The source of the quoted material is Space-Time-Matter by Hermann Weyl
I started trying to summarized the following development of the mathematical treatment of parameterized time. Then I realized there are many subtleties involved, and much of the discussion can probably be handled better using more modern terminology.
I would call his mathematical model of time a one-dimensional manifold formed of a strictly ordered infinite, unbounded set. It evidently has some topological structure. I'm not sure where the notions of homogeneity and congruence (ability to compare different lengths of time) fall into the taxonomy of mathematical concepts. I'm also not sure Weyl gives enough structure to call his manifold a continuum in the sense of Cantor. (A concept he took issue with.)
I have to qualify my characterization of his model as a set. What he develops is (apparently) a point set. But that is merely a model of continuum a priori.
Finally, I believe the group structure described at the end is that of a Lie Group.
So I ask: how might the structure described by Weyl be expressed in current mathematical language?
To be able to apply mathematical conceptions to questions of Time we must postulate that it is theoretically possible to fix in Time, to any order of accuracy, an absolutely rigorous now (present) as a point of Time---i.e., to be able to indicate points of time, one of which will always be the earlier and the other the later. The following principle will hold for this "order-relation". If $A$ is earlier than $B$ and $B$ is earlier than $C$, then $A$ is earlier than $C$. Each two points of Time, $A$ and $B$, of which $A$ is the earlier, mark off a length of time; this includes every point which is later than $A$ and earlier than $B$. The fact that Time is a form of our stream of experience is expressed in the idea of equality: the empirical content which fills the length of Time $AB$ can in itself be put into any other time without being in any way different from what it is. The length of time which it would then occupy is equal to the distance $AB$. This, with the help of the principle of causality, gives us the following objective criterion in physics for equal lengths of time. If an absolutely isolated physical system (i.e., one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general such a system is called a clock. Each period of the cycle lasts equally long.
The mathematical fixing of time by measuring it is based upon these two relations, "earlier (or later) times" and "equal times". The nature of measurement may be indicated briefly as follows:
In the following paragraph the sentence translated as "A difference arises, however, in the case of three point-pairs." Is originally:
"Anders wird die Sache aber, wenn wir zu drei Zeitpunkten übergehen."
"However things are different when we go to three time-points."
Unfortunately there are a large number of such errors in the translation.
Time is homogeneous, i.e., a single point of time can only be given by being specified individually. There is no inherent property arising from the general nature of time which may be ascribed to any one point but not to any other; or, every property logically derivable from these two fundamental relations belongs either to all points or to none. The same holds for time-lengths and point-pairs. A property which is based on these two relations and which holds for one point-pair must hold for every point-pair $AB$ (in which $A$ is earlier than $B$). A difference arises, however, in the case of three point-pairs. If any two time-points $O$ and $E$ are given such that $O$ is earlier than $E$, it is possible to fix conceptually further time-points $P$ by referring them to the unit-distance $OE$. This is done by constructing logically a relation $t$ between three points such that for every two points $O$ and $E$, of which $O$ is the earlier, there is one and only one point $P$ which satisfies the relation $t$ between $O$, $E$ and $P$, i.e., symbolically, $$ OP = t \cdot OE $$ (e.g., $OP = 2 \cdot OE$ denotes the relation $OE = EP$). Numbers are merely concise symbols for such relations as $t$, defined logically from the primary relations. $P$ is the "time-point with the abscissa $t$ in the co-ordinate system (taking $OE$ as unit length)". Two different numbers $t$ and $t^{*}$ in the same co-ordinate system necessarily lead to two different points; for, otherwise, in consequence of the homogeneity of the continuum of time-lengths, the property expressed by $$ t \cdot AB = t^{*} \cdot AB, $$ since it belongs to the time-length $AB = OE$, must belong to every time-length, and hence the equations $AC = t \cdot AB$, $AC = t^{*} \cdot AB$ would both express the same relation, i.e., $t$ would be equal to $t^{*}$. Numbers enable us to single out separate time-points relatively to a unit-distance $OE$ out of the time-continuum by a conceptual, and hence objective and precise, process. But the objectivity of things conferred by the exclusion of the ego and its data derived directly from intuition, is not entirely satisfactory; the co-ordinate system which can only be specified by an individual act (and then only approximately) remains as an inevitable residuum of this elimination of the percipient.
It seems to me that by formulating the principle of measurement in the above terms we see clearly how mathematics has come to play its rôle in exact natural science. An essential feature of measurement is the difference between the "determination" of an object by individual specification and the determination of the same object by some conceptual means. The latter is only possible relatively to objects which must be defined directly. That is why a theory of relativity is perforce always involved in measurement. The general problem which it proposes for an arbitrary domain of objects takes the form: (1) What must be given such that relatively to it (and to any desired order of precision) one can single out conceptually a single arbitrary object $P$ from the continuously extended domain of objects under consideration? That which has to be given is called the co-ordinate system, the conceptual definition is called the co-ordinate (or abscissa) of $P$ in the co-ordinate system. Two different co-ordinate systems are completely equivalent for an objective standpoint. There is no property, that can be fixed conceptually, which applies to one co-ordinate system but not to the other; for in that case too much would have been given directly. (2) What relationship exists between the co-ordinates of one and the same arbitrary object $P$ in two different co-ordinate systems?
In the realm of time-points, with which we are at present concerned, the answer to the first question is that the co-ordinate system consists of a time-length $OE$ (giving the origin and the unit of measure). The answer to the second question is that the required relationship is expressed by the formula of transformation $$ t = at' + b \text{; } (a > 0) $$ in which $a$ and $b$ are constants, whilst $t$ and $t'$ are the co-ordinates of the same arbitrary point $P$ in an "unaccented" and "accented" system respectively. For all possible pairs of co-ordinate systems the characteristic numbers, $a$ and $b$, of the transformation may be any real numbers with the limitation that $a$ must always be positive. The aggregate of transformations constitutes a group, as their nature would imply, i.e.,
- "identity" $t = t'$ is contained in it.
- Every transformation is accompanied by its reciprocal in the group, i.e., by the transformation which exactly cancels its effect. Thus, the inverse of the transformation $(a, b)$, viz., $t = at' + b$, is $\left(\dfrac{1}{a}, -\dfrac{b}{a}\right)$, viz., $t' = \dfrac{1}{a}t - \dfrac{b}{a}$.
- If two transformations of a group are given, then the one which is produced by applying these two successively also belongs to the group. It is at once evident that, by applying the two transformations $$ t = at' + b t' = a't" + b' $$ in succession, we get $$ t = a_{1} t" + b_{1} $$ where $a_{1} = a \cdot a'$ and $b_{1} = (ab') + b$; and if $a$ and $a'$ are positive, so is their product.
Sketch of beginnings of a rough draft
The following is my scribbling about the topic. I am not pretending that the use of mathematical terms is proper. It's not even close to a comprehensive answer. The main reason I'm including it here is to motivate a possible approach to the topic. In particular note that my "clock" is far more abstract than Weyl's. I make no appeal to physical reasoning. In fact my clock could be thought of as noting more than a selection of elements satisfying certain criteria.
The Temporal Continuum
The temporal continuum is an ideal form conceived as an infinite indivisible transcendental existence. Hopefully I will find time to unpack that metaphysical gibberish. For present purposes, take that to mean that we can think of time as isomorphic to the ideal Euclidean line. The line itself is indivisible. That is, no part can be separated from its adjacent parts. The line can be "marked" with infinite precision using dimensionless points. Whether those points of location are constituents of the line is not addressed.
Between any two distinct points on the line lies a segment of the line, which can be marked by a point distinct from the delimiting points, succeeding the first delimiter and preceeding the second delimiter. A representation of the segment can be rigidly transported along the line as a standard of comparison.
[Unfinished due to technological problems]
The temporal continuum model $\mathbb{X}$ is an unbounded, infinite, homogeneous, strictly ordered set of instants. The ordering is denoted $t_{1}<t_{2}$ and stated: instant $t_{1}$ is earlier than instant $t_{2}$. For any bounded set of instants, there is an earliest (also called the first, beginning, initial, etc.,) instant; and a latest (also called the last, final, ending, etc.,) instant. We shall call this ordering chronological.
In this context homogeneous means that instants have no intrinsic distinguishing attributes, and are only distinguished by their relative place in the continuum.
Between any two instants there is at least one additional instant which is later than the the first and earlier than the last.
Every pair of instants delimits a period. A clock is a device that enumerates instants in increasing chronological order. Each enumeration is called a tick of the clock. All periods delimited by a sequential pair of enumerations from the same clock are said to be equal in duration. Such a period is called a clock period of unit duration. This duration is also called a unit of time. The period formed by a sequence of $n$ adjacent clock periods is said to have a duration of $n$ units of time. Two periods formed of the same number of adjacent clock periods are said to be of equal duration. Indicating the enumeration of clock ticks using integer subscripts we say the period $P_{1}=\left[t_{0},t_{n}\right]$ has a duration of $n=n-0$ units, and for $0\le m\le n$ the period $P_{2}=\left[t_{m},t_{n}\right]$ has duration $p=n-m.$ If either of the inequalities hold, then $p<n$ and we say the duration of $P_{1}$ is greater than, or longer than that of $P_{2}$.
We now introduce the requirement that a clock shall have the ability to fractionally enumerate unit clock periods such that each unit period has the same number of fractional periods. Each fractional period delimited by a sequential pair of fractional enumerations is said to be of equal duration to that of any other such fractional period determined by the same clock. This method of uniform subdivision is assumed to be recursively applicable to an arbitrary number of levels.
Two clock which enumerate the same instants (possibly with offset enumeration) are said to be coinciding. Two coinciding clocks with identical enumeration are said to be synchronized. Coinciding clocks are said to be identical clocks. But it is possible to have identical clocks which do not coincide. Two clocks are identical if all unit periods have the same pattern of interleaving instants. That is, for example each unit tick of the first clock falls in the $n^{th}$ fractional period of each unit period of the second clock.