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 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)

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)

For a primitive recursive consistent and strong enough theory T we construct an independent statement which has some clear metamathematical meaning.

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is (...) a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa complexity measure associated withϕ[Blu67], [DW83]. (shrink)

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...) weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. (shrink)

In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a (...) much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. (shrink)

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)

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of (...) Rosser-type formalizations of Yablo’s paradox for each consistent but not Σ1-sound theory is dependent on the choice of a standard proof predicate. (shrink)

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular $\text{I}\Delta _{0}+\text{EXP}$ . The method (...) is adapted to obtain a similar completeness result for the Rosser logic. (shrink)

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the (...) proof, in particular I0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic. (shrink)

Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. (...) We argue, however, that the alternative argument raises a circularity problem. In sum, Gödel’s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Gödel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Gödel sentence. Hence, Gödel’s and its related theorem never can be a ground for Dialetheism. (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)

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)

The evolution of Mancur Olson’s views of his book, The Rise and Decline of Nations (1982), the middle of his three main books, is examined. It expands and extends to history and the world arguments presented in his The Logic of Collective Action (1965). While he never abandons the idea that the accumulation of interest groups in a democratic society may lead to its economic stagnation, how this comes about and can be overcome changes somewhat by the time of his (...) final book, Power and Prosperity (2000), which focuses on the problems of the transition economies and proper political governance. A sign of the greater complexity of his later views emerges in his analysis of the U.S. South, presented in his presidential address to the Southern Economic Association (1983). (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)

Duncan Foley’s Unholy Trinity: Labor, capital, and land in the new economy is the sixth in the series of Graz Schumpeter Lectures published by Routledge, all relatively slim volumes elucidating themes arguably related to Schumpeter, if just peripherally, and that usually summarize major arguments of the authors (previous authors were Stanley Metcalfe, Brian Loasby, Nathan Rosenberg, Ian Steedman, and Erich Streissler). In this one, which deals with questions of induced technological change in several sections, Foley attempts to provide an integration (...) of ideas that have evolved through his varied career, from high general equilibrium theorist (Foley, 1967), through deep student of Marxian economics (Foley, 1986), to complexity theorist (Foley, 1994; for more detailed discussion of his personal and intellectual path see Colander et al, 2004, Chapter 7). A central argument is that the classical political economists, not just Marx but also Adam Smith and Malthus with Ricardo (largely lumped together in his analysis), developed ideas that are well suited to representation and understanding using the methods of modern complexity economics, themes of dynamic out-of-equilibrium selforganization in systems with boundedly rational agents, in contrast to the requirements for full Walrasian general equilibrium that he studied early in his career. (shrink)

We consider Friedrich Hayek’s Road to Serfdom in light of global ideological and economic developments during the sixty years since its publication. Specific problems considered include socialism and planning, whether national socialism was really socialism, whether Hayek’s views could be labeled as social democratic and whether his critique of social democracy was too strong, and his discussion of the prospects for international economic order. While often right and enormously influential, Hayek himself agreed that some of his predictions did not become (...) true. (shrink)

E. Roy Weintraub’s How Economics Became a Mathematical Science (2002) presents a original, distinctive, and provocative perspective on the evolution of mathematical economics from the late nineteenth century to the late twentieth century. Its originality and distinctiveness and provocativeness extends as well to its view of the relationship between mathematics and economics. He reveals many little known facts and punctures many fallacious, if widespread ideas. At the same time, he ultimately leaves us hanging on certain crucial points with an ambivalence (...) that is never resolved. However, he is careful not to claim too much for his work: it is not the definitive history of mathematical economics nor does it claim to provide ultimate answers. (shrink)

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)

Roger Koppl (2009, p. 1) argues that “Austrian economics is a school of thought within the broader complexity movement in economics.” Is he correct? While there are many who have argued for some overlapping between the two, I shall argue that this is probably an overly strong statement. The main reason is that there are substantial elements and strands within Austrian economics that do not fit in with any of the multiple varieties of complexity theory, even though there are some (...) that clearly do. While I have some disagreements with Koppl’s argument, I think that he does a good job of identifying some of the main strands of Austrian economics that are consistent with complexity thought. The most important of these are associated with the work of Friedrich A. Hayek, who.. (shrink)

Flaws, biases, and ethical problems surrounding research and diagnosis may lead to inappropriate or inequitable treatments that exacerbate or fail to improve the misery that some individuals face due to their psychiatric conditions. Possible androcentric biases in the choice and definition of categories for diagnosis available in DSM-III-R may in turn influence the approaches of therapists to clients, particularly male therapists towards female clients. Androcentric bias in diagnosis, which may also be reflected in the values of the psychiatrist, may lead (...) to treatment regimens designed to make clients fit into roles, positions, and norms prescribed by a culture reflecting patriarchal values. Some acceptance of attempts by feminists to correct androcentrism are beginning to emerge in psychiatric diagnosis. Keywords: androcentrism, bias, feminist, psychiatry of biology CiteULike Connotea Del.icio.us What's this? (shrink)

We investigate how stochastic asset price dynamics with herding and financial constraints in heterogeneous agents’ decisions explain the presence of a period of financial distress (PFD) following the peak and preceding the crash of a bubble, documented by Kindleberger [2000, Appendix B] as common among most major historical speculative bubbles. Simulations show the PFD is due to agents’ wealth distribution dynamics, selling because of financial constraints after the bubble’s peak in relation to switching behavior of agents. An increase in switching (...) tendency increases the length of the PFD and decreases bubble amplitude, while increasing strength of interaction between the agents increases bubble amplitude. (shrink)

I support several of President Bush’s economic proposals: tort reform for medical malpractice suits, and in principle increased free international trade, increased use of market mechanisms for environmental protection, and tax simplification. However, President Bush’s proposal for social security reform is unnecessary and dangerous to the economic health of our country. The system is not broke and does not need to be “fixed.”.

abstract Cowen and I agree that rational‐expectations theory is unrealistic and that risk is difficult to quantify. However, we continue to disagree about the riskiness of consumption as opposed to investment. Since more investment might lead to a recession if investment is relatively risky, Cowen's use of rational‐expectations theory to buttress the Austrian school's claim that market economies can shift toward relatively more investment without experiencing macroeconomic disruption remains suspect.

One is the very founding document of oligopoly theory, Cournot’s seminal work of 1838. This is both because the specific model that he presented has been much studied for its ability to generate complex dynamics and also because of its more general foreshadowing of game theory. It has often been noted that the Cournot equilibrium is but a special case of the Nash (1951) equilibrium, the more general formulation used by modern industrial organization economists in studying oligopoly theory. Indeed, it (...) is sometimes even called the Cournot-Nash equilibrium. Although many of the models of complex oligopoly dynamics use the specific Cournot model, many use more general game theoretic formulations. We note simply as an aside here that Cournot’s work was the first to apply calculus to solving an economic optimization problem and also was the first to introduce supply and demand curves, albeit in the "Walrasian" form with price on the horizontal axis. (shrink)