entropy Formula
Published on February 16, 2010
The logical entropy formula
Arbitrage and Graphical Gridlock
Published on February 9, 2010
The arbitrage-free law (or Kirchhoff’s voltage law)
Recently I emailed a friend to complain when his organization used this 3 gear image as their logo. What was my complaint? Read on.
The basic idea of arbitrage is to “get something for nothing” by trading commodities or currencies around some circle ending up with more than one started with. The simplest example would be to have a commodity which can be bought and sold at two different prices. Then a clear profit is obtained by the arbitrage operation of buying low and selling high.
The circular pattern is that money is transformed into so many units of the commodity at the low price. Say $1 dollar buys 4 apples so each apple is priced at 25¢. Then these units of the commodity are transformed back into money at the high price, say at 50¢ per apple so the 4 apples would sell for $2. The arbitrage profit is the additional money received back over the money originally paid out to purchase the commodity, in this case $2 – $1 = $1. We assume that no transactions costs are involved with the exchanges.
Such an example of arbitrage profit could not be sustained for long since everyone would want to make the circular trades and get something for nothing. Those who were willing to sell apples at 25¢ would raise their prices and those who were willing to buy apples for 50¢ would offer less until an equilibrium price was reached somewhere between 25¢ and 50¢, say at 40¢. Then we would have a situation called “arbitrage-free” where circular trades would just break even (always assuming no transactions costs).
From Partition Logic to Information Theory
Published on February 6, 2010
Some basic analogies between subset logic and partition logic
A new logic of partitions has been developed that is dual to ordinary logic when the latter is interpreted as the logic of subsets rather than the logic of propositions. For a finite universe, the logic of subsets gave rise to finite probability theory by assigning to each subset its relative cardinality as a Laplacian probability. The analogous development for the dual logic of partitions gives rise to a notion of logical entropy that is related in a precise manner to Claude Shannon’s entropy. In this manner, the new logic of partitions provides a logical-conceptual foundation for information-theoretic entropy or information content. This post continues the earlier post which introduced some of the basic ideas and operations of partition logic.
Series-Parallel Duality: Part I: Combating series chauvinism
Published on February 5, 2010
Series chauvinism
This post describes the duality between the usual (series) addition and the dual parallel addition. This duality is normally considered in electrical circuit theory and combinatorics, but it has a much wider applications. In Part I of this post, the focus is on developing series-parallel dual formulas—in contrast to the usual focus on formulas using only the series sum.
From the viewpoint of pure mathematics, the parallel sum is “just as good” as the series sum. It is only for empirical and perhaps even some accidental reasons that so much mathematics is developed using the series sum instead of the equally good parallel sum. There is a whole “parallel mathematics” which can be developed with the parallel sum replacing the series sum. Since the parallel sum can be defined in terms of the series sum (or vice-versa), “parallel mathematics” is essentially a new way of looking at certain known parts of mathematics.
Exclusive promotion of the series sum and prejudice against the parallel sum is series chauvinism. Before venturing further into the parallel universe, we might suggest some exercises to help the politically incorrect reader to combat the heritage of series chauvinism. Anytime the series sum seems to occur naturally in mathematics with the parallel sum being seemingly invisible, it is an illusion due to series chauvinism. The parallel sum has a “parallel” role that has been unfairly neglected.
In Part II of the post, series-parallel duality is applied to financial arithmetic and is shown to underlie a certain duality of formulas that has long been noted in the literature on valuation and appraisal. Hence this instance of intellectual trespassing applies concepts best known from electrical circuit theory to the operations of financial arithmetic. Moreover in economic theory, the much-used duality of convex functions and their conjugates is the “integral” of series-parallel (SP) duality, or, to put it the other way around, SP duality is the “derivative” of convex duality.
Series-Parallel Duality: Part I: Combating series chauvinism continued »
Series-Parallel Duality: Part II: Finanical arithmetic
Published on February 5, 2010
Reciprocal formulas in financial arithmetic
In financial arithmetic and in the appraisal literature, it has been noticed that the basic formulas occur in pairs, one being the reciprocal of the other. For instance, one popular text on real estate appraisal presents the “Basic Functions of Compound Interest and Their Reciprocals” [Friedman, Jack P. and Nicholas Ordway 1988. Income Property Appraisal and Analysis. Englewood Cliffs: Prentice Hall, p. 70]. The functions could be presented as follows to bring out the underlying symmetry.
| Function | Reciprocal |
Principal Retired by Payment of One |
Payment to Retire Principal of One |
Principal Amortized by Payments of One |
Payments to Amortize a Principal of One |
Fund Accumulated by One per Period |
Payments to Accumulate a Fund of One |
The Six Functions of One
This Part II of the series-parallel duality post shows that these reciprocal formulas are an example of the SP duality normally associated with electrical circuit theory.
Series-Parallel Duality: Part II: Finanical arithmetic continued »
The Math of Double-Entry Bookkeeping: Part I (scalars)
Published on February 4, 2010
Mathematics and Accounting: Two disjoint universes?
Double-entry bookkeeping illustrates one of the most astonishing examples of intellectual insulation between disciplines—the very opposite of intellectual trespassing. Double-entry bookkeeping (DEB) was developed during the fifteenth century and was published in 1494 as a system by the Italian mathematician Luca Pacioli and was anticipated in a 1458 manuscript of the Croatian merchant Benedikt Kotruljević (that was only published in 1573). Double-entry bookkeeping has been used for over five centuries in commercial accounting systems. If the mathematical formulation of any field should be well understood, one would think it might be accounting. Remarkably, however, the mathematical formulation of double entry accounting—algebraic operations on ordered pairs of numbers—was first published only in 1982 and is still largely unknown both in mathematics and accounting.
The Math of Double-Entry Bookkeeping: Part I (scalars) continued »
The Math of Double-Entry Bookkeeping: Part II (vectors)
Published on February 4, 2010
Multi-dimensional double-entry accounting?
Although double-entry bookkeeping (DEB) has been used in the business world for 5 centuries, the mathematical formulation of the double entry method is almost completely unknown. In this post, the mathematical treatment of double-entry bookkeeping using scalars given in Part I is generalized to the multi-dimensional case using vectors. The success in maintaining the two-sided accounts, debits and credits, the double-entry principle, and the trial balance in both cases shows that the formulation captures the double-entry method in mathematical form.
The Math of Double-Entry Bookkeeping: Part II (vectors) continued »
The implication operation on partitions
Published on February 2, 2010
Partitions and equivalence relations
In a 2001 commemorative volume for my mathematical mentor, Gian-Carlo Rota, three of his associates noted that “the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join 
From propositional logic to subset logic to partition logic
Published on January 28, 2010
From propositional logic to subset logic
This note outlines the following sequence of ideas. First, ordinary propositional logic is reinterpreted as the logic of subsets of a universe set U, with the propositional case being isomorphic to the special case of U = 1. Then the category-theoretic duality between subsets of a set and partitions on a set is used to broach the idea of a dual logic of partitions. At the end of the note, a link is given to the final draft of my forthcoming paper in the Review of Symbolic Logic which develops the logic of partitions from the basic ideas up through the correctness and completeness theorems for a tableau system of (zeroth order) partition logic.
From propositional logic to subset logic to partition logic continued »