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SUPPLEMENT I
401
| an infinite improbability of the return of any indefinitely chosen point of matter to any point of position, occupied at any previous instant of time indefinitely, of a return, I say, taking place at any indefinite instant of subsequent time; hence, such a return must be excluded, without any fear as to error, since it must be considered that an infinite improbability merges into a sort of relative impossibility. This Theory indeed cannot be applied to the ordinary view. Hence, in this way it is clear, in my Theory of points of matter, there must be excluded from Nature both rest, which also we excluded above, & even return to the same point of position in which that point of matter once was situated. Therefore it comes about that all those first four cases will be excluded from Nature, & in them the analogy of time & space will be preserved accurately. | |
| No point of matter can come into any point of space that was once occupied by any other point; it is only in coexistence, which corresponds to this that the analogy is broken. | |
| 16. Finally, if we seek to find whether any point of matter is bound to occupy at some instant a point of position which was occupied by some other point of matter at some other instant, still the improbability will be infinitely infinite. For the number of existing points of matter is finite; & thus, if instead of the return of any point to points of position occupied by itself we consider the return to points that have been occupied by another, the number of cases increases in the ratio of unity to a number of points that is in every case finite, that is to say, in a finite ratio only. Hence, the improbability of the arrival of any point of matter indefinitely taken at a point of space that has been occupied at some time by any other point is still infinite; & this arrival must therefore be taken to be impossible. In this way, indeed, the sixth case, which depended on this return, is excluded; & much more so the seventh case, which involves the simultaneous arrival of a pair of points of matter at any the same point of position, that is to say, compenetration. The eighth case also is excluded for matter; for all things created together as a whole will continually last as a whole, & so will always have a common instant of time.[1] Only the fifth case, in which several points of matter connect the same instant of time with different points of position remains; & this is not only possible, but also necessary for all points of matter, seeing that they coexist. For it cannot be the case that the seventh & the eighth are excluded, unless straightway, on that very account, the fifth is included, as will be easily seen on consideration. Therefore in this point the analogy fails, namely, in that several points of matter can connect different points of space with the same instant of time, which is the fifth case; whereas it is impossible for the same point of space to be connected with several instants of time, which is the third case. This defect is necessarily induced by the exclusion of the seventh & eighth cases; for if either of the latter is included, the fifth might be excluded; just as if it were possible for points of matter, which had been created together, & do not perish, not to coexist; for then the same instant of time would in no way be connected with different points of position. | |
| Which of the cases are possible through Divine Omnipotence; use of the theorem give above on impenetrability. | |
| 17. At least six of the seven cases seem to be possible through Divine Omnipotence, that is to say, omitting the virtual extension of matter, about which there may be possibly some doubt; for in this case there must exist at the same time an absolutely infinite number of those real points of position; & this is impossible, if an existing thing that is infinite in number is contradictory in the modes. Moreover, since all points of position can exist one after another, arranged along any line, for instance, in continuous motion, & so can also all instants of continuous time, one after another in the duration of any thing, there will be reason for doubt as to whether all those points of position can also exist at the same time. This is a matter upon which I dare not make a definite statement. All I say is that this theory of mine with regard to the nature of space & continuity completely avoids all the chief difficulties that are obstacles in other theories; & that it is very suitable for the explanation of everything in connection with this matter. I will also add the remark that, if the arrival of any point of matter at a point of position, at which any point of matter has arrived at any instant, is excluded, & along with it compenetration is thus excluded, then real impenetrability of matter must necessarily follow, which will be of great service to us in our tenth book[2]. That is, unless repulsive forces prevent such a thing, any |
- ↑ This case also would never happen, if the duration were not something continuously permanent; in place of it, we should have to admit a kind of, so to speak, skipping existence; that is to say, as if any point of matter (and the same thing applies to all created entities) existed only in indivisible instants remote from one another, and in all intermediate instants possible did not exist at all. Coexistence, in this case, would be infinitely improbable, the argument being nearly the same, as in the case of the arrival of one point of matter at a point of space in which some other point had once been. In this case too, there would be no real continuum even in motion; different velocities could be explained much more easily; it would be much more evident in what way the very short life of an insect can be equivalent to the longest of lives, by means of the same number of existences coming in between the first & last instants. Indeed the exclusion of any coexistence would carry away with it all immediate physical influence altogether, & determinations; indeed, a continually fresh creation, & other inadmissible things of that sort, would be obtained.
- ↑ The reference is to Stay's "Philosophy," in which that most refined & learned author expounds my Philosophy. On what I have said above, I have plucked the fruit of the theorem, in which, in Art. 360 of this work, I dealt with impenetrability, & the apparent compenetration that would result, if there were no mutual forces.
