A relativistic Zeno effect
A Zenonian supertask involving an infinite number of identical colliding balls is generalized to include balls with different masses. Under the restriction that the total mass of all the balls is finite, classical mechanics leads to velocities that have no upper limit. Relativistic mechanics results in velocities bounded by that of light, but energy and momentum are not conserved, implying indeterminism. The notion that both determinism and the conservation laws might be salvaged via photon creation is shown to be flawed.
Achilles, the tortoise, and colliding balls
duced by Zeno of Elea around 460 b.c., was solved by mathematical advances in the nineteenth century. The techniques of Weierstrass, Dedekind and Cantor made it clear, according to this view, that Achil les' difficulty in traversing an infinite number of intervals while trying to catch up with the tortoise does not involve a contradiction, let alone a logical absurdity. Yet ever since the nineteenth century there have been dissidents claiming that the apparatus of Weierstrass et al. has not resolved the paradox, and that serious problems remain. It seems that these claims have received unexpected support from recent developments in math ematical physics. This support has however remained largely unnoticed by historians of philosophy, presumably because the relevant debates are cast in mathematical-technical terms that are only accessible to people with the relevant training. That is unfortunate, since the debates in question might well profit from input by philosophers in general and historians of philosophy in particular. Below we will first recall the Achilles paradox, and describe the way in which nineteenth-century mathematics supposedly solved it. Then we discuss recent work that contests this solution, reiterating the dis sident dogma that no mathematical approach whatsoever can even come close to solving the original Achilles. We shall argue that this dissat isfaction with a mathematical solution is inadequate as it stands, but that it can perhaps be reformulated in the light of new developments in mathematical physics.
Avocado Flowering 1
The avocado (Persea americana Mill.) has increased in importance over the years in many countries around the world. The United States, Mexico and Brazil lead in world commercial production, while Israel, South Africa, Australia, New Zealand, the Philippines, Spain, Cyprus, Canary Islands, Chile, Ecuador, Venezuela, Greece, and the Caribbean area are actively increasing commercial production (B.O. Bergh, personal communication; Gustafson 1976). The history of the avocado has been detailed by Gustafson (1973, 1975,1976) and others (Bergh 1975b, Knight 1980; Popenoe 1924; Ruehle 1958; Schroeder 1958; Williams 1976). Several reviews have been published on various aspects of the avocado, including its botany (Knight 1980; Purseglove 1968; Williams 1976), breeding (Bergh 1975b), horticulture (Hodgson 1930b; Ruehle 1958; Ryerson et al. 1924), cultivars (Campbell and Malo 1976), and pollination (Gustafson and Bergh 1966). A recent collection of papers discussing the horticultural aspects of avocado production is found in Sauls et al. (1976).