Our second seminar in the Keele Philosophy Forum Summer Series was givenInstallation by Roberts by Dr Phillip (Phil) Meadows on 19 May. (The paper can be downloaded Here!) The aim of the paper is to show that holes are not objects, but properties (or relations). The starting point is an argument that angles are not objects, but properties. The next step demonstrates that holes and angles are in the same category as far as their ontological status is concerned. These two steps suffice for one to conclude that holes are not objects, but properties or relations.

One interesting claim of the paper is that the argument can be made without a specific commitment to an account of the fundamental categories of things. This is important, since there are ongoing debates on this issue, so an argument which can be neutral towards the various positions involved in the debate will be quite strong.

To be sure, Phil adds, to show that holes are not objects, but properties, implies to assume at least an ontological distinction between objects and properties and, hence, an assumption that this distinction is correct. Yet, two influential current ontological systems accept this distinction (Armstrong’s two-category ontology and Lowe’s four-category ontology), so the argument will be neutral at least insofar as these systems are concerned.

Now, Phil takes properties (which include relations) to be about objects; in other words, objects are bearers of properties. He specifies, following Lowe, that objects are entities of order zero, whereas properties are entities of order greater than zero. He also mentions that, for Armstrong, the distinction object/property is captured by the distinction particular/universal. The implication, from Armostrong’s account, is that different things can have the same property, since different particulars can bear the same universal. In other words, on Armstrong’s account too, properties are ways objects are.

The claim that angles are not objects, but properties may seem strange for someone who thinks that an angle is simply the set of two lines which share one end. Phil distinguishes between an object shaped in an angular way, which is a way of being (referring to shape) of an object, and a figure which has an angle. When we talk about an agle in the latter way, we do not already assume that the angle is a property (or object), so the question whether angles are objects or properties makes sense. If we already assume that angles are properties, then, obviously, the question is no longer interesting.

When one says that a figure has a certain angle, one can take this to mean that there is a relation between two objects (the figure and the angle) or that the angle is a way of being of the figure. Given these possible interpretations, we do not yet know whether ‘angle’ in this sense refers to an object or to a property. This is a good starting point. To decide whether the angle is actually an object or a property, Phil says, we can first consider what it would mean for the angle to be an object. One way to explain this is to say that the angle is a constitutive part of the figure. The problem here is that there seems to be nothing beyond the angle that constitutes the angular figure. (Contrast this with the case of a statue which is constituted by a lump of bronze.) Hence, there seems to be no reason to count both the angle and the figure as objects. This goes against the view that an angle is an object.

We can add to this, Phil goes on to say, the fact that the angle depends for its existence on the angular object: once the object is obliterated, there is no angle as a way in which the object is. In Lowe’s system, this is accounted for in terms of both existential and identity dependence of properties on the particulars they characterise. Armstrong is committed to an immanent realism about universals, which means that he cannot fully account for this existential dependence of universals on particulars, but, Phil says, “this is perhaps so much the worse for his ontological system”. (P. 4.)

A further argument in support of the view that angles are properties is that it provides a more economic ontological account for the expression ‘x has an agle of n degress’, since it accounts for this in terms of an entity (the figure) that has a certain property (angular) that, in its turn, has a certain property (n degrees). This makes a second object (the angle) superfluous.

Phil concludes that, on the basis of these considerations, we have reasons to think of angles as properties, rather than objects. The question whether they are universals or particulars, he says, remains open, but will be discussed later on in the paper.

The next step is to show that angles and holes share identity and existence conditions to an extent which is sufficient to class them in the same ontological category. The strategy employed here is Casati’s and Varzi’s. Following this strategy, they class cavities, depressions and tunnels in the same ontological category, and Phil classes angles and holes in the same class.

If angles and holes must be classed in the same ontological category, the fact that angles seem to be better placed in the class of properties suggests the same must be the case for holes. One question Phil discusses in the final section of the paper is how this account of holes can cope with the fact that angles, tunnels, cavities, depressions and holes are countable. We can, for instance, unproblematically say ‘there are n holes in x’.

The difficulty of this question is given by the assumption that if an entity can be quantified over, it is an object, rather than a property. Once one tries to formulate an anti-realist argument about holes (or angles), one turns the hole into a property. The problem then is how to distinguish between objects which have two holes and objects which have what seems to be one hole, but which is the result of interlinked holes.

Phil claims to be able to sidestep this debate by questioning the assumption that quantifiable entities are objects. He thinks one can quantify over properties in both of the ontological systems with which his argument aims to be compatible. For instance, within Lowe’s system, a ball with two equal hemispherical regions of different shades of red can be characterised as having two numerically distinct particular colour properties, each representing a particular instance of the single colour universal ‘red’.

This seems problematic for Armstrong’s ontological system, since for Armstrong the ball is a particular and ‘red’ is universal. Thus, according to Phil, to account for the two shades of red the equal hemispherical regions of the ball have we cannot say that the ball stands in two numerically distinct instantion relations to the universal ‘red’. This is because we may end up with an infinite regress in the attempt to account for the instantiation relation. To avoid this problem, Phil adds, Armstrong conceives of the relation of instantiation of a universal by a particular as involving “no addition of being”. (P. 9.) But, then, instantiations can no longer be regarded as distinct countable entities.

To solve this problem, Phil suggests the following solution: we can say that the two halves of the ball are two numerically distinct particulars that separately instantiate the universal and that are component parts of the ball. The same can be done with an angular figure like ‘V’ composed of two lines.

Phil concludes the paper with a few cases to which he applies his account. The final section of the paper summarises the main philosophical moves of the paper.

There are many questions one can raise about the argument in this paper and discussion during the seminar went on for more than one hour. In what follows I will only mention one question and this has to do with the claim Phil makes that the argument in his paper is neutral in relation to Lowe’s and Armstrong’s ontological systems. In particular, I am not sure how the claims that angles are properties and that they are countable can be compatible with Armstrong’s ontological system.

To see what I mean, think again of the example of the ball with two equal hemispherical regions of different shades of red. On Armstrong’s account, we can say that the two halves of the ball are two numerically distinct particulars that separately instantiate the universal and that are component parts of the ball. But, then, we can only have one colour (‘red’), two particulars (which instantiate shades of red) and the object which is constituted by the two particulars. This suggests that, unless particulars are not regarded as objects, but as some different type of property which is distinct from universals, shades of red (and specific angles) are not properties or cannot be counted.

But, of course, this would mean that Phil’s account actually favours Lowe’s ontological system and, contrary to what Phil claims, it would not be equally compatible with both Lowe’s and Armstrong’s systems.


One thought on “Angles (and Holes)

  1. Sorin has written a very clear account of my paper and raised a worry about the extent to which my argument is genuinely neutral between Lowe’s and Armstrong’s ontologies. Sorin points out that while Lowe can fairly unproblematically give an account of why there can be true sentences about holes or angles which feature quantification (because we can count property instances of universals), Armstrong has more difficulty on this score. This is because instances of universals will, for Armstrong, be particulars which have all the hallmarks of objects, not properties. Moreover, we cannot count the instantiation relation as providing the truthmaker for our true sentences about holes.

    Sorin is quite right on this score. However, there is one alternative available to Armstrong that I did not get a chance to cover in the presentation of my paper. This is that Armstrong can count instances of holes or angles by counting particulars that instantiate the properties – but where these particulars are parts of the material thing that forms an angle or has a hole in it.

    Nonetheless, Sorin is right that my discussion is, in an important sense, not neutral between Lowe’s and Armstrong’s ontologies. It is not neutral in the sense that my view does not cohere equally well with both ontological systems. Lowe’s ontological system involves less revision of the common sense position that sentences like ‘There are two angles in x’ count the angles themselves, rather than particular parts of some angular figure. Nonetheless, my view is neutral in the sense that the view of holes I defend is logically compatible with both ontological systems: i.e. is not ruled out of court by either.

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