## A square is rounder than a rectangle

Sometimes (for example after imbibing my third whiskey) I am both intrigued and frustrated by the nature of shapes. Do shapes exist at all? Except, perhaps, as a property of a thing?

Without dimensions there can be no shapes. A point has no shape. In one dimension, shape is almost, but not quite, trivial. A one-dimensional shape is just a line. Both a point and a line are abstract things and do not exist physically. We perceive three physical dimensions but we are also constrained to experience nothing but 3 dimensions. We can imagine them, but there are no 1-D or 2-D things. Even a surface, which is always two-dimensional, is abstract. We talk about circular things but the concept of a circle is also an abstraction in an abstract two dimensions. Look as much as you like in the physical world but you can never find any 2-D circles in this 3-D world. Most shapes are two-dimensional. So how, I wonder, can some 3-D thing be described in terms of a 2-D circularity. If you rotate the abstract two-dimensional object called a circle in 3 dimensions, you can generate an abstract 3-D object called a sphere. It pre-supposes, of course that 3-D space exists within which rotation can occur. But what is a sphere? How do you rotate an abstract object? A square rotated gives a cylinder – not a cuboid. A point stretched into two dimensions, or twirled in three, remains a point and still imaginary. A line rotated gives just a line.

I find the word shape is diffusely defined in dictionaries – possibly because it is itself philosophically diffuse.

shape (n):

• the external form, contours, or outline of someone or something;
• a geometric figure such as a square, triangle, or rectangle;
• the graphical representation of an object or its external boundary, outline, or external surface.

Shape, it seems to me, has a connection with identity. Things without identity have no shape. All countable, physical things have shape as an attribute. But uncountable things – rain, mist, water, … – are devoid of shape. But any shape is also an abstraction which can be taken separate from the physical things. Abstract things and uncountable things can also be invested with shape as a descriptor, but this is both figurative and subjective. We can refer to the shape of an idea, or the shape of a history, or of a culture, but the meaning conveyed depends upon the physical things normally connected with such shapes. Even when we use the word shapeless we usually do not mean that it is devoid of shape but that the shape is not a standard recognised form. Shape emerges from existence though not necessarily from the existence of things. It is here that the distinction between form and substance originates. Shape needs existence but it is not difficult to imagine the concept of shapes existing in even a formless universe without substance.

In philosophy, shape is an ontological issue. There have been many attempts in philosophy to classify shapes. For example:

The shape of shapes

An important distinction to keep in mind is that between ideal, perfect and abstract geometric shapes on the one hand, and imperfect, physical or organic mind-external shapes on the other. Call the former “geometric shapes” and the latter “physical shapes” or “organic shapes”. This distinction can be understood as being parallel to types (classes, universals, general entities) and instances (individuals or particulars in the world). Geometric shapes typically have precise mathematical formalizations. Their exact physical manifestations are not, so far as I am aware, observed in mind-external reality, only approximated by entities exhibiting a similar shape. In this sense geometric shapes are idealizations or abstractions. This makes geometric shapes similar to types or universals. Their instances are inexact replicas of the shape type in question, but have similar attributes or properties in common, properties characterizing the type. By contrast, organic or physical shapes are irregular or uneven shapes of mind-external objects or things in the world. A planet is not perfectly spherical, and the branches of a tree are not perfectly cylindrical, for example. “Perfectly” is used here in the sense of coinciding with or physically manifesting the exact mathematical definitions, or precise symmetrical relations, of geometric shapes. Objects and physical phenomena in the world, rarely if ever, manifest or exhibit any concretization of geometric shapes, but this is not to say that it is not possible or that it does not obtain at times. Objects are not precisely symmetrical about a given axis, cube-shaped things do not have faces of exactly the same area, for example, and there is no concretization of a perfect sphere. ……………

With respect to the mind-external world, notice that if shapes are properties (of things), then we may have a situation in which properties have properties. At first glance this seems true because we predicate shape of objects in the world; we say that objects have a certain shape. We also describe types of shapes as having specific properties. If a shape is defined as having a particular number of sides (as with polygons), a particular curvature (as with curved shapes, such as the circle and the ellipse), specific relations between sides, or otherwise, then it should be apparent that we are describing properties of properties of things. We might be inclined to say that it is the shape that has a certain amount of angles and sides, rather than the object bearing the shape in question, but this is not entirely accurate. Shapes, conceived as objects in their own right (in geometric space), have sides, but in our spatiotemporal world, objects have sides, and surfaces, as well. When we divorce the shape from that which has the shape via abstraction, we use ‗side‘ for the former as much as we do for the latter. The distinction between geometric and physical space, between ideas and ideal or cognitive constructions and material mind-external particulars is significant.

My preferred definition of shape is:

shape is an abstract identity of form devoid of any substance

I take shapes to be forms both in two dimensions and in three. So, by this definition, I include spheres and cylinders and cuboids and pyramids to be shapes. Shape is about form – whether or not there is a thing it is attached to. We can have regular shapes where the regularity is abstract. We can have irregular shapes which cannot be described by any mathematical expression. And we can have shapeless shapes. We can compare shapes and discover the concept of similarity. We can even compare dissimilar shapes. I can conceive of the quality of form and talk about circularity or squareness or sphericality or even shapelessness.

I can have curvy shapes and I can have jagged shapes. My ping-pong ball is more spherical than my dimpled golf ball. They are both rounder than an orange but I have no doubt that an orange is rounder than a cucumber. Just as an apple is squarer than an orange. A fat person is rounder than a thin person. I know one cannot square a circle yet I have no difficulty – in my reason – to attributing and comparing levels of squareness and roundness of things. Some squashes are round and some are cylindrical. A circle squashed gives an ellipse and the shape of the earth is that of a squashed sphere. Circular logic is not a good thing. Logic is expected to be linear. A spherical logic is undefined.

And any square is rounder than a rectangle.