Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, (...) due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S22 + iWPHP, and use it to derive Fermat's little theorem and Euler's criterion for the Legendre symbol in S22 + iWPHP extended by the pigeonhole principle PHP. We prove the quadratic reciprocity theorem in the arithmetic theories T20 + Count2 and I Δ0 + Count2 (...) with modulo-2 counting principles. (shrink)

We investigate the “mathematical” strength of the theory I*2. In particular we prove the quadratic reciprocity law and Bertrand's postulate, using fragments of I*2 which employ some well-known number-theoretic functions.

The system called F is essentially a sub-theory of Frege Arithmetic without the ad infinitum assumption that there is always a next number. In a series of papers (Systems for a Foundation of Arithmetic, True” Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity) it was shown that F proves a large number of basic arithmetic truths, such as the Euclidean Algorithm, Unique Prime Factorization (i.e. the Fundamental Law of Arithmetic), and Quadratic Reciprocity, indeed a (...) sizable amount of arithmetic. In particular, F proves some (but not all) of the Peano Axioms; that is, F proves the axioms of a sub-theory - call it FPA - of second-order Peano-Arithmetic. This short technical note will demonstrate that the converse also holds, in the following sense. F has the same language as second-order Peano Arithmetic except that, in addition, it has a two-place predicate symbol “Μ”. Then it is possible to provide a definition, indeed a reasonable definition, for “Μ” such that FPA proves all the axioms of F. So F and FPA effectively have the same proof-theoretic strength. In particular FPA, which lacks the Successor Axiom stating that every natural number has a successor, is able to prove the Euclidean Algorithm, Unique Prime Factorization, and Quadratic Reciprocity, indeed (again) a sizable amount of arithmetic. (shrink)

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...) Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

In my article on proof [Philosophia Mathematica (3) 5 (1997), 153—165], I suggested or intimated that computer proofs of mathematical theorems had been found only for relatively simple or trivial theorems. I am obligated to Martin Davis and R. S. Boyer for the information that this suggestion or intimation is incorrect. For instance, a machine proof of quadratic reciprocity was published by D. M. Russinoff in J. Automated Reasoning 8 (1992), 3–21. A machine proof of the unsolvability of (...) the halting problem has been published by R S. Boyer and J. S. Moore. This and many other examples are presented in Chapter 1 of their book, A Computational Logic Handbook, Academic Press, 1990. A second edition of this book is expected shortly. (shrink)

On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or roughly 30,000 (...) lines. This count includes about 65 pages of elementary number theory that we had at the outset, developed by Larry Paulson and others; also about 50 pages devoted to a proof of the law of quadratic reciprocity and properties of Euler’s ϕ function, neither of which are used in the proof of the prime number theorem. The page count does not include the basic HOL library, or properties of the real numbers that we obtained from the HOL-Complex library. (shrink)

This article examines the research of Louis J. Mordell on the Diophantine equation y2-k=x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y^2-k=x^3$$\end{document} as it appeared in one of his first papers, published in 1914. After presenting a number of elements relating to Mordell’s mathematical youth and his (problematic) writing, we analyze the 1914 paper by following the three approaches he developed therein, respectively, based on the quadratic reciprocity law, on ideal numbers, and on binary cubic forms. This (...) analysis allows us to describe many of the difficulties in reading and understanding Mordell’s proofs, difficulties which we make explicit and comment on in depth. (shrink)

Augustin-Louis Cauchy publie une majorité de ses recherches arithmétiques entre 1829 et 1840. Celles-ci ne sont pourtant qu’évoquées dans certaines histoires de la théorie des nombres centrées sur les lois de réciprocité ou sur la théorie des nombres algébriques. Elles y sont décrites comme contenant quelques résultats similaires à ceux de Gauss, Jacobi ou Dirichlet mais de manière incomplète et désordonnée. L’objectif de cet article est de présenter une analyse des textes arithmétiques de Cauchy publiés entre 1829 et 1840 pour (...) montrer qu’ils contiennent au contraire un ensemble cohérent de résultats en lien avec les formes quadratiques \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4p^{\mu }=x^2+ny^2$$\end{document}, où \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} est un nombre premier et \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} un diviseur de \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p-1$$\end{document}. Nous discuterons également la forme particulière de ce corpus et la stratégie utilisée pour retrouver les lignes directrices du travail de Cauchy. Augustin-Louis Cauchy published most of his arithmetical research between 1829 and 1840. These are however only mentioned in some number theory history centered on reciprocity laws or on theory of algebraic numbers. They are described as containing some results similar to those of Gauss, Jacobi and Dirichlet but in a incomplete and disorganized way. The objective of this paper is to present an analysis of Cauchy’s arithmetical texts published between 1829 and 1840 to show that they contain a rather consistent set of results related to quadratic forms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4p^{\mu } = x ^2 + ny ^2 $$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} is a prime and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} a divisor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p-1 $$\end{document}. We will also discuss the particular form of this body of texts and the strategy we used to find the guidelines of the work of Cauchy. (shrink)

Economists and biologists have proposed a distinction between two mechanisms – “strong” and “weak” reciprocity – that may explain the evolution of human sociality. Weak reciprocity theorists emphasize the benefits of long-term cooperation and the use of low-cost strategies to deter free-riders. Strong reciprocity theorists, in contrast, claim that cooperation in social dilemma games can be sustained by costly punishment mechanisms, even in one-shot and finitely repeated games. To support this claim, they have generated a large body (...) of evidence concerning the willingness of experimental subjects to punish uncooperative free-riders at a cost to themselves. In this article, I distinguish between a “narrow” and a “wide” reading of the experimental evidence. Under the narrow reading, punishment experiments are just useful devices to measure psychological propensities in controlled laboratory conditions. Under the wide reading, they replicate a mechanism that supports cooperation also in “real-world” situations outside the laboratory. I argue that the wide interpretation must be tested using a combination of laboratory data and evidence about cooperation “in the wild.” In spite of some often-repeated claims, there is no evidence that cooperation in the small egalitarian societies studied by anthropologists is enforced by means of costly punishment. Moreover, studies by economic and social historians show that social dilemmas in the wild are typically solved by institutions that coordinate punishment, reduce its cost, and extend the horizon of cooperation. The lack of field evidence for costly punishment suggests important constraints about what forms of cooperation can or cannot be sustained by means of decentralised policing. (shrink)

One of the most notable features in recent Confucian political theory is the advocacy of political meritocracy. Though Confucian meritocrats’ controversial institutional design has been subject to critical scrutiny, less attention has been paid to their underlying normative claims. This paper aims to investigate the two justificatory conditions of Confucian political meritocracy—the service condition and the reciprocity condition—in light of classical Confucianism and with special attention to moral disagreement. Finding the normative argument for Confucian political meritocracy both incomplete (in (...) light of classical Confucianism) and implausible (under the circumstances of moral disagreement), it proposes Confucian constitutional democracy as an alternative that can meet the three conditions of the good Confucian polity—service, reciprocity, and remedy—by reconceiving the people's well-being in terms of their basic rights, as well as by promoting constitutional dialogue among the three branches of the government. (shrink)

This is a philosophical study by a group of international scholars discussing issues related to reciprocity in the globalized world. Concerned issues include ethical dimension and foundation of reciprocity and generosity, characteristics of reciprocity, different understandings of reciprocity across cultural traditions, the relationship between reciprocity and other human values such as justice, cooperation, friendship, etc.

사회 경제적 혜택의 분배 규범으로 롤즈가 제시하는 차등의 원칙의 근거를 검토한다. 통상 불확실 상황에서 합리적 선택 전략들 중 하나로 거론되는 최소치 극대화(maximin) 기준이 그 근거라고 간주되지만, 롤즈는 상호성(reciprocity)을 차등의 원칙의 선택 근거로 제시한다. 논문은 롤즈적 상호성의 내용, 관련된 이상들, 상호성으로부터 차등의 원칙으로 나아가는 추론을 검토한다. 논의 결과는 상호성에 관련된 이상들인 평등한 출발점이자 비교의 수준점, 도덕적으로 자의적 요소들에 의한 분배의 규제, 효율성, 자유롭고 평등한 개인들의 합의가 이질적이고 고유한 근거와 방향을 가진 이상들이기에 그것들 각각의 내용과 상대적 비중을 어떻게 보는지에 따라 차등의 (...) 원칙과 다른 분배 규범이 도출될 가능성이 있다는 것이다. (shrink)

Multitrait-multimethod (MTMM) analysis is one of the most frequently employed methods to examine the validity of psychological measures. Confirmatory factor analysis (CFA) is a commonly used analytic tool for examining MTMM data through the specification of trait and method latent variables. Most contemporary CFA-MTMM models either do not allow estimating correlations between the trait and method factors or they are restricted to linear trait-method relationships. There is no theoretical reason why trait and method relationships should always be linear, and (...) class='Hi'>quadratic relationships are frequently proposed in the social sciences. In this article, we present two approaches for examining quadratic relations between traits and methods through extended latent difference and latent means CFA-MTMM models (Pohl et al., 2008; Pohl and Steyer, 2010). An application of the new approaches to a multi-rater study of the nine inattention symptoms of attention-deficit/hyperactivity disorder in children (N = 752) and the results of a Monte Carlo study to test the applicability of the models under a variety of data conditions are described. (shrink)

Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings. Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + bY2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we (...) prove that except in some trivial cases, M[ x, y] with x2 + by2 = a can be embedded in a model of NOI. We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum. (shrink)

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

This article describes methods and ideas that emerged from a continuing enquiry into ‘metadesign’ that led us to think about the role of ‘consciousness’ in teams, communities and the biosphere. In the West the notion of ‘consciousness’ has been shaped by humanism, industrialization and some strident forms of individualism. These have encouraged us to see it in strongly anthropocentric and solipsistic terms. The global economic system also reflects this individualistic ideology, given that ‘growth’, is driven by personal avarice on a (...) collective scale. On the other hand, capitalism encourages enterprises to scale-up when they become successful. Both tendencies reduce the consciousness of organizations and communities. The article illustrates this idea by describing a ‘quadratic’ model of team consciousness that was developed to enhance practical developments that support personal, interpersonal, transpersonal and environmentally situated modes of awareness. (shrink)

The tendency to reciprocate – to return good for good and evil for evil – is a potent force in human life, and the concept of reciprocity is closely connected to fundamental notions of ‘justice’, ‘obligation’ or ‘duty’, ‘gratitude’ and ‘equality’. In _Reciprocity_, first published in 1986,_ _Lawrence Becker presents a sustained argument about reciprocity, beginning with the strategy for developing a moral theory of the virtues. He considers the concept of reciprocity in detail, contending that it (...) is a basic virtue that provides the basis for parental authority, obligations to future generations, and obedience to law. Throughout the first two parts of the book, Becker intersperses short pieces of his own narrative fiction to enrich reflection on the philosophical arguments. The final part is devoted to extensive bibliographical essays, ranging over anthropology, psychology, political theory and law, as well as the relevant ethics and political philosophy. (shrink)

Economists and biologists have proposed a distinction between two mechanisms – “strong” and “weak” reciprocity – that may explain the evolution of human sociality. Weak reciprocity theorists emphasize the benefits of long-term cooperation and the use of low-cost strategies to deter free-riders. Strong reciprocity theorists, in contrast, claim that cooperation in social dilemma games can be sustained by costly punishment mechanisms, even in one-shot and finitely repeated games. To support this claim, they have generated a large body (...) of evidence concerning the willingness of experimental subjects to punish uncooperative free-riders at a cost to themselves. In this article, I distinguish between a “narrow” and a “wide” reading of the experimental evidence. Under the narrow reading, punishment experiments are just useful devices to measure psychological propensities in controlled laboratory conditions. Under the wide reading, they replicate a mechanism that supports cooperation also in “real-world” situations outside the laboratory. I argue that the wide interpretation must be tested using a combination of laboratory data and evidence about cooperation “in the wild.” In spite of some often-repeated claims, there is no evidence that cooperation in the small egalitarian societies studied by anthropologists is enforced by means of costly punishment. Moreover, studies by economic and social historians show that social dilemmas in the wild are typically solved by institutions that coordinate punishment, reduce its cost, and extend the horizon of cooperation. The lack of field evidence for costly punishment suggests important constraints about what forms of cooperation can or cannot be sustained by means of decentralised policing. (shrink)

This paper explores ethical issues raised by quadratic voting. We compare quadratic voting to majority voting from two ethical perspectives: the perspective of utilitarianism and that of democratic theory. From a utilitarian standpoint, the comparison is ambiguous: if voter preferences are independent of wealth, then quadratic voting out- performs majority voting, but if voter preferences are polarized by wealth, then majority voting may be superior. From the standpoint of democratic theory, we argue that assess- ments in terms (...) of efficiency are too narrow. Voting institutions and political institutions more generally face a legitimacy requirement. We argue that in the presence of inequalities of wealth, any vote buying mechanism, including quadratic voting, will have a difficult time meeting this requirement. (shrink)

This paper addresses the question of whether the duties associated with public reason are conditional on reciprocity. Public reason is not a norm intended to stabilize commitment to justice, but a moral principle, albeit one that is conditional on reciprocity because grounded in the idea of mutual respect despite ongoing moral disagreement. We can build reciprocity into the principle by stipulating that unanimous acceptability is required only with respect to points of view accepting the principle. If compliance (...) with law is assured, then the duties of public reason associated with authorship of law should be considered conditional on reciprocity only in this ‘internal’ sense, which is not proportional but bi-lateral. (shrink)

In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched (...) by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found. (shrink)

This paper provides strong evidence challenging the self-interest assumption that dominates the behavioral sciences and much evolutionary thinking. The evidence indicates that many people have a tendency to voluntarily cooperate, if treated fairly, and to punish noncooperators. We call this behavioral propensity “strong reciprocity” and show empirically that it can lead to almost universal cooperation in circumstances in which purely self-interested behavior would cause a complete breakdown of cooperation. In addition, we show that people are willing to punish those (...) who behaved unfairly towards a third person or who defected in a Prisoner’s Dilemma game with a third person. This suggests that strong reciprocity is a powerful device for the enforcement of social norms involving, for example, food sharing or collective action. Strong reciprocity cannot be rationalized as an adaptive trait by the leading evolutionary theories of human cooperation (in other words, kin selection, reciprocal altruism, indirect reciprocity, and costly signaling theory). However, multilevel selection theories of cultural evolution are consistent with strong reciprocity. (shrink)

We study the Iterated Bilateral Reciprocity game in which the need for help arises randomly. Players are heterogeneous with respect to ‘neediness’ i.e. probability of needing help. We find bounds on the amount of heterogeneity which can be tolerated for cooperation (all players help when asked to help) to be sustainable in a collectivity. We introduce the notion of Generalised Reciprocity. Individuals make a costly first move to benefit another under the reasonable expectation that either the other or (...) somebody else will reciprocate. We hope that these tentative attempts at extending Axelrod’s seminal work on cooperation will inspire future efforts in the field of organisational culture and social theory more generally. (shrink)

The evidence of laboratory experiments of behavioral economists shows that individuals behave reciprocally. These data put into question the pure self-interest thesis of human motivation of the homo oeconomicus model and call for alternative models. Focusing on the explanation of reciprocal behavior in Trust Games, this article proposes two directions that economists and other social scientists might want to consider in order to establish a more solid foundation for economic theory. First, it presents models that economic theorists developed to explain (...) the laboratory evidence of reciprocal behavior. It highlights that all of these models subscribe to the Humean view that desires are at the source of any human motivation and suggests an alternative Kantian model where reasons have the capacity to motivate human action. Second, it emphasizes that a supplementary examination of the social background conditions would illuminate the analysis of the findings because of the connection between the 'local' and society-wide demands of reciprocity. (shrink)

This paper tries to reconcile reciprocity with a fundamentally 'subject-centred' ethic by interpreting the reciprocity condition as a consequence of the fact that justice is in part a relational value. Duties of egalitarian distributive justice are not grounded on the duty to reciprocate benefits already received, but limited by a reasonable assurance of compliance on the part of those able to reciprocate, because their point is to constitute a valuable relationship, one of mutual recognition as equals. We have (...) unconditional duty to help establish just global institutions, institutions which would allow us to share fairly in the burdens and benefits of global economic cooperation, but no unilateral duty to share fairly, where such institutions are not in place. Since non-contribution on the part of those unable to contribute involves no failure of recognition, the disabled do not fall outside the scope of distributive justice. (shrink)

Ernst Fehr and his collaborators have argued that traditional explanations of human cooperation cannot account for strong reciprocity. They provide substantial empirical evidence that strong reciprocity is an important phenomenon that cannot be explained by the traditional models of kin selection or reciprocal altruism. In this note, however, I argue that it will be difficult to test specific adaptive explanations of strong reciprocity because it is apparently unique to humans. Consequently, it is difficult to employ the comparative (...) method, which is one of biology’s best tools for testing adaptationist claims. (shrink)

Laland and colleagues have sought to challenge the proximate–ultimate distinction claiming that it imposes a unidirectional model of causation, is limited in its capacity to account for complex biological phenomena, and hinders progress in biology. In this article the core of their argument is critically analyzed. It is claimed that contrary to their claims Laland et al. rely upon the proximate–ultimate distinction to make their points and that their alternative conception of reciprocal causation refers to phenomena that were already accounted (...) for by standard theory. (shrink)

This paper explores the notion of reciprocity in the context of active pulmonary and laryngeal tuberculosis treatment and related control policies and practices. We seek to do three things: First, we sketch the background to contemporary global TB care and suggest that poverty is a key feature when considering the treatment of TB patients. We use two examples from TB care to explore the role of reciprocity: isolation and the use of novel TB drugs. Second, we explore alternative (...) means of justifying the use of reciprocity through appeal to different moral and political theoretical traditions. We suggest that each theory can be used to provide reasons to take reciprocity seriously as an independent moral concept, despite any other differences. Third, we explore general meanings and uses of the concept of reciprocity, with the primary intention of demonstrating that it cannot be simply reduced to other more frequently invoked moral concepts such as beneficence or justice. We argue that reciprocity can function as a mid-level principle in public health, and generally, captures a core social obligation arising once an individual or group is burdened as a result of acting for the benefit of others. We conclude that while more needs to be explored in relation to the theoretical justification and application of reciprocity, sufficient arguments can be made for it to be taken more seriously as a key principle within public health ethics and bioethics more generally. (shrink)

We axiomatize EFA in strictly mathematical terms, involving only the ring operations, without extending the language by either exponentiation, finite sets of integers, or polynomials.

This elegant treatise examines the nature of kindness through the fascinating lenses and contexts of ancient, medieval and contemporary philosophy, natural history, theories of mind, of natural selection, eco-psychology and sociobiology. It challenges the reader to consider the myriad potential consequences of human behavior, examining various iconographic moments from the history of art and science as a precursor to the concept and vital potentials for ecological conversion. Focusing on the fundamental mechanisms of reciprocity among humans, other species, communities and (...) nations, Tobias and Morrison lead readers on a remarkable journey whose itinerary, and the provocative questions explored, seek to affirm a pattern in evolution and in human thought that is emphatically oriented towards benevolence, not tyranny. Prosociality in all species - making others happy, kind gestures at any and every juncture of life - has, as a discipline of enquiry, enjoyed a social scientific renaissance during the last decade. Can natural selection move rapidly enough to meet that ultimate challenge? Can our species re-evolve in real time, moving from the ideas, to the ideals, to their applied engineering in a real world that is ecologically hemorrhaging? Which all the critical moral and cognitive changes in social communion such new human nature, as the Authors suggest, clearly requires? This groundbreaking work of ecological philosophy, with its roots in ancient Greek thought, represents a radical break with nearly every traditional scientific paradigm, in exploring the intuitive geography and dramatic questions of ourselves - each and every one of us - that will prove crucial to the survival of our species, and all those we co-habit this miraculous planet with. (shrink)

The concept of the reasonable plays an important role in Rawls's political philosophy, but there has been little systematic investigation of this concept or of the way Rawls employs it. This article distinguishes several different forms of reasonableness and uses them to explore Rawls's political liberalism. The discussion focuses on the idea, found especially in the most recent versions of this theory, of a family of liberal conceptions of justice each of which is regarded by everyone in a polity as (...) reasonable, even if only barely so. The idea of such a family is central to Rawls's notion of reciprocity and the view of political cooperation associated with it. This article questions whether the concept of the reasonable can play the role that Rawls intends. (shrink)

South Africa shared with the world the warning of a new strain of SARS-CoV2, Omicron, in November 2021. As a result, many high-income countries (HICs) instituted complete travel bans on persons leaving South Africa and other neighbouring countries. These bans were unnecessary from a scientific standpoint, and they ran counter to the International Health Regulations. In short, South Africa was penalized for sharing data. Data sharing during pandemics is commonly justified by appeals to solidarity. In this paper, we argue that (...) solidarity is, at best, an aspirational ideal to work toward but that it cannot ground an obligation to share data. Instead, low-and-middle income countries (LIMCs) should be guided by the principle of reciprocity, which states that we ought to return good for good received. Reciprocity is necessarily a conditional principle. LMICs, we argue, should only share data during future pandemics on the condition that HICs provide enforceable assurances that the benefits of data sharing will be equitably distributed and that LMICs won’t be penalized for sharing information. (shrink)

Reciprocity is ubiquitous in our lives, both as a way of rewarding and punishing others. Consequently, the social sciences have devoted many studies to this phenomenon. However, the concept of 'reciprocity' is quite polyvalent, and is used in many different ways across different disciplines - a situation potentially prone to equivocation, which hinders fruitful interdisciplinary work. At the same time, although philosophers often invoke 'reciprocity' in their work, there is a lack of conceptual clarification about what (...) class='Hi'>reciprocity actually means - an eminently philosophical task. This thesis fills such a gap, by offering a general framework that allows us to understand how 'reciprocity' is used across different sciences. Such a framework consists in a grammar of several uses/meanings of reciprocity. Moreover, the thesis fills a second gap, regarding the correct interpretation of the psychological role of the reciprocity-motive in economic experiments. There has been a long-standing tradition in economics which overemphasizes the role of inequality aversion and downplays the role of reciprocity. Contrary to this deeply ingrained trend, I argue that reciprocity is the main motive of punishment of free-riders in the public goods game. My argument includes the first systematic critique of an experiment which is currently quite popular in behavioural economics. I show that the latter does not deserve the credit which it has received so far, and thus many economists need to revise their endorsement of inequality aversion. By offering a renewed interpretation of the motives in economic experiments, the thesis highlights reciprocity as a distinctive motivation. The thesis also offers tools to rethink key aspects of economic methodology, and proposes a distinction between prediction, explanation and rationalization of behaviour, in order to achieve psychologically realistic explanations of behaviour. The disciplinary home of the thesis is in philosophy of social science, economics and moral psychology. (shrink)

Reciprocal libertarianism is a version of left-wing libertarianism that combines self-ownership with an egalitarian distribution of resources according to reciprocity. In this paper, I show that reciprocal libertarianism is a coherent and appealing view. I discuss how reciprocal libertarians can handle conflicts between self-ownership and reciprocity, and I show that reciprocal libertarianism can be realised in a framework of individual ownership of external resources or in a socialist scheme of common ownership (libertarian socialism). I also compare reciprocal libertarianism (...) with left-libertarian approaches: I argue that a reciprocity-sensitive version of left-libertarianism (reciprocal left-libertarianism) is coherent and morally superior to traditional left-libertarianism, on grounds of incorporating a distinctively solidaristic and recognition-oriented aspect of equality. The policy implications of reciprocal libertarianism will differ depending on which rights people can have over external resources, but all reciprocal libertarian views acknowledge the existence of social rights that people have as co-operators. (shrink)

A quadratic space is a generalization of a Hilbert space. The geometry of certain kinds of subspaces (“closed,” “splitting,” etc.) is approached from the purely lattice theoretic point of view. In particular, theorems of Mackey and Kaplansky are given purely lattice theoretic proofs. Under certain conditions, the lattice of “closed” elements is a quantum proposition system (i.e., a complete orthomodular atomistic lattice with the covering property).

In recent years, biologists and philosophers of science have argued that evolutionary theory should incorporate more seriously the idea of ‘reciprocal causation.’ This notion refers to feedback loops whereby organisms change their experiences of the environment or alter the physical properties of their surroundings. In these loops, in particular niche constructing activities are central, since they may alter selection pressures acting on organisms, and thus affect their evolutionary trajectories. This paper discusses long-standing problems that emerge when studying such reciprocal causal (...) processes between organisms and environments. By comparing past approaches to reciprocal causation from the early twentieth century with contemporary ones in niche construction theory, we identify two central reoccurring problems: All of these approaches have not been able to provide a conceptual framework that allows maintaining meaningful boundaries between organisms and environments, instead of merging the two, and integrating experiential and physical kinds of reciprocal causation. By building on case studies of niche construction research, we provide a model that is able to solve these two problems. It allows distinguishing between mutually interacting organisms and environments in complex scenarios, as well as integrating various forms of experiential and physical niche construction. (shrink)

In this paper, we present two proofs of the reciprocal butterfly theorem. The statement of the butterfly theorem is: Let us consider a chord PQ of midpoint M in the circle Ω(O). Through M, two other chords AB and CD are drawn, such that A and C are on the same side of PQ. We denote by X and U the intersection of AD respectively CB with PQ. Consequently, XM = YM. For the proof of this theorem, see [1]. The (...) reciprocal of the butterfly theorem has the following statement: In the circle Ω(O), let us consider the chords PQ, AB and CD which are concurrent in the point M≠O, such as the points A and C are on the same side of the line PQ. Let X and Y respectively be the intersections of the chord PQ with AD and BC respectively. If XM = YM, then M is the middle of the chord PQ. (shrink)