In [PEN-L 688] Dionisio Carmo-Neto wrote: __________________________________________________________________________ Hi fellows, I am writing a paper on Zeno's Paradox, attempting to make an application in economic theory. Does anyone know any particular article or book in your own language (other than English, Portuguese and Spanish) that could indicate me? I have made an extensive search, but many times the citations are made inside the text and not under a specific title. I will be very happy to thank anyone that can help me. __________________________________________________________________________ As luck would have it: Brian Knight of Greenwich University, specialises in this. In his inaugural lecture (ISBN 1 874529 477, from Greenwich University Dartford Campus, Oakfield Lane, Dartford, Kent DA1 2SZ) he aurveys the literature. He writes: 'Views on the significance of this paradox have varied greatly since...real analysis. According to A. N. Whitehead the Achilles paradox is a simple fallacy based upon ignorance of infinite numerical series...However as William James has pointed out the argument that because the infinite series of time intervals has a finite sum, therefore Achilles must catch the tortoise, misses Zeno's point entirely. Zeno would have granted that if the tortoise can be overtaken then it can be overtaken in a finite time. But that is all that we can deduce from the numerical series. We cannot infer that the tortoise has to be overtaken. The point is that Achilles needs to perform an infinite sequence of acts in order to overtake the tortoise, and it is this that is maintained to be impossible. G J Whitrow has re-cast the paradox in terms of a bouncing ball...A ball is imagined to be projected upwards from a level floor with velocity v, the coefficient of restitution between the floor and the ball being e...If v = 16 m/s and e=3/4 we find t[time to rest] = 4 secs. In these four seconds the ball is assumed to have bounced an infinite number of times. Each bounce takes threequarters of the time of the previous bounce, and these times become shorter and shorter but never zero. The set of bounced form an open sequence of events with non-zero duration. But at some stage there must be a bounce of zero duration if the ball is ever to come to rest. How is this transition from bouncing to rest to be accomplished? Each bounce is 'caused' bu the preceding one. How is the state of rest 'caused' by the open causal sequence?' Knight refers to a programme of research initiated by a man called Hayes who issued a 'Naive Physics Manifesto' in 1977 for 'the construction of a system to model the natural human view of the physical world, as opposed to the increasingly mathematical and remote scientific view...In 1983, James Allen in a seminal paper entitled 'Maintaining knowledge about Temporal Intervals' advanced a new theory of time to support the representation of the human view of time. This theory abandoned completely the notion of time points, maintaining that human reference to time is exclusively concerned with time intervals.' Knight's point of departure, working with Jixin Ma, is an attempt to re- introduce discrete time events into Allen's system. The problem is this (for example): if I switch off a light, does the act of switching take place at a definite point in time? If not, then there is the puzzling result that in one interval the light is on, in another the light is off, but there is no 'point' at which it is turned off. Tell us more about your project! Selected References ___________________ Allen, J.F. (1983) 'Maintaining Knowledge about Temporal Intervals' Communications of the ACM 26,123-154 Hayes, B. (1977) 'The Naive Physics Manifesto' in Expert Systems in the Micro-Electronic Age (ed D. Michie), Edinburgh University Press, Edinburgh James, W (1909) 'A Pluralistic Universe' Longmans Green, New York, 229 Pratchett, T. (1990) 'Pyramids' p175-177. London: Corgi. Whitehead, A.N. (1929) 'Process and Reality', The McMillan Company, New York Whitrow, G.J. (1980) 'The Natural Philosophy of Time', Oxford University Press, London. Knight has more references in the inaugural lecture. You can E-Mail him but I don;t have it to hand. Try perhaps [EMAIL PROTECTED] or maybe finger greenwich.ac.uk. If you don't get to him send me an E-Mail on [EMAIL PROTECTED] (*not* the address with this posting) and I will pick it up when I have access to the University's E-Mail list. Alan
