Thoughts on infinity, mostly infinite divisibility and the puzzle of traversing infinite halfway points between two points.

[first posted 12/18/12]

[some thoughts I wrote down when thinking about the subject (some editing), so it's kind of stream-of-thought but still focused. Some terms are used loosely, especially the word "mass." I know the paradox of half-way points has been solved by math but I haven't looked it up, if you know something about it and have insight feel free to share, or if you think I'm mistaken on a thought feel free to share.]

Infinity is an unattainable possibility from parts, it may be attainable by something infinite- perhaps by shared identity (infinity as attained by the corresponding concept of it vs. unattainable by the composition of finite parts since the composition of the finite parts will be a larger but finite whole). Shared identity means that the attaining of infinity by an infinite object doesn't mean that an object attains infinity as an act but by identity- by being what the meaning of the concept is (concept of infinity). Infinity is a possibility insofar as it's not logically impossible, but unattainable insofar as the attaining object is incompetent or insufficient. Infinite divisibility suggests that an object is infinitely divisible- divisible without end, not possibly not divisible after any number of divisions. This is true at least conceptually concerning the division of something's mass (or size?). But it is unattainable as each division decreases the mass of the part and will never collectively occupy a state of having been infinitely divided- each part's mass decreases without end as the number of parts increases without end, respectively, but the multiplication of the two (number of parts times amount of mass of each part) will equal the initial mass (conceptually, not considering a real occurrence of inefficiency). A finite object (material, or conceptually limited) will have a mass (limited correspondence to other objects). No matter how infinitely divided the mass is merely distributed. Insofar as points are concerned (as opposed to divisions with mass implications)- such as in the paradox of the impossibility of traversing an infinite number of points, if the point has no reference to a division of mass then it doesn't partake of the object it points to and thus poses no problem for traversing since there's nothing to be traversed. Insofar as it points to the object as an act of reference, the properties of the object in consideration are what determine if it may be traversed- and as a finite object the proposed problem of infinite points is not a problem since it doesn't correlate to an infinite mass (distance) to be traversed, even if the points don't just refer to the whole object but infinitely small (with respect to infinitely many points) (infinitely small doesn't necessarily imply zero, but it would seem that a point that is utterly without reference to any mass of the object pointed to, thus an utter point, points to nothing, but does it still refer to something? And thus these infinite references without mass can refer to an object with its given mass- in regard to a set domain of infinite points?). In this case an infinite number of points is traversed whenever any amount of mass (distance) is traversed, so instead of not being able to traverse an infinite set of points you actually can't help but do so.The question becomes whether on can traverse anything, since anything is infinitely divisible. And as an object of mass that can travel, by moving myself from one area of mass to another, that are at a distance from one another, I can traverse areas of mass (distance) and thus I can traverse infinite points as well. An object can be infinitely divided but its mass is merely distributed- neither decreased or increased; thus two different objects of different mass can be infinitely divided but will not change in correspondence of mass.

However, is mass merely relative? Two objects with different mass (size) that are separated from everything will be identical in attributes (considering mass (size) is the only difference) internally- its relation to itself, but not externally (its relation to other fixed objects, including each object to each other). If the ratio of difference between the two was afforded to identical items (except by comparative mass) that they correspond to (but in different domains- all objects associated to one object not associated with any objects associated with the other) then the respective realities, with no shared internal or external correspondence, would be the same- so is mass relative? It's the idea that if everything doubled in size (which seems to imply all properties) then nothing would be different, or that it wouldn't matter. As for infinite multiplication- you could conceptually multiply an item into double infinitely but this now affects other objects since it doubles the mass and increases the size- thus competing for space (domain). As you can infinitely divide all objects without difficulty you cannot infinitely multiply all objects (without multiplying the domain infinitely). Objects can be infinitely divided since the parts of the division already occupy an existing domain and thus don't affect it (change correspondence to it). Multiplying an object would seem internally consistent and thus beckons again whether mass is relative, since you could effectively double something by decreasing everything else in half. However, this consideration is not opposite. Infinitely dividing an object deals with its parts, internal, it is not about shrinking the object, thus multiplying an object would not be correlatingly contrary but rather synthesizing- connecting parts; but this would have resolution, not an infinite possibility as with division- assuming there are finite parts.

You can count infinitely with all sequential numbers as well as you can with all even or odd numbers alone. Something infinitely divided, in this consideration, starts as a whole object- and you divide without end; but you don't start with infinitely divided objects to synthesize to a whole (since arriving at a whole implies a finite process of synthesis- since you arrived; infinite division goes without end, infinite synthesis would not yield at a whole without failing to have been infinite; yet you also never arrive at infinity- infinitely divided). This is considering processes of division or synthesis, since it corresponds the parts of the process with time and the parts are infinite, the time will be infinite and thus never attained. But when the infinite division or synthesis is an identity it is immediate, precluded from time. So the concept of an infinitely divided or synthesized object is possible and attainable as an identity- that it is something (either inherently or as a considered domain) that is composed of or divisible into an infinite number of parts. Conceptually at least; if you could reduce matter into indivisible parts then the physical application of infinite divisibility may look different than its application to immaterial objects, but if there were indivisible objects they may not be so due to small size (whatever size, could you not conceive of smaller?) but their nature (having indivisible power as opposed to size) or because of there being no relative object that is competent to divide- such as an object smaller or stronger so as to divide, even though a smaller or stronger object is conceivable (this latter may be incidental indivisibility, the former could be inherent indivisibility, owing either to something like God's decree or to limitations of physics, or both correspondently.) The attainability of this possible infinity by identity can be conceived by a number, "one" for example, that is infinitely divisible but is immediately composed of the sum of all its divisibility.

Another consideration on synthesis would be the nature of an object. It seems more clear to conceive of infinite division of an object but less clear to synthesize two objects that differ in function. The parts of a chair are one thing, and synthesizing a chair and a table as a set is also not so unclear, or even all the things in a house as stuff that someone owns (although the correspondence is different- being stuff owned as opposed to a table and chair serving a purpose of a place to eat), or all things as stuff that exists; but even in these syntheses the parts of the whole seem distinctly different than the parts of these parts (a chair as part of the things you own versus all the parts that make up the chair to the point of infinite divisibility). It wouldn't make as much sense to talk about half of a chair as part of stuff you own (when you own the whole chair). Also, if someone asks you to bring them a book its not problematic, but if they ask for the leg of a chair (even for a feasible use) it will be more difficult since it must be removed and will likely damage an object (which may vary in concern based on one's value of or relations regarding the object; but being able to maintain synthesis is thus functionally important). So it seems to make more sense to talk about wholes when it bears a certain nature of being a whole, at least relative to its use as a whole considering the reference. To talk about the parts of a whole makes some sense when applicable, but to synthesize two wholes as a whole would make less sense (considering the situation . . . .) but as a set, more so. Maintaining the creative ability to think about the truth of something different from its use can be of value, to mistakingly think of some things as a whole (thus implied shared nature) when they bear distinct inherent and intended purposes may be hindering (to be able to do so is one thing, to do so mistakingly rather than understanding the more pertinent qualities and nature of the parts is another; and on the contrary to think of some things only as distinct parts to the extent of missing their synthesized function can also be problematic).


[additional thoughts I wrote down another day after thinking more about it, more succinct, didn't elaborate]

The infinite division is separate from the rate of motion of an object travelling a distance, and the competency of the combined rates of motion and time traveled to traverse a distance is correspondent to that distance which is the sum of the infinitely divided half-way points. Also, if considering an object's travel in correspondence to the infinite division- as the object approaches its end of distance the half-way points become infinitely small and thus the speed and time required to cross them becomes infinitely less, and given the finite distance altogether it is traversable.


*I'm not sure how clear it was, but the problem I'm assessing is the paradox that says that you cannot travel from one point to another, since in order to do so you will have to reach the half way point between those two points, and once you've reached that halfway point there will be another halfway point from there to the destination, and so on and so on, so that you never traverse all the halfway points between where you started and where you're going since at each halfway point you reach there is another halfway point from there to the destination, which is an infinite process and thus one cannot travel from one point to another.

*In short: an infinite number of halfway points can be traversed because they occupy a finite domain, and the rate of motion of the traversing object is separate from the rate of division (not slowing down according to the distance of halfway points) and is sufficient to reach the destination in a finite (and preferably pertinently short) amount of time.