Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...) calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use. (shrink)

The paper reviews statistical models for money, wealth, and income distributions developed in the econophysics literature since the late 1990s. By analogy with the Boltzmann-Gibbs distribution of energy in physics, it is shown that the probability distribution of money is exponential for certain classes of models with interacting economic agents. Alternative scenarios are also reviewed. Data analysis of the empirical distributions of wealth and income reveals a two-class distribution. The majority of the population belongs to the lower class, characterized by (...) the exponential (“thermal”) distribution, whereas a small fraction of the population in the upper class is characterized by the power-law (“superthermal”) distribution. The lower part is very stable, stationary in time, whereas the upper part is highly dynamical and out of equilibrium. (shrink)

This article argues that the neoclassical era in economics has ended and is being replaced by a new era. What best characterizes the new era is its acceptance that the economy is complex, and thus that it might be called the complexity era. The complexity era has not arrived through a revolution. Instead, it has evolved out of the many strains of neoclassical work, along with work done by less orthodox mainstream and heterodox economists. It is only in its beginning (...) stages. The article discusses the work that is forming the foundation of the complexity era, and how that work will likely change the way in which we understand economic phenomena and the economics profession. (shrink)

Much empirical analysis and econometric work recognizes that there are nonlinearities, regime shifts or structural breaks, asymmetric adjustment costs, irreversibilities and lagged dependencies. Hence, empirical work has already transcended neoclassical economics. Some progress has also been made in modeling endogenously generated cyclical growth and fluctuations. All this is inconsistent with neoclassical general equilibrium. Hence there is growing evidence of Kuhnian anomalies. It therefore follows that there is a Kuhnian crisis in economics and further research in nonlinear dynamics and complexity can (...) only increase the Kuhnian anomalies. This crisis can only deepen. However, there is an ideological commitment to general equilibrium that justifies “free enterprise” with only minimal state intervention that may still sustain neoclassical economics despite the growing evidence of Kuhnian anomalies. Thus, orthodox textbook theory continues to ignore this fact and static neoclassical theory remains a dogma with no apparent reformulation to replace it. (shrink)

This book is about the economics profession, or more precisely, the process by which economic thinking changes. We believe that this process is important because economics is currently at a turning point; it is changing from a static approach to understanding, in which deductive reasoning is the key method used, to a complexity approach to understanding, in which inductive and deductive methods are used simultaneously, and the full complexity of the system is acknowledged and dealt with. The change is ongoing (...) and has many levels and dimensions, most of which have not coalesced to the degree that they have reached the lay public. But anyone involved in economic research recognizes the changes, although they do not necessarily understand how they all fit together. (shrink)

The first is that we are wrong to suggest that the mainstream is no longer limited to a restrictive orthodoxy of beliefs and assumptions that discourages dissenting voices. In developing his argument, Vernengo claims that our characterization of a cutting edge branch of the mainstream that does not hold to a neoclassical orthodoxy is misleading. Although he states that he accepts our characterization of the economics profession as a complex adaptive system, with many competing views, he sees the cutting edge (...) as a sham. He argues that the true role of the cutting edge is to allow the mainstream to “sound reasonable when talking about reality, while orthodoxy provides authority to the cutting edge.” He calls this an “organized hypocrisy” and calls us naive about the sociology of the economics profession. Because of this naiveté on our part he believes that we are giving bad advice to advocate that heterodox economists should think of themselves as economists first and heterodox economists second. (shrink)

We attempt to clarify divisions made by us in previous work (Colander et al., 2004a,b) between “orthodox, mainstream, and heterodox” in economics, following very useful remarks in Dequech (2007-08), whom we thank. We also provide specific advice for heterodox economists, namely: worry less about methodology, focus on being economists first and heterodox economists second, and prepare ideas to leave the incubator of heterodoxy to enter the mainstream economic debate.

This book is about the economics profession, or more precisely, the cutting edge of the economics profession. Economics is currently at a turning point; it is changing from a static approach to understanding, in which deductive reasoning is the key method used, to a complexity approach to understanding, in which inductive and deductive methods are used simultaneously, and the full complexity of the system is acknowledged and dealt with. The change is just beginning, but the groundwork is currently being laid. (...) This book is about that groundwork and those economists who are developing it. They are the cutting edge of economics. Those who are doing cutting edge work are researchers who are pushing and testing the boundaries of the profession in such a way that it draws the attention of the elite in the profession. The cutting edge has the potential of changing mainstream economics and ultimately what is considered the orthodoxy. (shrink)

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex, then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the (...) purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦. That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦. (shrink)

A deep theme of Austrian economics has been that of spontaneous order or selforganization of the economy. The origin of this theme dates to the putative founder of the Austrian School, Carl Menger, with his theory of the spontaneous emergence of money for transactions purposes in primitive economies being archetypal example (Menger, 1892). Menger drew this approach from the Scottish Enlightenment figures David Hume, Adam Ferguson, and Adam Smith, with the latter’s Wealth of Nations (1776) particularly important. The most important (...) developer of this idea within the tradition after Menger was F.A. Hayek (1948), who would identify this self-organization phenomenon with emergence, later expanding upon this into the broader concept of complexity (Hayek, 1952, 1967). Caldwell (2004) argues that this became an increasingly important focus of Hayek’s thought in the later years of his life. Among those examining this development in more detail besides Caldwell have been Koppl (2006, 2009), Rosser (2010a),1 and Lewis (2010). This essay will consider more thoroughly the relationship between the concepts of emergence and complexity and the roles that they have played in Austrian economics as well as more broadly in philosophy and science. An important point is that both of these concepts do not possess precise meanings; they are “terms of art” within philosophy. However, while closely linked through the general idea of a whole being “greater than.. (shrink)

According to Bikas Chakrabarti (2005, p. 225), the term econophysics was neologized in 1995 at the second Statphys-Kolkata conference in Kolkata (formerly Calcutta), India by the physicist H. Eugene Stanley, who was also the first to use it in print (Stanley, 1996). Mantegna and Stanley (2000, pp. viii-ix) define “the multidisciplinary field of econophysics” as “a neologism that denotes the activities of physicists who are working on economics problems to test a variety of new conceptual approaches deriving from the physical (...) sciences.” The list of such problems has included distributions of returns in financial markets (Mantegna, 1991; Levy and Solomon, 1997; Bouchaud and Cont, 1998; Gopakrishnan, Plerou, Amaral, Meyer, and Stanley, 1999; Sornette and Johansen, 2001; Farmer and Joshi, 2004) the distribution of income and wealth (Drăgulescu and Yakovenko, 2001; Bouchaud and Mézard, 2000; Chatterjee, Yarlagadda, and Charkrabarti, 2005), the distribution of economic shocks and growth rate variations (Bak, Chen, Scheinkman, and Woodford, 1993; Canning, Amaral, Lee, Meyer, and Stanley, 1998), the distribution of firm sizes and growth rates (Stanley, Amaral, Buldyrev, Havlin, Leschhorn, Maass, Salinger, and Stanley, 1996; Takayasu and Okuyama, 1998; Botazzi and Secchi, 2003), the distribution of city sizes (Rosser, 1994; Gabaix, 1999), and the distribution of scientific discoveries (Plerou, Amaral, Gopakrishnan, Meyer, and Stanley, 1999; Sornette 1 and Zajdenweber, 1999), among other problems, all of which are seen at times not to follow normal or Gaussian patterns that can be described fully by mean and variance. The main sources of conceptual approaches from physics used by the econophysicists have been from models of statistical mechanics (Spitzer, 1971), geophysical models of earthquakes (Sornette, 2003), and “sandpile” models of avalanches, the latter involving self-organized criticality (Bak, 1996). An early physicist to assert the essential identity of statistical methods used in physics and the social sciences was Majorana (1942). A common theme among those who identify themselves as econophysicists is that standard economic theory has been inadequate or insufficient to explain the non-Gaussian distributions empirically observed for various of these phenomena, such as “excessive” skewness and leptokurtotic “fat tails” (McCauley, 2004).. (shrink)

considerable publicity in some of the leading general science journals such as Science and Nature. While most of this research has appeared in physics journals, some has appeared in economics journals as well, more often when at least one author is an economist. Strong claims have been made by some advocates regarding its reputed superiority to economics (McCauley, 2004), with arguments that in fact the teaching of microeconomics and macroeconomics as they are currently constituted should cease and be replaced by (...) appropriate courses in mathematics, physics, and some other harder sciences. The lack of invariance principles in economics and the failure of economists to deal properly with certain empirical regularities are held against it in this line of argument, although most econophysicists do not go as far as McCauley in proposing the complete replacement of economics as such by econophysics. (shrink)

“When demand and supply are in stable equilibrium, if any accident should move the scale of production from its equilibrium position, there will be instantly brought into play forces tending to push it back to that position; just as if a stone hanging by a string is displaced from its equilibrium position… But in real life such oscillations are seldom as rhythmical as those of a stone hanging freely from a string; the comparison would be more exact if the string (...) were supposed to hang in the troubled waters of a mill-race, whose stream was at one time allowed to flow freely and at another partially cut off. Nor are these complexities sufficient to illustrate all the disturbances with which the economist and the merchant alike are forced to concern themselves. (shrink)

We use NF to designate the system known as Quine's New Foundations, and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic”. This system ML is just the systemPproposed by Wang in [4], and essentially includes NF as a part.The pripcipal results of the present paper are:A. In NF the axiom of infinity is equivalent to the definability (...) of an ordered pair having the same type as its constituents.B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML. (shrink)

distraction that leads innocent Post Keynesians into “classical sin.” Davidson (1994, 1996) argues that core Post Keynesian (PK) ideas such as that insufficient aggregate demand arise from fundamental uncertainty in a monetary economy do not depend on nonlinearity or complexity, that these core concepts are axiomatically and ontologically true, and that the inability of agents to forecast well in dynamically complex situations reflects mere epistemological problems of insufficient computational abilities. Thus complex dynamics is merely a classical stalking horse. This writer (...) (Rosser, 1990, 1998, 2001) disagrees with the argument presented above and its relatives (Mirowski, 1990; Carrier, 1993). Dynamic complexity provides a foundation for fundamental uncertainty in Keynesian and PK models, and this applies to most of the various sub-branches of PKE besides Davidson’s “fundamentalist” or “Keynes-Post Keynesian”2 approach. The argument will be considered regarding three subdivisions of Post Keynesianism as identified by Hamouda and Harcourt (1988): the aforementioned fundamentalist Keynesianism, Sraffian (or neo-Ricardian), and Kaleckian (or Kaleckian- Robinsonian).3 Following King (2002, chap. 10), I admit to being more in sympathy with those he describes as “synthesizers” than with the more partisan sectarians of these approaches.4 I shall describe how each sub-branch has been analyzed using ideas of.. (shrink)

Ever since the collapse of Soviet-bloc socialism, and the associated breakup of the Soviet Union itself, it has been accepted by the vast majority of political economists that Friedrich A. Hayek and his fellow Austrians, notably his mentor, Ludwig von Mises, were the unequivocal victors in the famous “socialist calculation debate” that had raged for a good seven decades. It was over. The anti-socialist, Austrian position had won. Market capitalism was triumphant in both theory and practice. The combination of lack (...) of incentives and inevitably dispersed and asymmetric information had doomed command central planning of economies, especially in an increasingly complicated modern world economy (Rosser and Rosser, 2004, Chap. 3). (shrink)

Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates (...) these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. (shrink)

We examine the “new comparative economics” as proposed by Djankov et al. (2003) and their use of the concept of an institutional possibilities frontier. While we agree with their general argument that one must consider a variety of institutions and their respective social costs, including legal systems and cultural characteristics, when comparing the performance of different economic systems, we find various complications and difficulties with the framework they propose. We propose that a broader study of clusters of institutions and such (...) newly emerging forms as the new traditional economy may be better suited as ways to approach the study of comparative economics in the era after the breakdown of the old comparison of market capitalism and command socialism that came to an end with the breakup of the Soviet Union. (shrink)

Relationships are studied between the non-observed economy, income inequality, corruption, social capital measured as trust, and various institutional quality, policy, and macroeconomic variables for a global data set of countries for two time periods accounting for social interactions. Tentative support is found for positive relations between the non-observed economy and income inequality, the non-observed economy and corruption, and a negative relation between corruption and trust. No significant relation was found between the nonobserved economy and tax rates, contrasting with previous studies (...) finding significant relations of opposite signs. Data difficulties and weak robustness tests suggest limits to our results. (shrink)

James Madison University Harrisonburg, VA 22807 USA Tel: 540-568-3212 Fax: 540-568-3010 E-mail: [email protected]..

In 1991 this author published a book entitled, From Catastrophe to Chaos: A General Theory of Economic Discontinuities with Kluwer Academic Publishers. Due to the con troversial and unusual nature of this book’s content, there was considerable difficulty in getting publishers to agree to publish it prior to its being accepted by Kluwer. Initially conceived as a heterodox challenge to established economic thinking, this book became viewed by many readers as a reference volume on applications of nonlinear dynamics in general (...) to many fields of economics. Now, in 2000 Volume 1 of the second edition has appeared from Kluwer also, subtitled, Mathematics, Microeconomics, Macroeconomics, and Finance. It covers in expanded form the material contained in the first eight chapters of the first edition. What is this book about and how does the new edition differ from the first one? The underlying philosophy presented is that discontinuities are important objects of study in economics, indeed in many disciplines beyond economics as well. Although discontinuities may arise dynamically due to random exogenous shocks, the more truly interesting ones are those that arise endogenously from dynamical economic systems. In dynamical systems discontinuities can arise endogenously in the presence of nonlinearities in the systems. Thus, the more general topic of nonlinear dynamics in.. (shrink)

Three principles of dialectical analysis are examined in terms of nonlinear dynamics models. The three principles are the transformation of quantity into quality, the interpenetration of opposites, and the negation of the negation. The first two of these especially are interpreted within the frameworks of catastrophe, chaos, and emergent dynamics complexity theoretic models, with the concept of bifurcation playing a central role. Problems with this viewpoint are also discussed.

We present three arguments regarding the limits to rationality, prediction, and control in economics, based on Morgenstern’s analysis of the Holmes-Moriarty problem. The first uses a standard metamathematical theorem on computability to indicate logical limits to forecasting the future. The second provides possible nonconvergence for Bayesian forecasting in infinite dimensional space. The third shows the impossibility of a computer perfectly forecasting an economy with agents knowing its forecasting program. Thus, economic order is partly the product of something other than calculative (...) rationality. The joint presentation of these existing results should introduce the reader to implications of these concepts for certain shared concerns of Keynes and Hayek. (shrink)

This essay reviews the arguments made for a New Austrian theory of business cycles by Tyler Cowen, based on risk analysis and assuming rational expectations. This contrasts with the Old Austrian view that questions measurable risk in economic analysis. The way risk is applied to analyze business cycles suffers from serious inconsistencies. The use of rational expectations is mistaken in the face of economic complexity as understood by the traditional Austrians. However, Cowen is commended for his open-mindedness, even as this (...) leads him away from a strictly Austrian perspective. (shrink)

This paper examines the rising competition between computational and dynamic conceptualizations of complexity in economics. While computable economics views the complexity as something rigorously defined based on concepts from probability, information, and computability criteria, dynamic complexity is based on whether a system endogenously and deterministically generates erratically dynamic behavior of certain kinds. On such behavior is the phenomenon of emergence, the appearance of new forms or structures at higher levels of a system from processes occurring at lower levels. While the (...) two concepts can overlap, they represent substantially different perspectives. A competition of sorts between them may become more important as new, computerized market systems emerge and evolve to higher levels of complexity of both kinds. (shrink)