Important characteristics of Euclidean time include the seven properties of immediacy, universality, continuity, staticity, unidirectionality, infinity, and linearity.

Time has a special and mysterious quality about it having to
do with it occurring *now*, which we may name **immediacy**. Of
an infinity of times, only the immediate instant is *now*. But the
referent of *now* is constantly changing, so that it points to
some different time, for example, *now*. What is constant about this
ever-changing temporal pointer? Not the things that occur in time,
which includes just about everything. Depending on our point of view
we may call these things ``states'', or ``events'', or ``elements'' or
even ``features'' or ``feature-values''.

In considering the mystery of immediacy, one may notice that its
special, elusive character is not so special from an extra-temporal
perspective (as in the later examination of a record of any particular
temporal sequence). No event in the record is singled out as in any
way different from any other event in the record, except that each
occurs at its own unique time. The only remnant of *now*-ness left in
the record is the property of each event that it occurs at its own
individual immediate time, which is *now* relative to
the event itself. This could be taken as evidence that records of
temporal sequences have not recorded the essence of this principle of
immediacy. However, for the purposes of examining records of events,
immediacy is the principle that events occur
``at'' particular times; that is, that times are associated
with events.

The other properties require less discussion. **Universality** is
the principle that everything (in the universe under discussion)
occurs at some time. **Continuity** in real time may be axiomatized
as the statement that between every two (distinct) times there is
another time. **Staticity** is the principle that an event (in this
context, more properly called a ``state'') may extend across (be
associated with) a contiguous sequence of times. **Infinity** is
the principle that time does not begin or end.^{A.1}**Unidirectionality** specifies not just that the two directions are
distinct from each other (this follows from linearity) but that one
direction is special, in that certain sequences of events may not be
reversed; they can not in principle occur in the reverse order.

**Linearity** is defined in any mathematics-for-linguists
text. The important properties here are that for a given element e in
a sequence,

A: all other distinct elements either precede or follow^{A.2}e.

B: no element may both precede and follow e.

We may define classes of time other than Euclidean time by eliminating
certain of these properties. Thus **discrete time** drops the
continuity principle, but retains infinity, linearity, immediacy,
universality, staticity, and unidirectionality. **Finite time**, of
which **linguistic time** is an instance, further eliminates the
requirement of infinity. One might also drop the requirement of
staticity, so that events have duration of only one time unit.

Linear, linguistic representations have the character of non-static,
finite, discrete time. Thus, for example, linear sequences of feature
matrices, which that form phonological representations in SPE, have
the properties of immediacy (the association of elements with
particular locations in the ordering), universality (the claim that
all elements are thus associated with some location), and linearity
(the complete ordering of all the locations), but not of continuity,
infinity, or staticity. Staticity^{A.3} deserves
special treatment in this case: geminate segments might be considered
as states, since the same feature matrix repeats itself at two
successive times. However, I take it as essential about SPE
representations that repeating identical matrices are precisely that:
repetitions; they are NOT the same matrix extended across two times.
In this respect, SPE is unlike both its predecessors and its
successors.

The linearity of time gives an order to temporal
events/states/elements^{A.4}. In static finite-temporal
representations, this ordering is represented by in some way adding to
the description of each element of the sequence a uniquely ordered
index. This is the implicit effect of writing forms in a sequence on
paper: the physical layout provides an analog of the temporally
ordered indexing of all the elements in the string. If this implicit
ordering is done explicitly, by numbering the elements with an
(ordered) index, then the actual location of elements in a particular
physical representation becomes immaterial. Words in a sentence, for
example, could be written on separate pieces of paper with their
respective indices and tossed into the air: their linear ordering is
still fully specified since their (ordered) indices are specified. It
makes no difference whether elements are concretely put in order (for
example, by printing them in a spatial sequence on a teletype
machine) or ordered abstractly by associating unique indices/times -
for which the set of indices has the properties of finite time - with
the elements (for example, by storing numbers with the elements in
unordered, random-access, computer memory). In either case their
ordering is fully specified; that is, concrete and abstract ordering
are equivalent for our purposes.

This view of ordering adds no new descriptive machinery to linguistic representations. The features of finite time are implicit in any finite, linearly ordered, non-reversible sequence of discrete elements. Sequences of linguistic elements may therefore be described in terms of discrete time. If one wishes to add to a system of linguistic representation explicit machinery which indexes each element with a distinct ``time'' (or number, say), this makes no further claims than those already implicitly made by the unindexed, concrete representation of the same sequences. Thus this view merely makes explicit and formal the formerly implicit and informal machinery which has always been used in linguistic representations.