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