David Ellerman

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts.

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. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. 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).

This chapter gives the partitional treatment of quantum measurement along with a number of other applications such as: commuting and non-commuting observables, von Neumann’s two types of quantum processes, the collapse postulate, quantum jumps, Feynman’s rules about adding amplitudes or probabilities (and the resulting “state reduction principle”), the partition version of the principle of identity of indistinguishables, Weyl’s interesting imagery for measurement, and the indistinguishability of like particles. Then the Yoga is used to systematically extend the concepts of ‘classical’ logical entropy to the quantum logical entropy which is then shown to naturally measure the results of quantum measurement. Finally the Yoga is again applied to relate group representations on sets (i.e., a group actions) and group representations on vector spaces over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {C}$$\end{document} which turn out to have such important applications in particle physics.

TheRohrlich, Fritz basic non-classical notion in quantum mechanics (QM) is the notion of superposition; entanglement is a particularly vexing special case. This chapter focuses on slowly developing the notion of a superposition state starting at the logical level of superposition applied to the notion of a set. To mathematically represent a superposition set, one must go beyond the one-dimensional notion of a 0, 1-vector representing which elements of the universe set are in the subset. A two-dimensional matrix will do the job with the 0, 1-vector along the diagonal and the off-diagonal elements of 1 representing which elements are superposed in the sense of being in the same equivalence class of an equivalence relation (which is just another way to look at a partition). Those 0, 1-matrices are the incidence matrices of equivalence relations. When such logical representations of equivalence relations are normalized by dividing through by the trace (sum of the diagonal elements), one arrives at a density matrix–which is the partitional way to represent a quantum state. The three main distinctive elements of QM math are the notions of a quantum state, a quantum observable, and a (projective) measurement applying an observable to a state. This chapter gives the partition-based treatment of quantum states via density matrices.

This chapter gives the partitional treatment of the concept of a quantum observable. This is done using a bit of mathematical folklore that we call the “Yoga of Linearization.” It is a method to transform set concepts into the corresponding vector space concepts. Starting with a real-valued numerical attributeNumerical attribute f:U→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:U\rightarrow \mathbb {R} $$\end{document} on a set U (like weight, height, age, etc.), we apply the Yoga to obtain the notion of a real-valued operator on a vector space over a field containing R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {R}$$\end{document}. If U is taken as a basis set for a vector space over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {C}$$\end{document}, then this procedure obtains a Hermitian operator, i.e., a quantum observable. But the Yoga also has an interesting intermediate step to transform set concepts into vector space concepts over the two element field Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {Z}_{2}$$\end{document}. This intermediate step gives a simplified pedagogical (or toy) model of QM over sets. Hence we develop that model to illustrate results in standard QM over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {C} $$\end{document} such as the double-slit experiment and Bell’s Theorem.

ThisGalileo new approach to understanding and interpreting quantum mechanics (QM) is based on the development of the notion of a partition on a set (or, equivalently, an equivalence relation on a set or a quotient set). The notion of a partition is not some ad hoc notion designed to make yet another interpretation of quantum mechanics. It is as fundamental a notion as that of a subset. But the math (and logic) of subsets was developed long before the corresponding partition math. There is a recent sequence of developments that constitute a revival or renaissance of partitions outside of the standard treatment in enumerative combinatorics. Standard logic is the Boolean logic of subsets (propositional logic being a special case). Subsets and partitions are dual concepts in category theory. The first stage in the revival was the development of the logic of partitions dual to the logic of subsets. Then just as finite Boole-Laplace probability theory starts as the quantitative version of the logic of subsets, so the quantitative version of the logic of partitions was the second stage resulting in the new foundations for information theory based on the notion of logical entropy. The third stage was based on the realization that the distinctive mathematics of QM is the vector (particularly Hilbert) space version of the mathematics of partitions. That is the subject of this book.

This book presents a new ‘partitional' approach to understanding or interpreting the math of standard quantum mechanics (QM). The thesis is that the mathematics (not the physics) of QM is the Hilbert space version of the math of partitions on a set and, conversely, the math of partitions is a skeletonized set level version of the math of QM. Since at the set level, partitions are the mathematical tool to represent distinctions and indistinctions (or definiteness and indefiniteness), this approach shows how to interpret the key non-classical QM notion of superposition in terms of (objective) indefiniteness between definite alternatives (as opposed to seeing it as the sum of ‘waves'). Thus, the book develops a new mathematical, or indeed, logical, approach to the century-old problem of interpreting quantum mechanics, ensure it is of interest to philosophers of science as well as mathematicians and physicists.

A basic duality arises throughout the mathematical and natural sciences. Traditionally, logic is thought to be based on the Boolean logic of subsets, but the development of category theory in the mid-twentieth century shows the duality between subsets and partitions (or equivalence relations). Hence, there is an equally fundamental dual logic of partitions. At a more basic or granular level, the elements of a subset are dual to the distinctions (pairs of elements in different blocks) of a partition. The quantitative version of subset logic is probability theory (as developed by Boole), and the quantitative version of the logic of partitions is information theory re-founded on the notion of logical entropy. The subset side of the duality uses a one-sample (or one-element) approach, e.g., the mean of a random variable; the partition side uses a two-sample (or pair-of-elements) approach. This paper gives a new derivation of the variance (and covariance) based on the two-sample approach, which positions the variance on the partition and information theory side of the duality and thus dual to the mean.

The purpose of this chapter is to summarize the new foundations for information theory presented in the book and to point out work yet to be done on the topic. The claim is that logical information theory fills the gap left by the Shannon theory of giving a definition of information as being about distinctions, differences, distinguishability, and diversity (as opposed to “uncertainty” etc.) with logical entropy as its direct measure—and with Shannon entropy as its requantification for the purposes of coding and communications theory. Shannon himself said that the “information theory” bandwagon created by the science press and popularizers had gone far beyond the actual accomplishments of communications theory—so the mathematical theory of logical entropy should not be seen as contrary to Shannon’s claims about his concept of entropy. The linearization methodology showed how to extend the ‘classical’ logical information theory to the quantum version where it was shown that quantum logical entropy is naturally related to ‘measuring’ quantum measurement and to providing a distance measure between quantum states. Finally, it is emphasized that the surface has only been scratched to relate logical entropy to all the fields that touch on information theory.

This chapter develops the multivariate (i.e., three or more variables) entropies. The Shannon mutual information is negative in the standard probability theory example of three random variables that are pair-wise independent but not mutually independent. When we assume metrical data in the values of the random variable (e.g., a real-valued variable), then there is a natural notion of metrical logical entropy and it is twice the variance—which makes the connection with basic concepts of statistics. The twice-variance formula shows how to extend logical entropy to continuous random variables. Boltzmann entropy is analyzed to show that Shannon entropy only arises in statistical mechanics as a numerical approximation that has attractive properties of analytical tractability. Edwin Jaynes’s Method of MaxEntropy uses the maximization of the Shannon entropy to generalize the indifference principle. When other constraints rule out the uniform distribution, the Jaynes recommendation is to choose the distribution that maximizes the Shannon entropy. The maximization of logical entropy yields a different distribution. Which solution is best? The maximum logical entropy solution is closest to the uniform distribution in terms of the ordinary Euclidean notion of distance. The chapter ends by giving the transition to coding theory.

This book presents a new foundation for information theory where the notion of information is defined in terms of distinctions, differences, distinguishability, and diversity. The direct measure is logical entropy which is the quantitative measure of the distinctions made by a partition. Shannon entropy is a transform or re-quantification of logical entropy for Claude Shannon’s “mathematical theory of communications.” The interpretation of the logical entropy of a partition is the two-draw probability of getting a distinction of the partition (a pair of elements distinguished by the partition) so it realizes a dictum of Gian-Carlo Rota: ProbabilitySubsets≈InformationPartitions\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac {Probability}{Subsets}\approx \frac {Information}{Partitions}$$ \end{document}. Andrei Kolmogorov suggested that information should be defined independently of probability, so logical entropy is first defined in terms of the set of distinctions of a partition and then a probability measure on the set defines the quantitative version of logical entropy. We give a history of the logical entropy formula that goes back to Corrado Gini’s 1912 “index of mutability” and has been rediscovered many times.

This chapter is focused on developing the basic notion of Shannon entropy, its interpretation in terms of distinctions, i.e., the minimum average number of yes-or-no questions that must be answered to distinguish all the “messages.” Thus Shannon entropy is also a quantitative indicator of information-as-distinctions, and, accordingly, a “dit-bit transform” is defined that turns any simple, joint, conditional, or mutual logical entropy into the corresponding notion of Shannon entropy. One of the delicate points is that while logical entropy is always a non-negative measure in the sense of measure theory (indeed, a probability measure), we will later see that for three or more random variables, the Shannon mutual information can be negative. This means that Shannon entropy can in general be characterized only as a signed measure, i.e., a measure that can take on negative values.

In this chapter, all the compound notions of simple, joint, conditional, and mutual logical entropy are defined and then the corresponding notions of Shannon entropy are derived via the dit-bit transform. Moreover, a number of other notions of divergence, cross entropy, and Hamming distance are developed for logical entropy along with the corresponding notions for Shannon entropy. And finally, a number of intriguing parallels between the two entropies and related inequalities are developed which allow some inequalities directly relating the two entropies.

Are corporations the problem? Can reforms in the area of corporate responsibility (e.g., more stakeholder governance) lead to any real changes? The goal of the paper is to analyse debates concerning the Citizens United case, corporate personhood, the stakeholder theory, the affected interests principle and, finally, deeper fallacies with respect to the rights of capital embedded in Marxism and conventional economic theories of capital and corporate finance. The last analysis considers another institution at the root of the problems of the current economic system: the renting of human beings in the employment relationship – which has also corrupted the original idea of a corporation that dates back to medieval times.

Classical liberalism tends to respond to the criticism of any voluntary market contract by promoting a wider choice of options and increased information and bargaining power so that no one would seem to be ‘forced’ or ‘tricked’ into an ‘unconscionable’ contract. Hence, at first glance, the strict logic of the classical liberal freedom-of-contract philosophy would seem to argue against ever abolishing any mutually voluntary contract between knowledgeable and consenting adults. Yet the modern liberal democratic societies have abolished (i.e., treated as invalid) at least three types of historical contracts: the voluntary slavery or perpetual servitude contract, the coverture marriage contract, and an undemocratic constitution to establish an autocratic government. Thus, the rights associated with those contracts are considered as inalienable. This paper analyzes these three contracts and shows that there is indeed a deeper democratic or Enlightenment classical liberal tradition of jurisprudence that rules out those contracts. The ‘problem’ is that the same principles imply the abolition of the employment contract, the contract for renting human beings, which is the foundation for the economic system that is often (but superficially) identified with classical liberalism itself. Frank Knight is taken throughout as the exemplary advocate of the economics of conventional classical liberalism.

The World Bank, the leading multilateral development agency, begins its mission statement with a dedication to helping people help themselves, and Oxfam, a leading nongovernmental organization (NGO) working on development, states that its “main aim is to help people to help themselves.”1 Perhaps the most successful example of development assistance in modern history was the Marshall Plan, which “did what it set out to do—help people help themselves” (Stern 1997). American of‹cial assistance to developing countries began with Harry Truman’s “Point Four” program in 1949, which was conceived as a worldwide “program of helping underdeveloped nations to help themselves.” 2 John Dewey gave perhaps the best statement of the theme: The best kind of help to others, whenever possible, is indirect, and consists in such modi‹cations of the conditions of life, of the general level of subsistence, as enables them independently to help themselves. (Dewey and Tufts 1908, 390)3 The idea and the rhetoric of “helping people help themselves” has been with us throughout the postwar period of of‹cial development assistance. For instance, we are all familiar with the ancient Chinese saying that if you give people ‹sh, you feed them for a day, but if you teach them how to ‹sh—or rather, if you enable them to learn how to ‹sh— then they can feed themselves for a lifetime.4 There is broad agreement— at least as a statement of high purpose—that helping people help themselves is the best methodology for development assistance in the developing countries as well as for other types of helping relationships.

There are fundamentally two mathematical logics. One the Boolean logic of subsets, usually presented today in the special case of propositional logic, which has many sublogics and extensions, the most important being the intuitionistic logic usually modeled by the open subsets of a topological space. The other co-fundamental mathematical logic is the topic of this book, the logic of partitions. We are using ”logic” in a mathematical sense as being about basic mathematical objects, subsets of a universe set or partitions on a universe set.1 Logic, in this mathematical sense, is not about propositions, although, as with any mathematical theory, it involves propositions about the basic objects, e.g., that an element is in a subset or that a distinction is made by a partition. Moreover, by taking the universe set to be the one element set 1, with two subsets ∅ and 1, there is a special case of subset logic, namely propositional logic, that can be interpreted as being about propositions with ∅ representing falsehood and 1 representing truth.

Democratic ownership in the sphere of economic production is often contrasted with capitalist ownership along the lines of transferability of legal rights, such as profit and governance rights. If capitalist ownership implies full transferability of legal rights in production, democratic ownership anchors legal rights with the current generation of workers in the firm. A worker cooperative is generally considered the best practical proxy for a democratic firm (Ellerman, 2021; Erdal, 2012); however, worker cooperatives remain rare in contemporary economies, while the capitalist form of enterprise continues to dominate the markets despite contrary predictions by intellectual giants like John Stuart Mill. Why did democratic ownership fail to achieve scale? Worker cooperatives are most commonly established from scratch or by fully converting a capitalist firm. In this chapter, we argue that cooperatives have faced challenges because they did not introduce a mechanism that would allow for the gradual conversion of capitalist ownership into democratic ownership. We introduce an alternative proposal that can help scale democratic ownership using the mechanism of leveraged gradual cooperative conversions. In the literature, this solution is described as the “Cooperative ESOP”, since it uses the ESOP leveraged financing mechanism, which is attached to a cooperative vehicle (Ellerman et al., 2022 and Appendix). We argue that the mechanism of gradual and leveraged conversion, with the aid of institutional support, can help democratic ownership grow in our economies and become mainstream.

The short answer is whenever the actual activity of the “cooperative” is not carried out by the members but by employees. The problem is, of course, not in cooperation per se but in the hiring, employing, renting, or leasing of people to carry out the supposedly “cooperative” activities of the “cooperative” (Ellerman, 2021). Consider the case of a typical consumer cooperative. What is the cooperative activity carried out by the consumer‑members? They do not consume cooperatively; that would be a commune or kibbutz. They shop and consume as individuals or as individual families. They do not carry out the activity of the consumer cooperative business—which is conducted by the hired managers and employees of the business. The whole notion of the consumer‑members cooperating together in some joint activity is a beautiful fiction, but a fiction nevertheless. Of course, there may be some overlap between employees and consumers, but we are analyzing functional roles, i.e., the roles people have qua consumers and qua workers. Moreover, the number of consumers will far exceed the number of employees.

From the 1970’s, there has been almost a half-century of development of employee-owned firms. There has been a wide variety of legal/capital structures that have been tried but too little analysis of which legal forms work or don’t work over the longer term, e.g. the transition from one generation of employee-owners to the next generation of employee owners. This paper provides a critical analysis of the major forms. These include the forms based on common ownership (Yugoslav self-managed firms and the UK EOTs), the older plywood cooperatives in the US Northwest along with the Spanish Sociedades Laborales, and the American Employee Stock Ownership Plans (US ESOPs). Finally, we advocate a variation on the US ESOP model, called the ‘European ESOP’, that seems to avoid some pitfalls and combines the best aspects of the past forms of employee ownership.

There seems to be two rather different philosophies of aid to development and poverty relief. (1) The progressive/social-democratic approach is for the government or aid agencies to do more and more good things for people. (2) The classical-liberal approach is to change the underlying conditions so that people are empowered to do good things for themselves. In this paper, we analyze Alexis de Tocqueville’s approach to these questions in his First Memoir and his (unfinished) Second Memoir on Pauperism.

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps in other sciences, not to mention in philosophy.

Consent is, of course, a necessary condition for a legitimate institution or practice, but is it a sufficient condition? From ancient times down to the modern day, there have been arguments that consent is sufficient for the legitimacy of ancient and modern forms of slavery or servitude. Even today, consider the simple question of what was wrong with historical slavery in America or elsewhere. Perhaps the most common answer is that it was without consent; it was coercion on an institutional scale. It is easy to argue historically that slavery and associated forms of servitude had, at best, only the “fiction of consent” (Chakravarty, 2022). But what about a “happy slave” (Herzog, 1989) who gives genuine consent? If the lack of consent was the basic reason for the illegitimacy of historical slavery, then the implication is that a similar institution or practice would be acceptable if there were genuine, informed, and knowledgeable consent.

This first chapter treats (informally) the sins of omission and commission about property rights and contracts in neoclassical microeconomic theory. The analysis is developed at both the descriptive and normative levels, and then the Fundamental Theory of Property Theory connects the two levels. That theorem is the property theoretic counterpart of the price theoretic theorem that a competitive equilibrium, under certain assumptions, is Pareto optimal. The property theoretic version states that when the market mechanism operates with no property externalities (no thefts or conversions of property) and no breaches of contracts, then the appropriation of newly created assets and liabilities by market participants (producers and consumers) satisfies the juridical principle of imputation of legal responsibility in accordance with factual responsibility.

The abolition of slavery abolished not only the involuntary ownership of other people (workers) but also voluntary contractual forms of lifetime servitude. But that system of lifetime servitude was replaced by the current system of voluntary renting, hiring, employing, or leasing workers, i.e., the employment system. Hence the name “Neo-Abolitionism” for the idea of abolishing the employer–employee contract in favor of each firm being a workplace democracy. The three arguments against the human rental system are modern versions of old arguments that descend from the Reformation and Enlightenment in the Abolitionist and Democratic Movements. The first argument derives from noting that the old inalienable rights argument based on de facto inalienability of responsibility and decision-making that ruled out the long-term contract of lifetime servitude also applies against the shorterterm contract to rent oneself out. The second argument is the old labor or natural rights theory of private property (in the fruits of one’s labor) that is violated when the employer legally appropriates the positive and negative fruits of the employees working in a firm. And the third argument applies to the firm the democratic arguments against the subjection contract that alienates the rights of self-governance in favor of a democratic contract of delegation.