Formalised knowledge systems, including universities and research institutes, are important for contemporary societies. They are, however, also arguably failing humanity when their impact is measured against the level of progress being made in stimulating the societal changes needed to address challenges like climate change. In this research we used a novel futures-oriented and participatory approach that asked what future envisioned knowledge systems might need to look like and how we might get there. Findings suggest that envisioned future systems will need (...) to be much more collaborative, open, diverse, egalitarian, and able to work with values and systemic issues. They will also need to go beyond producing knowledge about our world to generating wisdom about how to act within it. To get to envisioned systems we will need to rapidly scale methodological innovations, connect innovators, and creatively accelerate learning about working with intractable challenges. We will also need to create new funding schemes, a global knowledge commons, and challenge deeply held assumptions. To genuinely be a creative force in supporting longevity of human and non-human life on our planet, the shift in knowledge systems will probably need to be at the scale of the enlightenment and speed of the scientific and technological revolution accompanying the second World War. This will require bold and strategic action from governments, scientists, civic society and sustained transformational intent. (shrink)

John Rawls famously holds that the basic structure is the 'primary subject of justice.'1 By this, he means that his two principles of justice apply only to a society's major political and social institutions, including chiefly the constitution, the economic and legal systems, and (more contentiously) the family structure.2 This thesis — call it the basic structure restriction — entails that the celebrated difference principle has a narrower scope than one might have expected. It doesn't apply directly to choices (...) that individuals make within the basic structure. Individuals can live up to the demands of justice simply by obeying whatever rules are set by, and by doing what is necessary to sustain, the basic .. (shrink)

This is a review article about John Searle's most recent book The Rediscovery of the Mind, which criticizes it for not going far enough in its departure from orthodox materialistic views of the brain and mind. It argues that Searle's two central propositions, consciousness is irreducible and consciousness cannot cause anything that cannot be explained by the causal behaviour of neurons, are incompatible; and suggests that it is reasonable and scientifically respectable to reject the latter rather than the former.

Do the demands of egalitarian justice extend to the international realm? Some believe that a positive answer follows from a simple line of reasoning: where a child happens to be born is a morally arbitrary fact; accordingly, it shouldn’t unduly influence her life prospects, as will inevitably be the case unless economic inequalities between countries are ironed out. I argue that this style of argument overlooks an important problem concerning the extent to which a person can unilaterally impose enforceable obligations (...) on others by choosing to have a child. I maintain that properly egalitarian duties of distributive justice only arise when this problem of unilateral imposition has been solved. I further argue that solving the problem requires robust political institutions that do not exist in the international context—institutions instantiating a version of what John Rawls calls pure procedural justice, to ensure that the aggregate result of individual reproductive choices is acceptable to all. (shrink)

In response to the proposal by Walter Jaeschke contained in the preceding paper, the Nineteenth Century Theology Group of the American Academy of Religion discussed plans, at the annual meeting of the Academy on 15–17 November 1979, to complete a new English study edition of Hegel’s Lectures on the Philosophy of Religion, and has agreed to sponsor its publication by Scholars Press in the AAR Texts & Translations Series. An Editorial Committee has been formed with the following membership: Robert F. (...) Brown, Richard Crouter, James O. Duke, Francis S. Fiorenza, Joseph Fitzer, Peter C. Hodgson, Walter Jaeschke, Darrell Jodock, O. Kem Luther, Dale M. Schlitt, John C. Shelley, James Yerkes. Many of these persons will be involved in the editing and translating process. (shrink)

Kyle Johannsen soutient que, pour être pleinement convaincante, la théorie de la justice de John Rawls doit incorporer la conception de l’équité associée avec l’égalitarisme des chances de G.A. Cohen. Il maintient également que, lorsqu’on modifie ainsi la théorie de Rawls, on voit que les principes choisis dans la position originelle doivent être ce que Cohen appelle des «règles de régulation». Je rétorque que la conception de l’équité qu’adopte Rawls est idéalement adaptée aux besoins de sa théorie, et que (...) l’incorporation de la conception de l’équité que défend Cohen rendrait indéterminé le résultat de la position originelle. (shrink)

Ferdinand Christian Baur, one of the great innovators in the study of the New Testament, argued that each of its books reflects the interests and tendencies of its author in a particular religio-historical milieu. A critique of the writings must precede any judgments about the historical validity of individual stories about Jesus in the Gospels. Thus Baur could move beyond the impasse created by Strauss's Life of Jesus. Baur demonstrated that the Gospel of John is not a historical document (...) comparable to the Synoptic Gospels and cannot be used to reconstruct the teaching of Jesus, and that the Synoptic Gospels must be read critically and selectively. He applied the same principles to the Epistles, arguing that only four are genuinely Pauline.Baur's Lectures on New Testament Theology, delivered in Tübingen during the 1850s, summarize thirty years of his research. The lectures begin with an Introduction on the concept, history, and organization of New Testament theology. Part One is devoted to the teaching of Jesus, which Baur finds most reliably in Matthew. Part Two contains the teaching of the Apostles in three chronological periods. The first period presents the theological frameworks of the Apostle Paul and the Book of Revelation; the second period, the frameworks of Hebrews, the Deutero-Pauline Epistles, James and Peter, the Synoptic Gospels and Acts; and the third period, those of the Pastoral Epistles and the Gospel of John. (shrink)

Responses to my article on Dawkins and God have fallen into two classes: those that challenge my criticism of Dawkins’ atheism, and those that challenge my criticism of the morality on display in some Bible stories. I will briefly respond to those in the first class, and then those in the second class. P. J. Moss suggests I am attracted to “the Cartesian notion of mind body dualism,” and do not have regard to “the work of those philosophers of mind (...) who … see the task of the philosopher as posing the problem into a precise enough form so that it admits of scientific resolution;” and he commends the work of John Searle. I am indeed attracted to a kind of dualism. However, it is not the Cartesian dualism of “two distinct realms” rejected by Searle, but rather a dualism that accepts, as Searle does, that there are two categories of empirical reality, subjective and objective, which are mutually irreducible The Rediscovery of the Mind, pp19, 98), and that there are features of subjective reality that cannot be fully understood in terms of objective reality. In a major work published in 2001, Rationality in Action, Searle even leaves open as a reasonable possibility a view I support, namely that consciousness may be able to cause things that cannot be fully explained by the causal behaviour of neurons, and he also supports a non Humean notion of the self, as an entity that can, as a whole, consciously try to do things: see my review in Journal of Consciousness Studies 9, 92 94. In any event, my argument against Dawkins does not depend on acceptance of dualism, just on the undoubted fact that science does not yet have the first idea what objective features are necessary and sufficient to give rise to subjectivity. Robert McLaughlin makes out a reasoned case against my three suggested errors in Dawkins. It would take a book to deal fully with points of the kind he raises, but I have to be brief here. (shrink)

This paper responds to Professor John McMurtry, primarily to his critique of my recent book, Economics as Moral Science. Although agreeing with my attribution of a "moral a priorism" to orthodox or neo-classical economics, McMurtry takes issue with my "conversion thesis", that an a priori, ethically committed theory can be transformed into a testable empirical science of actual behaviour through the application of institutional constraints to individual motivations. McMurtry views such a thesis as "logically possible but morally abhorrent". In (...) so doing, he ascribes a version of economic determinism to me which, he claims, leads me to mistakenly understand the neo-classical paradigm as circumscribing the "boundaries of reality itself" and thereby entrenching the life-destructive values presupposed by this paradigm. I reject such a reading of Economics as Moral Science and explain the manner in which it is inconsistent both with the theoretical substance and practical agenda of my work. I propose that the irreducible basis of disagreement remains one wherein I believe that a more radical reform of the capitalist market order is mandatory to establish more defensible moral ideals than does McMurtry. My reply closes by recommending a constitutional partitioning of material goods such that we may more securely act outside the ethical constraints of neo-classical theory. (shrink)

My paper responds to certain themes of Professor John McMurtry's recent book, Unequal Freedoms: The Global Market as an Ethical System. Although I am in general sympathy with McMurtry's penetrating critique of conventional market theory and practice, I find Unequal Freedoms ambivalent on the critical question of whether endorsing and enacting the life-value code McMurtry proposes would require only a mitigation of the principles and definitive activities of the competitive market system or whether significant reforms within the system would (...) have to be deep structural ones. It is argued that the second alternative is inescapable. I defend my perspective from the point of view of a philosophical analysis of orthodox or neo-classical theory-construction about the competitive market order. In particular, I examine three fundamental principles of such modelling: the maximization of the satisfaction of self-interest, the unboundedness of consumer desire for material goods, and the distribution of monetized wealth. In order for McMurtry's "civil commons" to survive, the practice of each of these principles would need to be radically modified. However, in doing so, competitive capitalism would lose its essential identity as a socio-economic order. (shrink)

In his book Economics as a Moral Science , Bernard Hodgson argues that economics is not value neutral as is often claimed, but is a value-laden discipline. In the long argument for this in his book, Hodgson never discusses or even mentions corporations. This article explains that corporations are absent from Hodgson’s discussion because he considers only the consumption side of general equilibrium theory (GET), and it shows that if Hodgson had included corporations and the production side, his overall argument (...) would have been more complete and convincing. This article shows that Hodgson’s methodology, when applied to the production side of GET, has value implications for CEOs of large corporations, for shareholders and members of Boards of Directors, and for legislators and regulators of business. Hodgson’s claim that economics must consider the ability of economic agents to create or change the institutional, cultural, and organizational conditions of their own economic actions is supported and expanded. (shrink)

The Alumni Office at Bradford University has been operating a PC based Alumni System using Dbase III Plus. This system is now breaking down due to the workload being placed upon it and a new system is required. The production of a new system has been undertaken by Martin Charlesworth, Timothy Hodgson and Ioanis Trikukis as an M.Sc. project. This report relates to that project. A system has been produced as a joint effort whereby each team member has produced elements (...) of the whole system. The result has been a system, which while not fuIIy complete, is in a state which can be used by the Alumni Office. The production of the system involved an analysis of the current system, the design of a new system and the implementation of the design. These stages are detailed in this report. The project has met its aims and its organisation has proved to be an effective method of working. (shrink)

This paper achieves two general objectives. It first analyses Bernard Hodgson's "Economic As Moral Science" as a path-breaking internal critique of neo-classical economic theory, and it then demonstrates that the underlying neo-classical paradigm he presupposes suffers from a deeper-structural myopia than his standpoint recognizes. EMS mainly exposes the a priori moral prescriptions underlying orthodox consumer choice theory - namely, its classical utilitarian ground and four or, as argued here, five hidden universal categorical-ought prescriptions which the theory presupposes as instrumental imperatives: (...) comparability evaluations by all consumer judgements; non-satiety of consumer desire; consistency and transitivity of consumer preferences; diminishing rate of marginal substitution by consumer choice; and an unlimited aggregate growth of commodity production, or "the liberal growth ethic". The article argues that Hodgson's refutation of the neo-classical claims of "value neutral scientific method" is sound, that his bridging of the Humean reason-desire divide by the "rational review" of wants is resonantly demonstrated, and that his argument for conversion of an "a priori-cum-normative-cum-idealized" neoclassical theory into scientific status is logically plausible but morally abhorrent. The principal objection to Hodgson's magisterial exposé of neo-classical doctrine's moral a priorism is that the latter's normative presuppositions are profoundly deranged at a level that he himself assumes as given. In consequence, there is theoretical closure at three levels: to the underlying "life economy" of non-priced and non-profit production and distribution of goods otherwise in short supply; to the "civil commons" infrastructure sustaining these non-commodity systems of social and ecological production and distribution; and to the systemic despoiling of both by monetized market mechanisms which are falsely assumed as the defining limits of "the economy". (shrink)

One of the paradoxes of recent political and economic theory is that, in spite of a period of extended economic difficulty, there has been a growing consensus concerning the virtues of the market economy. In particular, there has been a trend in socialist theory to argue that not only are socialism and the market not incompatible, but that some version of market socialism is the only feasible, practicable, and ethically and politically desirable form of socialism. Notable proponents of this view (...) with whom this paper will be particularly concerned are Selucky, Nove and Hodgson. I will not, in this paper, address the question of whether the market and socialism are necessarily incompatible. Neither will I examine the whole gamut of political and ethical issues surrounding the relation of the market to democracy, freedom, individual rights, and so on – not because they are unimportant, but because they require more extended attention than I could give here. My concern will rather be with some of the economic arguments to which defenders of market socialism have appealed. (shrink)

The Orders of Discourse: Philosophy, Social Science, and Politics, JOHN GUNNELLHow Economics Forgot History: The Problem of Historical Specificity in Social Science, GEOFFREY HODGSON.

In lieu of an abstract, here is a brief excerpt of the content:Keeping Philosophy in Mind:Shadworth H. Hodgson's Articulation of the Boundaries of Philosophy and ScienceThomas W. StaleyIntroductionShadworth H. Hodgson's (1832–1912) contributions to Victorian intellectual discourse have faded from prominence over the past century. However, despite his current anonymity, Hodgson's case is important to an understanding of the historical split between philosophy and science in late nineteenth century Britain. In particular, his example illuminates the specific role played by developing concepts (...) of the mind in a period of deep transition in the British intellectual community. His virtual disappearance from the record, I contend, is indicative also of the disappearance of his form of philosophy as a vital social project. His ideas, rather than being rejected, were instead largely neglected—they became lost as philosophical discussion moved elsewhere, both conceptually and institutionally. However, in the inaugural volume of Mind in 1876, Hodgson's ideas initially found a prominent place in a central dialogue over the position of philosophy with respect to science. There, and throughout his career, Hodgson engaged in the defense of an intellectual tradition he regarded as under threat from psychological science. Both his particular formulation of an ongoing role for philosophy and his own uneasy position in the intellectual environment of late Victorian Britain shed significant light on an under-examined issue [End Page 289] of the period: the fraught relationship between philosophy and science as competing enterprises engaged in the study of the human mind."Philosophy" and "science" are easily construed as partnered intellectual complexes in modern Western society. However, closer scrutiny of the historical record shows that the two terms—whether as theoretical referents or actors' categories—have often reflected divergent priorities and that transitions from "philosophical" to "scientific" terms have been contentious. In examining the birth of natural science out of natural philosophy during the Enlightenment, the historian Andrew Cunningham has emphasized how the reordering of intellectual fields created new conflictive relationships between philosophy, science, and religion.1 He reminds us that the process of scientific secularization was but one of a set of constitutive shifts among these human activities, and that the earlier natural philosophy thus played by fundamentally different rules than its successor. A similar story can be told about the late nineteenth-century transition I am considering: in a nutshell, to borrow a phrase from Cunningham, men did not start doing mental science until they stopped doing moral philosophy. However, this time, philosophy already stood in competition with existing scientific institutions, sparking a more severe contest for authority. As mental science began to develop along the lines of the natural disciplines, Hodgson and others feared little would remain as a distinctive domain for philosophy as such, moral or otherwise. His efforts on the part of philosophy thus represent one front in an intellectual "turf war"—what recent scholars have called "boundary work," shoring up the institutional positions of different intellectual enterprises competing for authority in the world.2 Gieryn's formative analysis of boundary work was based on the contemporaneous example of John Tyndall's campaign to promote scientific over religious authority in Victorian Britain, but this was much more than a two-way struggle. The early career of Mind bears witness to a closely related, but less discussed, battle for influence between philosophy and science, in which the journal itself served as a critical "boundary object," a medium for enforcement of the distinctions between these two intellectual fields.3 [End Page 290]Thomas Dixon has recently surveyed the origins of this shift. Like Cunningham, Dixon emphasizes the constitutive differences in scope and orientation of the two intellectual enterprises, salient among them the secularization of the new human sciences.4 This process—begun tentatively by Thomas Brown and brought to fruition by Bain, Spencer, and Darwin—involved wholesale shifts in the scholarly linguistic complex, from a set of several mental categories rooted in Christian philosophical categories—affections, passions, sentiments, etc.—to a novel, explicitly "scientific," and singular grouping of emotions. While the older tradition dominant until mid-century had understood the broad range of mental phenomena as largely rational and voluntary, the new emotions were formulated as more circumscribed in purview, subconscious or autonomic in action... (shrink)

This introduction presents an overview of key concepts discussed in this book, which focuses on how citizenship is addressed in the context of education or, more specifically, learning, which is understood as central to the government of individuals and societies in Europe. Educational research has provided numerous critical responses to the citizenship education introduced in the UK and elsewhere. Governance is a form of governing commensurate with the decentralisation associated with neoliberalism, but articulated in terms of transparency, accountability, and social (...) justice in accordance with Third Way thinking. The concern with voice, narrative, and accounting for oneself in the name of democracy and social justice has become central to the discourses and practices of citizenship in the learning society. The active way in which individuals are asked to relate to their European heritage is measured as a form of participation, and therefore contributes evidence of active citizenship. (shrink)

Recent European and member state policy shows innovation to be a current guiding logic of government. This article offers an analysis of how innovation, seen partly in terms of learning but more significantly in terms of research, forms part of the discourses and practices of government today. Research is now something that all actors must engage with and so constitutes the individual’s self-understanding. Both the European and UK policies that I discuss speak of a shift away from excessive measurement and (...) control in order to open the way for greater flexibility, mobility, and thus productive innovation. The openness and mobility required by these policies exemplify a shift in governmental rationality that places responsibility on the individual, not only in relation to their own conduct and performance, but also to that of society as a whole. The demand for competitiveness becomes a demand for innovation and specialisation. The notion of the Big Society is discussed as an example of the operation of the logic of innovation. (shrink)

Social scientists have paid insufficient attention to the role of law in constituting the economic institutions of capitalism. Part of this neglect emanates from inadequate conceptions of the nature of law itself. Spontaneous conceptions of law and property rights that downplay the role of the state are criticized here, because they typically assume relatively small numbers of agents and underplay the complexity and uncertainty in developed capitalist systems. In developed capitalist economies, law is sustained through interaction between private agents, courts (...) and the legislative apparatus. Law is also a key institution for overcoming contracting uncertainties. It is furthermore a part of the power structure of society, and a major means by which power is exercised. This argument is illustrated by considering institutions such as property and the firm. Complex systems of law have played a crucial role in capitalist development and are also vital for developing economies. (shrink)

With precision and profundity born of 30 years of devoted study, John Boyle offers an essential introduction to St. Thomas Aquinas on Scripture, shedding helpful light on the goals, methods, and commitments that animate the Angelic Doctor's engagement with the sacred page. Because the genius of St. Thomas's approach to the Bible lies not so much in its novelty but rather in the fidelity and clarity with which he recapitulates the riches of the preceding interpretive Tradition, this initiation into (...) St. Thomas's vision of Scripture is itself an orientation to the Church's vision of Scripture, from the Fathers through and beyond the Middle Ages. St. Thomas's embeddedness within the Church's Tradition and his own historical context is integral to his approach to Scripture, yet it sets him at some distance from modern readers, for whom his interpretive vision may seem perplexing or even impenetrable. In this Primer, Boyle first provides an acclimation to this medieval context through a survey and explanation of pertinent technical terminology used by St. Thomas and characteristic of the scholastic theology of the time. With an eye to the medieval practice of considering Scripture according to the fourfold division of causes, Boyle builds on this initial foundation by exploring in turn St. Thomas's accounts of the end or use of Scripture (final cause), its divine and human authorship (efficient cause), its order and division (material cause), and its literary styles or genres (formal cause). Drawing on writings from across St. Thomas's corpus, but especially his work On the Commendation and Division of Sacred Scripture and the prologues to his biblical commentaries, Boyle masterfully elucidates both the hermeneutical principles and deep wisdom of the Angelic Doctor's approach to Scripture, imparting invaluable guidance not only for reading and understanding St. Thomas and other great masters of the Tradition, but also-and ultimately-for understanding Scripture in light of this Tradition and reading it with greater benefit and joy. (shrink)

This paper introduces a conjecture that laws of nature may be of different kinds, in particular that there may, in addition to laws which constrain outcomes, be laws which empower systems to direct or select outcomes and laws which guide systems in such selections. The paper defends this conjecture by suggesting that it is not excluded by anything we know, is plausible, and is potentially of great explanatory power.

This chapter focuses on the work of Ralph Waldo Emerson and Henry David Thoreau, and their interpretation by Stanley Cavell. Cavell's philosophy has sought to reclaim the work of Emerson and Thoreau for and as philosophy, and particularly as American philosophy. In Cavell's Emersonian moral perfectionism, perfectionism is understood as ateleological, and not in terms of perfectibility. Cavell therefore seeks to distinguish this American expression of perfectionism from what he identifies as a dimension or tradition of the moral life that (...) spans the course of Western thought, found in teleological form in, for example, Plato and Aristotle. Emerson's essay expresses a disappointment in America's passive acceptance of its European inheritance and in the implications of this for citizenship and democracy. America, or American man, must undertake its own labour to produce its own knowledge of itself and the world. (shrink)

This chapter begins by providing some historical background to European integration. It draws attention to the way that history has been used to promote a European identity since the European Union and, with it, European citizenship were created in 1992. The framing of the relationship between globalisation and its socioeconomic challenges has made the need to attend to questions of citizenship, particularly through education, self‐evident. The shift in the mode of governance has not only entailed using education as a means (...) of educating citizens about Europe: in the current context, education, or specifically, learning, has become a central aspect of how citizenship is understood. Examples of recent educational and cultural policy highlight terms that are central to producing a particular mode of governance. These examples illustrate the practices of accountability that have emerged in recent years that require governments, institutions, organisations, and individuals to speak about themselves in particular ways. (shrink)

Educational research tends to conduct its analysis according to fixed identity categories and concepts, and in its concern for voice, empowerment, and inclusion, offers ways in which researchers can articulate an account of themselves and their practice. The researcher herself then can be seen as being constituted as a particular subject in the current context of the entrepreneurial lifelong learning citizen, whose virtues are evidenced through the continual accumulation of skills and competencies. The concern with social justice, representation, and ethics (...) found in methodological discussion effects a shoring up of identity and a process of risk management in the sense of a thorough audit of who the researcher is. Considerations of citizenship and of educational research then begin to become inseparable because the understanding of both citizenship and research is seen to entail a particular relationship to the self. (shrink)

This chapter provides examples of European and local programmes and policies deriving from the education and cultural policies, and focuses on the ecological subject. These examples further illustrate not only the way in which the citizen is addressed, but also the construction of citizenship in a particular relationship to space and time. To begin the analysis of space in the construction of European citizenship, the chapter focuses on Foucault's account of governmentality, which shows the historical shift in the object of (...) government from territory to population, and thus the emergence of liberalism. More recent accounts draw attention to the subsequent shift to environments. In light of the environmental self‐understanding of the active learning citizen, the chapter considers the relationship to history seen in the promotion of heritage in European cultural policy. This entails placing the citizen in a particular spatial and temporal relationship to Europe. (shrink)

In 1933, the year that Hitler took power in Germany, Martin Heidegger accepted the Rectorship of the University of Freiburg. Rectoral Address is selected for its pertinence to this period of European history, which remains so central to the understanding of what constitutes Europe's heritage. This chapter explores the relationship between the sociopolitical events that took place in Germany in 1933 and philosophical texts relating to them. It provides some contextualisation of the post‐war events that ensued from Heidegger's Address and (...) political decisions in terms of the re‐reading of his philosophy and the Heidegger affair. Karl Liowith's account gives a positive interpretation of Heidegger's work, in the sense of situating him within a lineage of European philosophers who responded to the decline of Europe, diagnosing and responding to its nihilism, during the nineteenth and early twentieth centuries. This is followed by an introduction to Heidegger's Being and Time and subjectivity. (shrink)

Plato's texts have been subject to re‐reading in recent years, reflecting new ways in which philosophy has sought to understand the relationship between the author, the reader, and the text. This chapter begins by restating the allegory of the Stanley Cavell in The Republic, before turning to Cavell's reading of this in relation to the opening of the text. It further illustrates the idea of education as a finding of voice, which Cavell articulates through Emersonian moral perfectionism with reference to (...) his work on philosophy as autobiography. The educative relationship to the self and to the other, and the responsibility one takes for one's voice, is then illustrated further by way of Michel Foucault's account of Socratic parrhesia, a practice evident in The Republic and in other texts of Plato to which Foucault refers. Cavell ventures the term sublimation in relation to his questioning of the allegory of the Cave. (shrink)

This chapter illustrates how the language of research, citizenship, and education operates in two particular ways. First, by highlighting the way in which the discourses identified in policy and practice are taken up in educational research, and then by indicating how the particular mode of subjectivation detailed thus far is evident in certain forms of educational research. Both have implications for the critique educational research might provide. As a first example, the way in which educational research has responded to the (...) introduction of citizenship education in England is illustrated in the chapter by a review of research, policy, and practice covering the period 1995‐2005, commissioned by the British Educational Research Association. The review is taken to exemplify the field of education policy sociology. This is followed by discussion, in a second example, of what might be termed a genre of educational research concerned with voice and social justice. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...) first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)