Page:BoscovichSpaceTime.djvu/4

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 * if the same point of matter connects the same instant of time with several points of position distant from one another by some interval, then we shall have replication. Fourthly, if it connects the instant with a continuous series of points of position contained within some continuous interval, we shall have something which several of the Peripatetics admitted, calling it virtual extension; by virtue of which an indivisible particle of matter, quite without parts, could occupy divisible space. There are four other combinations, when several points are considered. That is to say, fifthly, if several points connect the same instant of time with several points of position; in this is involved coexistence. Sixthly, if they connect the same point of space with several instants of time; as would be the case when different points of matter were forced successively into the same position. Seventhly, if they connect the same point of space with the same instant of time; in this is involved compenetration. Eighthly, if they have no instant of time, & no point of space, common to them; as would be the case, if they did not coexist, nor, any of them, occupied the positions that had been occupied by any of the others at any time.
 * ||rowspan=2 valign=top|The relations of these cases to one another of them are possible, & how.
 * 14. Out of these eight cases, the third corresponds to the first, the fourth to the second, the sixth to the fifth, the eighth to the seventh. The third case, namely replication, is usually considered to be naturally impossible. Many think that the fourth case holds good for the rational soul, which they consider to have its seat in some divisible space; for instance, the Peripatetics think that it pervades the whole of the body, other philosophers think it is situated in a certain part of the brain, or in some juice of the nerves; so that, since it is indivisible, the whole of it must be in the whole of the space, & the whole of it in any part of the space. Just in the same way as the same indivisible Divine Nature is as a whole in the whole of space, & as a whole in any part of space, being necessarily present everywhere, & coexisting with & accompanying created things wherever created things are. Others admit this same case for matter, & consider that particles of matter can be extended in a similar manner, as we have said; although they are simple, & although they are devoid of parts, not only parts that are really separated, but also such as are distinct & only separable. I do not consider that this supposition can be entertained, for the reason that, whenever we perceive with our senses matter occupying positions distinct from one another, we see that it is also separable, although we may have to use a very great force; here, parts are separated which were at a distance from one another. Indeed, by no other argument can we exclude replication from Nature, than that we never see any portion of matter, as far as can be perceived by the senses, occupying two positions at the same time. The idea of Virtual extension of matter goes infinitely further beyond the idea of simple replication.
 * ||rowspan=2 valign=top|Rest & return to the same position are infinitely improbable in Nature; hence arises a very great analogy between them.
 * 15. If the second case of rest, & the first case of return to the same position could be obtained naturally, then indeed there would be a certain defect in the analogy between space & time. But it seems to me that I can prove that neither ever happens in Nature; & so they cannot be obtained naturally; this is my argument. If a point of matter at any instant of time is at a certain point of space, & we do not know where it is at some other instant, let us inquire how much more probable it is that it should be somewhere else than at the same point as before. The former will be more probable than the latter in the proportion of the number of all the other points of space to that single point. There are an infinite number of these points in any straight line, the number of lines in any plane is infinite, & the number of planes in the whole of space is infinite. Hence, the number of other points of space is an infinity of the third order; & thus the probability is infinitely greater with an infinity of the third order, when we are concerned with any other particular instant of time. Now let us deal indefinitely with all the instants of infinite time; then the first probability will decrease in proportion as the number of instants increases, at any of which it might at least be possible that the point was in the same place as before. Moreover, there are an infinite number of instants, the infinity being of the same order as that of the number of possible points in an infinite line. Hence, still considering indefinitely all the instants of infinite time, it is infinitely more improbable that the point should be in the same position as before, than that it should be somewhere else. Now consider, not a single point of position occupied at a single particular instant, but any point of position occupied at any indefinite instant; then still the probability of return to any one of these points of position will increase as the number of them increases; & this number, in a time that is also infinite, is an infinity of the same order as the number of lines in any plane. Hence the improbability of this case, in which any particular point of matter returns at some indefinite instant of time to some indefinite point of position, in which it was assumed to be at some other indefinite instant of time, remains an infinity of the first order. Moreover, this, for all points of matter, which are finite in number, will decrease in the finite ratio of this number to infinity (which would not be the case with the usual theory, in which the number of points of matter is taken to be an infinity of the third order). Hence we are still left with
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 * ||rowspan=2 valign=top|Rest & return to the same position are infinitely improbable in Nature; hence arises a very great analogy between them.
 * 15. If the second case of rest, & the first case of return to the same position could be obtained naturally, then indeed there would be a certain defect in the analogy between space & time. But it seems to me that I can prove that neither ever happens in Nature; & so they cannot be obtained naturally; this is my argument. If a point of matter at any instant of time is at a certain point of space, & we do not know where it is at some other instant, let us inquire how much more probable it is that it should be somewhere else than at the same point as before. The former will be more probable than the latter in the proportion of the number of all the other points of space to that single point. There are an infinite number of these points in any straight line, the number of lines in any plane is infinite, & the number of planes in the whole of space is infinite. Hence, the number of other points of space is an infinity of the third order; & thus the probability is infinitely greater with an infinity of the third order, when we are concerned with any other particular instant of time. Now let us deal indefinitely with all the instants of infinite time; then the first probability will decrease in proportion as the number of instants increases, at any of which it might at least be possible that the point was in the same place as before. Moreover, there are an infinite number of instants, the infinity being of the same order as that of the number of possible points in an infinite line. Hence, still considering indefinitely all the instants of infinite time, it is infinitely more improbable that the point should be in the same position as before, than that it should be somewhere else. Now consider, not a single point of position occupied at a single particular instant, but any point of position occupied at any indefinite instant; then still the probability of return to any one of these points of position will increase as the number of them increases; & this number, in a time that is also infinite, is an infinity of the same order as the number of lines in any plane. Hence the improbability of this case, in which any particular point of matter returns at some indefinite instant of time to some indefinite point of position, in which it was assumed to be at some other indefinite instant of time, remains an infinity of the first order. Moreover, this, for all points of matter, which are finite in number, will decrease in the finite ratio of this number to infinity (which would not be the case with the usual theory, in which the number of points of matter is taken to be an infinity of the third order). Hence we are still left with
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