Is it correct to assume that if there is only one object in the void (really empty void), then its location (the answer to the question “where” this object is) cannot be established?
You need at least two objects to be able to tell where each one is.
Or is it theoretically possible to specify where an object is in absolute void (in a certain area of void, top, left, etc.)?
In an "absolute void", which is just a thought experiment since there are no "absolute voids" in nature, the position of a single object would be very well defined, and there is a unique choice for that position: "0", the "origin" (which you can name any way you want).
The object would have to be the "origin" of that "absolute void". Not that it would be terribly useful, since there is nothing else for which we would need that "origin" in order to establish a position. But it seems like the only possible choice, if one insists on giving a "position" to that object in a void. What else could it be? It's either undefined or "the origin".
This is actually similar to a "pointed space" in topology. We just have a space with an identified point. Since there is nothing else, the only thing that matters is to differentiate the space from that singular point.
There exist no "landmarks" or "grid lines" in empty space. That means the idea of locating a single object in empty space is meaningless: in empty space, each possible "location" is indistinguishable from any other.
This is another way of saying that the laws of physics contain no position or direction dependence. If they did, both linear and angular momentum would not be conserved.
This is true when using the laws of physics to describe the universe we happen to inhabit. If you are talking philosophically, then you get to make up any rules you want, and that universe which you then describe is under no obligation to resemble the one we inhabit.
You can define your position relative to the Earth. We can define the Earth's position relative to the Sun, and the Sun's position relative to the Milky Way, and the Milky Way's position relative to other galaxies, or any combination of these.
We don't have any absolute position for anything.
So if the only object in existence is a single object in a void, there's nothing for its position to be relative to.
If you have 2 objects, you can establish their position relative to each other, but you still can't establish an absolute position for either.
If you mean a void in the sense of an absence of matter in a particular area or a vacuum, but there is other matter in existence outside of that, then we can most definitely establish the position of the object in the void relative to the boundary of the void or objects outside the void.
The question can really only be answered by fleshing out additional presumptions, because there's a lot of metaphysical play in this question.
If there's but a single object in the void, where is the observer? If there is no observer, how can one speculate to the knowledge of the object in the void? If there is an observer, then he would have to lie outside this void you speak of. If there's a void with an observer outside, then how is it the observer knows the void with the object exists? Why should we not presume the void can have an overlay of a coordinate system? Does the void instantiate space-time principles? Which ones? If the void is an empty space-time, can it be modeled as a Minkowski space? Then there are merelogical considerations. Does this thing have parts, and are the parts capable of moving to some degree independent of each other? Doesn't a thing with parts serve as two distinct things thus establishing a coordinate system? One can go on with these language games because empirically there are no voids with single objects with them suggesting that you are not reflecting on things in space, but models of things in space that you choose to construct with language, so what is the true purpose of your Wittgensteinian language game?
If there is a Universe containing only a single object then you cannot specify its location relative to anything else- that is true. You can arbitrarily label its position with any coordinate you like, but that tells you nothing about where the object is in an absolute sense, so you cannot establish the position of a single object.
When you say you need at least two objects to establish a position, you are misleading yourself in the following sense. Suppose your two objects are spheres a certain distance d apart. You can say that sphere 1 is d from sphere 2 and vice versa, but that doesn't 'establish' their positions. Indeed, if you connected the two spheres with a rod, like a dumbbell, you would now have a single object again, and we have already agreed it is impossible to establish the position of a single object. To go further, if you have n particles you still cannot 'establish' their location in any absolute sense- all you can do is specify their positions relative to each other.
A funny nuance to this question is "what do you mean by location anyway?"
Space is a funny concept. It did not always mean what you and I believe it means. Or at least what I believe it to mean, but its pretty much a cultural norm these days which is why you didn't feel the need to specify it further. As an example of this, consider Aristotle's Physics. He had a very different view of what space is:
In effect, Aristotle is suggesting that the place of a thing is the surface which contains that thing. Thus, consider a woman standing on a corner. She is surrounded by an envelope of air. Her place is the surface boundary between the air and her body.
Aristotle's space was inseparable from the concept of the object defining the container. If one has a jar, the space it contains is that up to the surface of the jar. In such a viewpoint, "a single object in empty space" is a very simple concept, for the space is not actually empty. The only space that can be talked of is the space which is indeed full of the object. In fact, he argues that "empty space" cannot exist, for if empty space did exist, one could put an object there and then two objects would exist in the same place; in his philosophy, two objects could not occupy the same space at the same time.
Another approach to the question can be seen in Chinese thought with yin and yang. In your question you mention "You need at least two objects to be able to tell where each one is." In Chinese thought, this is challenged. Your phrasing is a dual phrasing - there are two things which are separable from the other. In Chinese thought, yin and yang are viewed in a non-dual way, inseparable from one another. They would not have seen yin and yang as two objects, but rather two aspects of the same object. In the Taoist cosmology, all starts from wuji(無極), which is translated as "without limit" or more literally as "without ridgepole" in reference to the rafter at the top of a house which defines its space. This then becomes taiji(太極), which is translated as "supreme ultimate" or more literally "great ridgepole." Taiji would be the conception of such a single object in empty space in their thinking, but it would also be seen as the linguistic vehicle to capture the idea of all space.
In modern mathematics, we might use topology to answer your question. Topology is probably the broadest mathematical concept which can capture the idea of a space as it captures the idea of neighbors to that space. A topological space is a set X, and a topology 𝜏 over X. A topology over a set X is a set of subsets of X that has the following properties:
With these definitions, we can consider a few topological sets which could be meaningful for defining locations. There's the null topology, X={} and 𝜏={∅}, which has no locations at all, and thus we cannot define a location at all. There's a trivial topological space, given a single point a, X={} and 𝜏={∅, {a}} which has defines no spatial properties other than having a point at which one can put the object. And then there's the discrete and indiscrete typologies. Given a X={a₁, a₂, ..., aₙ}, the discrete topology is the power set of X (all possible subsets), and the indiscrete topology is 𝜏={∅, {a₁, a₂, ..., aₙ}} (only the empty set and X). The discrete topology would capture an empty space where all points can be distinguished using topological properties, defining what it would mean for a point to be in empty space. The indiscrete topology is one where no two points are distinguishable topologically. We can say the points exist, but we can't say anything about them. This latter case is probably close to what you are thinking of, but its just one of many possible topological definitions.
Of course, in modern physics, the spaces we work with are typically infinite differentiable manifolds. These are a very specific kind of topology with properties that let us do calculus on them. Modern physics, from high school physics all the way through Quantum Field Theory and General Relativity, is built on these. These are built on discrete spaces, so it is very easy to define the location of a single object. However, they also exhibit several key symmetries, such as a translational symmetry which roughly says that the laws of physics will work the same if you just shift everything's coordinates by the same amount.
So part of why the question is tricky to answer is that there are many competing definitions for what "empty space" or "location" might mean in the first place. Paradoxes arise when we are not consistent in our views, but typically vanish when we stick to one view.
Location is normally specified by choosing an arbitrary point of reference or origin.
In a universe with only a single object, obvious choice would place the singleton object at the origin. This would indeed allow you to unambiguously state the location of your single object: It's at the center. In such a universe, you cannot say that the object is in the left or right of the space, because those all imply an alternate choice of origin.
The remaining origins are virtual, and come down to being defined in relation to the object itself. In the case of a single object, this is circular, since the origin is defined in relation to the object which is located in relation to the origin which is defined in relation to...
Your distance metric would also be inherently virtual. You can only measure distance between two virtual objects, or a virtual object and a (the) real one. You cannot talk about real distances between two real objects, because, well... You know. But interestingly, you can say that the origin is some undefined distance away. So it's the size of the displacement that is virtual, not the displacement itself.
With the off-center origins, you can define the origin so as to leave the object on the left or right. However, you could not really say "how much to the right" because you would only have virtual distances to measure it in. You would be saying that it's to the "right of the center by the amount that the center is to the right of it", which is vacuous. But you could say that the object is to the right of the origin, without specifying how much to the right, and the fact that it is to the right would not be vacuous. Actually, the concept of left or right is likewise virtual with only one real object, so you would have to narrow down more and say that the object is not at the center as opposed to being at the center. You cannot say how far off center, or which direction.
Lastly, the exception to all these is exotic spaces. It's possible to conceive a space that is inherently non-isotropic, such that some feature of the space itself provides a choice of origin. It could even provide two such features, so you could define a real distance. The point of origin could then be defined in relation to the space, and the object located in relation to the origin, accomplishing your goal. However, this seems to me a trivial variation on your initial question, since you could say that "feature of the space" is just another type of object. Indeed, if there are no objects in the universe and its non-isotropic space has only one feature, the same argument as above applies. Therefore I think the best you can do is expand the question to being about location in space where [number of objects] + [number of space features] = 1 in which case it is still not possible to talk about any real distance, only virtual, and the singleton object/feature cannot lie at any real distance away from the center of the universe. But it can lie at not-center.
There's a lot of back and forth here in comments, so I'll provide a second answer to clarify a response.
If a void is to be declared meaningfully capable of having something, then it must have at least two things to be free of logical contradiction and exist as a generalized non-surd being (MW). Thus, when you say 'I have a void with only one thing', it logically follows that you have two things despite what you say. You can say whatever you want, but it doesn't make it true. That's the difference between use and mention. Logical analysis is what philosophy does to explore the differences between appearance and actuality.
Fundamental to thought and therefore language there are dimensionless things and things with dimension. Let's call them points and spaces. We can use other terms such as elements and sets, vectors and vector spaces, whatever. They're isomorphisms. What's important is we understand the logical consequences of the relationship between them.
What is the relationship? Generally, we put dimensionless things in dimensions, or points in spaces, but we can do the opposite as when we map a space onto a point which happens in some systems. Abstraction is moving from a set of points to a point, and specification is moving from a point to a set of points.
The question you have asked: Is it possible to have a space that has only a point, and nothing else? My contention is that any claim in which a space is not a point (IS-NOT-A relationship) entails that the space has two points, the point you define, and the trivial point that is implied to exist by definition of what it means to say a space is not a point and has the property of having points (HAS-A relationship). Therefore, let's accept a space is not a point, and a space with points is non-controversial.
The question stands: Is it possible to claim 'A space has only one point'?
Let's use sets. A set with no elements is the empty set. A set with many elements is unproblematic. If you declare a set with one element, does it imply there are no other elements in it? No. Your declaration (your use of language to say 'This set has a single element' logically entails that your set has at least two elements, the element you declare and the empty set. We don't count the empty set, because that might confuse our counting system, but we recognize its existence. (Actually if we consider the naturals as starting with zero, we do count it.) And if we remove the declared element, the empty set is still recognized to exist. Let's try with "void language".
A void with no objects is the empty void. A void with many objects is unproblematic. If you declare a void with one object, does it imply there are no other objects in it? No. Your declaration (your use of language to say 'This void has only a single object' logically entails that your set has at least two objects, the object you declare and the empty object, the empty void itself. We don't count the empty void, because that might confuse our counting system, but we recognize its existence. (Actually if we consider the naturals as starting with zero, we do count it.) And if we remove the declared object, the empty void is still recognized to exist. Again.
A space with no points is an empty space. A space with many points is unproblematic. If you declare a space with one point, does it imply there are no other points in it? No. Your declaration (your use of language to say 'This space has only a single point' logically entails that your space has at least two points, the point you declare and the empty point, the empty space itself. We don't count the empty point, because that might confuse our counting system, but we recognize its existence. (Actually if we consider the naturals as starting with zero, we do count it.) And if we remove the declared object, the empty space is still recognized to exist.
Some claimed that there are pointless topologies defined in lattices. But lattices are logically entailed to contain points. You can treat the lattice as a primitive and decompose it into points through compositional natural language treating the system as more fundamental, but the points eventually exist. And if you declare the lattice empty of points, then the lattice itself IS a point. Same with vectors. You can say there are vector spaces where vectors are primitives, but vectors are directions intuitively, and directions require an intial and end point. You can say vectors have no points, but then vectors ARE points.
The contention is simple. There are HAS-A relationships and IS-A relationships, and if something "contains" only one thing, it must be an IS-A relationship. If it isn't an IS-A relationship, then HAS-A relationship requires the existence of a two things. The HAS-A(subject,object) always implies HAS-A(subject,object-complement) otherwise there is no difference between HAS-A and IS-A. This is recognized in Venn Diagrams which topologically model spaces in the difference between a set being a universal set and a set being a subset.
It doesn't make a difference that you SAY your set, space, and void has a single inhabitant. That's the appearance. In actuality, through logical consequence, your set, space, and void has at least two inhabitants other wise your container is its inhabitant and the language 'has a' contradicts the 'is a'. To argue that 'I said has one' makes it true that it has one is a category error.
Ontologically, this is the essence of Meinong's jungle and the recognition of Sosein which says that while there is a difference between concrete objects and abstract objects, abstract objects are implied to exist. It is also intuitively true of sets with the following arguments provided by this source:
- The set A is a subset of the set B if and only if every element of A is also an element of B. If A is the empty set then A has no elements and so all of its elements (there are none) belong to B no matter what set B we are dealing with. That is, the empty set is a subset of every set.
- Another way of understanding it is to look at intersections. The intersection of two sets is a subset of each of the original sets. So if {} is the empty set and A is any set then {} intersect A is {} which means {} is a subset of A and {} is a subset of {}.
- You can prove it by contradiction. Let's say that you have the empty set {} and a set A. Based on the definition, {} is a subset of A unless there is some element in {} that is not in A. So if {} is not a subset of A then there is an element in {}. But {} has no elements and hence this is a contradiction, so the set {} must be a subset of A.
