NrG poses a very difficult and vexing set of questions. NrG writes:
For example, nothing in my position precludes the existence of a humble and completely isolated object in a far off region of the universe related to nothing else at all. This is possible because objects have both their endo-differential structure and their exo-relational structure. At the level of endo-differences and relations, the object still produces differences, but these are differences that remain, as it were, internal to the system of the object.
I guess my question is simply this: “How can we not have something exo- for every object?” Or to put this another way, how can we have a closed set without something exterior to the set itself?
I think this simple question gets at something fundamental pertaining to issues of space and time– issues that I haven’t fully worked out –so a few remarks are in order.
I think the first point would be that mathematical necessity does not imply natural necessity. There are all sorts of necessary truths in the domain of mathematics that do not, as far as we know, obtain anywhere in nature. Moreover, what is possible in the domain of natural necessity can often depart widely from what maths would dictate. I tried to illustrate this point recently with respect to the Barber of Seville paradox. In the Barber of Seville paradox we are asked who would cut the Barber’s hair if the Barber of Seville cut everyone’s hair except for those who cut their own hair. At the level of the formal, it is impossible for the Barber to get his hair cut. This is a variant of Russell’s paradox. However, the really existing Barber has no difficulty getting his hair cut despite this formal rule. Similarly, mathematically the One is such that it is self-identical and the same in all cases. However, when I say “that is one car”, the mathematical ontologist might say I am mistaken as cars are composed of many different parts and therefore are not one. I think examples like this illustrate why we need to take great care in distinguishing being from mathematical formalization. Mathematical entities are types of beings in my view, but it is a mistake to suppose that all of being is exhausted by mathematics or that other entities are governed by the same formal structures as mathematical entities are. This is why I am not a Badiouian. I take it that any ontology worth its salt must primarily be a discourse on reality or existence and that there is a difference between maths and existence.
However, there are some formal mathematical analogues that can be used to thematize what I’m talking about. Gaussian manifolds or multiplicities are a good example. Gauss made significant contributions to non-Euclidean geometry. Prior to the development of these sorts of geometries it was held that figures could only be described with reference to an embedding space. The fascinating thing about Gauss’s manifolds and Riemann spaces was that they allowed for the description of a space– for example the surface properties of a sphere –without reference to an embedding space outside the surface. In other words, Gauss rendered the intrinsic metric of an entity thinkable. DeLanda has an excellent discussion of these issues in the first chapter of Intensive Science & Virtual Philosophy, but I also have a post developing these issues from a couple of years ago.
Now why are Gaussian manifolds significant in relation to NrG’s question? If they are significant, then this is because Gaussian manifolds render manifolds thinkable without the sort of distinguishing set NrG is calling for. In other words, Gaussian manifolds give an example of how it is possible to think the endo-relations or internal structure of a thing without reference to anything else.
However, there are further points to be made regarding the nature of space. I confess that I still have a lot to work out regarding the nature of time and space, but I will minimally say at this point that I reject the thesis that time and space are indifferent containers in which objects reside. My thesis is that space and time are products of objects, not the reverse. This requires a lot of work in mereology and other disciplines that I’m only now beginning. When NrG asks how it is possible to have a closed set without an exterior, I suspect that he has spatial relations in mind as his primary example. The first point to be made in this connection is that not all objects– in my understanding of objects –are in space or are of the spatial variety. The number 52 is an object in my ontology, but it is not in space, nor, perhaps, even in time. In this connection, it is important to avoid treating physical objects as the paradigm of objectness as such. There are all sorts of objects of which only a subset are physical object.
However, physical objects are particularly interesting. When I make reference to “exo-relations” or inter-ontic relations (inter-object relations), I am referring to real connections between objects. Insofar as I argue that objects are defined by their affects or their capacity to affect and be affected, I am also committed to the thesis that exo-relations only exist in those instances where an object is either affecting another object or affected by another object. Affectation, in its turn, is marked by a change in quality in the object. In this respect, relations like “being-to-the-left-of” or “being-above”, etc., are not proper exo-relations in my book as there is no affection taking place. While it is certainly true that a mind regarding two objects in a spatial reason can say they are related in this way, there is no real relation between these two objects. Consequently, in evaluating exo-relations the issue is that of the real relations between objects. My body, for example, shares genuine exo-relations between the planet earth, the air pressure of the planet, and its temperature because all of these other entities genuinely affect me at a qualitative level. Thus, if the temperature changes my body will either become very hard from freezing or charred cinders from burning. Likewise, if the air pressure of the planet were different, then my body would either implode or explode. Finally, I am as tall as I am due to gravitational constraints. Were I born on Mars it’s likely that I would be much taller, while if I were born on a more massive planet it’s likely that I would be a lot shorter. The qualities produced through these affectations are rendered possible on the basis of the affects that belong to my body qua its internal structure. In other words, the internal structure of my body as a system of affects presides over the ways in which these other objects can affect me qualitatively.
One of the aims of Onticology is to investigate these different types of relations in processes of translation. My height is the result of a translation of gravity, diet, etc. What is the structure of this translation process? When Deleuze repeats Spinoza’s declaration that “we do not know what a body can do!” he is getting at these qualitative transformations that result from exo-relations between objects.
These issues can, in many respects, be traced back to questions of space and time. General relativity has shown us that space-time is an effect of the movement of objects, rather than objects being in space and time as formally structured containers. In other words, objects precede space-time, not the reverse. The more massive an object is, the greater the curvature of space-time. Moreover, this curvature is limited by the speed of light. Thus, for example, were the sun to suddenly be completely destroyed, we would not experience the gravitational effects of this destruction for about eight minutes because this is the amount of time it takes for light to travel to earth. This is one reason that we are able to claim that two objects can be spatially unrelated. If enough time has not elapsed for light to travel to the other object, then there is no gravitational relation between these objects. All of this, I think, requires us to raise a number of questions as to how objects produce their own space-time structures and to think multiple space-times that in many instances are not related. This requires forays into topology and the conceptualization of very different sorts of space-time structure.