Shape is a Non-Quantifiable Physical Dimension

In the natural-scientific community it is often taken for granted that, sooner or later, all basic physical property dimensions can be quantified and turned into a kind-of-quantity; meaning that all their possible determinate properties can be put in a one-to-one correspondence with the real numbers. By using some transfinite mathematics, the paper shows this tacit assumption to be wrong. Shape is a very basic property dimension; but, since it can be proved that there are more possible kinds of determinate shapes than real numbers, shape cannot be quantified. There will never be a shape scale the way we have length and temperature scales. This is the most important conclusion, but more is implied by the proof. Since every n-dimensional manifold has the same cardinality as the real number line, all shapes cannot even be represented in a three-dimensional manifold the way perceivable colors are represented in socalled color solids. 1 Shape as a Physical Dimension Let me start almost from scratch. The word ‘shape’ (possible synonyms: ‘form’ and ‘figure’) is in everyday life mostly used to refer to the twoand three-dimensional finite outlines of surfaces and things. Such shapes are closed shapes, but in science there is often talk also of open shapes, e.g., parabolas and hyperbolas. In what follows, when nothing to the contrary is said, the extension of the term ‘shape’ will include both closed and open shapes with finite spatial extension. Moreover, not only geometrical such shapes, but any arbitrary such shape whatsoever. If a determinate shape (at a certain region in space) by means of translation, rotation, and uniform scaling can be made congruent with another determinate shape at another region, then these two shapes are two instances of the same (kind of) determinate shape. In other words, shapes are invariant to location, rotation, and size. Shape so delineated is a property dimension on a par with basic physical property dimensions such as length, mass, and temperature. Some determinate shapes also have second-order properties such as being symmetric, being regular, and being polyhedral; and some have mathematical properties such as specific eccentricities. Ever since the birth of modern science, shapes have been of considerable significance in several disciplines. In astronomy, the shapes of the orbits of planets and asteroids have always been of central interest; and in classifications of crystals, plants and animals, shape has always been one among the features used. Famously, the fact that the DNA molecule has the shape of a double helix is crucial when it comes to understanding how it functions, but the same is true of most protein macromolecules. Structural biology has even become a special branch of molecular biology. In medicine, shape can play quite a role when it comes to evaluating what is seen by means of X-ray pictures, computer tomography, magnetic resonance imaging, and functional neuroimaging. In short, the natural sciences and the life sciences have always been referring to shapes without quantifying this feature in the way the basic property dimensions of mathematical physics have been quantified, i.e., that each and every possible determinate of the property dimension in question is (via a conventionally chosen standard unit) put in a one-to-one correspondence with a real number. That shape, despite not being quantified, is commonly regarded as a physical property dimension with many different determinate shapes is obvious. In ordinary language, determinate shapes are represented by means of words such as ‘round’, ‘elliptical’, ‘triangular’, and ‘star-shaped’; but we can also represent determinate shapes by means of pictures. As a matter of fact, shape is today in the natural sciences an important but nonetheless non-quantified physical dimension; I will show that this is not a contingent fact. (The paper develops thoughts earlier put forward in [1].) The quantifications of length, mass, temperature, etc. are quantifications of all the possible determinate properties of the physical dimension in question; for instance, there are no possible determinate mass properties outside of the quantity dimension mass. Therefore, with respect to shape, the quantification problem to be dealt with here is whether or not all possible shapes can be quantified. Surely, at least one subset of shapes, the ellipses, can be linearly ordered and mapped onto a part of the number line. This is done by means of their eccentricity (e). If the length of the semimajor axis of the ellipse is called a and the length of the semiminor axis b, then each and every ellipse can be given an eccentricity value larger than zero but lesser than one according to the formula: e = √ (1 – b/a). Such orderings can be very useful and important for specific purposes in science, but this is beside the quantification problem now at issue. What has been done with respect to ellipses cannot, I claim, be done for all possible shapes. Whereas it makes good sense to say ‘this ellipse has an eccentricity of 0.73’, it is impossible to give every shape a number related to a standard unit (call it ‘morphe’), and make sense of sentences such as ‘this shape has a morphe of 2.31’. In my non-quantifiability proof, I will in detail only discuss finite 2D-shapes; if these cannot be quantified neither can the more complex finite 3D-shapes. 2 Shapes and their Segments Every 2D-shape has some two-dimensional extension in a real or an abstract space, and the shape outline can always be divided into a number of segments. Furthermore, since the shape/line in question constitutes a continuum, it must contain at least as many infinitesimally small segments as there are real numbers. The next thing to be noted is that there are different kinds of shape segments. Using an idea from Hoffman and Richards 2, I will do some of the reasoning on the assumption that there are five basic different kinds of such segments, but I will later comment on this assumption. The five kinds of segments are: