2. Atoms

The smallest thing has no parts.

The word “atom” , as defined by the early Greek and Indian philosophers who invented the term, means literally uncuttable or indivisible. The atom is therefore the smallest thing, since it has no parts. For a universe to be atomic (i.e. to have parts which are atomic) means that the process of creating parts cannot occur indefinitely: there are small things which cannot be subdivided.

In the physical universe, the name “atom” now represents a particular kind of particle. Unfortunately, that particle was subsequently found to contain parts (atoms have electrons, neutrons, and protons as parts). Oddly enough, the name stuck to the particle, despite the fact that the particle ceased to earn its name. The quest to find increasingly small particles has continued, and today it remains an open question whether the physical universe is atomic or not.

In addition to physical atoms, we may wish to consider perceptual atoms . In the perceptual universe, the smallest difference in perception is referred to in psychology as the “just noticeable difference” . From a physical point of view, the just noticeable difference may be bounded below by the firing of a single neuron (although there are also numerous chemical changes that are associated with this event). In other words, if we assume that our perception is mediated by the electrical interaction of the neurons in our brains, then the smallest unit of information which we are able to perceive is the firing of a neuron. From a perceptual point of view, however, this neuronal firing is the smallest observable change, so it might be a candidate for the title of “perceptual atom” . However, this perceptual atom is so small relative to perception as a whole that perception, even if it is not continuous, is a discrete approximation of continuity.

The determination of whether or not concepts are atomic is complicated because the scientific study of concepts and the conceptual universe is contentious: verbal report of mental states is notoriously unreliable. Hence, in order to measure concepts, we will measure their near analogues, symbols (not the perception of them, but the conception of them). The smallest units of symbolic meaning are morphemes, which in many cases correspond to words (or more technically, lexemes). Morphemes are atomic in that they are not composed of smaller meaningful parts, as with other parts of speech. Although the representation of a concept is not atomic (it consists of letters or sounds and has a distributed representation in the brain), and the object referenced by a concept may or may not be atomic, is a central thesis of this book that concepts are atomic.

Something cannot have a dimensionality less than its parent thing; it occupies a nonzero interval on every dimension which the parent occupies.

Can something have a dimensionality less than the whole of which it is a part? Although one could imagine something that occupied an arbitrarily small extent along one of the dimensions of the parent thing, to posit that something has no extent along one of its dimensions leads to a large number of Zeno-like paradoxes.

One of the older of a number of conundrums related to this topic asks how many points exist on a line (where points are assumed to be zero-dimensional things and lines are assumed to be one-dimensional things). Although one branch of modern mathematics provides a ready answer, it is debatable whether this answer is truly substantive. In particular, although we have named the answer, it may not be that we have actually defined the answer in meaningful terms. The name given to the answer, an “uncountable infinity of points” , is not a number like other numbers. For example, it does not grow when other numbers are added to it. In some sense, then, it is not a number at all, at least in the sense of the original question.

A similar problem is posed by understanding space as composed of zero-dimensional points. Maintaining that volumes are composed of points corresponds to the mathematical notion of point sets. Point sets assume that things are composed of points, or atoms which are of a lower dimensionality than the larger whole which they occupy. Unfortunately, this understanding forces points that lie on the boundary between one object and another to be associated with either one object or the other. This poses problems because the boundaries of objects (and hence the objects themselves) become characteristically different: the object possessing this boundary is said to be “closed” , and the object lacking this boundary is said to be “open” . One of the many problems which result from this view is that two closed objects cannot touch each other, since between any two points are an infinite number of other points.[16]

This does not mean that all talk of points, lines, and other such objects is discounted, but it does mean that none of their dimensions will be allowed to have an extent of zero. In other words, points are taken to correspond to atoms whose extent is (only) arbitrarily small: perhaps infinitesimal, but still nonzero. Boundaries, on the other hand, are free to have a lower dimensionality, since they do not exist as parts in the space that they divide.

[16] Mathematically inclined readers who are interested in approaches to mathematics which do not rely on the notions of infinity and point-sets may be interested in intuitionist mathematics and point-free topologies. Several introductory references may be found at http://www.cognitivesettheory.com/links