The Concept of Nature / The Tarner Lectures Delivered in Trinity College, November 1919

by Alfred North Whitehead

If an event e be cogredient with a duration d, and d′ be any duration which is part of d. Then d′ belongs to the same time-system as d. Also d′ intersects e in an event e′ which is part of e and is cogredient with d′.

Let P be any event-particle lying in a given duration d. Consider the aggregate of events in which P lies and which are also cogredient with d. Each of these events occupies its own aggregate of event-particles. These aggregates will have a common portion, namely the class of event-particle lying in all of them. This class of event-particles is what I call the ‘station’ of the event-particle P in the duration d. This is the station in the character of a locus. A station can also be defined in the character of an abstractive element. Let the property σ be the name of the property which an abstractive set possesses when (i) each of its events is cogredient with the duration d and (ii) the event-particle P lies in each of its events. Then the group of σ-primes, where σ has this meaning, is an abstractive element and is the station of P in d as an abstractive element. The locus of event-particles covered by the station of P in d as an abstractive element is the station of P in d as a locus. A station has accordingly the usual three characters, namely, its character of position, its extrinsic character as an abstractive element, and its intrinsic character.

It follows from the peculiar properties of rest that two stations belonging to the same duration cannot intersect. Accordingly every event-particle on a station of a duration has that station as its station in the duration. Also every duration which is part of a given duration intersects the stations of the given duration in loci which are its own stations. By means of these properties we can utilise the overlappings of the durations of one family—that is, of one time-system—to prolong stations indefinitely backwards and forwards. Such a prolonged station will be called a point-track. A point-track is a locus of event-particles. It is defined by reference to one particular time-system, α say. Corresponding to any other time-system these will be a different group of point-tracks. Every event-particle will lie on one and only one point-track of the group belonging to any one time-system. The group of point-tracks of the time-system α is the group of points of the timeless space of α. Each such point indicates a certain quality of absolute position in reference to the durations of the family associated with α, and thence in reference to the successive instantaneous spaces lying in the successive moments of α. Each moment of α will intersect a point-track in one and only one event-particle.

This property of the unique intersection of a moment and a point-track is not confined to the case when the moment and the point-track belong to the same time-system. Any two event-particles on a point-track are sequential, so that they cannot lie in the same moment. Accordingly no moment can intersect a point-track more than once, and every moment intersects a point-track in one event-particle.

Anyone who at the successive moments of α should be at the event-particles where those moments intersect a given point of α will be at rest in the timeless space of time-system α. But in any other timeless space belonging to another time-system he will be at a different point at each succeeding moment of that time-system. In other words he will be moving. He will be moving in a straight line with uniform velocity. We might take this as the definition of a straight line. Namely, a straight line in the space of time-system β is the locus of those points of β which all intersect some one point-track which is a point in the space of some other time-system. Thus each point in the space of a time-system α is associated with one and only one straight line of the space of any other time-system β. Furthermore the set of straight lines in space β which are thus associated with points in space α form a complete family of parallel straight lines in space β. Thus there is a one-to-one correlation of points in space α with the straight lines of a certain definite family of parallel straight lines in space β. Conversely there is an analogous one-to-one correlation of the points in space β with the straight lines of a certain family of parallel straight lines in space α. These families will be called respectively the family of parallels in β associated with α, and the family of parallels in α associated with β. The direction in the space of β indicated by the family of parallels in β will be called the direction of α in space β, and the family of parallels in α is the direction of β in space α. Thus a being at rest at a point of space α will be moving uniformly along a line in space β which is in the direction of α in space β, and a being at rest at a point of space β will be moving uniformly along a line in space α which is in the direction of β in space α.

I have been speaking of the timeless spaces which are associated with time-systems. These are the spaces of physical science and of any concept of space as eternal and unchanging. But what we actually perceive is an approximation to the instantaneous space indicated by event-particles which lie within some moment of the time-system associated with our awareness. The points of such an instantaneous space are event-particles and the straight lines are rects. Let the time-system be named α, and let the moment of time-system α to which our quick perception of nature approximates be called M. Any straight line r in space α is a locus of points and each point is a point-track which is a locus of event-particles. Thus in the four-dimensional geometry of all event-particles there is a two-dimensional locus which is the locus of all event-particles on points lying on the straight line r. I will call this locus of event-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect ρ. Thus ρ is the instantaneous rect in M which occupies at the moment M the straight line r in the space of α. Accordingly when one sees instantaneously a moving being and its path ahead of it, what one really sees is the being at some event-particle A lying in the rect ρ which is the apparent path on the assumption of uniform motion. But the actual rect ρ which is a locus of event-particles is never traversed by the being. These event-particles are the instantaneous facts which pass with the instantaneous moment. What is really traversed are other event-particles which at succeeding instants occupy the same points of space α as those occupied by the event-particles of the rect ρ. For example, we see a stretch of road and a lorry moving along it. The instantaneously seen road is a portion of the rect ρ—of course only an approximation to it. The lorry is the moving object. But the road as seen is never traversed. It is thought of as being traversed because the intrinsic characters of the later events are in general so similar to those of the instantaneous road that we do not trouble to discriminate. But suppose a land mine under the road has been exploded before the lorry gets there. Then it is fairly obvious that the lorry does not traverse what we saw at first. Suppose the lorry is at rest in space β. Then the straight line r of space α is in the direction of β in space α, and the rect ρ is the representative in the moment M of the line r of space α. The direction of ρ in the instantaneous space of the moment M is the direction of β in M, where M is a moment of time-system α. Again the matrix of the line r of space α will also be the matrix of some line s of space β which will be in the direction of α in space β. Thus if the lorry halts at some point P of space α which lies on the line r, it is now moving along the line s of space β. This is the theory of relative motion; the common matrix is the bond which connects the motion of β in space α with the motions of α in space β.

Motion is essentially a relation between some object of nature and the one timeless space of a time-system. An instantaneous space is static, being related to the static nature at an instant. In perception when we see things moving in an approximation to an instantaneous space, the future lines of motion as immediately perceived are rects which are never traversed. These approximate rects are composed of small events, namely approximate routes and event-particles, which are passed away before the moving objects reach them. Assuming that our forecasts of rectilinear motion are correct, these rects occupy the straight lines in timeless space which are traversed. Thus the rects are symbols in immediate sense-awareness of a future which can only be expressed in terms of timeless space.

We are now in a position to explore the fundamental character of perpendicularity. Consider the two time-systems α and β, each with its own timeless space and its own family of instantaneous moments with their instantaneous spaces. Let M and N be respectively a moment of α and a moment of β. In M there is the direction of β and in N there is the direction of α. But M and N, being moments of different time-systems, intersect in a level. Call this level λ. Then λ is an instantaneous plane in the instantaneous space of M and also in the instantaneous space of N. It is the locus of all the event-particles which lie both in M and in N.

In the instantaneous space of M the level λ is perpendicular to the direction of β in M, and in the instantaneous space of N the level λ is perpendicular to the direction of α in N. This is the fundamental property which forms the definition of perpendicularity. The symmetry of perpendicularity is a particular instance of the symmetry of the mutual relations between two time-systems. We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced.

The theory of perpendicularity in the timeless space of any time-system α follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let ρ be any rect in the moment M of α and let λ be a level in M which is perpendicular to ρ. The locus of those points of the space of α which intersect M in event-particles on ρ is the straight line r of space α, and the locus of those points of the space of α which intersect M in event-particles on λ is the plane l of space α. Then the plane l is perpendicular to the line r.