Is space one-dimensional?

While the answer is No, space is three-dimensional, it is instructional to compare space with time as people speak about it.

The philosopher JME McTaggart made a well-known distinction between an A-series, which is “the series of positions [in time] running from the far past through the near past to the present, and then from the present to the near future and the far future,” and a B-series, which is the “series of positions [in time] which runs from earlier to later” (Mind 17 (1908), p.458).

In general usage, a world line is “the sequential path of personal human events (with time and place as dimensions) that marks the history of a person”. So a personal world line shows places where we were before the present, the place we are presently located, and (perhaps) places we may be in the future. It also shows the dates we were at each location. From this it may be determined the times at and between locations, and the distances too.

A world line exhibits a temporal B-series and, if the present time is indicated, an A-series as well. But a world line also exhibits spatial versions of A-series and B-series: the series of positions in space “running from the far past through the near past to the present, and then from the present to the near future and the far future” as a spatial A-series and the series of positions in space “which runs from earlier to later” as a spatial B-series. A world line shows that the path we take through space is analogous to the path we take through time. Since we “know” that time is one-dimensional, space must be one-dimensional, too.

The problem is that time is confused with change. McTaggart wrote, “It would, I suppose, be universally admitted that time involves change”. No, that should not be admitted. Both space and time involve change and stasis. For example, speed is the change in spatial position divided by the change in temporal position. Both space and time may involve stasis as well. Even an object that does not change spatial position has parallel cases in which an object does not change temporal position.

This may happen in several senses. The strongest sense is that of relativity theory, in which time at a certain speed (that of light) or in certain cases (depending on forces) does not result in measurable duration and so time stops. Another sense is that when we measure duration (as with any other measurement) there is a beginning to that measurement so if the measurement of time has not begun, then there is no change in time. For example, before a stopwatch is started, no duration is measured. A third sense in which time stops is that time is measured by cycles and at the end of each cycle, time has returned to its starting point and so has not changed position.

Neither space or time are one-dimensional.