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This is a result Post Date: Tue, 5 Aug 2008 0:56:54 +0000 Therefore there is no reason for saying that there is any natural form which limits division. 164 The distinction between a natural and a mathemati cal body is not admissible in what relates to division. This is a result of the nature of extension, which is real in natU ral bodies, and ideal in mathematical. Autor of the post: Undefined But, if they are not Post Date: Tue, 5 Aug 2008 1:11:30 +0000 That the parts in natural bodies are not actual but possible, may be under stood in two ways ; it may mean that they are not actually separated ; or, that they are not distinct. That they are not separated has no bearing on the question ; for division may be conceived without separating the parts. But, if they are not distinct, the division is impossible ; for it can not even be conceived where the things are not distinct. Autor of the post: Undefined It consists in this Post Date: Tue, 5 Aug 2008 1:22:45 +0000 165 This distinction seems to have originated in the attempt to avoid the necessity of admitting infinite divisi bility in natural bodies. But the difficulty still remaining with regard to mathematical bodies, the philosophical mys tery still subsists. It consists in this, that no limit can be assigned to division so long as there is any thing extended ; and, on the other hand, if, in order to assign this limit, we come to simple points, then it is impossible to reconstitute extension. Autor of the post: Undefined 166 THERE are two strong Post Date: Tue, 5 Aug 2008 1:41:37 +0000 The difficulty arises from the very nature of extended things, whether realized or only conceived ; the real order escapes none of the difficulties of the ideal. If ideal extension cannot be constituted out of unextended points, neither can real extension ; if ideal extension has no limit to its divisibility until we come to simple points, the same is also true of real extension ; for in both it is a result of the essence of extension, and inseparable from it. 166 THERE are two strong arguments against the exist ence of unextended points : the first is, that we must sup pose them infinite in number, for otherwise it does not seem possible to arrive at the simple, starting from the extended : the second is, that even supposing them infinite in number they are incapable of producing extension. Autor of the post: Undefined The unextended is zero Post Date: Tue, 5 Aug 2008 2:00:43 +0000 These argu ments are so powerful as to excuse all the aberrations of the contrary opinion, which, however strange they may seem, are not more strange than the simple forming exten sion, and the smallest portion of matter containing an infin ite number of parts. 167 It does not seem possible to arrive at unextended points unless by an infinite division. The unextended is zero in the order of extension, and in order to arrive at zero by a decreasing geometrical progression it must be continued ad infinitum. Autor of the post: Undefined If we separate the interior Post Date: Tue, 5 Aug 2008 2:17:59 +0000 Mathematical calculation presents a sensible image of this. When two parts are united they must have a side where they touch, and another whera they are not in contact. If we separate the interior side from the exterior we have two new sides, one which touches and another which does not. Autor of the post: Undefined From the moment that we Post Date: Tue, 5 Aug 2008 2:31:05 +0000 Continuing the division the same thing happens again ; we must, therefore, pass through an infinite series in order to arrive at the unextended, which is equivalent to saying that we shall never arrive there. To continue the division ad infinitum we must suppose infinite parts, and consequently the existence of an actual infinite number. From the moment that we suppose this infinite number to exist it seems to become finite, since we already see a limit to the division, and also other numbers greater than it. Autor of the post: Undefined The imagination loses itself Post Date: Tue, 5 Aug 2008 2:50:29 +0000 Let us suppose that this infinite number of parts is found in a cubic inch; there are numbers which are greater than this which we suppose infinite ; a cubic foot, for example, will contain 1,728 times the infinite number of parts contained in the cubic inch. Thus the opinion of unextended points seeking to avoid infinite division, runs into it ; just as its adversaries trying to escape from unextended points are forced to acknowledge their existence. The imagination loses itself and the under standing is confused. Autor of the post: Undefined Let us imagine two points Post Date: Tue, 5 Aug 2008 3:06:54 +0000 168 The other objection is not less unanswerable. Sup pose we have arrived at unextended points, how shall we re constitute extension ? The unextended has no dimensions ; therefore, no matter how many unextended points we may take, we can never form extension with them. Let us imagine two points to be united, as neither of them alone occupies any place, neither will they both together. Autor of the post: Undefined 169 It is certain Post Date: Tue, 5 Aug 2008 3:22:56 +0000 We can not say that they penetrate each other ; for penetration can not exist without extension. We must admit that these parts being zero in the order of extension, their sum can never give extension, no matter how many of them we may add together. 169 It is certain that a sum of zeros can give only zero for the result, but mathematicians admit that there are certain expressions equal to zero, which multiplied by an infinite quantity will give a finite quantity for the product. 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