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Traditional mathematics is the language of precision. Statements are either true or false. But, in reality there are few things that are truly simply true or false. Life is full of shades of grey. Capturing these shades of grey has been problematic using traditional mathematics. Mathematics can model the uncertainty itself but rarely incorporates it into the model. However, an emerging area of mathematical inquiry known as the mathematics of uncertainty seeks to overcome some of these problems by integrating the uncertainly, the shades of grey, directly into the model. Fuzzy theory is one of those emerging ideas. This paper is simply an overview of some of the basic mechanics of fuzzy theory and recent economic applications in both micro and macro-economics.

Although modern scientific and technological advances derive computational power primarily from the classical evidence-based bottom-up cognition as founded by Greek philosopher Aristotle in his philosophy of science with a bivalent logic, the classical cognition, however, has met stiff challenges during the last few decades because of uncertainty faced by many new scientific endeavors. The holistic top-down nature of nanotechnology and brain modeling are just two of many examples. This new development points to the need for a critical review of the historical origins and distinctions of both top-down and bottom-up cognitions. This paper reviews the philosophy of science as founded by Aristotle (300BC), the Platonic realism as founded by Aristotle's teacher Plato (400BC-300BC), and the YinYang philosophy as founded by the ancient Chinese philosopher Laozi (or Laotze) (600BC). It is suggested that the long-standing unsettled dispute between Aristotle and Plato features a major source of uncertainty for both logic and mathematics. The authors hence propose a number of controversial philosophical 207 New Math. and Nat. Computation 2008.04:207-229. Downloaded from www.worldscientific.com by DUKE UNIVERSITY on 08/10/15. For personal use only. 208 W.-R. Zhang et al.and logical issues for debate. We advocate YinYang as an inspiration and unifying force for both top-down inductive and bottom-up deductive reasoning. We attempt to use an equilibrium-based YinYang bipolar dynamic logic (BDL) to bridge the gap between Aristotle and Plato as well as between logic and mathematics. Furthermore, we present a taxonomy for YinYang scientific computing with a classification of logical and statistical models for further discussion; we suggest that YinYang can be used as a catalyst for resolving certain "terminological difficulties" regarding truth, polarity, intuitionism, para-consistency, and fuzziness for equilibrium and harmony. A number of critical points are enumerated and discussed. An open challenge is posted.

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