On May 31, 2006, at 9:01 AM, Ralph G Selfridge wrote: > The sort of crazy results you can get if tolerence is not > identically zero > is a squence of numbers, each of which 'equals' its neighbours but the > first and last are not equal.
>On May 31, 2006, at 9:34 PM,Eugene McDonnell wrote : >Paul Berry tells this story: >In an early talk Ken was explaining the advantages of tolerant >comparison. A member of the audience asked incredulously, “Surely you >don’t mean that when A=B and B=C, A may not equal C?” Without >skipping a beat, Ken replied, “Any carpenter knows that!” and went on >to the next question. In 1902, in a book called "Science and Hypothesis" Henri Poincare wrote about physical and mathematical continuum : " It has, for instance, been observed that a weight A of 10 grammes " and a weight B of 11 grammes produced identical sensations, that the " weight B could no longer be distinguished from a weight C of 12 grammes, " but that the weight A was readily distinguished from the weight C. " Thus the rough results of the experiments may be expressed by the " following relations: " A=B, B=C, A < C, " which may be regarded as the formula of the physical continuum. " " But here is an intolerable disagreement with the law of contradiction, " and the necessity of banishing this disagreement has compelled us to " invent the mathematical continuum. We are therefore forced to conclude " that this notion of mathematical continuum has been created entirely " by the mind, but it is experiment that has provided the opportunity. " We cannot believe that two quantities which are equal to a third are not " equal to one another, and we are thus led to suppose that A is different " from B, and B from C, and that if we have not been aware of this, it is " due to the imperfections of our senses. [...] It is to escape " this contradiction contained in the empiric data that the mind is led " to create the concept of a continuum formed of an indefinite number " of terms. ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
