i seek in this paper to explore Spinoza's physical ontology; the structure of space and extended objects. Spinoza inherits his physics from Descartes, but the talk of physical objects as modes instead of substances creates a very different context for the physics.
two components of Spinoza's discussion of extended substance are important here. first, physical objects are spoken of as mere modes of the one extended substance, God, or nature. second, Spinoza's physical digression and some letters speak the manner in which particular objects are individuated.
the solution i favor is called by Jonathan Bennett a field metaphysic. however, Bennett's description of the metaphysic relies crucially on modern continuum topology, especially the notion of a continuum as a certain kind of metric topology, a set of points with some additional structure.
so the goal here is to motivate and elaborate the field metaphysic, and then to explore ways of making it plausible without the full-blown modern spatial topology. however, a certain modesty is necessary here. there is ultimately no way to make sense of metric geometry without what we know of as point-set topology. the goal then cannot be to make the field metaphysic fully consistent without the use of modern topology, for this is not possible.
we know that if one starts with geometry, adds a metric, and a few other uncontested assumptions, the ending point will be the full blown point-set metric topology understood today. the more modest goal here will then to see how far one can get. we know the sorts of misconceptions Descartes and Spinoza shared about the nature of a continuum; the goal will be to take those misconceptions for given and see how elaborately the field metaphysic can be developed.
any further, and the misconceptions themselves would be exposed, and indeed, in that way was in fact developed the modern conception of a continuum.
Spinoza's physical ontologySpinoza most famously holds to a monism, a doctrine in which there is one substance. the notion of substance from Aristotle was supposed to account for object individuation, for physical properties rooted in essences, and so forth. when Spinoza abandons the Aristotelian notion of a plurality of substances (which was carried over into scholastic philosophy, and perhaps into Descartes as well) we must look carefully to find a new solution to these problems which the traditional notion of substance was intended to solve.
the one-substance doctrine is perhaps most clearly expressed where it is first proved in the ethics: except God, no substance can be conceived...in nature there is only one substance...an extended thing and a thinking thing are either attributes of God, or affections of God's attributes (Ethics, 1p14 and 1p14c1, translated by Edwin Curley). here Spinoza gives his first answer to the problem of objects are modal on God.
moreover, these physical extended modes have a structure, for we are told that the order and connection of ideas is the same as the order and connection of things (ethics, 2p7). Spinoza has an account for the persistence through time of an object. if some [parts] are removed, and...others of the same nature take their place (ethics, 2l4) the individual remains. the parts can also become larger or smaller, as long as they keep the same ratio of motion and rest to each other as before (ethics, 2l6).
the lemmas of the physical digression here amount to a rich account of what establishes object identity through time: what it is that allows us to say that the object at time t1 is the same object as we see later at time t2 . but they do not give us a very rich account of object identity from a synchronic perspective: what distinguishes the object here at x1 from that other object at x2? for the aristotelian, they are different substances. merely replacing substance by mode and attempting to keep the aristotelian account would not be satisfactory.
for a richer account then, Jonathan Bennett has proposed a particular interpretation of the one-substance/things-as-modes language which is happily consonant with modern ideas as well as a plausible rendering of Spinoza (extended substance, in a study of Spinoza's ethics, Hackett, 1984). first i will discuss the manner in which a thoroughly modern field metaphysic might be articulated.
the modern field metaphysic
the mathematical structure of a modern field metaphysic begins with a topology; for our purposes it will be sufficient to assume the standard metric euclidean topology. then with each point in the space a set of numbers is associated. in newtonian mechanics, each point has an associated mass density; after Maxwell one wants to add a charge density and an electric and magnetic field strength.
the physical correlate of this mathematical structure is known as a field; it might be thought of as an all-pervasive presence which has different local character at each point. one might choose to understand electric forces in terms of a field, but see mass as composed differently; nothing forces a choice of taking everything or nothing to be specified in a field-like manner.
however, it is part of a physical field theory that the space of points making up the field is the same for all the different fields one might take. As a result, a set of fields can always be thought of as a single field with a more complex value at each point.
despite intriguing hints in Newton's principia philosophiae, the field idea remained dormant until the nineteenth century; the development of electrodynamics required a mathematical structure of a field and it was natural to take this as something real. With the later development of continuum mechanics, it became clear that all of classical physics can be articulated in field-theoretic terms.
a field metaphysic then is structured similarly. we take space to be a collection of points under some metric topology, with properties of various physical quantities specified at each point. in modern classical physics, the metric topology would need to conform to Einstein's so-called field equations, and the values of the field at each point would be specified by mass and charge density, and the electrodynamic field strength. (as i restrict the term in this essay, the term field equations in general relativity is strictly a misnomer.)
it is clear that this can help to understand Spinoza's ontology. We take the field (space, extension) to be basic, and the field variables are simply its local properties: modes. speaking crudely, space is a little thicker here, thinner there, and so forth.
a field metaphysic also tells a story about the individuation of objects. what we know as a table, or chair, or human being, is a localized pattern of thickening of space. thanks to the dynamical laws, certain such patterns persist for significant periods of time. but the individuation of one object from another is a matter of the field properties in different locations; that is, the individuation is a matter of convention and human understanding, it is not out there in the world, but is an additional layering of interpretation on the various local field properties.
thus far Jonathan Bennett. however, one serious problem remains: Spinoza, like most of his day, did not have modern topology. instead, space was taken to be described by euclidean geometry, with a metric, but the notion that geometric lines, planes, and spaces were made up of points was not accepted. the notion would seem impossible for the simple reason that nobody could say how an infinity of points could build up to a line, and so forth.
since a field metaphysic is specified in terms of the values of field variables at each point, if space is not just a suitable union of points, then the field metaphysic would not be plausible as a candidate explanation of Spinoza's views. my goal in what follows will be to explore how Spinoza might have held to some thing very close to the modern field metaphysic without being committed to the modern continuum topology.
a field metaphysic in Spinozabefore tackling directly the problem of the point structure of continuous space, i would like to describe some of the further details in a spinozistic metaphysic that need not be parts of all field metaphysics.
the subjectivity of individuation in the basic field metaphysic is certainly one Spinoza is happy with. he adds an additional component as well, in fleshing out the nature of the ways objects are just persisting patterns in the field. an object is a composite of small objects, which have some determinate relation to each other (proportion of motion and rest). as long as the sub-objects continue to have the same mutual relations, the object itself continues to exist. to understand what those mutual relations are, one has to look to biology, for example.
in addition, while the sub-objects need not also continue to exist for the object they compose to persist, if they vanish it must be by their replacement with others of the same nature (ethics, 2l4). the sub-objects must themselves therefore be made up of sub-sub-objects, and so forth. following this line, results in a conception of a hierarchical structure of the objects that make up reality. it seems to me to be a nice addition to the basic field metaphysic's understanding of object identity and persistence.
this hierarchical description is not found full blown in the Ethics, but in addition to the passages cited above, it is very strongly suggested in a letter: on the question of whole and parts, i consider things as parts of a whole to the extent that their natures adapt themselves to one another so that they are in the closest possible agreement [read, proportion of motion and rest]. In so far as they are different from one another, to that extent each one forms in our mind a separate idea and is therefore considered as a whole, not a part (letter 32, emphasis added). the letter goes on to liken the place of man in nature to that of the blood corpuscles in a man and otherwise suggests the hierarchical understanding very closely. note as well the clear treatment of the make-up of objects as a subjective concern as indicated by my emphasis.
in the physical digression in book ii, Spinoza briefly mentions the simplest bodies (ethics, 2a2'') at one point. if we take this in an absolute sense, it would destroy the picture i've built thus far; the simplest bodies would not be made up of smaller bodies and would represent ontologically independent bodies. their existence would render suspect the subjective understanding of object individuation that the field metaphysic implies, and we would need an account of their persistence through time as based on something other than the proportion of motion and rest of constituent parts.
rather than force such a task on Spinoza, which he does not seem to address, i would prefer to read the simplest bodies as only referring to those bodies which are most simplest under some relative consideration. in any (finite, human) consideration, we cannot see all the way down the chain of composition; we are at any time only aware of some smallest sort of body. such a gloss would be entirely consistent with his usage of the term in the passage in question.
making it worknow we are ready to confront the principal problem i seek to address: how can Spinoza have held anything like the modern field metaphysic, given that he would presumably have rejected the notion of space as composed of a collection of indivisible points?
space for Spinoza is not just a collection of points, but it could be thought of as a collection of regions of space, each with extent. (indeed, Descartes seems to have seen extended substance in just such terms.) so the natural modification of the modern field metaphysic would be to speak of regions of space with field properties instead of points; instead of the modern mass and charge densities at a point, for example, we would have the mass or charge of a region.
one problem is apparent right from the outset with this account however. properties might not be homogeneous over even small regions; we have left out the possibility of properties which vary continuously. if it is satisfactory to say that the property varies over a region by having a determinate value at each point in the region, then the modern form of the field metaphysic would be acceptible. but this is the only way to account for contiuous variation. no account could satisfactorily deal with such properties, however, without having within it the full topological sophistication which conceives of a continuum as composed of points.
so we will take it that properties do not in fact vary continuously over spatial extents. but then the regions we have described look like fundamental persistent objects. and yet, as Descartes pointed out, we can conceive of the region as divisible in principle: as composed of subregions. how can this be reconciled?
the regions can and do change; new regions can be created, old regions destroyed. such a process requires an account, but once it is given, the worry about the persistence as atoms of the regions is settled. the conceptual divisibility of regions is unproblematic, because we do not assume the regions to be static. so the remaining difficulty will be solved by considering the manner in which changes in regioun boundaries occur over time.
however it does give a clue, i think, to the correct solution of the difficulty. at any moment in time indeed space is made up of regions. suppose for the moment that time is made up of moments of time. in that case, we would have no difficulty at all; the modal properties of the field metaphysic are now fully grounded in the regions in existence at each moment, and the moments of time which make up all of time.
however, the assumption that time is made up of moments of time is just as problematic as the notion that space is made up of points; only by solving the topological issues can one provide a complete account. however, if we assume that region boundaries move in a continuous fashion, then there is no problem.
whatever their understanding of space, it is clear that all concerned accept the possibility of motion: of continuous variation through time. and so the final understanding of the field metaphysic, which indeed i believe Spinoza can accept, would treat space as made up of regions, each of which has determinate physical properties (one is thicker or thinner). the region boundaries themselves move over time. a region can vanish by progressive shrinking of its boundary; a new regions can appear by the inverse process.
we cannot expect that Spinoza had a fully coherent notion of motion, given the lack of a good understanding of the continuous. if time is discrete, however, we can provide an even clearer notion. if a region boundary moves between one discrete moment and the next, then we can simply consider that there is a discrete homogeneous region (the difference between the earlier and later boundaries) which undergoes a discrete change in that moment of time.
the region boundaries are then also modes of extension. but what exactly is a region? here we point to geometry for the answer. a full understanding of this metaphysic would need to understand what motion of the region boundaries is, and so forth. in addition, i have assumed that regions have sharp boundaries and are internally homogeneous.
are the regions the simplest bodies? it would seem in one sense that they are. however, they are not atomic, because they appear and disappear. they have no independent status. there is no existence of the region apart from the property (of thickness or thinness) which obtains over that region.
to account properly for motion and for continuous spatial variation instead of sharply-defined regions, one would proceed by considering smaller and smaller regions, and taking the limit as the regions vanish. of course, this procedure is precisely that of the calculus. but without taking that procedure, one cannot adequately account for continuous variation and motion at all.
in conclusioni believe then that this is an understanding of the field metaphysic which Spinoza could himself have accepted. indeed, it still has problems, but they are problems that will be inherent in any pre-topological treatment. if we begin solving those problems, we inevitably result in deriving the results of modern topology.
things i wrote for my m.a.