چکیده:
One of the proofs for the finiteness of dimensions is the proof of equidistance. Some great scholars, such as Khwaja, have not accepted it, while others, like Sadr al-Mutalahin, have found it stable or valid. The basis of the proof lies in the intersection between two infinite lines or a finite line with an infinite one that, through a rotation toward the infinite line, meets it at an intersection. By denying the state of intersection and deviation from parallelism, a contradiction occurs via reductio ad absurdum, caused by the assumption of infinity. The impossibility of such an intersection is due to the lack of a first point of contact at infinity; or because equidistance is an accident, and an accident requires a beginning. Authors see the proof as logically sound, but due to the curvature of space, the states of two lines are not limited to parallelism and intersection as they are on a plane; therefore, the reason of equidistance is not applicable in real space, and with this reasoning, the finiteness of dimensions is not proven.
خلاصه ماشینی:
The authors find the argument logically sound, but due to the curvature of space, the states of two lines in contrast to a plane are not limited to parallelism and intersection; therefore, the argument of Masamatah is not applicable in real space, and with this reasoning, the finiteness of dimensions is not proven.
In the current case, it has also been said that the finitude of dimensions itself (the antecedent) is not impossible, but rather it is a characteristic of an infinite line that it does not have a starting point for intersection, or that intersection with it is altogether impossible.
If we want to apply this matter to the proof by reduction to absurdity (dalil musamatah), we must say that the intersection of two lines, the exhaustion of cases between parallelism and intersection for two lines, and the existence of an infinite dimension are all correct individually; however, when all are present together, an impossibility occurs.
The purpose of the proof of divergence is the contact of line with line, and according to Khwaja, this matter is gradual and "has no beginning," and in Lahiji's expression, it comes into existence with the slightest rotation from the state of parallelism; however, the whole point is this: the slightest rotation cannot find the possibility of existence, not because the angle is infinitely divisible, but because such a thing cannot exist in any of the segments of time.
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