Is Shape Physical?
Shape is included on the traditional list of primary qualities, but that doesn’t settle the question of whether it is physical. Indeed the question is seldom raised: is the shape of an object a physical property of it? Nor is it an easy question to answer—largely because of vagueness in the word “physical”. Is geometry a physical science? It is not usually so described, since it deals in abstract objects. Such objects are ideal: perfect circles, perfect rectangles, etc. No physical object (so called) exhibits these properties, so it seems reasonable to say that perfect shapes are not physical. Shapes are mathematical abstractions imperfectly exemplified by physical objects. We apply geometry in physics, but the objects of geometry are not themselves physical entities (in so far as we know that means). Moreover, there are no laws of shape recognized in physics, as there are laws of gravity and electromagnetism: shape doesn’t act as a force producing motion. A physics textbook doesn’t have a chapter on the laws governing shape. Shape is irrelevant to gravity and electromagnetism: the mass of an object is relevant to its gravitational force but the shape is not. Physics would be essentially the same if every object had the same shape. So there seems every reason to deny that shape is physical. But does that imply that it is mental? Not at all, and the idea seems obviously wrong (short of generalized idealism). Shape is neither mental nor physical. It may be causally consequential but it is not thereby a physical property. Someone who believed that the world is fundamentally geometrical would not ipso facto be a physicalist (or materialist). Nor would it be correct to say that size, number, motion, and dimensionality are physical properties: they too belong on the mathematical side of things. Maybe it is necessary truth that anything that has such properties is a physical thing, but that by itself doesn’t entail that they themselves are physical properties (the same might be said of colors). One is then left wondering what indisputably is a physical property, if shape isn’t one. Is mass a physical property? But mass is defined in terms of inertia, which is a dispositional mathematical property (measured by how much force is needed to initiate motion). What about electric charge? But that too is dispositional and mathematical, and has historically been regarded as clearly non-physical (like gravity). Is it perhaps that no property is physical but that objects are what fall under that (alleged) concept? But in virtue of what—don’t we need some viable notion of a physical property? The very idea of the “physical” starts to slip through our fingers once we focus hard on these questions. It is entirely conceivable that the whole subject called “physics” has no intelligible notion of the physical—and that this is no objection to the science known by that name. Here we reach a conclusion that has persistently threatened the would-be philosophical physicalist: we simply have no workable general notion of the physical. Shape might have seemed to provide at least a paradigm of the physical, but that has turned out to be a frail reed. All we are left with is the idea that a physical property is what the subject called “physics” talks about, but that is a variable and pragmatic matter. We have no clear idea of what a physical property is intrinsically—unless we decide to stipulate that everything is to be counted as physical, the term being equivalent to “real”.
I say all this to make a metaphysical point, viz. that we should stop trying to divide the world up into the physical and the non-physical. We can talk about what is mind-dependent and what is mind-independent, but we should drop the assumption that anything can be usefully described as “physical”. This means that falling under that term is no measure of ontological primacy or clarity: we can’t contrast other types of putative property with physical properties and hope to formulate a useful distinction. We can’t characterize mental or moral or mathematical properties as “non-physical” and hope to join a genuine metaphysical debate: for there is simply no such thing as the category of “the physical”. We might have supposed that shape would give us a firm foothold, but shape turns out not to be a good candidate for fixing the notion of the physical. The word “physical” has an everyday use (or several such uses) but as a theoretical term it lacks any clear definition, as has frequently been pointed out. That is why we hesitate when asked if shape is physical—as we do when asked if color is physical (or beauty or moral rightness). Let me put it more forcibly: if we can’t say whether shape is physical, we may as well retire the term from serious theoretical employment. It’s like asking whether shapes are holy: the term belongs to an outmoded theoretical framework and survives mainly as a term of approbation. Are circles more holy than rectangles? Are circles less physical than irregular figures? Such questions have no meaning, because “holy” and “physical” lack determinate content. I would say that shapes are clearly not physical if we mean by “physical” something like “tangible and concrete” or “relating to the body” (as the OED has it): for shapes are abstract, and they are not peculiar to the body. Nor are they physical in the sense that they are perceived by the senses: they may sometimes be imperceptible to the senses, and they may never be perceived as they really are. Shapes are just not intuitively physical (though intuitions about the concept of the physical are notoriously slippery). So the world contains properties that are not mental and not physical, and these properties are among the most salient in our experience. We can intelligibly ask whether everything real reduces to the standard list of primary qualities, but that is not the same question as whether everything real reduces to the physical—which is pretty much vacuous. I can imagine a “shape-ist” metaphysics in which geometrical form is taken to be the most basic property in the universe, but it would be misguided to describe this as a form of physicalism. Maybe everything comes down to the motion of shaped objects, but it is not helpful to describe this as a type of physicalism. The doctrine known as “physicalism” survives largely on the lack of clarity in the term: we should abolish it and speak directly of specific kinds of properties such as shape and color. Whether these properties count as physical is an empty question best ignored. 
 The familiar (and good) point that gravity is not physical according to traditional notions of the physical (deriving from mechanism) is the usual way of questioning the utility of the term “physical”; my point here is that even shape poses problems for that term. In what sense precisely is shape to be deemed physical? Would Plato and Pythagoras so characterize it? It is hugely tendentious to count circularity as a physical property and is not remotely warranted by tradition—you may as well declare that beauty is a physical property because physical things (whatever that may mean) have it.