Computational Economics in the Era of Natural Computationalism

After a brief review of natural computationalism, this introductory chapter presents a new skeleton of computational economics and finance (CEF) along with an overview of the handbook. It begins with a conventional pursuit focusing on the algorithmic or numerical aspect of CEF such as computational efforts devoted to rational expectations, (dynamic) general equilibrium, and volatility. It then moves toward an automata- or organism-based perspective of CEF, involving nature-inspired intelligence, algorithmic trading, automated markets, network- and agent-based computing, and neural computing. As an alternative way to introduce this novel skeleton, the chapter starts with a view of computation or computing, addressing what computational economics intends to compute and what kinds of economics make computation so hard, and then it turns to a view of computing systems in which the Walrasian kind of computational economics is replaced by the Wolframian kind due to computational irreducibility.

COUNTING ON EPIC: MATHEMATICAL POETRY AND HOMERIC EPIC IN ARCHIMEDES' CATTLE PROBLEM

In 1773, the celebrated enlightenment thinker G.E. Lessing discovered in Wolfenbüttel's Herzog August Library a manuscript which contained a previously unknown Ancient Greek poem. The manuscript identified the author as Archimedes (c.287-212 BCE), and the work became known as the Cattle Problem (henceforth CP). On the surface, its twenty-two couplets capitalise on Homer's depiction of the ‘Cattle of the Sun’ in Book 12 of the Odyssey and its numerical aspect. A description of the related proportions of black, white, brown and dappled herds of cattle, which are then configured geometrically on Sicily, creates a strikingly colourful image. The author's decision to encode a number into the figure of the Cattle of the Sun styles the poem as a response to, and expansion of, Homer's scene. Reading through the work, though, it becomes clear that the mathematics is more complex than that of Homer's Odyssey.

On the Composition of Amos 1-2 LXX

For an adequate and readily verifiable description of a given text a count of the number of its words is of essential value. This article on the literary composition of Amos 1-2 LXX uses this numerical aspect as a main criterion. It is found that the author-translator frequently balances small and larger segments of his text; he favours round figures and square numbers, achieves clear and simple proportions within his composition, such as, i.a., the golden mean, and so on. He is not only a capable translator, but throughout this passage shows himself a skilful and competent craftsman, ambitious and well trained in the numerical features of ancient literary composition.

Abduction as Incomplete Parameter Estimation

Abduction is a kind of logical inference, and has been studied in computer science and artificial intelligence (Fin- lay and Dix 1996). Recently, Sawa and Gunji (2010) introduced a diagram to represent three types of inference: i.e. deduc- tion, induction, and abduction, which are articulated by C.S.Peirce. Sawa-Gunji’s representation provides a new approach to a numerical aspect of abduction. In the present paper, we show that Sawa-Gunji's representation of abduction is consistent with Finlay-Dix's one, and integrate the two representations. Both parameter estimation and abduction occupy a similar position on the integrated representation, although they are not completely corresponding. We present "incomplete" pa- rameter estimation as a sort of "simulated abduction", which is a numerical aspect of abduction. It is applied to a first-order autoregressive (AR(1)) model. As a result of numerical analyses on AR(1), the incompletely estimated parameter (IEP) follows a Cauchy distribution, which has a power law of the slope -2 in the tail, although conventionally estimated parameter is normally distributed. It is shown that the Cauchy distribution of the IEP is based on structure of ratio distribution of normal random variables generated from the AR(1). This research suggests that the distribution of the IEP is not based on a mech- anism of system itself, but on relationship between data structure on the given system (i.e. the given AR(1) process) and one on the system observer (i.e. the estimator of the AR(1) parameter).

Abduction as Incomplete Parameter Estimation

Abduction is a kind of logical inference, and has been studied in computer science and artificial intelligence (Fin- lay and Dix 1996). Recently, Sawa and Gunji (2010) introduced a diagram to represent three types of inference: i.e. deduc- tion, induction, and abduction, which are articulated by C.S.Peirce. Sawa-Gunji’s representation provides a new approach to a numerical aspect of abduction. In the present paper, we show that Sawa-Gunji's representation of abduction is consistent with Finlay-Dix's one, and integrate the two representations. Both parameter estimation and abduction occupy a similar position on the integrated representation, although they are not completely corresponding. We present "incomplete" pa- rameter estimation as a sort of "simulated abduction", which is a numerical aspect of abduction. It is applied to a first-order autoregressive (AR(1)) model. As a result of numerical analyses on AR(1), the incompletely estimated parameter (IEP) follows a Cauchy distribution, which has a power law of the slope -2 in the tail, although conventionally estimated parameter is normally distributed. It is shown that the Cauchy distribution of the IEP is based on structure of ratio distribution of normal random variables generated from the AR(1). This research suggests that the distribution of the IEP is not based on a mech- anism of system itself, but on relationship between data structure on the given system (i.e. the given AR(1) process) and one on the system observer (i.e. the estimator of the AR(1) parameter).