Prince_James: “ Can it truly be said that anything divided ad infinitum would be accorded to nothingness and not simply the "infinitely small"? Though we could never reach the infinitely small by definition that its infinite status would make it impossible to be reached, it must nonetheless be a the ultimate end, as every number - or every portion of space - is infinitely divisible and never would reach zero (nothingness) in the process. As a number's quotient cannot be zero if that number is divided by another number, one could never reach zero through the process of division………….”.
Re: That is one way of analyzing it. It’s also possible to use the idea of the most infinitely small (i.e. nothingness) to definite the infinitely small. Accordingly, the infinitely small, in this case, is not one specific entity, but an attribute of an infinite number of entities whose relative degree of it increases progressively by approaching the limit of the series. Can the process of dividing ad infinitum reach the ultimate limit of the series? Yes! But only if we assume the process of division is at once infinite and needs no time at all to be carried out single step by single step. Using an infinite process to reach the limit and to close an infinite series of elements is precisely the well-known ‘mental jump’ mentioned earlier. This procedure is very important for identifying the limits. But every precaution should be taken against the temptation of concluding falsely that just because this infinite series has a beginning and a limit, then the infinitely smallest of its elements must be lurking somewhere between them. That is because such a conclusion is contradictory from the start and logically equivalent to asserting that the one and the same series of elements is both infinite and non-infinite at the same time! In brief, a limit of an infinite series must not become a member of it under any circumstances.
Prince_James: “…Whereas I do agree that if we take this as true that it would not lose its positionary aspect, I must disagree with you on one point, namely, that it could so lose its extension in space. But this also hinges upon my objection to the conception of a dimension not united with the two others. For instance, consider a truly two dimensional square suspended in three dimensional space. Now, according to the viewpoint that there can exists two dimensions apart from a third, the square has literally no notion of "depth". That is to say, it has no extension in the third dimension, so it is literally flat. But I ask, if then looked at on the corner, would not it have no existence? Or even if one looks at it so that the square is viewed from above and its flatness ought to be seen, I ask whether this would even be possible? For if it has no depth, how does it retain this flatness? For there would be nothing to separate it from the nothingness above and beneath it. That is to say, such an square could not exist. The only way to satisfy its existence as a "virtually two dimensional square" would be to postulate the existence of an infinitely small existence in the third, to allow for the least amount of depth possible. Or to bring it back to our philosophic discussion, I postulate that the ultimate point of space must be infinitely small in all dimensions, not lacking in any, and thus retains both space and position, rather than simply position………..”.
Re: As explained earlier, the abstract geometrical point can be obtained by supposing the process of division is infinite as a given. As for the square above, it would be as you’ve described, if and only if it was made of matter of some sort. But if it’s truly Euclidean and constructed mentally upon the abstract void, then it needs no depth and the considerations above do not apply. That is because the whole Euclidean geometry is abstract and all its basic entities are idealized; and hence there should be no difficulty in building squares and the like using idealized right angles and segments of abstract lines.
Prince_James: “…As a beginning question: What sort of contradictions and paradoxes do you propose come about when you consider infinity in a numerical sense? That is to say, that such things as half of infinity cannot be defined, that every number is equal in distance to the end of infinity? For if we investigate the concept of infinity, we find that many of these concepts, which at first glance, seem absurd, are in fact rather reasonable. That half of infinity cannot be defined points to the fact that infinity cannot be divisible by virtue of having no boundary and that boundary is the source of all divisibility in a true sense, whereas 1 and 1010234182108 being equal to the end of infinity demonstrates, once again, this same point - that infinity is always an infinite distance from the finite. That being said, the infinite must indeed be composed of an infinite series of finite parts, so that numbers cannot end is very appropriate and also points to the fact that finite space can always be compared to finite space regardless of the size of one side or another, that is to say, 1132431231 and 023108320820381038108861912379122312987319871561 can interact regardless of the fact that the latter number is astronomically higher than the first. That all finite things would also equal finite things, but the sum of all finite things would be infinite, is also here found………..”.
Re: I would say that infinity presents no real problems in mathematics, because mathematicians over the centuries have developed very effective methods and procedures for identifying contradictions and paradoxes in their subject and for dealing with them explicitly and properly. The real trouble with infinity is in philosophy, cosmology, and theology. Here, the concept of infinity is poorly defined and vague; and it means different things for different people. And to answer your question, the frequent mistake in this regard is to start a discussion by treating the Universe, for instance, as something genuinely infinite; but then half way during the discussion, the Universe becomes a closed totality which is treated implicitly and without pointing that out as if it were finite and limited and all of its element, including the limits, are present and real and actual! In short, the notorious ‘mental jump’ has been made, but implicitly and without the necessary precautions. And so treating the Universe as both infinite & non-infinite at the same time is the most frequent contradiction in this field.
Prince_James: “…. Let me give a better description of what I mean "things which can have a fullest sense ".Now there seems to me two things. Things which cannot have a perfection as they have no logically greatest point, and things which can because they do. An example of the former would be smell. What is "the perfect smell"? Or for that matter, "the perfect colour"? And even if these two absurdities existed - which they do not - would they be necessary? But necessity is needed for perfection, otherwise you are right, it is arbitrary, and clearly nothing in this existence can be arbitrary if it is a perfection……..”.
Re: I would like to disagree with you, here, and say yes, there is a perfect smell! And that perfect smell is the ideal perfume. The absolutely perfect perfume is, of course, an ideal that can never be actualized. Nevertheless, it’s the limit that the perfume manufacturers must set their eyes on it, if they want to improve their actual perfumes year, after year, after year. Without this perfect ideal before their eyes, those industrialists will go sluggish at first, then stagnant, and finally will start to regress and go backwards. Ideals are a driving force of fundamental importance, even though they are as far as infinity. I agree with the rest of your comments.