Time and Reality
Dimension is sometimes defined as the number of independent directions one has freedom to move in. For the 3-dimensional space we seem (!) to exist in, it is easy to realise we can move in three dimensions and it is rather easy to invent a technique to measure distances, e.g. by using arm-lengths, number of steps, meters or other standardized units.
However, time is different, we are not free at all to move in time and moreover, time seems to be passing in only one direction, from past to future as we defined them. Also, everybody experiences time differently and even one person has a different perception of the passing of time during different activities or periods of the day, or even just during daytime or at night. Of course, formally one may proclaim time as a fourth dimension [mathematically speaking] but then this neglects common human observations. For time measuring, we have to rely on mechanical tools, using the rotation of earth, the orbit around the sun or some vibration frequency in a quartz crystal, we even use atomic clocks to measure as precise as possible. In most Physics time is used as a parameter, varying again over the real numbers hence sort of representing a line, so it is viewed as a fourth dimension even though it does not correspond to some usual understanding of dimension. So it is certainly fair to say that the 4-dimensional model for the space we live in is a rather formal ad hoc construction.
There is no evidence that measurement of time can be done by points on a real line, in fact there is no proof that two events happening are happening on “moments” that may be compared, let alone be expressed with respect to the same unit. For example, one might just as well assume time is a real vector space of dimension 2, or any number even.
What is harmless to assume (but it is an assumption!) is that time is defined by a totally ordered set, which means for any two elements it is possible to decide that one comes before the other or conversely, if both come before the other then they coincide. This ordering of time induced the concept of causality and the cause is happening before the consequence. Causality is detected via observation, it may not really “exist “ in reality. There is a natural problem here: is the passing of time created by observing?
Clearly time is related strongly to change in the situation of the universe and change and observation are also related but less strict. Another problem is the definition of a “moment”, we think of this as a point on the time-line, so assuming already the one-dimensionality and a linear geometric representation. However, points do not exist in reality, we said before [see the previous issue], hence a moment as a period of zero duration does not exist in reality. In reality, changes take non-zero time (at least when when they are observed) which means events happen in time intervals and these may be viewed as the time ”between” two moments using the order relation defined on the (abstract mathematical set of moments). Thus, in our observed reality we may think of time as a totally ordered set of moments with events happening in time intervals. Observing anything also takes non-zero time so can we observe several things at the same moment? Well, certainly not by the same observer, but we may use several observing tools so what about it then?
Obviously, it is impossible to observe on the same moment a moment has zero duration but in Physics one may talk mathematically about observing operators by saying the operators are diagonalized (meaning that the matrix describing the operation is a diagonal matrix with only non-zero elements on the left downward diagonal). But again there is a problem, it is called the Heisenberg uncertainty principle, stating that the position and impulse of a particle (viewed as operators in some Hilbert space) cannot be diagonalized simultaneously, this follows from a theorem in Linear Algebra, stating that two matrices can only be diagonalized simultaneously (by one base change) if they commute (meaning AB=BA). Hence the uncertainty principle expresses the non-commutativity of the operators one wants to measure.
In next section I will construct a deeper non-commutativity by making the geometry non-commutative.
Back to time now. The universe can be seen as a set of states, a state for every moment in the totally ordered time set. What we observe as change is then the passing from one state to another and observing the change gives the impression of the duration of the change which corresponds to the interval in time between the moments of the states.
Important note: we cannot observe a state of the universe that is at a moment of zero duration. You could say that the universe is a book with pages in states “numbered” by moments in time (“numbered “could really mean numbered if the set of moments in time is taken to be the real numbers as in most Physics) but we can only observe changes in intervals between moments. This exactly phrases the impossibility for us to grasp reality and the fact that we have to rely on (incomplete) observations. Between every two states, there is a transition function connecting the states to the changes that happened between them. There are several remaining questions, for example: is there for a set of changes between two states always an intermediate (in the total order of time) state where a preselected change is not present? In other words is there for every, even the tiniest, change a state where that change is the only one that happened? or again i.o.w. is there a state for every change? The answer is not relevant for the theory developed, but it is an interesting assumption one can make. The transition functions depend on all events in the universe, so we cannot hope to once know them explicitly. We have seen that the geometry of the reality depends on all events in it, so we may see the above construction as a dynamical geometry with a totally ordered set (Time) and at each moment a geometry with connecting geometric morphisms for every pair of moments. The geometry of the (reality of the) universe is then this dynamical geometry, it will be different from the geometry of the observed reality we impose in some actual models.
In the next section I will propose a non-commutative approximation for the geometry of reality having for its commutative shadow the usual space-time manifold. These notions are abstract but as explained in foregoing section they are essential for interpretations of scientific and philosophical so-called truths.
Professor Emeritus Fred Van Oystaeyen