*The Philosophy of Space and Time*

by Hans Reichenbach (German: 1927, English: 1958)

pseudo-problems arise if we look for truth where definitions are needed.

–pp. 15

This is the classic work on its subject, but I have only now read it, having fortuitously come across a used copy in Bruised Books in Pullman. I come to this book late, deterred by the prejudice in my field that books on general relativity written before the 70s are boring and obsessed with transformation rules, but I wish I hadn’t waited so long, because it is a masterpiece.

Reichenbach puts great emphasis on what he calls “coordinative definitions”. For example, suppose you use ordinary measuring devices to test Euclidean geometry over large distances (e.g. summing the angles of a triangle or comparing the diameter of a circle to its circumference), and you find Euclidean relations to be violated. How to explain this? Perhaps the space you inhabit is non-Euclidean. On the other hand, perhaps there is some weird physical effect (“universal forces”) that causes all material bodies to shrink/expand/distort in the same way as they move from one place to another. How can you tell which is the case? That you cannot tell was already known, but Reichenbach drives home the idea that *there is no answer to this question*. To compare lengths at different regions requires a *definition* of congruence, of how the comparison is to be done. We can *define* the congruence by moving ideally rigid rods from one place to another. (Real rods are not ideal, but only because of differential forces that can be identified and corrected for.) It is an *empirical fact* (not a necessary truth) that one finds this comparison to be independent of the path taken to transport the rod, which makes this definition useful. Defining rigid rods to maintain their length (i.e. no universal forces), the question of the geometry of space becomes well defined. It is an empirical question and can only be settled by experiment. Introducing flexibility in the definition of congruence, it is possible to visualize non-Euclidean geometries.

In the same way we might as well regard ideal clocks as keeping constant rates–the idea that everything in the universe might be slowing down together is not only untestable but meaningless. The definition of simultaneity via synchronization of distant clocks is likewise arbitrary up to travel times of the fastest signal. Reichenbach distinguishes the scientific theory of relativity, which makes empirically testable claims about the world, from the epistemological theory of relativity, our recognition of how much our statements depend on coordinative definitions, which is independent of what experimenters may find. He constructs a metric space using only light signals and clocks. By construction, the family of coordinates singled out are related by Lorentz transformations (by construction, since the speed of light has been defined constant) (also rotations, although he implicitly discounts these). The empirical claim of the theory of relativity is that the resulting “light geometry” matches the geometry constructed with rigid rods, as in Newtonian physics it would not.

Reichenbach is a famous advocate (along with Leibniz) of the causal theory of time and space. Time is the measure of causality: that events before can affect events after is the ultimate meaning of “before” and “after”. Causal relations remain absolute (frame/coordinate-independent) in the theory of relativity. That spacelike-separated events, being causally disconnected from each other, have no invariant time order is not a mathematical curiosity, but what a philosopher should demand. Our experience of time is prior to our experience of space–distance ultimately boils down to the time it takes for spatial points to influence each other–and in relativity too proper times appear more naturally (as time experienced on the worldline of a genidentical object) than proper lengths (which require introducing some convention of simultaneity, although to be fair a rigid rod will always pick out a rest frame throughout its length). Even the three-dimensionality of space Reichenbach convincingly argues is ultimately a statement about causal relations. One requires–as a coordinative definition–that causal influence be local and spread from one point to another coving all the intermediaries in expected time order. We find empirically that causality acts like this if we map our space to a three-dimensional manifold. Having found this one answer, we know it is unique, because no different-dimensional manifold will be topologically equivalent, meaning that any map to this other manifold will violate neighborhood relations and break the desired causal behavior. By similar arguments, all aspects of the topology of space are unambiguously defined (although still an empirical matter!) by demanding the absence of “causal anomalies”. As an application, Reichenbach argues that this is why we think of the 3D space with *n* particles as “real” space rather than the 3n-dimensional configuration space. It occurs to me then that if there were natural forces of the form “push particle 1 here if particle 2 is there”, our intuition would be the reverse.

Reichenbach believes he can also deduce the asymmetry of time relations from the asymmetry of causal relations. If the ball at time A is before the ball at time B, then a mark applied at A will carry over to B, but not vice versa. I am sympathetic to such attempts, but I don’t think this argument works. One needs to know the time order to say whether the mark is being applied or removed. I would have thought it more consistent with the author’s general procedure to introduce a coordinative definition that some process or other is only said to go one way (with the proviso that this can only be pulled off if we find empirically that one time order can be found under which it always goes that way). His fuller treatment of time order came later in his book *The Direction of Time*, which I have not yet read.

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