Explaining, Seeing, and Understanding in Thought Experiments

Explaining, Seeing, and Understanding in Thought Experiments James Robert Brown Department of Philosophy University of Toronto In this paper, I analyze the relation between visualization and explanatory understanding in thought experiments. I discuss two thought experiments from special relativity. Both cases lead to a paradox, which can, however, be explained (away) as non-contradictory. The resolution in one of the two cases sheds light on the nature of thought experiments more generally by indicating what sort of thing it is that we see when we are observing events in the mind’s own laboratory. And it will in turn shed light on the notions of explanation and understanding as they arise in thought experiments. Theories often run into paradoxes. Some of these are outright contradictions, sending the would-be champions of the theory back to the drawing board. Others are paradoxical in the sense of being bizarre and unexpected. The latter are sometimes mistakenly thought to be instances of the former. That is, they are thought to be more than merely weird; they are mistakenly thought to be self-refuting. Showing that they are not self- contradictory but merely a surprise is often a challenge. Notions of explanation and understanding are often at issue. For instance, we might explain—or explain away—a paradox by invoking some mechanism pro- vided by the theory and showing how it does not really lead to a logically incoherent result. This is what I’m going to do here. I want to look at two paradoxes, both from special relativity. Neither are paradoxes in the sense of outright contradictions. But showing that they are merely paradoxes in the sense of being unexpected and bizarre is something of a challenge. In both cases the paradox arises in a thought I am grateful to the organizers and participants of the Leiden conference for a stimulating discussion and valuable comments on the talk that gave rise to this paper. I also thank Henk de Regt and an anonymous reader for providing several valuable suggestions for im- proving this paper. Finally, I am grateful to SSHRC for ongoing ªnancial support. Perspectives on Science 2014, vol. 22, no. 3 ©2014 by The Massachusetts Institute of Technology doi:10.1162/POSC_a_00138 357 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00138 by guest on 28 September 2021 358 Explaining, Seeing, Understanding experiment, which complicates the issue considerably, but also provides a valuable opportunity. The resolution in one of the two cases will, I claim, shed light on the nature of thought experiments more generally by indicating what sort of thing it is that we see when we are observing events in the mind’s own laboratory. And it will in turn shed light on the notions of explanation and understanding as they arise in thought experiments. Special Relativity Einstein was perhaps the greatest thought experimenter ever. Galileo was arguably his equal, but none is his superior. Much of his theorizing was motivated by thought experiments. This was certainly true of both special and general relativity. To begin, let’s review some of the basics of special relativity, starting from the two standard postulates. And as we do so, let us keep in mind that special and general relativity contain several thought experiments, some of which are central to the creation of new principles and laws, while others play a more pedagogical role. Postulate 1: Laws of nature are the same in every inertial frame. Postulate 2: The speed of light is constant; it has the same value, c, in every frame. There is probably no need to comment on these postulates, since special relativity is widely known, but we may need to be reminded of the deªni- tion of simultaneity. Distant events e1 and e2 in frame F are simultaneous if and only if light from the two events meets at a spatial mid-point in F. Since the second postulate says in effect that light travels at a constant speed c, this deªnition should strike us a perfectly correct. We are in for a surprise, however, when we try to mix this concept of simultaneity with the ªrst postulate, the postulate that asserts that all inertial frames are equivalent. Consider a train running along a track. Suppose we have two frames, which we take to be attached to the train and to the track, respec- tively. We assume that the train is moving at velocity v in the track frame. We also assume that the observer in the track frame is midway between e1 and e2, which are separated events (light ºashes). We further assume that the observer receives signals from each event at the same time. Then by the earlier deªnition, events e1 and e2 are simultaneous in the track frame. Some time passes, however, while the light signals come to the mid- points—the train has moved forward. An observer midway on the train Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00138 by guest on 28 September 2021 Perspectives on Science 359 Figure 1. Events e1 and e2 are simultaneous in the track frame. Figure 2. Event e2 is earlier than e1 in the train frame. frame receives the signal from e2 before e1. So, the event e2 was earlier than e1 in the train frame. The upshot of this is simple but profound: the simultaneity of distant events is relative to a frame. (In case this seems too shocking, remember the equivalence of all frames and the fact that an event happens in every frame.) This is the ªrst of many profound discoveries in special relativity. From this conclusion and a bit of algebra we can derive the length contraction and time dilation formulae. (I will state them below but not derive them, since they can be found in any text on special relativity.) So far this is standard stuff and all part of the normal presentation of special relativity. Einstein developed the theory in the way I have pre- sented it (Einstein 1905) and most expositors follow him. The use of simple thought experiments to bring out the details of special relativity is clever but so far not uncommon or in any way exceptional. The philosoph- ical lessons to be learned from these thought experiments will arise later. Time Dilation and the Twin Paradox In Newtonian physics and in common sense, the time interval between two events is the same no matter what reference frame in which the interval is measured. Not so in special relativity. We’ll begin with the notion of proper time which applies to events happening at the same place in the same frame. First, we need a deªnition: Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00138 by guest on 28 September 2021 360 Explaining, Seeing, Understanding The proper time is the temporal interval between those events that happen at the same spatial location in a given frame. When a frame is in motion relative to another, time is dilated (stretched out, slowed down). The temporal interval between events is longer than the proper time. It is lengthened by a factor, known as the time dilation factor, which can be easily derived from the initial postulates of special relativity: 1 . 1 − vc22/ If a clock (which is at rest in frame F) registers an interval of time ⌬t, then a clock that is moving at velocity v will register an interval ⌬tЈϭ⌬t 1 − vc22/ , according to any observer in F. The limits should be noted: As the relative velocity goes to zero, the two clocks will coincide, that is, they will run at the same rate. And as the relative velocity increases to the speed of light, the moving clock will slow down and approach stopping when the limit, v ϭ c, is approached (though never reached). The impor- tant thing to take from this is that when v is not zero, then ⌬tЈϽ⌬t. The twin paradox grows out of this. Actually, there are two paradoxes. The ªrst is merely that sort of paradox that comes as a great surprise. Imagine a pair of twins, one stays at home while the other goes on a long trip by rocket at a very high speed, then returns home. Everything will be slower for the moving twin, all clocks on the rocket, all biological processes, and so on. When the travelling twin gets back, she will have aged, say, ªve years, while the stay-at- home twin will have aged most of a normal lifetime. This is a serious shock to common sense and a violation of pre- relativistic physics, but it seems to be correct. We may ªnd this stagger- ing, but we can and do get used to it. However, there might be a second paradox, the sort that we cannot live with, since it would be an outright contradiction. The second paradox—if it is indeed a paradox—arises from the follow- ing simple consideration. The principle of relativity (the ªrst postulate) demands the equivalence of all frames. That in turn suggests that the so- called travelling twin could be considered the stay-at-home and the stay- at-home could be considered the traveler. The same reasoning about moving things temporally running more slowly is now applied the other way; the twin who previously came back younger will this time be the older. Now we do seem to have an outright contradiction, because the theory Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_a_00138 by guest on 28 September 2021 Perspectives on Science 361 seems to say that each twin is both older and younger than the other.

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